fun fractions homework

Fractions Worksheets

Welcome to the fractions worksheets page at Math-Drills.com where the cup is half full! This is one of our more popular pages most likely because learning fractions is incredibly important in a person's life and it is a math topic that many approach with trepidation due to its bad rap over the years. Fractions really aren't that difficult to master especially with the support of our wide selection of worksheets.

This page includes Fractions worksheets for understanding fractions including modeling, comparing, ordering, simplifying and converting fractions and operations with fractions. We start you off with the obvious: modeling fractions. It is a great idea if students can actually understand what a fraction is, so please do spend some time with the modeling aspect. Relating modeling to real life helps a great deal too as it is much easier to relate to half a cookie than to half a square. Ask most students what you get if you add half a cookie and another half a cookie, and they'll probably let you know that it makes one delicious snack.

The other fractions worksheets on this page are devoted to helping students understand the concept of fractions. From comparing and ordering to simplifying and converting... by the time students master the material on this page, operations of fractions will be a walk in the park.

Most Popular Fractions Worksheets this Week

Fraction Circles

Fraction circle manipulatives are mainly used for comparing fractions, but they can be used for a variety of other purposes such as representing and identifying fractions, adding and subtracting fractions, and as probability spinners. There are a variety of options depending on your purpose. Fraction circles come in small and large versions, labeled and unlabeled versions and in three different color schemes: black and white, color, and light gray. The color scheme matches the fraction strips and use colors that are meant to show good contrast among themselves. Do note that there is a significant prevalence of color-blindness in the population, so don't rely on all students being able to differentiate the colors.

Suggested activity for comparing fractions: Photocopy the black and white version onto an overhead projection slide and another copy onto a piece of paper. Alternatively, you can use two pieces of paper and hold them up to the light for this activity. Use a pencil to represent the first fraction on the paper copy. Use a non-permanent overhead pen to represent the second fraction. Lay the slide over the paper and compare the two circles. You should easily be able to tell which is greater or lesser or if the two fractions are equal. Re-use both sheets by erasing the pencil and washing off the marker.

Adding fractions with fraction circles will involve two copies on paper. Cut out the fraction circles and segments of one copy and leave the other copy intact. To add 1/3 + 1/2, for example, place a 1/3 segment and a 1/2 segment into a circle and hold it over various fractions on the intact copy to see what 1/2 + 1/3 is equivalent to. 5/6 or 10/12 should work.

• Small Fraction Circles Small Fraction Circles in Black and White with Labels Small Fraction Circles in Color with Labels Small Fraction Circles in Light Gray with Labels Small Fraction Circles in Black and White Unlabeled Small Fraction Circles in Color Unlabeled Small Fraction Circles in Light Gray Unlabeled
• Large Fraction Circles Large Fraction Circles in Black and White with Labels Large Fraction Circles in Color with Labels Large Fraction Circles in Light Gray with Labels Large Fraction Circles in Black and White Unlabeled Large Fraction Circles in Color Unlabeled Large Fraction Circles in Light Gray Unlabeled

Fraction Strips

Fractions strips are often used for comparing fractions. Students are able to see quite easily the relationships and equivalence between fractions with different denominators. It can be quite useful for students to have two copies: one copy cut into strips and the other copy kept intact. They can then use the cut-out strips on the intact page to individually compare fractions. For example, they can use the halves strip to see what other fractions are equivalent to one-half. This can also be accomplished with a straight edge such as a ruler without cutting out any strips. Pairs or groups of strips can also be compared side-by-side if they are cut out. Addition and subtraction (etc.) are also possibilities; for example, adding a one-quarter and one-third can be accomplished by shifting the thirds strip so that it starts at the end of one-quarter then finding a strip that matches the end of the one-third mark (7/12 should do it).

Teachers might consider copying the fraction strips onto overhead projection acetates for whole class or group activities. Acetate versions are also useful as a hands-on manipulative for students in conjunction with an uncut page.

The "Smart" Fraction Strips include strips in a more useful order, eliminate the 7ths and 11ths strips as they don't have any equivalents and include 15ths and 16ths. The colors are consistent with the classic versions, so the two sets can be combined.

• Classic Fraction Strips with Labels Classic Fraction Strips in Black and White With Labels Classic Fraction Strips in Color With Labels Classic Fraction Strips in Gray With Labels
• Unlabeled Classic Fraction Strips Classic Fraction Strips in Black and White Unlabeled Classic Fraction Strips in Color Unlabeled Classic Fraction Strips in Gray Unlabeled
• Smart Fraction Strips with Labels Smart Fraction Strips in Black and White With Labels Smart Fraction Strips in Color With Labels Smart Fraction Strips in Gray With Labels

Modeling fractions

Fractions can represent parts of a group or parts of a whole. In these worksheets, fractions are modeled as parts of a group. Besides using the worksheets in this section, you can also try some more interesting ways of modeling fractions. Healthy snacks can make great models for fractions. Can you cut a cucumber into thirds? A tomato into quarters? Can you make two-thirds of the grapes red and one-third green?

• Modeling Fractions with Groups of Shapes Coloring Groups of Shapes to Represent Fractions Identifying Fractions from Colored Groups of Shapes (Only Simplified Fractions up to Eighths) Identifying Fractions from Colored Groups of Shapes (Halves Only) Identifying Fractions from Colored Groups of Shapes (Halves and Thirds) Identifying Fractions from Colored Groups of Shapes (Halves, Thirds and Fourths) Identifying Fractions from Colored Groups of Shapes (Up to Fifths) Identifying Fractions from Colored Groups of Shapes (Up to Sixths) Identifying Fractions from Colored Groups of Shapes (Up to Eighths) Identifying Fractions from Colored Groups of Shapes (OLD Version; Print Too Light)
• Modeling Fractions with Rectangles Modeling Halves Modeling Thirds Modeling Halves and Thirds Modeling Fourths (Color Version) Modeling Fourths (Grey Version) Coloring Fourths Models Modeling Fifths Coloring Fifths Models Modeling Sixths Coloring Sixths Models
• Modeling Fractions with Circles Modeling Halves, Thirds and Fourths Coloring Halves, Thirds and Fourths Modeling Halves, Thirds, Fourths, and Fifths Coloring Halves, Thirds, Fourths, and Fifths Modeling Halves to Sixths Coloring Halves to Sixths Modeling Halves to Eighths Coloring Halves to Eighths Modeling Halves to Twelfths Coloring Halves to Twelfths

Ratio and Proportion Worksheets

The equivalent fractions models worksheets include only the "baking fractions" in the A versions. To see more difficult and varied fractions, please choose the B to J versions after loading the A version. More picture ratios can be found on holiday and seasonal pages. Try searching for picture ratios to find more.

• Picture Ratios Autumn Trees Part-to-Part Picture Ratios ( Grouped ) Autumn Trees Part-to-Part and Part-to-Whole Picture Ratios ( Grouped )
• Equivalent Fractions Equivalent Fractions With Blanks ( Multiply Right ) ✎ Equivalent Fractions With Blanks ( Divide Left ) ✎ Equivalent Fractions With Blanks ( Multiply Right or Divide Left ) ✎ Equivalent Fractions With Blanks ( Divide Right ) ✎ Equivalent Fractions With Blanks ( Multiply Left ) ✎ Equivalent Fractions With Blanks ( Multiply Left or Divide Right ) ✎ Equivalent Fractions With Blanks ( Multiply or Divide Right ) ✎ Equivalent Fractions With Blanks ( Multiply or Divide Left ) ✎ Equivalent Fractions With Blanks ( Multiply or Divide in Either Direction ) ✎ Are These Fractions Equivalent? (Multiplier 2 to 5) Are These Fractions Equivalent? (Multiplier 5 to 15) Equivalent Fractions Models Equivalent Fractions Models with the Simplified Fraction First Equivalent Fractions Models with the Simplified Fraction Second
• Equivalent Ratios Equivalent Ratios with Blanks Only on Right Equivalent Ratios with Blanks Anywhere Equivalent Ratios with x 's

Comparing and Ordering Fractions

Comparing fractions involves deciding which of two fractions is greater in value or if the two fractions are equal in value. There are generally four methods that can be used for comparing fractions. First is to use common denominators . If both fractions have the same denominator, comparing the fractions simply involves comparing the numerators. Equivalent fractions can be used to convert one or both fractions, so they have common denominators. A second method is to convert both fractions to a decimal and compare the decimal numbers. Visualization is the third method. Using something like fraction strips , two fractions can be compared with a visual tool. The fourth method is to use a cross-multiplication strategy where the numerator of the first fraction is multiplied by the denominator of the second fraction; then the numerator of the second fraction is multiplied by the denominator of the first fraction. The resulting products can be compared to decide which fraction is greater (or if they are equal).

• Comparing Proper Fractions Comparing Proper Fractions to Sixths ✎ Comparing Proper Fractions to Ninths (No Sevenths) ✎ Comparing Proper Fractions to Ninths ✎ Comparing Proper Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Proper Fractions to Twelfths ✎

The worksheets in this section also include improper fractions. This might make the task of comparing even easier for some questions that involve both a proper and an improper fraction. If students recognize one fraction is greater than one and the other fraction is less than one, the greater fraction will be obvious.

• Comparing Proper and Improper Fractions Comparing Proper and Improper Fractions to Sixths ✎ Comparing Proper and Improper Fractions to Ninths (No Sevenths) ✎ Comparing Proper and Improper Fractions to Ninths ✎ Comparing Proper and Improper Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Proper and Improper Fractions to Twelfths ✎ Comparing Improper Fractions to Sixths ✎ Comparing Improper Fractions to Ninths (No Sevenths) ✎ Comparing Improper Fractions to Ninths ✎ Comparing Improper Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Improper Fractions to Twelfths ✎

This section additionally includes mixed fractions. When comparing mixed and improper fractions, it is useful to convert one of the fractions to the other's form either in writing or mentally. Converting to a mixed fraction is probably the better route since the first step is to compare the whole number portions, and if one is greater than the other, the proper fraction portion can be ignored. If the whole number portions are equal, the proper fractions must be compared to see which number is greater.

• Comparing Proper, Improper and Mixed Fractions Comparing Proper, Improper and Mixed Fractions to Sixths ✎ Comparing Proper, Improper and Mixed Fractions to Ninths (No Sevenths) ✎ Comparing Proper, Improper and Mixed Fractions to Ninths ✎ Comparing Proper, Improper and Mixed Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Proper, Improper and Mixed Fractions to Twelfths ✎
• Comparing Improper and Mixed Fractions Comparing Improper and Mixed Fractions to Sixths ✎ Comparing Improper and Mixed Fractions to Ninths (No Sevenths) ✎ Comparing Improper and Mixed Fractions to Ninths ✎ Comparing Improper and Mixed Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Improper and Mixed Fractions to Twelfths ✎
• Comparing Mixed Fractions Comparing Mixed Fractions to Sixths ✎ Comparing Mixed Fractions to Ninths (No Sevenths) ✎ Comparing Mixed Fractions to Ninths ✎ Comparing Mixed Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Mixed Fractions to Twelfths ✎

Many of the same strategies that work for comparing fractions also work for ordering fractions. Using manipulatives such as fraction strips, using number lines, or finding decimal equivalents will all have your student(s) putting fractions in the correct order in no time. We've probably said this before, but make sure that you emphasize that when comparing or ordering fractions, students understand that the whole needs to be the same. Comparing half the population of Canada with a third of the population of the United States won't cut it. Try using some visuals to reinforce this important concept. Even though we've included number lines below, feel free to use your own strategies.

• Ordering Fractions with Easy Denominators on a Number Line Ordering Fractions with Easy Denominators to 10 on a Number Line Ordering Fractions with Easy Denominators to 24 on a Number Line Ordering Fractions with Easy Denominators to 60 on a Number Line Ordering Fractions with Easy Denominators to 100 on a Number Line
• Ordering Fractions with Easy Denominators on a Number Line (Including Negative Fractions) Ordering Fractions with Easy Denominators to 10 + Negatives on a Number Line Ordering Fractions with Easy Denominators to 24 + Negatives on a Number Line Ordering Fractions with Easy Denominators to 60 + Negatives on a Number Line Ordering Fractions with Easy Denominators to 100 + Negatives on a Number Line
• Ordering Fractions with All Denominators on a Number Line Ordering Fractions with All Denominators to 10 on a Number Line Ordering Fractions with All Denominators to 24 on a Number Line Ordering Fractions with All Denominators to 60 on a Number Line Ordering Fractions with All Denominators to 100 on a Number Line
• Ordering Fractions with All Denominators on a Number Line (Including Negative Fractions) Ordering Fractions with All Denominators to 10 + Negatives on a Number Line Ordering Fractions with All Denominators to 24 + Negatives on a Number Line Ordering Fractions with All Denominators to 60 + Negatives on a Number Line Ordering Fractions with All Denominators to 100 + Negatives on a Number Line

The ordering fractions worksheets in this section do not include a number line, to allow for students to use various sorting strategies.

