problem solving lesson 1.9

Go Math Interactive Mimio Lesson 1.9 Problem Solving Multiplication and Division

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Technological Graphing

9.1: It Begins With Data (10 minutes)

CCSS Standards

Building On

Routines and Materials

Required Materials

• Statistical technology

The mathematical purpose of this activity is to gain familiarity with entering data into a spreadsheet and to prepare students for finding statistics using technology.

Arrange students in groups of 2. If students are using the digital version of the materials, show them how to open the GeoGebra spreadsheet app in the math tools. If students are using the print version of the materials, they can access the GeoGebra spreadsheet app at www.geogebra.org/spreadsheet . If they use a different technology, you may need to provide them with alternate instructions.

Make sure students input the data in one column, even though the data is represented in two columns in the task statement.

Student Facing

Open a spreadsheet window and enter the data so that each value is in its own cell in column A.

• How many values are in the spreadsheet? Explain your reasoning.
• If you entered the data in the order that the values are listed, the number 7 is in the cell at position A1 and the number 5 is in cell A5. List all of the cells that contain the number 13.
• In cell C1 type the word “Sum”, in C2 type “Mean”, and in C3 type “Median”. You may wish to double-click or drag the vertical line between columns C and D to allow the entire words to be seen.

Student Response

For access, consult one of our IM Certified Partners .

Activity Synthesis

The goal is to make sure that students know how to type data into a spreadsheet and to locate values in the spreadsheet by row and column. The locations will be referenced with spreadsheet functions in upcoming activities. Here are some questions for discussion.

• “What value is in cell A7?” (14)
• “What was interesting or challenging about this activity?” (I never knew that you could describe each cell in a spreadsheet using the row and column labels.)

9.2: Finding Spreadsheet Statistics (15 minutes)

Building Towards

Instructional Routines

• MLR1: Stronger and Clearer Each Time

The mathematical purpose of this activity is to calculate statistics, create data displays, and to investigate how those change when values are added or removed from the data set. Monitor for students discussing the relationship between outliers and the measure of center.

Keep students in the same groups. They will continue working using the spreadsheet they started in the previous activity.

Tell students that statistics are values that are calculated from data, such as the mean, median, or interquartile range.

Tell students that after they change the value in A1 to change the mean in the first set of questions, they should continue to use the changed value for the second set of questions rather than reset them to the values from the warm-up.

Note that GeoGebra is like any other computer program. It needs directions written in a specific way for it to execute a command. For example, if students forget to type the = symbol or don’t capitalize “Sum,” the formula won’t work. Ask students to pause after typing the formulas and ensure that cells D1, D2, and D3 display numbers for each statistic. If not, ask students to delete the contents of the cell and retype the formula, ensuring that they start with an = symbol and capitalize Sum, Mean, and Median.

Using the data from the warm-up, we can calculate a few statistics and look at the data.

• Next to the word Sum, in cell D1, type =Sum(A1:A20)
• Next to the word Mean, in cell D2, type =Mean(A1:A20)
• Next to the word Median, in cell D3, type =Median(A1:A20)
• What are the values for each of the statistics?
• Change the value in A1 to 8. How does that change the statistics?
• What value can be put into A1 to change the mean to 10.05 and the median to 9?

We can also use Geogebra to create data displays.

• Click on the letter A for the first column so that the entire column is highlighted.
• Click on the button that looks like a histogram to get a new window labeled One Variable Analysis .
• Click Analyze to see a histogram of the data.
• What does the value for n represent?
• What does the value for \(\Sigma x\) represent?
• What other statistics do you recognize?
• Adjust the slider next to the word Histogram. What changes?
• Click on the button to the right of the slider to bring in another window with more options. Then, click the box next to Set Classes Manually and set the Width to 5. What does this do to the histogram?
• Click the word Histogram and look at a box plot and dot plot of the data. When looking at the box plot, notice there is an x on the right side of box plot. This represents a data point that is considered an outlier. Click on the button to the right of the slider and uncheck the box labeled Show Outliers to include this point in the box plot. What changes? Why might you want to show outliers? Why might you want to include or exclude outliers?

The purpose of this discussion is for students to create data displays using technology and to analyze what happens to the displays and the statistics when changes are made to the data set. Here are some questions for discussion.

• What happened to the statistics when you changed the value for A1 to 8 in the spreadsheet?” (When it was changed to 8, the mean increased slightly but the median stayed the same.)
• “Why did the mean increase?” (The sum of the data increased but the number of numbers stayed the same so the mean had to increase.)
• “Why did the median stay the same?” (Changing a 7 to an 8 in the data set did not change the middle numbers, 8 and 9, in the data set).
• “What did you notice when you changed the width of the classes for the histogram?” (This changed the intervals for each bar to a width of 5 and the data was resorted into those intervals.)