• Ordering Positive Fractions Ordering Positive Fractions with Like Denominators Ordering Positive Fractions with Like Numerators Ordering Positive Fractions with Like Numerators or Denominators Ordering Positive Fractions with Proper Fractions Only Ordering Positive Fractions with Improper Fractions Ordering Positive Fractions with Mixed Fractions Ordering Positive Fractions with Improper and Mixed Fractions
• Ordering Positive and Negative Fractions Ordering Positive and Negative Fractions with Like Denominators Ordering Positive and Negative Fractions with Like Numerators Ordering Positive and Negative Fractions with Like Numerators or Denominators Ordering Positive and Negative Fractions with Proper Fractions Only Ordering Positive and Negative Fractions with Improper Fractions Ordering Positive and Negative Fractions with Mixed Fractions Ordering Positive and Negative Fractions with Improper and Mixed Fractions

Simplifying & Converting Fractions Worksheets

Rounding fractions helps students to understand fractions a little better and can be applied to estimating answers to fractions questions. For example, if one had to estimate 1 4/7 × 6, they could probably say the answer was about 9 since 1 4/7 is about 1 1/2 and 1 1/2 × 6 is 9.

• Rounding Fractions with Helper Lines Rounding Fractions to the Nearest Whole with Helper Lines Rounding Mixed Numbers to the Nearest Whole with Helper Lines Rounding Fractions to the Nearest Half with Helper Lines Rounding Mixed Numbers to the Nearest Half with Helper Lines
• Rounding Fractions Rounding Fractions to the Nearest Whole Rounding Mixed Numbers to the Nearest Whole Rounding Fractions to the Nearest Half Rounding Mixed Numbers to the Nearest Half

Learning how to simplify fractions makes a student's life much easier later on when learning operations with fractions. It also helps them to learn that different-looking fractions can be equivalent. One way of demonstrating this is to divide out two equivalent fractions. For example 3/2 and 6/4 both result in a quotient of 1.5 when divided. By practicing simplifying fractions, students will hopefully recognize unsimplified fractions when they start adding, subtracting, multiplying and dividing with fractions.

• Simplifying Fractions Simplify Fractions (easier) Simplify Fractions (harder) Simplify Improper Fractions (easier) Simplify Improper Fractions (harder)
• Converting Between Improper and Mixed Fractions Converting Mixed Fractions to Improper Fractions Converting Improper Fractions to Mixed Fractions Converting Between (both ways) Mixed and Improper Fractions
• Converting Between Fractions and Decimals Converting Fractions to Terminating Decimals Converting Fractions to Terminating and Repeating Decimals Converting Terminating Decimals to Fractions Converting Terminating and Repeating Decimals to Fractions Converting Fractions to Hundredths
• Converting Between Fractions, Decimals, Percents and Ratios with Terminating Decimals Only Converting Fractions to Decimals, Percents and Part-to- Part Ratios ( Terminating Decimals Only) Converting Fractions to Decimals, Percents and Part-to- Whole Ratios ( Terminating Decimals Only) Converting Decimals to Fractions, Percents and Part-to- Part Ratios ( Terminating Decimals Only) Converting Decimals to Fractions, Percents and Part-to- Whole Ratios ( Terminating Decimals Only) Converting Percents to Fractions, Decimals and Part-to- Part Ratios ( Terminating Decimals Only) Converting Percents to Fractions, Decimals and Part-to- Whole Ratios ( Terminating Decimals Only) Converting Part-to-Part Ratios to Fractions, Decimals and Percents ( Terminating Decimals Only) Converting Part-to-Whole Ratios to Fractions, Decimals and Percents ( Terminating Decimals Only) Converting Various Fractions, Decimals, Percents and Part-to- Part Ratios ( Terminating Decimals Only) Converting Various Fractions, Decimals, Percents and Part-to- Whole Ratios ( Terminating Decimals Only)
• Converting Between Fractions, Decimals, Percents and Ratios with Terminating and Repeating Decimals Converting Fractions to Decimals, Percents and Part-to- Part Ratios Converting Fractions to Decimals, Percents and Part-to- Whole Ratios Converting Decimals to Fractions, Percents and Part-to- Part Ratios Converting Decimals to Fractions, Percents and Part-to- Whole Ratios Converting Percents to Fractions, Decimals and Part-to- Part Ratios Converting Percents to Fractions, Decimals and Part-to- Whole Ratios Converting Part-to-Part Ratios to Fractions, Decimals and Percents Converting Part-to-Whole Ratios to Fractions, Decimals and Percents Converting Various Fractions, Decimals, Percents and Part-to- Part Ratios Converting Various Fractions, Decimals, Percents and Part-to- Whole Ratios Converting Various Fractions, Decimals, Percents and Part-to- Part Ratios with 7ths and 11ths Converting Various Fractions, Decimals, Percents and Part-to- Whole Ratios with 7ths and 11ths

Multiplying Fractions

Multiplying fractions is usually less confusing operationally than any other operation and can be less confusing conceptually if approached in the right way. The algorithm for multiplying is simply multiply the numerators then multiply the denominators. The magic word in understanding the multiplication of fractions is, "of." For example what is two-thirds OF six? What is a third OF a half? When you use the word, "of," it gets much easier to visualize fractions multiplication. Example: cut a loaf of bread in half, then cut the half into thirds. One third OF a half loaf of bread is the same as 1/3 x 1/2 and tastes delicious with butter.

• Multiplying Two Proper Fraction Multiplying Two Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ ✎ Multiplying Two Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Proper Fractions with No Simplifying (Printable Only) Multiplying Two Proper Fractions with All Simplifying (Printable Only) Multiplying Two Proper Fractions with Some Simplifying (Printable Only)
• Multiplying Proper and Improper Fractions Multiplying Proper and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Improper Fractions with No Simplifying (Printable Only) Multiplying Proper and Improper Fractions with All Simplifying (Printable Only) Multiplying Proper and Improper Fractions with Some Simplifying (Printable Only)
• Multiplying Two Improper Fractions Multiplying Two Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Improper Fractions with No Simplifying (Printable Only) Multiplying Two Improper Fractions with All Simplifying (Printable Only) Multiplying Two Improper Fractions with Some Simplifying (Printable Only)
• Multiplying Proper and Mixed Fractions Multiplying Proper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Mixed Fractions with No Simplifying (Printable Only) Multiplying Proper and Mixed Fractions with All Simplifying (Printable Only) Multiplying Proper and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying Two Mixed Fractions Multiplying Two Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Mixed Fractions with No Simplifying (Printable Only) Multiplying Two Mixed Fractions with All Simplifying (Printable Only) Multiplying Two Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying Whole Numbers and Proper Fractions Multiplying Whole Numbers and Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Proper Fractions with No Simplifying (Printable Only) Multiplying Whole Numbers and Proper Fractions with All Simplifying (Printable Only) Multiplying Whole Numbers and Proper Fractions with Some Simplifying (Printable Only)
• Multiplying Whole Numbers and Improper Fractions Multiplying Whole Numbers and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Improper Fractions with No Simplifying (Printable Only) Multiplying Whole Numbers and Improper Fractions with All Simplifying (Printable Only) Multiplying Whole Numbers and Improper Fractions with Some Simplifying (Printable Only)
• Multiplying Whole Numbers and Mixed Fractions Multiplying Whole Numbers and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Mixed Fractions with No Simplifying (Printable Only) Multiplying Whole Numbers and Mixed Fractions with All Simplifying (Printable Only) Multiplying Whole Numbers and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying Proper, Improper and Mixed Fractions Multiplying Proper, Improper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper, Improper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper, Improper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper, Improper and Mixed Fractions with No Simplifying (Printable Only) Multiplying Proper, Improper and Mixed Fractions with All Simplifying (Printable Only) Multiplying Proper, Improper and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying 3 Fractions Multiplying 3 Proper Fractions (Fillable, Savable, Printable) ✎ Multiplying 3 Proper and Improper Fractions (Fillable, Savable, Printable) ✎ Multiplying Proper and Improper Fractions and Whole Numbers (3 factors) (Fillable, Savable, Printable) ✎ Multiplying Fractions and Mixed Fractions (3 factors) (Fillable, Savable, Printable) ✎ Multiplying 3 Mixed Fractions (Fillable, Savable, Printable) ✎

Dividing Fractions

Conceptually, dividing fractions is probably the most difficult of all the operations, but we're going to help you out. The algorithm for dividing fractions is just like multiplying fractions, but you find the inverse of the second fraction or you cross-multiply. This gets you the right answer which is extremely important especially if you're building a bridge. We told you how to conceptualize fraction multiplication, but how does it work with division? Easy! You just need to learn the magic phrase: "How many ____'s are there in ______? For example, in the question 6 ÷ 1/2, you would ask, "How many halves are there in 6?" It becomes a little more difficult when both numbers are fractions, but it isn't a giant leap to figure it out. 1/2 ÷ 1/4 is a fairly easy example, especially if you think in terms of U.S. or Canadian coins. How many quarters are there in a half dollar?

• Dividing Two Proper Fractions Dividing Two Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Proper Fractions with No Simplifying (Printable Only) Dividing Two Proper Fractions with All Simplifying (Printable Only) Dividing Two Proper Fractions with Some Simplifying (Printable Only)
• Dividing Proper and Improper Fractions Dividing Proper and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Improper Fractions with No Simplifying (Printable Only) Dividing Proper and Improper Fractions with All Simplifying (Printable Only) Dividing Proper and Improper Fractions with Some Simplifying (Printable Only)
• Dividing Two Improper Fractions Dividing Two Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Improper Fractions with No Simplifying (Printable Only) Dividing Two Improper Fractions with All Simplifying (Printable Only) Dividing Two Improper Fractions with Some Simplifying (Printable Only)
• Dividing Proper and Mixed Fractions Dividing Proper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Mixed Fractions with No Simplifying (Printable Only) Dividing Proper and Mixed Fractions with All Simplifying (Printable Only) Dividing Proper and Mixed Fractions with Some Simplifying (Printable Only)
• Dividing Two Mixed Fractions Dividing Two Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Mixed Fractions with No Simplifying (Printable Only) Dividing Two Mixed Fractions with All Simplifying (Printable Only) Dividing Two Mixed Fractions with Some Simplifying (Printable Only)
• Dividing Whole Numbers and Proper Fractions Dividing Whole Numbers and Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Proper Fractions with No Simplifying (Printable Only) Dividing Whole Numbers and Proper Fractions with All Simplifying (Printable Only) Dividing Whole Numbers and Proper Fractions with Some Simplifying (Printable Only)
• Dividing Whole Numbers and Improper Fractions Dividing Whole Numbers and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Improper Fractions with No Simplifying (Printable Only) Dividing Whole Numbers and Improper Fractions with All Simplifying (Printable Only) Dividing Whole Numbers and Improper Fractions with Some Simplifying (Printable Only)
• Dividing Whole Numbers and Mixed Fractions Dividing Whole Numbers and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Mixed Fractions with No Simplifying (Printable Only) Dividing Whole Numbers and Mixed Fractions with All Simplifying (Printable Only) Dividing Whole Numbers and Mixed Fractions with Some Simplifying (Printable Only)
• Dividing Proper, Improper and Mixed Fractions Dividing Proper, Improper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper, Improper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper, Improper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper, Improper and Mixed Fractions with No Simplifying (Printable Only) Dividing Proper, Improper and Mixed Fractions with All Simplifying (Printable Only) Dividing Proper, Improper and Mixed Fractions with Some Simplifying (Printable Only)
• Dividing 3 Fractions Dividing 3 Fractions Dividing 3 Fractions (Some Whole Numbers) Dividing 3 Fractions (Some Mixed) Dividing 3 Mixed Fractions

Multiplying and Dividing Fractions

This section includes worksheets with both multiplication and division mixed on each worksheet. Students will have to pay attention to the signs.

• Multiplying and Dividing Two Proper Fractions Multiplying and Dividing Two Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Proper Fractions with No Simplifying (Printable Only) Multiplying and Dividing Two Proper Fractions with All Simplifying (Printable Only) Multiplying and Dividing Two Proper Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Proper and Improper Fractions Multiplying and Dividing Proper and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Improper Fractions with No Simplifying (Printable Only) Multiplying and Dividing Proper and Improper Fractions with All Simplifying (Printable Only) Multiplying and Dividing Proper and Improper Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Two Improper Fractions Multiplying and Dividing Two Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Improper Fractions (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Improper Fractions with No Simplifying (Printable Only) Multiplying and Dividing Two Improper Fractions with All Simplifying (Printable Only) Multiplying and Dividing Two Improper Fractions (Printable Only)
• Multiplying and Dividing Proper and Mixed Fractions Multiplying and Dividing Proper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Mixed Fractions with No Simplifying (Printable Only) Multiplying and Dividing Proper and Mixed Fractions with All Simplifying (Printable Only) Multiplying and Dividing Proper and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Two Mixed Fractions Multiplying and Dividing Two Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Mixed Fractions with No Simplifying (Printable Only) Multiplying and Dividing Two Mixed Fractions with All Simplifying (Printable Only) Multiplying and Dividing Two Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Whole Numbers and Proper Fractions Fractions Multiplying and Dividing Whole Numbers and Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Proper Fractions with No Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Proper Fractions with All Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Proper Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Whole Numbers and Improper Fractions Multiplying and Dividing Whole Numbers and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Improper Fractions with No Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Improper Fractions with All Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Improper Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Whole Numbers and Mixed Fractions Multiplying and Dividing Whole Numbers and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Mixed Fractions with No Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Mixed Fractions with All Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Proper, Improper and Mixed Fractions Multiplying and Dividing Proper, Improper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper, Improper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper, Improper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper, Improper and Mixed Fractions with No Simplifying (Printable Only) Multiplying and Dividing Proper, Improper and Mixed Fractions with All Simplifying (Printable Only) Multiplying and Dividing Proper, Improper and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing 3 Fractions Multiplying/Dividing Fractions (three factors) Multiplying/Dividing Mixed Fractions (3 factors)

Adding Fractions

Adding fractions requires the annoying common denominator. Make it easy on your students by first teaching the concepts of equivalent fractions and least common multiples. Once students are familiar with those two concepts, the idea of finding fractions with common denominators for adding becomes that much easier. Spending time on modeling fractions will also help students to understand fractions addition. Relating fractions to familiar examples will certainly help. For example, if you add a 1/2 banana and a 1/2 banana, you get a whole banana. What happens if you add a 1/2 banana and 3/4 of another banana?

• Adding Two Proper Fractions with Equal Denominators and Proper Fraction Results Adding Two Proper Fractions with Equal Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Proper Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Equal Denominators, Proper Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Equal Denominators, Proper Fractions Result, and Some Simplifying (Printable Only)
• Adding Two Proper Fractions with Equal Denominators and Mixed Fraction Results Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Result, and Some Simplifying (Printable Only)
• Adding Two Proper Fractions with Similar Denominators and Proper Fraction Results Adding Two Proper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Proper Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Similar Denominators, Proper Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Similar Denominators, Proper Fractions Result, and Some Simplifying (Printable Only)
• Adding Two Proper Fractions with Similar Denominators and Mixed Fraction Results Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Result, and Some Simplifying (Printable Only)
• Adding Two Proper Fractions with Unlike Denominators and Proper Fraction Results Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Result, and Some Simplifying (Printable Only)
• Adding Two Proper Fractions with Unlike Denominators and Mixed Fraction Results Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Result, and Some Simplifying (Printable Only)
• Adding Proper and Improper Fractions with Equal Denominators Adding Proper and Improper Fractions with Equal Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Equal Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Equal Denominators and No Simplifying (Printable Only) Adding Proper and Improper Fractions with Equal Denominators and All Simplifying (Printable Only) Adding Proper and Improper Fractions with Equal Denominators and Some Simplifying (Printable Only)
• Adding Proper and Improper Fractions with Similar Denominators Adding Proper and Improper Fractions with Similar Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Similar Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Similar Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Similar Denominators and No Simplifying (Printable Only) Adding Proper and Improper Fractions with Similar Denominators and All Simplifying (Printable Only) Adding Proper and Improper Fractions with Similar Denominators and Some Simplifying (Printable Only)
• Adding Proper and Improper Fractions with Unlike Denominators Adding Proper and Improper Fractions with Unlike Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Unlike Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Unlike Denominators and No Simplifying (Printable Only) Adding Proper and Improper Fractions with Unlike Denominators and All Simplifying (Printable Only) Adding Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Printable Only)

A common strategy to use when adding mixed fractions is to convert the mixed fractions to improper fractions, complete the addition, then switch back. Another strategy which requires a little less brainpower is to look at the whole numbers and fractions separately. Add the whole numbers first. Add the fractions second. If the resulting fraction is improper, then it needs to be converted to a mixed number. The whole number portion can be added to the original whole number portion.

• Adding Two Mixed Fractions with Equal Denominators Adding Two Mixed Fractions with Equal Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Equal Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Equal Denominators and No Simplifying (Printable Only) Adding Two Mixed Fractions with Equal Denominators and All Simplifying (Printable Only) Adding Two Mixed Fractions with Equal Denominators and Some Simplifying (Printable Only)
• Adding Two Mixed Fractions with Similar Denominators Adding Two Mixed Fractions with Similar Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Similar Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Similar Denominators and Some Simplifying Adding Two Mixed Fractions with Similar Denominators and No Simplifying (Printable Only) Adding Two Mixed Fractions with Similar Denominators and All Simplifying (Printable Only) Adding Two Mixed Fractions with Similar Denominators and Some Simplifying (Printable Only)
• Adding Two Mixed Fractions with Unlike Denominators Adding Two Mixed Fractions with Unlike Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Unlike Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Unlike Denominators and No Simplifying (Printable Only) Adding Two Mixed Fractions with Unlike Denominators and All Simplifying (Printable Only) Adding Two Mixed Fractions with Unlike Denominators and Some Simplifying (Printable Only)

Subtracting Fractions

There isn't a lot of difference between adding and subtracting fractions. Both require a common denominator which requires some prerequisite knowledge. The only difference is the second and subsequent numerators are subtracted from the first one. There is a danger that you might end up with a negative number when subtracting fractions, so students might need to learn what it means in that case. When it comes to any concept in fractions, it is always a good idea to relate it to a familiar or easy-to-understand situation. For example, 7/8 - 3/4 = 1/8 could be given meaning in the context of a race. The first runner was 7/8 around the track when the second runner was 3/4 around the track. How far ahead was the first runner? (1/8 of the track).

• Subtracting Two Proper Fractions with Equal Denominators and Proper Fraction Results Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Two Proper Fractions with Similar Denominators and Proper Fraction Results Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Two Proper Fractions with Unlike Denominators and Proper Fraction Results Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Equal Denominators and Proper Fraction Results Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Similar Denominators and Proper Fraction Results Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Unlike Denominators and Proper Fraction Results Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Equal Denominators and Mixed Fraction Results Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Similar Denominators and Mixed Fraction Results Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Unlike Denominators and Mixed Fraction Results Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Mixed Fractions with Equal Denominators Subtracting Mixed Fractions with Equal Denominators, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Equal Denominators, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Equal Denominators, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Equal Denominators, and No Simplifying (Printable Only) Subtracting Mixed Fractions with Equal Denominators, and All Simplifying (Printable Only) Subtracting Mixed Fractions with Equal Denominators, and Some Simplifying (Printable Only)
• Subtracting Mixed Fractions with Similar Denominators Subtracting Mixed Fractions with Similar Denominators, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Similar Denominators, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Similar Denominators, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Similar Denominators, and No Simplifying (Printable Only) Subtracting Mixed Fractions with Similar Denominators, and All Simplifying (Printable Only) Subtracting Mixed Fractions with Similar Denominators, and Some Simplifying (Printable Only)
• Subtracting Mixed Fractions with Unlike Denominators Subtracting Mixed Fractions with Unlike Denominators, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Unlike Denominators, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Unlike Denominators, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Unlike Denominators, and No Simplifying (Printable Only) Subtracting Mixed Fractions with Unlike Denominators, and All Simplifying (Printable Only) Subtracting Mixed Fractions with Unlike Denominators, and Some Simplifying (Printable Only)

Adding and Subtracting Fractions

Mixing up the signs on operations with fractions worksheets makes students pay more attention to what they are doing and allows for a good test of their skills in more than one operation.

• Adding and Subtracting Proper and Improper Fractions Adding and Subtracting Proper and Improper Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Proper and Improper Fractions with Similar Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Proper and Improper Fractions with Equal Denominators and Some Simplifying (Printable Only) Adding and Subtracting Proper and Improper Fractions with Similar Denominators and Some Simplifying (Printable Only) Adding and Subtracting Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Printable Only)
• Adding and Subtracting Mixed Fractions Adding and Subtracting Mixed Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Mixed Fractions with Similar Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Mixed Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Mixed Fractions with Equal Denominators and Some Simplifying (Printable Only) Adding and Subtracting Mixed Fractions with Similar Denominators and Some Simplifying (Printable Only) Adding and Subtracting Mixed Fractions with Unlike Denominators and Some Simplifying (Printable Only) Adding/Subtracting Three Fractions/Mixed Fractions

All Operations Fractions Worksheets

• All Operations with Two Proper Fractions with Equal Denominators and Proper Fraction Results All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and No Simplifying (Printable Only) All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and All Simplifying (Printable Only) All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and Some Simplifying (Printable Only)
• All Operations with Two Proper Fractions with Similar Denominators and Proper Fraction Results All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and No Simplifying (Printable Only) All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and All Simplifying (Printable Only) All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and Some Simplifying (Printable Only)
• All Operations with Two Proper Fractions with Unlike Denominators and Proper Fraction Results All Operations with Two Proper Fractions with Unlike Denominators, Proper Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Unlike Denominators, Proper Fractions Results and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Unlike Denominators, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Unlike Denominators, Proper Fractions Results and No Simplifying (Printable Only) All Operations with Two Proper Fractions with Unlike Denominators, Proper Fractions Results and All Simplifying (Printable Only) All Operations with Two Proper Fractions with Unlike Denominators, Mixed Fractions Results and Some Simplifying (Printable Only)
• All Operations with Proper and Improper Fractions with Equal Denominators All Operations with Proper and Improper Fractions with Equal Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Equal Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Equal Denominators and No Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Equal Denominators and All Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Equal Denominators and Some Simplifying (Printable Only)
• All Operations with Proper and Improper Fractions with Similar Denominators All Operations with Proper and Improper Fractions with Similar Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Similar Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Similar Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Similar Denominators and No Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Similar Denominators and All Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Similar Denominators and Some Simplifying (Printable Only)
• All Operations with Proper and Improper Fractions with Unlike Denominators All Operations with Proper and Improper Fractions with Unlike Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Unlike Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Unlike Denominators and No Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Unlike Denominators and All Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Printable Only)
• All Operations with Two Mixed Fractions with Equal Denominators All Operations with Two Mixed Fractions with Equal Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Equal Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Equal Denominators and No Simplifying (Printable Only) All Operations with Two Mixed Fractions with Equal Denominators and All Simplifying (Printable Only) All Operations with Two Mixed Fractions with Equal Denominators and Some Simplifying (Printable Only)
• All Operations with Two Mixed Fractions with Similar Denominators All Operations with Two Mixed Fractions with Similar Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Similar Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Similar Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Similar Denominators and No Simplifying (Printable Only) All Operations with Two Mixed Fractions with Similar Denominators and All Simplifying (Printable Only) All Operations with Two Mixed Fractions with Similar Denominators and Some Simplifying (Printable Only)
• All Operations with Two Mixed Fractions with Unlike Denominators All Operations with Two Mixed Fractions with Unlike Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Unlike Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Unlike Denominators and No Simplifying (Printable Only) All Operations with Two Mixed Fractions with Unlike Denominators and All Simplifying (Printable Only) All Operations with Two Mixed Fractions with Unlike Denominators and Some Simplifying (Printable Only)
• All Operations with 3 Fractions All Operations with Three Fractions Including Some Improper Fractions All Operations with Three Fractions Including Some Negative and Some Improper Fractions

Operations with Negative Fractions Worksheets

Although some of these worksheets are single operations, it should be helpful to have all of these in the same location. There are some special considerations when completing operations with negative fractions. It is usually very helpful to change any mixed numbers to an improper fraction before proceeding. It is important to pay attention to the signs and know the rules for multiplying positives and negatives (++ = +, +- = -, -+ = - and -- = +).

• Adding with Negative Fractions Adding Negative Proper Fractions with Unlike Denominators Up to Sixths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Negative Proper Fractions with Unlike Denominators Up to Twelfths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Negative Mixed Fractions with Unlike Denominators Up to Sixths, Proper Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Adding Negative Mixed Fractions with Unlike Denominators Up to Twelfths, Proper Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Adding Negative Proper Fractions with Denominators Up to Sixths, Proper Fraction Results and Some Simplifying (Printable Only) Adding Negative Proper Fractions with Denominators Up to Twelfths, Proper Fraction Results and Some Simplifying (Printable Only) Adding Negative Mixed Fractions with Denominators Up to Sixths and Some Simplifying (Printable Only) Adding Negative Mixed Fractions with Denominators Up to Twelfths and Some Simplifying (Printable Only)
• Subtracting with Negative Fractions Subtracting Negative Proper Fractions with Unlike Denominators Up to Sixths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Negative Proper Fractions with Unlike Denominators Up to Twelfths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Negative Mixed Fractions with Unlike Denominators Up to Sixths, Mixed Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Negative Mixed Fractions with Unlike Denominators Up to Twelfths, Mixed Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Negative Proper Fractions with Denominators Up to Sixths, Proper Fraction Results and Some Simplifying (Printable Only) Subtracting Negative Proper Fractions with Denominators Up to Twelfths, Proper Fraction Results and Some Simplifying (Printable Only) Subtracting Negative Mixed Fractions with Denominators Up to Sixths and Some Simplifying (Printable Only) Subtracting Negative Mixed Fractions with Denominators Up to Twelfths and Some Simplifying (Printable Only)
• Multiplying with Negative Fractions Multiplying Negative Proper Fractions with Denominators Up to Sixths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Negative Proper Fractions with Denominators Up to Twelfths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Negative Mixed Fractions with Denominators Up to Sixths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Negative Mixed Fractions with Denominators Up to Twelfths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Negative Proper Fractions with Denominators Up to Sixths, Proper Fraction Results and Some Simplifying (Printable Only) Multiplying Negative Proper Fractions with Denominators Up to Twelfths, Proper Fraction Results and Some Simplifying (Printable Only) Multiplying Negative Mixed Fractions with Denominators Up to Sixths and Some Simplifying (Printable Only) Multiplying Negative Mixed Fractions with Denominators Up to Twelfths and Some Simplifying (Printable Only)
• Dividing with Negative Fractions Dividing Negative Proper Fractions with Denominators Up to Sixths, Mixed Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Negative Proper Fractions with Denominators Up to Twelfths, Mixed Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Negative Mixed Fractions with Denominators Up to Twelfths, Mixed Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Dividing Negative Mixed Fractions with Denominators Up to Twelfths, Mixed Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Dividing Negative Proper Fractions with Denominators Up to Sixths, Proper Fraction Results and Some Simplifying (Printable Only) Dividing Negative Proper Fractions with Denominators Up to Twelfths, Proper Fraction Results and Some Simplifying (Printable Only) Dividing Negative Mixed Fractions with Denominators Up to Sixths and Some Simplifying (Printable Only) Dividing Negative Mixed Fractions with Denominators Up to Twelfths and Some Simplifying (Printable Only)

Order of Operations with Fractions Worksheets

The order of operations worksheets in this section actually reside on the Order of Operations page, but they are included here for your convenience.