Select students who were previously identified as discussing the relationship between outliers and the measures of center. Ask, “what is the relationship between outliers and the measures of center?” (When outliers are present the median is the preferred measure of center because it is less impacted by outliers than the mean.)

9.3: Making Digital Displays (10 minutes)

• MLR2: Collect and Display

The mathematical purpose of this activity is for students to create data displays and calculate statistics using technology. Students plot the survey data they collected from a statistical question in a previous lesson.

Arrange students in groups of two. Tell them that they will be using technology to create data displays and calculate statistics for data they collected from a survey question in a previous lesson.

Use the data you collected from the numerical, statistical question from a previous lesson. Use technology to create a dot plot, boxplot, and histogram for your data. Then find the mean, median, and interquartile range for the data.

Are you ready for more?

A stem and leaf plot is a table where each data point is indicated by writing the first digit(s) on the left (the stem) and the last digit(s) on the right (the leaves). Each stem is written only once and shared by all data points with the same first digit(s). For example, the values 31, 32, and 45 might be represented like:

\(\displaystyle \begin{array}{r|l l} 3 & 1 & 2\\ 4 & 5 \end{array}\) Key: 3 | 1 means 31

A class took an exam and earned the scores:

86, 73, 85, 86, 72, 94, 88, 98, 87, 86, 85, 93, 75, 64, 82, 95, 99, 76, 84, 68

Use technology to create a stem and leaf plot for this data set.

How can we see the shape of the distribution from this plot?

What information can we see from a stem and leaf plot that we cannot see from a histogram?

What do we have more control of in a histogram than in a stem and leaf plot?

Anticipated Misconceptions

Students may lose one data display when they begin to create the next one. Explain to students that it is important to copy their solutions into a more permanent place so they can refer to it later.

The goal of this activity was for students to create graphs and find statistics using technology. Here are some questions for discussion.

• “What were some challenges that you faced using technology and how did you overcome them?” (I was not sure what buttons to press to get to the spreadsheet. I checked with my partner and figured it out.)
• “What width did you use for your histogram? Why?” (I used 5 because my data set has values ranging from 1 to 42. I could have used 10 but then I would have only had 5 bars.)
• “What is the appropriate measure of center for your data set?” (The median was appropriate because my data set has a skewed distribution.)
• “Which display allows you to calculate the IQR the most easily?” (The box plot because it displays Q1 and Q3.)
• “Can you find the median using your histogram?” (No, the data is grouped into intervals, so a histogram cannot be used to find the middle value for the median.)

Lesson Synthesis

The goal of this lesson is for students to display and investigate data using technology. Here are some questions for discussion.

• “How do you create data displays using technology?” (You type the data into the spreadsheet and then click the appropriate buttons.)
• “What are some advantages of using technology to display data and calculate statistics?” (You can easily switch between different data displays and you can change the intervals on histograms without having to sort through the data again. The advantage of having the technology calculate the statistics is that I can see how the statistics change as I enter or make changes to the data.)
• “When do you think it is appropriate to use technology to display data or to calculate statistics?” (Graphing technology makes it easier to determine the shape of a distribution. I might use it to determine the most appropriate measure of center for a data set. Using technology to calculate statistics makes sense to do in most situations because statistics are calculated using algorithms that can get complicated when there are many values in the data set. The chance of making a mistake while calculating statistics by hand makes using technology a good choice.)

9.4: Cool-down - What Are These Values? (5 minutes)

Student lesson summary.

Data displays (like histograms or box plots) are very useful for quickly understanding a large amount of information, but often take a long time to construct accurately using pencil and paper. Technology can help create these displays as well as calculate useful statistics much faster than doing the same tasks by hand. Especially with very large data sets (in some experiments, millions of pieces of data are collected), technology is essential for putting the information into forms that are more easily understood.

A statistic is a quantity that is calculated from sample data as a measure of a distribution. Mean and median are examples of statistics that are measures of center. Mean absolute deviation (MAD) and interquartile range (IQR) are examples of statistics that are measures of variability. Although the interpretation must still be done by people, using the tools available can improve the accuracy and speed of doing computations and creating graphs.