• Order of Operations with Fractions 2-Step Order of Operations with Fractions 3-Step Order of Operations with Fractions 4-Step Order of Operations with Fractions 5-Step Order of Operations with Fractions 6-Step Order of Operations with Fractions
• Order of Operations with Fractions (No Exponents) 2-Step Order of Operations with Fractions (No Exponents) 3-Step Order of Operations with Fractions (No Exponents) 4-Step Order of Operations with Fractions (No Exponents) 5-Step Order of Operations with Fractions (No Exponents) 6-Step Order of Operations with Fractions (No Exponents)
• Order of Operations with Positive and Negative Fractions 2-Step Order of Operations with Positive & Negative Fractions 3-Step Order of Operations with Positive & Negative Fractions 4-Step Order of Operations with Positive & Negative Fractions 5-Step Order of Operations with Positive & Negative Fractions 6-Step Order of Operations with Positive & Negative Fractions

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Fraction Worksheets

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Use These Easy Homemade Spinners to Practice Fractions

Mastering fractions one spin at a time.

Many elementary teachers have told us that fractions are one of the hardest math skills for kids to master. That’s why we created these free fraction games, to go along with our free multiplication games. Our packet of free fractions worksheets includes 11 printable games. Students can play the games to practice writing and recognizing fractions, coloring and producing fractions, fraction equivalents, and fraction-decimal conversion.

Here’s how it works

1. save and print our free fractions worksheets printable packet ..

It includes 11 free fractions worksheets for different levels, from beginners to advanced. Plus: answer keys!

2. Choose the fraction set or skill that you want students to practice and distribute that sheet.

The packet includes games to help recognize and write fractions, produce and color fractions, practice fraction equivalents, and work on fraction-decimal conversion.

3. Give each student a pencil and a paper clip to create a simple spinner.

Have students spin for a fraction to write, reproduce, or convert, depending on the worksheet.

4. As students become more confident with their fractions skills, give them the more challenging fractions worksheets.

This is the fractions to decimals conversion game.

5. Watch the video to see our fraction games in action!

Ready to try it yourself? Get our  free fractions worksheets printable packet  now!

Number lines are terrific for estimation, too.  Find 18 estimation activities for your classroom here.

10 frames are another incredibly useful math tool.  Here are some fantastic and fun 10 Frame Activities you’ll want to try next.

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40 Fun Fraction Games and Activities for Kids

Math wouldn't be half as fun without activities and online games like these. Continue Reading

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10 Must Have 3rd Grade Fractions Activities

The fractions unit in one of the most critical math focuses in 3rd grade. Students should come to third grade with a basic understanding of fractions of a shape based on 2nd grade geometry standards. But, in 3rd grade, students are introduced to fractions and quickly move into other key fraction standards: fractions on a number line, comparing and equivalent fractions, mixed numbers, and fractions greater than 1. The foundations laid in 3rd grade are built upon in 4th and 5th grades as students move to multiplying and dividing fractions and decimals and percents. It’s critical that students build a deep understanding of fractions and what they represent so they have a strong foundation for later years. These hands-on fraction activities have been must-haves in my 3rd grade classroom over the last several years.

Please note: all fraction resources featured here focus on fractions of a shape and fractions on a number line. Fractions of a set are  not covered. This is intentional as fractions of a set are ratios rather than fractional parts of whole numbers.

Hands-On Fractions Strips

Commercially purchased fractions strips are widely available and there’s nothing wrong with using them. However, I’ve found that my students use them without building an understanding of what each piece of the strip represents. They become a cheat sheet, almost, or a crutch. Instead, I have students make their own during our first fractions lesson. I introduce our fractions unit by having students create their own fractions strips and turning them into a reference poster. This one, hands-on lesson lays the foundation for our fractions unit.

During this fraction introduction, students explore sizes of denominators to compare fractions and explore equivalence. Because students create each unit fraction themselves, and use one strip to represent the whole, key foundational concepts to understanding fractions are established. The fraction strips can also be used to represent fractions on a number line by adding the number line below. The reference poster can continue to be used as a tool throughout the rest of the fractions unit. This hands-on fractions introduction lesson will make all the difference in your fractions unit and it’s detailed for you, including a video tutorial, in the linked post!

Fractions Greater than One Lesson

One of the most difficult parts of our 3rd grade fractions unit is working with fractions greater than 1. I use soft sugar cookies from the store as part of my hands-on, interactive lesson. Through this one lesson, we revisit concepts such as numerator, denominator, comparing, and equivalence. But, the focus of our lesson is on fractions greater than one, or improper fractions. Students are able to connect what they already know about fractions as they learn more about fractions where the numerator is larger than the denominator.

I have a video tutorial for this lesson in my Fractions Greater than 1 post.

Math Learning Center Apps

I rely on the Fractions and Number Lines Apps from the Math Learning Center so often during our fractions unit! These quick tools allow me to display and manipulate circles and rectangles for fractions of a shape, and use number lines to explore fractions less than, equal to, and greater than 1. Both are very customizable so you can use them in a variety of your lessons. Each app is available as a web app you can link to (and link to precreated versions) and is available as an app for Chrome and iOS. I’ll do my best to detail how I use each tool, but spend some time exploring each of them yourself!

The Fractions App gives you a blank slate to begin. Using the toolbar on the bottom, you choose whether to use a rectangle or a circle and then you choose the amount of parts in the shape. Once the shape is displayed on the screen, you can select to have the fraction displayed as you add shading to the parts of the whole. This tool is perfect for giving visual models of fractions of a shape and allowing students to explore fractions. It’s also great for visual models when comparing fractions or determining equivalence. Students can build two shapes of the same size as they work independently. Or, you can use the tool during your lessons.

The Number Lines app allows you to choose the spacing on your line and whether or not you want numbers and fractions displayed. Without numbers, it’s virtually an open number line that can be used for any part of your lesson. If you choose to display the fraction, you can select the denominator up to 12.

The masking tool allows you to display only specific numbers. By clicking on one of the blue boxes, that number will be displayed. The jump tool shows the movement from one hash to the next. The unit fraction for each jump can be turned on or off.

I have only ever used the web version of both of these apps but I find them invaluable during my fractions unit. I also use the number line app during our measurement unit when we’re measuring to the nearest quarter-inch.

Digital 3rd Grade Fractions Resources

I teach my math lesson in two different rotations, allowing me to differentiate my instruction and provide differentiated scaffolding. You can read more detail in my Math Block Structure post.  When students are not with me, it’s crucial that they are engaging in high-quality, standards-based practice. I utilize a variety of websites like Moby Max, Freckle, and Khan Academy to give my students ongoing practice at their independent and instructional levels. I also utilize those websites for intentional standards-based practice. But I also use my digital lessons and digital task cards to give students specific standards-based independent practice. This allows me to have a better look at what students complete independently and what they still need support in.

After we’ve spent a couple days working through the standard together, I assign my digital lesson. The digital lesson walks through the skill from concrete representations to more abstract concepts like word problems. Students read lesson information, explore content specific vocabulary, and apply their learning to word problems. These digital lessons reinforce the same content I’m teaching within my instruction.

On another day, students complete digital task cards on the standard. The task cards give students independent practice of the skill in its most typical formats. As a teacher, I love that I can quickly assess students’ understanding of the skill by spending just a quick minute going through their work.

I typically do not assess my students’ mastery of the standard until a full week after I taught it- in short, the Friday following the week of instruction. This allows me to continue to support and reteach those students that need additional practice. It’s also during this second week that I send homework on this particular skill. I like to send homework when students should be able to complete the work relatively independently. To guide my support during this second week, I give my Google Forms quiz at the end of the first week- often on the same day I assign the task cards. Because most of the form is self-grading, I can spend my time planning my reteaching instruction for the following week.

Each of these 3 components are available in my Digital Fractions Resources . The bundle includes units for each of the fractions skills and standards: understanding fractions and unit fractions, fractions on a number line, fractions greater than one (sometimes referred to as improper fractions), whole numbers as fractions, comparing fractions, and equivalent fractions. The bundle is broken into 3 files based on the standards. You can check it out by clicking the image below.

3rd Grade Fractions Assessments

Throughout the year, I spiral back to standards to continue to check in on students’ mastery. I’ve found that after we’ve moved on from a skill, some students forget some things without ongoing exposure. There’s also those kids that don’t master it the first time that need continued instruction and practice. I use my 3rd Grade Fractions Assessments in Google Forms as both a formative assessment tool and as continued practice throughout the year. I love to assign these on e-learning days (we have several scheduled throughout the year and then also for snow days). It doesn’t take students very long, they get immediate feedback, and it’s easy for me to quickly grade the open-ended portions. I can then use the results to form reteaching groups.

My 3rd Grade Fractions Assessments include two Google Forms for each standard- perfect for pre and post assessments or as an assessment or spiraled practice later in the year. It also includes 4 Google Forms that review all of the fractions standards on each Form. These are the versions I use most often for spiral review practice.

With versions for each fractions skill and standard, and for spiral review, these Google Forms assessments are my go-to’s throughout the year!

Fractions of a Shape Game

5 in a Row is a fun, fast paced game that students literally BEG to play! In this free Fractions of a Shape game , students identify the fraction represented in the given shape and match it on their game boards. You can adjust the timing of the game to give students more or less time to play. 5 in a Row keeps kids engaged with its speed. Because it’s self-running, you have a quick second to check an email or manage the paper stack on your desk. To play, all you need is Powerpoint! Students play on printed gameboards (you choose to laminate and reuse or not). You launch the slideshow, click a button to randomize the slides, and push start. That’s it! you can download my free Fractions of a Shape 5 in a Row by clicking the image below.

You may also be interested in my Fractions & Mixed Numbers on a Number Line 5 in a Row! 4 different versions are included to provide and remove scaffolding.

Pizza Pandas from Arcademics

If you aren’t familiar with Arcademics.com, formerly known as Arcademic Skill Builders, you should take a few minutes to explore it! They offer individual and group based online games for a variety of skills. Pizza Pandas is a fun, fast paced game that has students matching fractions into pizza representations of fractions of a shape. This would be a fun and engaging game to add to your small groups since students could play against each other live.

Fractions on a Number Line Song from NUMBEROCK

NUMBEROCK has a multitude of math songs available for a variety of grade levels and math standards. There are several videos that would work well with your fractions unit, but my personal favorite fractions resource is their Fractions on a Number Line song. It’s catchy and my students love singing it as they work with fractions on a number line. You can watch Fractions on a Number Line , and so many of their other videos, on their YouTube channel. They also have the video hosted on Vimeo for easy sharing, along with lyrics and other helpful information, on the NUMBEROCK site .

Fractions Task Cards

I love to use task cards in my classroom. One of my favorite ways is Musical Solve. I put task cards up around the room, put on some fun music, and have the kids walk from card to card. I put extras up in the room so there’s typically a buffer if a kid stops at one they’ve already done before. I’ve also seen people do this on the kids’ desks but I usually use spaces around the room or just a desk corner. In all of my 3rd grade Mustache You Task Cards, I include 36 cards . This allows for larger class sizes while still giving a few cards that are available to be done whole group. These 3.NF.1 Fractions of a Shape Task Cards are free in my TpT store. Just click to head over and download them!

I also have Fractions on a Number Line task cards . It includes 3 sets: 1) identifying the fraction given on a number line; 2) partitioning a number line and locating the fraction; 3) fractions greater than 1 on a number line. Throughout the sets, students will also have practice working with whole numbers as fractions. As with my Fractions of a Shape task cards, each of the sets has 36 cards making it perfect for practice throughout several days, or as both a whole group guided practice and independent practice activity. You can get my Fractions on a Number Line Task Cards by clicking the image below.

Fraction Models Task Cards

This  HUGE  freebie includes 392 fraction task cards! With 98 pages, with 4 task cards per page, you have fraction task cards for anything you might need. You can use the task cards to match visual models (fractions of a shape and fractions on a number line) with the corresponding fraction. You can sort the models and/or fractions to the corresponding label (less than 1, equal to 1, greater than 1, and equal to another whole).

Sign up at using this form to receive this giant task card set to your inbox immediately!

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Find is a  powerful formative assessment tool during your fractions unit. And, it’s quick and easy to implement. Give students an open number line with one given fraction. Ask them to find another. What that other fraction is, however, is where the power of Find comes in. The number lines aren’t partitioned so students have to reason through where the fraction should go. The difficultly of Find can be adjusted based on where you are in your unit: identifying fractions on a number line, or when working with mixed numbers and/or fractions greater than one. Fractions on a Number Line is one of the hardest part of our fractions unit in 3rd grade, and Find really helps me hone in on where students need support. It also gives you an indication of their number sense. To take things a step further, ask students to explain their reasoning on either the bottom or the back of the task card. You’ll get valuable information on their thinking which will help you adjust your instruction as you teach.

You can download a set of 10 pages of Find Task Cards for free by filling out the form below. The task cards will be sent after you confirm your email.

Free Fractions on a Number Line Activity

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These 3rd grade fractions resources help my students master fractions each year. Use any of these fraction activities in your classroom? I’d love to hear what you like best in the comments! Looking for resources for other 3rd grade math standards? Check out my 3rd Grade Math Games post.

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Fraction activities to try at home

Teaching fractions can be hard. Really hard. Games are a great way to help make those numerators and denominators stick. Here are some fun fraction games that are sure to boost your child’s understanding and critical thinking skills.

Build a fraction wall

You can use sticks, swim noodles or even strips of paper to compare fractions. Have a go at different ways to make one whole, one half and so on. You can also use these resources to look at equivalent fractions.

Hands on fractions with sticks

Here’s a wonderful activity to try when you are out and about, with some sticks at hand. For more hands-on fraction activities, why not download our  Making Maths Stick activity pack ?

Lego towers

Visualising the quantities represented by fractions is easier (and a lot more fun) when you use LEGO! Lay your blocks out side by side or build towers. How many different fractions can you build?

Fraction Hopscotch

It’s hopscotch—with a fraction games twist! Draw a hopscotch board on the pavement (or outline one with tape on your hallway floor). Label the squares with fractions instead of whole numbers. Children throw a marker and jump to where it lands, then name the equivalent fractions for that square.

Fraction picnic

Get a group of children to each plan a picnic and choose the food they want to bring along. Then the picnic has to be shared fairly amongst everyone. Is everything easy to share everything in the basket? Probably not!

Fraction connect

We all know Connect Four. Here’s a version with fractions! You need to decide on the fractions to be used and label the counters on both sides.The goal is to match not only your colours, but the fractions themselves. For instance, you need four one-fourths in a row, but only three one-thirds. This makes for some lovely strategy.

Fraction Pictionary

Can you draw a fraction—without using any numbers? That’s the challenge of this fractions game. Children can draw single objects divided to represent fractions. They can try more creative approaches too – for example, they might draw three apples and two oranges to represent three-fifths, or a pizza cut into fifths.

Winner keeps all four

Deal two cards each, a numerator and denominator. Then make the largest fraction with the two cards and determine whose fraction is the largest. The winner keeps all four cards, and play continues until the cards are gone.

For example, here 23 is larger than 28.

Domino fractions

Dominoes are like ready-made fractions! Pick two domino tiles and add (or subtract, or multiply, or divide) them. Turn it into a race to see who can calculate the answer first.

We hope you enjoy trying these fun fraction activities. Maths is always more enjoyable and memorable when it’s turned into a game!

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Teach Fractions Through Word Problems

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Teaching fractions can often seem like a daunting task. You may hear many the groan or sigh when you open a book to the section on fractions. This does not have to be the case. In fact, most students will not dread a topic once they feel confident working with the concept. 

The concept of a “fraction” is abstract. Visualizing apart versus a whole is a developmental skill not fully grasped by some students until middle or high school. There are a few ways to get your class embracing fractions, and there are a number of worksheets you can print out to nail the concept home for your students.

Make Fractions Relatable

Children, in fact, students of all ages prefer a hands-on demonstration or an interactive experience to pencil-and-paper math equations. You can get felt circles to make pie graphs, you can play with fraction dice, or even use a set of dominoes to help explain the concept of fractions.

If you can, order an actual pizza. Or, if you happen to celebrate a class birthday, well perhaps make it a "fraction" birthday cake. When you engage the senses, you have a higher engagement of the audience. Also, the lesson has a great chance of permanence, too.

You can print fraction circles so your students can illustrate fractions as they learn. Have them touch the felt circles, let them watch you create a felt circle pie representing a fraction, ask your class to color in the corresponding fraction circle. Then, ask your class to write the fraction out.

Have Fun with Math

As we all know, not every student learns the same way. Some children are better at visual processing than auditory processing. Others prefer tactile learning with hand-held manipulatives or may prefer games.

Games make what could be a dry and boring topic more fun and interesting. They provide that visual component that might make all the difference. 

There are plenty of online teaching tools with challenges for your students to use. Let them practice digitally. Online resources can help solidify concepts.

Fraction Word Problems

A problem is, by definition, a situation that causes perplexity. A primary tenet of teaching through problem-solving is that students confronted with real-life problems are forced into a state of needing to connect what they know with the problem at hand. Learning through problem-solving develops understanding.

A student's mental capacity grows more complex with time. Solving problems can force them to think deeply and to connect, extend, and elaborate on their prior knowledge. 

Common Pitfall

Sometimes you can spend too much time teaching fraction concepts, like "simplify," "find the common denominators," "use the four operations," that we often forget the value of word problems. Encourage students to apply their knowledge of fraction concepts through problem-solving and word problems. 

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12 Fun Hands-On Activities for Teaching Fractions Your Kids Will Absolutely Love!

by Sara Ipatenco

Fractions are an essential part of math curriculum for students of all ages. However, many students struggle with fractions, and, let’s be honest, fractions aren’t the easiest thing for teachers to teach either. But never fear! Fractions don’t have to cause a struggle for you or your students if you introduce a few fun activities into your math lessons.

Here are some fun, hands-on ideas to get you started!

1. Use dough

Dough is a great tool for fine motor skill development, but it is also useful when teaching fractions. Give each student a ball of dough and have them flatten it. Call out some fractions, such as one-half or one-fourth, and have them use a plastic knife to cut their dough to show each fraction.

2. Cut paper plates

Have your students decorate several paper plates to look like their favorite kind of pie or cake. Then cut the plates into various fractions, such as into two pieces for halves. You can do several activities once the plates are cut. Give your students examples of fractions and they have to find the corresponding pie. For example, you could say one-third, and your students will need to find their pie cut into thirds and hold up one piece. Your students can also play this game with partners or in small groups. You might also have your students label each part with the corresponding fraction.

3. Have a snack

Kids love snacks! Use that to your advantage by teaching fractions with food. Provide your students with some paper labels of the fractions you’re working on and then have them sort foods, such as pre-sliced fruit, into groups so they match each fraction label.

4. Sort small items

Decide what fraction you need your students to work on and then provide work mats with circles divided into that number of sections. Provide small items, such as cereal, novelty erasers, or beads, and have your students show you what fractions look like if they are sorted into groups. This lesson can also help reinforce different fractions that are actually equal.

5. Fold paper

One of the easiest activities to teach fractions only requires regular paper! Simply give your students several pieces of blank paper and have them fold them into equal parts. Once folded, your students can label each section to visually represent each fraction. Make it even more fun by assigning a different color of paper to each fraction so your students have another way of visually recognizing and identifying fractions.

6. Play dominoes

Dominoes are perfect for learning and practicing fractions because they are already divided in half, which looks just like fractions. Your students can use dominoes to translate the dots into numerical fractions and then identify whether the fractions are proper or improper. The lesson can go further for older students because they can turn the improper fractions into their proper form.

7. Draw with sidewalk chalk

Take your math class outside and practice drawing fractions with sidewalk chalk. You can do this in two ways. Give your students a fraction, such as one-fourth, and have them draw a picture depicting that fraction. Another option is to show them the picture of a fractional part and have them write the numerical equivalent. Partner your students up and have them give each other fraction quizzes, allowing them to draw or write their answers with sidewalk chalk.

8. Write story problems

Story problems are an essential part of any math curriculum, and you don’t have to stop using them when you get to fractions. In fact, having your students write their own story problems is an effective way to help cement the concept of fractions in their brains. Make it even more fun (and let them do your grading for you) by asking your students to switch papers and solve each other’s word problems.

9. Make some pizza

This is one of those fractions activities that’ll make you hungry! Paper pizza can be (almost) as enjoyable as actual pizza – and it can help your students learn their fractions. Give each student a blank paper pizza crust and have them divide it into a certain number of pieces, such as 4 or 6. Then have them use paper to create toppings for their pizza. Explain that each slice of their pizza has to be different than the other slices. When they are done making their pizza, they can write down what fraction of the pizza has pepperoni or black olives.

10. Play with Legos

All kids love Legos, and they lend themselves well to learning fractions. Ask your students to make a stack of Legos using two colors and then have them identify which fraction of their tower is each color. Students can sort the Legos into piles depending on the number of studs on each block to show fractions. A block with eight studs, for example, can be considered a whole while a block with four studs would be one-half. Also, check out this list of 18 outstanding math activities with Lego .

11. Use paint chips

Paint chips are a low-prep way to teach and practice fractions. A quick trip to the home improvement store is all it takes to get this lesson ready to go. Grab a stack of several paint chips – most are separated into three or four sections. Have your students label each section of the paint chip with one-third or one-fourth. Extend the lesson and have students draw a picture and write the words for each fraction.

12. Name fractions

Have each student draw and decorate their name in large letters however they would like. Then ask students to draw boxes under each letter of their name and write a fraction of what each letter is compared to how many letters are in their name. For example, for a five-letter name, each letter would be one-fifth. Challenge students to then write down what fraction of their name is consonants and what fraction is vowels. Older students should reduce any unsimplified fractions. Partner your students up and see if they can come with some addition problems to go with their fractions.

When fractions are fun, you’ll get more buy-in from your students, which will make it more enjoyable for everyone. Even better, these hands-on activities will show your students the fractions in the real world around them.

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A revolutionary, bold educational endeavor for Belize

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When 14-year-old Jahzhia Moralez played a vocabulary game that involved jumping onto her friend like a backpack, she knew Itz'at STEAM Academy wasn’t like other schools in Belize. Transferring from a school that assigned nearly four hours of homework every night, Moralez found it strange that her first week at Itz'at was focused on having fun. 

“I was very excited,” Moralez says. “I want to be an architect or a vet, and this school has the curriculum for that and other technology-based stuff.”

The name “Itz’at” translates to “wise one” in Maya, honoring the local culture that studied mathematics and astronomy for over a thousand years. Launched in September 2023, Itz’at STEAM Academy is a secondary school that prepares students between the ages of 13 and 16 to build sustainable futures for themselves and their communities, using science, technology, engineering, arts, and mathematics (STEAM). The school’s mission is to create a diverse and inclusive community for all, especially girls, students with special educational needs, and learners from marginalized social, economic, and cultural groups.

The school’s launch is the culmination of a three-year project between MIT and the Ministry of Education, Culture, Science, and Technology of Belize. “The Itz’at STEAM Academy represents a revolutionary and bold educational endeavor for us in Belize,” a ministry representative says. “Serving as an institution championing the pedagogy of STEAM through inventive and imaginative methodologies, its primary aim is to push the boundaries of educational norms within our nation.”

Itz’at is one of the first Belizean schools to use competency-based programs and individualized, authentic learning experiences. The Itz’at pedagogical framework was co-created by MIT pK-12 — part of MIT Open Learning — with members of the ministry and the school. The framework’s foundation has three core pillars: social-emotional and cultural learning, transdisciplinary academics, and community engagement.

“The school's core pillars inform the students' growth and development by fostering empathy, cultural awareness, strong interpersonal skills, holistic thinking, and a sense of responsibility and civic-mindedness,” says Vice Principal Christine Coc.

Building student confidence and connecting with community

The teaching and learning framework developed for Itz’at is rooted in proven learning science research. A student-centered, hands-on learning approach helps students develop critical thinking, creativity, and problem-solving skills. 

“The curriculum places emphasis on fostering student competence and cultivating a culture where it's acceptable not to have all the answers,” says teacher Lionel Palacio.

Instead of measuring students’ understanding through tests and quizzes, which focus on memorization of content, teachers assess each stage of students’ project-based work. Teachers are reporting increased student engagement and deeper understanding of concepts.

“It’s like night and day,” says Moralez’s father, Alejandro. “I enjoy seeing her happy while working on a project. She’s not too stressed.”

The transdisciplinary approach encourages students to think beyond the boundaries of traditional school subjects. This holistic educational experience reinforces students’ understanding. For example, Moralez first learned about conversions in her Quantitative Reasoning course, and later applied that knowledge to convert centimeters to kilometers for a Belizean Studies project.

Students are also encouraged to consider their roles in and outside of school through community engagement initiatives. Connections with outside organizations like the Belize Zoo and the Belize Institute of Archaeology open avenues for collaboration and mutual growth.

“We have seen a positive impact on students’ confidence and self-esteem as they take on challenges and see the real-world relevance of their learning,” says Coc. 

Assignments that engage in real-world problem-solving are practical, offering students insight into future careers. The school aims to create career pathways to strengthen Belize’s existing industries, such as agriculture and food systems, while also supporting the development of new ones, such as cybersecurity.

Students’ sense of belonging is readily apparent to teachers, which positively correlates with their learning. “There's a noticeable companionship among students, with a willingness to assist one another and an openness to the novel learning approach,” says Palacio.

Parents see the impact of the safe learning environment that Itz’at creates for their children. Izaya Lovell, for example, gets to embrace his whole self. “I get to speak my mother tongue, Kriol,” he says. “I can be like my dad — get dreads and grow out my hair. I can play sports and be physical.”

Izaya’s mother, Odessa Lovell, says her son was a completely different person after one month of studying at Itz’at. “He’s so independent, he’s saving money, and he’s doing things on his own,” she says.

A vision for Belize

The development of Itz’at emerged from a 2019 agreement between MIT's Abdul Latif Jameel World Education Lab (J-WEL) and the ministry for the implementation of a STEAM laboratory school in Belize, with funding from the Inter-American Development Bank. MIT had a proven track record of projects and partnerships that transformed education globally. For example, MIT collaborated with administrators in India, which trained 3,300 teachers to launch a large-scale education system focusing on hands-on learning and competencies in values, citizenship, and professional skills that would prepare Indian students for further academic studies or the workforce. The Belize program is the first time that groups across the Institute have come together to develop a school from the ground up, and MIT pK-12 led the charge.

“One of the key aspects of the project has been the approach to co-design and co-creation of the school,” says Claudia Urrea, principal investigator for the Itz’at project at MIT and senior associate director of MIT pK-12. “This approach has not only allowed us to create a relevant school for the country, but to build the local capacity for innovation to sustain beyond the time of the project.”

Working with an extended team at MIT and stakeholders from the ministry, the school, parents, the community, and businesses, Urrea oversaw the development of the school’s mission, vision, values, governance structure, and internship program. The MIT pK-12 team — Urrea; Emily Glass, senior learning innovation designer; and Joe Diaz, program coordinator — led a collaborative effort on the school’s pedagogical framework and curriculum. Other core MIT team members include Brandon Muramatsu, associate director of special projects at Open Learning, and Judy Perry, director of the MIT Scheller Teacher Education Program, who created operational guidance for finances, policies, and teacher professional development. By sharing insights with J-WEL, the MIT pK-12 team is fueling shared thinking and innovations that improve students’ learning and pathways from early to higher education to the workforce. 

Like the students, this is the Belizean teachers’ first experience with project-based learning. The MIT team shared the skills, mindsets, and practical training needed to achieve the school’s core values. The professional development training was designed to build their capacity, so they feel confident teaching this model to students and future educators. 

Itz’at currently has 64 students, with plans to reach full capacity of 300 students by 2026. The goal is to continue to build capacity toward STEAM education in the country, expand the possibilities available to students after graduation, and foster a robust school-to-career pipeline. 

“The opening of this school marks a pioneering milestone not just within Belize but also across the broader Central American and Caribbean regions,” a ministry spokesperson says. “We are excited about the future of Itz’at STEAM Academy and the success of its students.”

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Lambda Calculus

COMP 105 Assignment

Due Wednesday, April 17, 2019 at 11:59PM

Lambda calculus is not just a universal model of computation—it is also a language you can use to communicate with educated people around the world. In this assignment,

• You use lambda calculus to write simple functions
• You implement lambda calculus using substitution, reduction, and alpha-conversion

Substitution, reduction, and alpha-conversion are found all over programming-language semantics, not just in lambda calculus.

TL;DR: download the template solution and compile with compile105-lambda . Everything will be fine.

Behind the curtain: For the first part, coding in lambda calculus, you will code things from scratch. For the second part, implementing lambda calculus, you will extend an interpreter I’ve written. But because you can work with ML modules now, you won’t be stuck modifying a huge pile of code. Instead, you’ll define several modules, for both implementation and testing, and you’ll use several of my interfaces.

The ML module system is nice, but Moscow ML’s module bureaucracy is not at all nice. I’ve hidden the bureaucracy behind a shell script, compile105-lambda . This script lives in /comp/105/bin , and if you run use comp105 at the command line, you have access to it. But if something goes wrong, you may wish to know about the pieces of the assignment. Here are the source codes: 1

church.lam Your solutions to the first part solution.sml Your module implementing terms, substitution, and reduction client.sml Your module demonstrating term functions string-tests.sml Test cases for your classmates’ code subst-tests.sml Test cases for substitution link-lambda.sml Instructions for linking your code with mine link-lambda-a.sml More instructions for linking your code with mine link-lamstep.sml Even more instructions for linking your code with mine

Using these sources, the compile105-lambda script will create binaries:

./run-solution-unit-tests Runs some of your unit tests ./run-client-unit-tests Runs more unit tests ./run-string-tests Runs more unit tests ./run-subst-tests Runs the last of your unit tests ./linterp Runs your complete interpreter (normal-order reduction) ./lamstep Runs your interpreter, showing each reduction ./linterp-applicative Runs your complete interpreter (applicative-order reduction)

Learning about the lambda calculus

There is no book chapter on the lambda calculus. Instead, we refer you to these resources:

The edited version of Raúl Rojas’s “ A Tutorial Introduction to the Lambda Calculus ” is short, easy to read, and covers the same points that are covered in lecture:

• Free and bound variables
• Capture-avoiding substitution
• Addition and multiplication with Church numerals
• Church encoding of Booleans and conditions
• The predecessor function on Church numerals
• Recursion using the Y combinator

Rojas doesn’t provide many details, but he covers everything you need to know in 9 pages, with no distracting theorems or proofs.

When you want a short, easy overview to help you solidify your understanding, Rojas’s tutorial is your best source.

I have written a short guide to coding in Lambda calculus . It shows how to translate ML-like functions and data, which you already know how to program with, into lambda calculus.

When you are solving the individual programming problems, this guide is your best source.

Prakash Panangaden’s “ Notes on the Lambda-Calculus ” cover the same material as Rojas, but with more precision and detail. Prakash is particularly good on capture-avoiding substitution and change of bound variables, which you will implement.

Prakash also discusses more theoretical ideas, such as how you might prove inequality (or inequivalence) of lambda-terms. And instead of just presenting the Y combinator, Prakash goes deep into the ideas of fixed points and solving recursion equations—which is how you achieve recursion in lambda calculus.

When you are getting ready to implement substitution, Prakash’s notes are your best source.

I have also written a short guide to reduction strategies . It is more useful than anything that could be found online in 2018. As a bonus, it also explains eta-reduction , which is neglected by other sources.

When you have finished implementing substitution and are ready to implement reduction, this guide is your best source.

Wikipedia offers two somewhat useful pages: 2

The Lambda Calculus page covers everything you’ll find in Rojas and much more besides. (If you wish, you can read what Wikipedia says about reduction strategies and evaluation strategies. But do not expect to be enlightened.)

The Church Encoding page goes into more detail about how to represent ordinary data as terms in the lambda calculus. The primary benefit relative to Rojas is that Wikipedia describes more kinds of arithmetic and other functions on Church numerals.

You need to know that the list encoding used on Wikipedia is not the list encoding used in COMP 105 . In order to complete the homework problems successfully, you must use the list encoding described in the guide to coding in lambda calculus .

Introduction to the lambda interpreter

You will implement the key components of an interactive interpreter for the lambda calculus. This section explains how to use the interpreter and the syntax it expects. A reference implementation of the interpreter is available in /comp/105/bin/linterp-nr .

The syntax of definitions

Like the interpreters in the book, the lambda interpreter processes a sequence of definitions. The concrete syntax is very different from the “bridge languages” in the book. Every definition must be terminated with a semicolon. Comments are line comments in C++ style, starting with the string // and ending at the next newline.

The interpreter supports four forms of definition: a binding, a term, the extended definition “ use ”, and an extended definition “ check-equiv ”.

A binding is something like a val form in μ Scheme. A binding has one of two forms: either

In both forms, every free variable in the term must be bound in the environment—if a right-hand side contains an unbound free variable, the result is a checked run-time error. The first step of computation is to substitute for each of the free variables: each occurrence of each free variable is replaced by that variable’s definition.

In the first form, where noreduce appears, no further computation takes place. The substituted right-hand side is simply associated with the name on the left, and this binding is added to the environment.

The noreduce form is intended only for terms that cannot be normalized, such as

The noreduce form is also needed for definitions that use terms like bot  and  Y .

If noreduce is absent, the interpreter substitutes for free variables, then reduces the term on the right until there are no more beta-redexes or eta-redexes. (You will implement the two reduction strategies presented in class.) If reduction doesn’t terminate, the interpreter might loop.

Loading files with use

The use extended definition loads a file into the interpreter as if it had been typed in directly. It takes the form

Comparing normal forms with check-equiv

The check-equiv form immediately reduces two terms to normal form and compares them for equivalence. It has the form

And here are some examples:

Unlike the check-expect in the other interpreters, check-equiv is not “saved for later”—the given terms are normalized right away.

Terms as definitions

As in the book, a term can be entered at the read-eval-print loop, just as if it were a definition. Every free variable in the term is checked to see if it is bound in the environment; if so, each free occurrence is replaced by its binding. Free variables that are not bound in the environment are permissible; they are left alone. 3 The term is reduced to normal form (if possible) and the result is printed.

The syntax of terms

A lambda term can be either a variable, a lambda abstraction, an application, or a parenthesized lambda term. Precedence is as in ML.

A lambda abstraction abstracts over exactly one variable; it is written as follows:

Application of one term to another is written:

The lambda interpreter is very liberal about names of variables. A name is any string of characters that contains neither whitespace, nor control characters, nor any of the following characters: \ ( ) . = / . Also, the string use is reserved and is therefore not a name. But a name made up entirely of digits is OK; the lambda calculus has no numbers, and names like 105 have no special status.

As examples, all the following definitions are legal:

A short example transcript

A healthy lambda interpreter should be capable of something like the following transcript:

For more example definitions, see the predefined.lam file distributed with the assignment.

Software provided for you

Both capture-avoiding substitution and normal-order reduction can be tricky to implement. 4 So that you may have a solid foundation on which to write your lambda code, I provide an interpreter linterp-nr . Running use comp105 should give you access to that interpreter.

Even with a correct interpreter, lambda code can be hard to debug. So I also provide an interpreter called lamstep-nr , which shows every reduction step. Some computations require a lot of reduction steps and produce big intermediate terms. Don’t be alarmed.

All questions and problems

There are four problems on programming with Church numerals, which you’ll do on your own.

There are four problems on implementing the lambda calculus, which you can do with a partner. Your solutions will go into a Standard ML module, which you will link with the rest of the interpreter.

Reading comprehension

These problems will help guide you through the reading. We recommend that you complete them before starting the other problems below. You can download the questions .

(NOT ON THE READING.) Throughout the term, your code’s functional correctness has been assessed by automated testing. The automated test scripts are intended not only to assign a grade but to identify the most important fault in the code. Please answer these two questions:

How did you benefit from the feedback you received about functional correctness?

What were the drawbacks, if any, of the feedback you received about functional correctness?

Syntax of lambda terms . In this assignment, or in Rojas or Panangaden, read about the concrete syntax of lambda-terms . Now define, in Standard ML, an algebraic data type term that represents the abstract syntax of terms. Your data type should have one value constructor for a variable, one for a lambda abstraction, and one for an application.

You are ready for exercise 5, and you have a foundation for exercises 6 and 8.

Recognizing redexes . Read about redexes in Wikipedia . (You will then follow up with Panangaden .)

Wikipedia mentions two kinds of redex. What are their names?

In Panangaden, Definition 1.7 defines a redex. Which of the two redexes mentioned in Wikipedia is being defined here?

Your code will have to recognize redexes, and it starts with knowing the form of each kind. As of Spring 2019, both forms are shown in Wikipedia. But if Wikipedia changes, one form can be found in Panangaden; for the other, look in the last section of my guide to reduction strategies.

For each of the two kinds of redex, use the concrete syntax for our lambda interpreter ( see above ) to show what form every redex of that kind takes.

For each of the two kinds of redex, use your algebraic data type from the preceding question to write a pattern that matches every redex of that kind.

You are getting ready for exercise 8 (reductions).

Practicing reduction . Read about reduction on Wikipedia. Then in Panangaden , be sure you have an idea about each of these concepts:

Capture-avoiding substitution (Definition 1.3)

Reduction (Definition 1.5), including the example reduction (Example 1.3)

Redex , contractum , and normal form (Definitions 1.7 and 1.8)

Showing each reduction step, reduce the following term to normal form. At each step, choose a redex and replace the redex with its contractum. Do not expand or replace the names ZERO and NONZERO .

The term contains more than one redex, but no matter which redex you choose at each step, you should reach the normal form after exactly four reductions.

You are preparing to complete exercise 8.

Reduction: the general case . For each kind of redex, repeat the general form of the redex from question 2(c) 3(c) above, then show what syntactic form the redex reduces to (in just a single reduction step).

When to reduce . Read my handout on reduction strategies . Using the concrete syntax accepted by the interpreter (and defined above) , write a lambda term that contains exactly two redexes, such that normal-order reduction strategy reduces one redex, and applicative-order reduction strategy reduces the other redex.

You are (finally!) ready for exercise 8.

Understanding Church numerals . You may recognize the practice reduction above as a computation that tells if a Church numeral is zero. Read about Church numerals, either on pages 9 and 10 of Panangaden or in Section 2 of Rojas (“Arithmetic”). Then, say whether each of the following lambda-calculus terms is a Church numeral. If so, write the corresponding decimal representation. If not, write “not a Church numeral”.

Programming in the lambda calculus (individual problems)

These problems give you a little practice programming in the lambda calculus. Most functions must terminate in linear time , and you must do these exercises by yourself. You can use the reference interpreter linterp-nr .

Lambda-calculus programs work at the same intellectual level as assembly-language programs. Therefore, every new helper function must be well named and must be accompanied by a contract. Detailed guidance can be found below .

Helper functions listed in the assignment are exempt from the contract requirement, as are the helper functions in predefined.lam .

Complete all four problems below, and place your solutions in file church.lam .

Not counting code copied from the lecture notes, my solutions to all four problems total less than fifteen lines of code. And all four problems rely on the same related reading.

Related reading for lambda-calculus programming problems 1 to 4:

My guide Coding in Lambda Calculus should explain everything you need to know to write functional programs in lambda calculus. If not, or if the explanations there are a little too terse, consult the additional readings below.

Basic techniques can be found in Wikipedia on Church Encoding and in section 2 of Panangaden , which is titled “Computing with Lambda Calculus” (from page 8 to the middle of page 10). These basics are sufficient for you to tackle problems 1 and 2.

Another alternative is Section 2 of Rojas’s tutorial, entitled “arithmetic.” Rojas doesn’t mention Church numerals by name, but that’s what he’s working with. You may find the examples useful and the presentation more accessible than what you see from Panangaden.

On problems 3 and 4 only, if you have the urge to write a recursive function, you may use a fixed-point combinator. My guide ends with a few pages on recursion. You may also wish to consult the first paragraph under “Fixed-Point Combinators” on page 10 of Panangaden . This explanation is by far the best and simplest explanation available—but it is very terse. For additional help, consult the examples on page 11.

I recommend against the Wikipedia “main article” on fixed-point combinators: the article is all math all the time, and it won’t give you any insight into how to use a fixed-point combinator.

1. Church Numerals—parity . Without using recursion or a fixed-point combinator , define a function even? which, when applied to a Church numeral, returns the Church encoding of true or false , depending on whether the numeral represents an even number or an odd number.

Your function must terminate in time linear in the size of the Church numeral.

Ultimately, you will write your function in lambda notation acceptable to the lambda interpreter, but you may find it useful to try to write your initial version in Typed μScheme (or ML or μ ML or μScheme) to make it easier to debug.

Remember these basic terms for encoding Church numerals and Booleans:

You can load these definitions by typing use predefined.lam; in your interpreter.

2. Church Numerals—division by two . Without using recursion or a fixed-point combinator , define a function div2 which divides a Church numeral by two (rounding down). That is, div2 applied to the numeral for 2 n returns  n , and div2 applied to the numeral for 2 n  + 1 also returns  n .

We don’t know if this one can be done in linear time, but it is sufficient if your function terminates in time quadratic in the size of the Church numeral.

Hint : Think about function split-list from the Scheme homework , about the implementation of the predecessor function on natural numbers, and about the “window” example from recitation.

3. Church Numerals—conversion to binary. Implement the function binary from the Impcore homework . The argument and result must be Church numerals. For example,

For this problem, you may use the Y combinator. If you do, remember to use noreduce when defining binary , e.g.,

This problem, although not so difficult, may be time-consuming. If you get bogged down, go forward to the list-selection problem ( nth ), which can benefit from similar skills in recursion, fixed points, and Church numerals. Then come back to this problem.

Your function must terminate in time quadratic in the size of the Church numeral.

EXTRA CREDIT . Write a function binary-sym that takes three arguments: a name for zero, a name for one, and a Church numeral. Function binary-sym reduces to a term that “looks like” the binary representation of the given Church numeral. Here are some examples where I represent a zero by a capital  O  (oh) and a one by a lower-case  l  (ell):

It may help to realize that l l O l is the application (((l l) O) l) —it is just like the example E 1 E 2 E 3 … E n in the first section of Rojas’s tutorial .

Function binary-sym has little practical value, but it’s fun. If you write it, please put it in your church.lam file, and mention it in your README file.

4. Church Numerals—list selection . Write a function nth such that given a Church numeral n and a church-encoded list xs of length at least n +1, nth n xs returns the n th element of xs :

To get full credit for this problem, you must solve it without recursion. But if you want to define nth as a recursive function, use the Y combinator, and use noreduce to define nth .

Provided xs  is long enough, function  nth must terminate in time linear in the length of the list. Don’t even try to deal with the case where xs is too short.

Hint: One option is to go on the web or go to Rojas and learn how to tell if a Church numeral is zero and if not, and how to take its predecessor. There are other, better options.

Implementing the lambda calculus (possibly with a partner)

For problems 5 to 8 below, you may work on your own or with a partner. These problems help you learn about substitution and reduction, the fundamental operations of the lambda calculus. The first problem also gives you a little more practice in using continuation-passing to code an algebraic data type, which is an essential technique in lambda-land.

For each problem, you will implement types and functions described below. When you are done, the compile105-lambda script will link your code with mine to build a complete lambda interpreter. To simplify the configuration, most of the functions and types you must define will be placed in a module called SealedSolution , which you will implement in a single file called solution.sml . The module must be sealed with this interface:

You can download a template solution .

5 . Evaluation—Coding terms .

In your file solution.sml , create an ML type definition for a type term , which should represent a term in the untyped lambda calculus. Using your representation, define functions lam , app , var , and cpsLambda .

Compile this file by running compile105-lambda (with no arguments), then run any internal unit tests by running ./run-solution-unit-tests .

My solution is under 15 lines of ML code.

Related reading: The syntax of lambda terms in this homework.

6 . Evaluation—Substitution . In file solution.sml , implement capture-avoiding substitution on your term representation. In particular,

Define function freeIn of type string -> term -> bool , which tells if a given variable occurs free in a given term. (If you adapt your solution to the pair problem on the ML homework, or my model solution to that problem, acknowledge your sources!)

Define function freeVars of type term -> string list , which returns the variables free in a given term. The list must have no duplicates.

Define function subst of type string * term -> term -> term . Calling subst ( x , N ) M returns the term M [ x  ↦  N ] (“ M  with x goes to  N ”).

Function subst obeys these algebraic laws, 5 in which x  and  y stand for variables, and N  and  M stand for terms:

• subst ( x , N ) x = N
• subst ( x , N ) y = y , provided y is different from  x
• subst ( x , N ) ( M 1 M 2 ) = ( subst ( x , N ) M 1 ) ( subst ( x , N ) M 2 )
• subst ( x , N ) ( λ x . M ) = ( λ x . M )
• subst ( x , N ) ( λ y . M ) = λ y .( subst ( x , N ) M ) , provided x  is not free in  M or y  is not free in  N , and also provided y is different from  x

If none of the cases above apply, then subst ( x , N )  M should return subst ( x , N )  M ′ , where M ′  is a term that is obtained from  M by renaming bound variables. Renaming a bound variable is called “alpha conversion.”

You need to rename bound variables only if you encounter a case that is like case (e), but in which x  is free in  M and y  is free in  N . In such a case, subst ( x ,  N )  ( λ y . M ) can be calculated only by renaming  y , which is bound in the lambda abstraction, to some new variable that is not free in M  or  N .

To help you implement subst , you may find it useful to define this helper function:

• Function freshVar , which is given a list of variables and produces a variable that is different from every variable on the list

By using freshVar on the output of freeVars , you will be able to implement alpha conversion.

To test this problem, you have three possible approaches:

The next problem demands a minimal set of test cases. You can stick with this set and choose not to worry about further testing.

You can add Unit tests to your solution.sml file. You would then compile it by running compile105-lambda with no arguments, and run the binary ./run-solution-unit-tests that results.

You can also build and run the full interpreter ./linterp , again by running compile105-lambda without arguments. But be warned: you may see some alarming-looking terms that have extra lambdas and applications. This is because the interpreter uses lambda to substitute for the free variables in your terms. Here’s a sample:

Everything is correct here except that the code claims something is in normal form when it isn’t. If you reduce the term by hand, you should see that it has the normal form you would expect.

My solution to this problem is just under 40 lines of ML code.

Related reading:

Panangaden describes free and bound variables in Definition 1.2 on page 2. He defines substitution in Definition 1.3 on page 3. (His notation is a little different from our ML code, but the laws for subst are the same.)

In his Definition 1.3, case 6, plus Definition 1.4, Panangaden explains the “change of bound variables” that you need to implement if none of the cases for subst apply.

Page 470 of your book defines an ML function freshName which is similar to the function freshVar that you need to implement. The freshName on page 470 uses an infinite stream of candidate variables. You could copy all the stream code from the book, but it will probably be simpler just to define a tail-recursive function that tries an unbounded number of variables.

Don’t emulate function freshtyvar on page 517. It’s good enough for type inference, but it’s not good enough to guarantee freshness in the lambda calculus.

7 . Substitution tests . As shown in the previous problem, function subst has to handle five different cases correctly. It also has to handle a sixth case, in which none of the laws shown above applies, and renaming is required. In this problem, you create test cases for your subst function. They should go into a file subst-tests.sml , which should look like this:

To run these tests, run compile105-lambda without arguments, then run the resulting binary ./run-subst-tests .

8 . Evaluation—Reductions . In this problem, you use your substitution function to implement two different reduction strategies, called reduceN and reduceA .

A reduction strategy is a function that takes a term  M and produces a one of the following two values:

• Reduction.DOESN'T_STEP , if it is not possible to reduce  M
• Reduction.ONE_STEPS_TO  N , if  M  reduces to  N in a single step

The relation “ M  reduces to  N in a single step” is written M  →  N , and it is explained in the handout on reduction strategies as well as in many other sources on the lambda calculus.

Each function takes a term and tries to perform a single reduction step, using any rule that applies: Beta , Eta , Mu , Nu , or Xi . (The rules are shown in the handout on reduction strategies.) Each function is specified as follows:

Function reduceN implements normal-order reduction: it tries the leftmost, outermost redex first. In other words, it prefers Beta over Nu and Nu over Mu ).

Function reduceA implements applicative-order reduction: it uses the Beta rule only when the argument is normal form. In other words, it prefers Mu over Beta .

Both functions must also implement Eta reduction.

To compile and test this code, run compile105-lambda without arguments, then test using ./linterp (normal-order reduction), ./lamstep (normal-order reduction, showing each step), and ./linterp-applicative (applicative-order reduction). You may also wish to consult the hints below.

I’ve written two solutions to this problem. One solution uses only first-order functions: it implements reduceN and reduceA directly, by extensive case analysis. My first-order solution is about 30 nonblank lines of ML code. The other solution uses higher-order functions to define reduceN and reduceA . It implements each rule as its own function, then combines them using the >=> operator described below. My higher-order solution is about 25 nonblank lines of ML code.

Start with my guide, “ Reduction Strategies for Lambda Calculus .”

For implementation, read the Hints on the implementation of reduction section below.

Consider consulting Panangaden , who describes the reduction relation in Definition 1.5. Although he treats it as a mathematical relation, not a computational rule, you may find his definitions helpful. But some commentary is required:

Rules α  (change of variables) and ρ  (reflexivity) have no computational content and should therefore play no part in reduceN or reduceA . (Rule  α plays a part in subst .)

Rule τ  (transitivity) involves multiple reductions and therefore also plays no part in reduceN or reduceA .

The remaining rules are used in both reduceN and reduceA , but with different priorities.

Rule β is the key rule, and in normal-order reduction, rule  β is always preferred.

In applicative-order reduction, rule μ (reduction in the argument position) is preferred.

In normal-order reduction, rule ν (reduction in the function position) is preferred over rule  μ but not over rule  β .

Finally, Panangaden omits rule  η , which like rule  β is always preferred:

• λ x . M x  →  M , provided x  is not free in  M

You must implement the η rule as well as the other rules.

If you want to know more, or you want a different formulation, go (cautiously) to Wikipedia. Wikipedia describes some individual reduction rules in the Reduction section of the lambda-calculus page. And it briefly describes applicative-order reduction and normal-order reduction, as well as several other reduction strategies, in the reduction strategies section of the lambda-calculus page.

Hints on the implementation of reduction

The return type of reduceA and reduceN is term Reduction.result , where Reduction.result is defined by this interface, which also defines some useful helper functions:

The helper functions rmap , nostep , and >=> are used to implement the second of two possible implementation options:

The first-order option is simply to take a term, break its representation down by cases, and in each case, define a right-hand side that combines all the rules for that case, including the Eta rule. The advantage of this option is it’s concrete, and the programming techniques are ones you’ve been using all along—break down the data by cases, apply the rules. The disadvantage is that there are a lot of cases, and the logic on the right hand side is complicated. Once you’ve written the code, it will be hard to understand and hard to debug. Students choosing this option often forget cases or botch cases.

The higher-order option is to define each rule as its own function, then to compose the functions using the >=> operator in the Reduction module. 6 The advantage of this option is that the construction of the reduction strategy makes it crystal clear what is going on and in what order—it becomes very hard to forget or botch a case. This option also makes it easy to implement and test one rule at a time. The disadvantage of this option is that it is abstract, and it is aggressively higher-order: you are using the >=> arrow to compose simple functions into more complex functions. Understanding the “reducer” abstraction well enough to implement it will take a little time.

Notes on the higher-order option

If you want to try the REDUCTION interface and the higher-order option, here are some notes:

The type called 'a reducer is really a partial reducer: a function of this type implements some sequence of rules. Function nostep implements no rules, and the composition arrow  >=> combines two functions to implement a combined sequence of rules. Your implementation task breaks down into two steps: first, define rule functions; second, compose them.

The rmap function is the classic mapping idea (called “homomorphic”) which you have already seen in List.map and Option.map . It is especially useful in conjunction with curry , as in

You may also find a use for flip .

The composition arrow is mean to be used as an infix operator. In your solution file, copy these definitions:

The most beautiful code emerges if you define functions beta , eta , nu , mu , and  xi , then compose them:

But there’s a problem here: the nu , mu , and xi rules all need the capability of doing general reduction on a subterm, which means they have to be mutually recursive with the reducer. Mutually recursion can be handled in several ways, but the easiest is to define the individual rule functions inside the reducer, in a let binding. This easy way does, however, duplicate code. If you want to avoid the duplication, you can do something like this:

Overall, I think the higher-order option is worth the extra effort needed to understand the reducer type and its composition: when you split each rule into its own function, it’s much, much easier to get them all right. And it’s easy to reuse the same functions in multiple reduction strategies.

Debugging support

As shipped, the lambda-calculus interpreter reduces each term repeatedly, until it reaches a normal form. But when you are debugging reduction strategies, you may find it helpful to see the intermediate terms. The compile105-lambda script should produce an executable program ./lamstep , which will show the results of every reduction step. You can compare this interpreter with the reference version, called lamstep-nr .

Even more debugging support

If the ./lamstep interpreter doesn’t provide enough information (or provides too much), here is a way to print a status report after every n reductions:

I have defined a status function size that prints the size of a term. You can print whatever you like: a term’s size, the term itself, and so on. Here is how I show the size of the term after every reduction. Some “reductions” make terms bigger!

More Extra Credit

Solutions to any of the extra-credit problems below should be placed in your README file. Some may be accompanied by code in your solution.sml file.

Extra Credit. Normalization. Write a higher-order function that takes as argument a reduction strategy (e.g., reduceA or reduceN ) and returns a function that normalizes a term. Your function should also count the number of reductions it takes to reach a normal form. As a tiny experiment, report the cost of computing using Church numerals in both reduction strategies. For example, you could report the number of reductions it takes to reduce “three times four” to normal form.

This function should be doable in about 10 lines of ML.

Extra Credit . Normal forms galore. Discover what Head Normal Form and Weak Head Normal Form are and implement reduction strategies for them. Explain, in an organized way, the differences between the four reduction strategies you have implemented. (If you choose the higher-order option for implementing reduction strategies, this extra credit is easy. Otherwise, not so much.)

Extra Credit . Typed equality. For extra credit, write down equality on Church numerals using Typed uScheme, give the type of the term in algebraic notation, and explain why this function can’t be written in ML. (By using the “erasure” theorem in reverse, you can take your untyped version and just add type abstractions and type applications.)

What and how to submit: Individual work

Using script submit105-lambda-solo , submit

• The names of the people with whom you collaborated
• Any extra credit you may have earned
• File cqs.lambda.txt , containing your answers to the reading-comprehension questions
• File church.lam containing your solutions to the Church-numeral problems, including possibly the binary-sym extra credit

As soon as you have the files listed above, run submit105-lambda-solo to submit a preliminary version of your work. Keep submitting until your work is complete; we grade only the last submission.

What and how to submit: Pair work

Using script submit105-lambda-pair , submit

README Collaborators, extra credit, and so on solution.sml Your module implementing terms, substitution, and reduction subst-tests.sml Test cases for substitution

As soon as you have the files listed above, and all the code compiles, run submit105-lambda-pair to submit a preliminary version of your work. Keep submitting until your work is complete; we grade only the last submission.

Avoid common mistakes

Common mistakes with church numerals.

Here are some common mistakes to avoid when programming with Church numerals:

Don’t forget names and contracts for helper functions.

Don’t forget a semicolon after each definition.

Don’t forget the question mark in the name of even? .

When using a fixed-point combinator to define a function, don’t forget to use noreduce in the definition form.

Don’t use the list representation or primitives from Wikipedia. We will test your code using the representation and primitives from Coding in Lambda Calculus , which you will also find in the file predefined.lam .

Don’t include any use directives in church.lam .

Don’t copy predefined terms from predefined.lam . We will load the predefined terms before running your code.

To make sure your code is well formed, load it using

If you want to build a test suite, put your tests in file test.lam and run

Common mistakes with the lambda interpreter

Here are some common mistakes to avoid in implementing the interpreter:

Don’t forget the Eta rule :

Here is a reduction in two eta steps:

Your interpreters must eta-reduce when possible.

Don’t forget to reduce under lambdas (the Xi rule).

Don’t forget that in an application M 1   M 2 , just because M 1 is in normal form doesn’t mean the whole thing is in normal form. If  M 1 doesn’t step, you must try to reduce  M 2 .

If you are using the first-order implementation option, don’t clone and modify your code for reduction strategies; people who do this wind up with wrong answers. The code should not be that long; use a clausal definition with nested patterns, and write every case from scratch.

Do make sure to use normal-order reduction, so that you don’t reduce a divergent term unnecessarily.

Don’t try to be clever about a divergent term; just reduce it. (It’s a common mistake to try to detect the possibility of an infinite loop. Mr. Turing proved that you can’t detect an infinite loop, so please don’t try.)

When implementing freshVar , don’t try to repurpose function freshTyvar from section 7.6. That function isn’t smart enough for your needs.

How your work will be evaluated

Your ML code will be judged by the usual criteria, emphasizing

• Correct implementation of the lambda calculus
• Names and contracts for helper functions
• Structure that exploits standard basis functions, especially higher-order functions, and that avoids redundant case analysis

Your lambda code will be judged on correctness, form, naming, and documentation, but not so much on structure. In particular, because the lambda calculus is such a low-level language, we will especially emphasize names and contracts for helper functions .

This is low-level programming, and if you don’t get your code exactly right, the only way we can recognize and reward your learning is by reading the code. It’s your job to make it clear to us that even if your code isn’t perfect, you understand what you’re doing.

Try to write your contracts in terms of higher-level data structures and operations. For example, even though the following function does some fancy manipulation on terms, it doesn’t need much in the way of a contract:

Documenting lambda calculus is like documenting assembly code: it’s often sufficient to say what’s happening at a higher level of abstraction.

Although it is seldom ideal, it can be OK to use higher-level code to document your lambda code. In particular, if you want to use Scheme or ML to explain what your lambda code is doing, this can work only because Scheme and ML operate at much higher levels of abstraction. Don’t fall into the trap of writing the same code twice—if you are going to use code in a contract, it must operate at a significantly higher level of abstraction than the code it is trying to document.

In more detail, here are our criteria for names:

And here are our criteria for contracts:

Files link-lambda.sml and link-lambda-a.sml are copied into your directory by the compile105-lambda script. The others are created by you. ↩

They were more useful in 2017 then they are now—as always, Wikipedia pages are subject to change without notice. ↩

Try, for example, (\x.\y.x) A B; . ↩

I have botched capture-avoiding substitution multiple times. ↩

The laws, although notated differently, are identical to the laws given by Prakash Panangaden as Definition 1.3. ↩

This operator is an example of “Kleisli composition,” which is an advanced form of function composition. ↩

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A History of Moscow in 13 Dishes

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Time to Do: To do everything listed below will take approximately 3 days.

1)      What to do in Moscow – Red Square

2)      What to do in Moscow – The Kremlin

It started as a humble wooden garrison on a hill and turned into one of the biggest and best known fortresses in the world. This is both a very impressive tourist attraction and also the seat of power for Russia with the office of the president inside its walls. It is home to several 500 hundred year old cathedrals, the world renowned Faberge Eggs and also the world’s biggest sapphire (not to mention a 160 carat diamond!). If you want to know more I have a separate post on visiting the Red Square and the Kremlin in Moscow .  Nearest metro: Aleksandrovksy Sad- Dark Blue, Red or Grey lines.

3)      What to do in Moscow – Novodevichy Convent and Cemetery

4)      What to do in Moscow – State Tretyakov Gallery

This is the best collection of Russian Art in the world and was started in the 19 th century by Pavel Tretyakov. If you go and you see how many great paintings he managed to get by himself you can’t help but be impressed. The collection today has 130,000 items! Like most museums it is not possible to see everything in a day and as I don’t know much about art I looked at the more famous painters and some of the paintings and icons are amazing. Even being from poor old Ireland I recognised some of the paintings and here they were in Moscow! Artists such as Repin, Vereshchagin, Surikov, Serov etc etc.  They may not be household names to people not in the know in the west but they still are very impressive. The older paintings of landscapes etc are hard to get excited about but the late 19 th century stuff of poverty and social issues are excellent. The State Tretyakov Gallery is not to be confused with the New Tretyakov Gallery (modern art) which is situated close by. Even to spend 3hrs to half a day here is well worth it. Admission is 400 Ru and 200Ru for photography. http://www.tretyakovgallery.ru/en/

5)      What to do in Moscow – Moscow Metro

If you are visiting Moscow you will undoubtedly be riding the metro unless you are crazy enough to contend with the randomness and traffic of the streets. Moscow’s Metro stations are one of the fanciest in the world. They have tributes to war, peace, women, Ukraine, agriculture, the list goes on and each metro is unique. There are no modern contemporary stations built of glass or modern art but they still are a sight to behold. It is well worth giving yourself more time when going somewhere so you can get off when you see a fancy stop. For more info and the best stations see my other post on the beautiful Moscow Metro.

6)      What to do in Moscow – Park Pobedy / Victory Park

This park doesn’t appear high on many lists but you should definitely pay it a visit. Like many memorials in Moscow it commemorates war and this one is victory in WWII or the Great Patriotic War as they call it in Russia. The large park is great to walk around but the main highlight and close to the metro station is the central avenue which has a fountain for everyday of the war and 10cm for each day on a 142mtr (520ft) high obelisk. Behind the obelisk is the Great Patriotic Museum which goes through artefacts and history of the war including plenty of surviving examples of guns, artillery, coats etc etc. There are some English translations for the main parts but not everything is covered. Admission 250Ru. Just outside Park Pobedy metro is the Triumphal Arch which is a huge arch in the middle of the road to commemorate Napoleon’s defeat in 1812. It was this same hill where Napoleon waited in vain to receive the keys to the Kremlin. Nearest Metro: Park Pobedy – Dark Blue line.

7)      What to do in Moscow – Christ the Saviour Cathedral

8)      What to do in Moscow – Peter the Great Statue and Statue Park (Fallen Monument Park)

The statue park is over the Moskva River to the east in Park Rayon Yakimanka and contains weird statues as well as old USSR statues of the heroes.  Nearest Metro: Kropotkinskaya on the red line or Polyanka on the grey line. Both attractions are in-between the two.

9)      What to do in Moscow – Bolshoi Theatre

The most famous theatre in Moscow and one of foremost theatres in the world and even adorns the 100 Rubles banknote. The company also has the largest ballet in the world. Originally built in 1824 and after a very expensive upgrade in 2011 the Bolshoi is very grand indeed (Bolshoi means grand or big in Russian). The theatre is worth a look from the outside but you cannot get in unless you buy a ticket which without the help of touts outside you must do it well in advance online. It is only a 5 min walk north of the Kremlin so very central. http://www.bolshoi.ru/en/ . Nearest Metro: Teatralnaya on the dark green line

10)   What to do in Moscow – Moscow’s Viewpoint

One of the highest points in Moscow is on the south west of the city close to Moscow State University. This point is located in front of one of Stalin’s huge ‘7 sisters’ buildings and is immediately above where the 1980 Olympic Stadium is.  The view point has free viewing telescopes and it is possible to see both the new and old sides of the city. This is not an unbelievable view or anything but it is good and if you get off at Vorobyevy Gory metro station then there is a pleasant 10min walk up the hill through a park which runs alongside the Moskva River. The viewpoint is on (street) Ul. Kosygina and the Nearest Metro is Vorobyevy Gory on the Red line.

Map of Moscow

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Gallery of What to do in Moscow

About Ross Travellingforfun

When I visited Moscow I visited the Fallen Monument Park. I saw the boat with rabbits as pictured above and I wonder if you know what the story is behind it. Thank you!

Judith, the short answer is I don’t know. I tried to find out but couldn’t find anything on it. They do have works from other artists there so maybe it has some obscure meaning. Sorry I can’t be of help.

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How many equal parts of a whole

Slice a pizza, and we get fractions:

The top number says how many slices we have. The bottom number says how many equal slices the whole pizza was cut into .

Have a try yourself:

Equivalent fractions.

Some fractions may look different, but are really the same, for example:

It is usually best to show an answer using the simplest fraction ( 1 / 2 in this case ). That is called Simplifying , or Reducing the Fraction

Numerator / Denominator

We call the top number the Numerator , it is the number of parts we have . We call the bottom number the Denominator , it is the number of parts the whole is divided into .

Numerator Denominator

You just have to remember those names! (If you forget just think "Down"-ominator)

Adding Fractions

It is easy to add fractions with the same denominator (same bottom number):

Another example:

Adding Fractions with Different Denominators

But what about when the denominators (the bottom numbers) are not the same?

We must somehow make the denominators the same.

In this case it is easy, because we know that 1 / 4 is the same as 2 / 8 :

There are two popular methods to make the denominators the same :

• Least Common Denominator , or
• Common Denominator

(They both work nicely, use the one you prefer.)

Other Things We Can Do With Fractions

We can also:

• Subtract Fractions
• Multiply Fractions
• Divide Fractions

Visit the Fractions Index to find out even more.