Fun teaching resources & tips to help you teach math with confidence

Math Strategies: Problem Solving by Working Backwards

As I’ve shared before, there are many different ways to go about solving a math problem, and equipping kids to be successful problem solvers is just as important as teaching computation and algorithms . In my experience, students’ frustration often comes from not knowing where to start. Providing them with strategies enables them to at least get the ideas flowing and hopefully get some things down on paper. As in all areas of life, the hardest part is getting started! Today I want to explain how to teach  problem solving by working backwards .

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–>Pssst! Do your kids need help making sense of and solving word problems? You might like this set of editable word problem solving templates ! Use these with any grade level, for any type of word problem :

Solve a Math Problem by Working Backwards: 

Before students can learn to recognize when this is a helpful strategy, they must understand what it means. Working backwards is to start with the final solution and work back one step at a time to get to the beginning.

It may also be helpful for students to understand that this is useful in many aspects of life, not just solving math problems.

To help show your students what this looks like, you might start by thinking about directions. Write out some basic directions from home to school:

• Start: Home
• Turn right on Gray St.
• Turn left on Sycamore Ln.
• Turn left on Rose Dr.
• Turn right on Schoolhouse Rd.
• End: School

Ask students to then use this information to give directions from the school back home . Depending on the age of your students, you may even want to draw a map so they can see clearly that they have to do the opposite as they make their way back home from school. In other words, they need to “undo” each turn to get back, i.e. turn left on Schoolhouse Rd. and then right on Rose Dr. etc.

In math, these are called inverse operations . When using the “work backwards” strategy, each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backwards they will need to subtract. And if they multiply working forwards, they must divide when working backwards.

Once students understand inverse operations , and know that they must start with the solution and work back to the beginning, they will need to learn to recognize the types of problems that require working backwards.

In general, problems that list a series of events or a sequence of steps can be solved by working backwards.

Here’s an example:

Sam’s mom left a plate of cookies on the counter. Sam ate 2 of them, his dad ate 3 of them and they gave 12 to the neighbor. At the end of the day, only 4 cookies were left on the plate. How many cookies did she make altogether?

In this case, we know that the final cookie amount is 4. So if we work backwards to “put back” all the cookies that were taken or eaten, we can figure out what number they started with.

Because cookies are being taken away, that denotes subtraction. Thus, to get back to the original number we have to do the opposite: add . If you take the 4 that are left and add the 12 given to the neighbors, and add the 3 that Dad ate, and then add the 2 that Sam ate, we find that Sam’s mom made 21 cookies .

You may want to give students a few similar problems to let them see when working backwards is useful, and what problems look like that require working backwards to solve.

Have you taught or discussed problem solving by working backwards  with your students? What are some other examples of when this might be useful or necessary?

Don’t miss the other useful articles in this Problem Solving Series:

• Problem Solve by Drawing a Picture
• Problem Solve by Solving an Easier Problem
• Problem Solve with Guess & Check
• Problem Solve by Finding a Pattern
• Problem Solve by Making a List

So glad to have come across this post! Today, word problems were the cause of a homework meltdown. At least tomorrow I’ll have a different strategy to try! #ThoughtfulSpot

I’m so glad to hear that! I hope you found some useful ideas!! Homework meltdowns are never fun!! Best of luck!

This is really a great help! We have just started using this method for some of my sons math problems and it helps loads. Thanks so much for sharing on the Let Kids Be Kids Linkup!

That’s great Erin! I hope this is a helpful method and makes things easier for your son! 🙂

I’ve not used this method before but sounds like a good resource to teach. Thanks for linking #LetKidsBeKids

I hope this proves to be helpful for you!

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Problem Solving - Organize Data - Lesson 2.1

Use Picture Graphs - Lesson 2.2

Make Picture Graphs - Lesson 2.3

Use Bar Graphs - Lesson 2.4

Make a Bar Graph - Lesson 2.5

Solve Problems Using Data - Lesson 2.6

Use and Make Line Plots - Lesson 2.7

Number Patterns - Lesson 1.1

Round to Nearest Ten or Hundred - Lesson 1.2

Estimate Sums - Lesson 1.3

Mental Math Strategies for Addition - Lesson 1.4

Use Properties to Add - Lesson 1.5

Use the Break Apart Strategy to Add - Lesson 1.6

Use Place Value to Add - Lesson 1.7

Estimate Differences - Lesson 1.8

Mental Math Strategies for Subtraction - Lesson 1.9

Use Place Value to Subtract - Lesson 1.10

Combine Place Values to Subtract - Lesson 1.11

Describe Plane Shapes - Lesson 12.1

Describe Angles in Plane Shapes - Lesson 12.2

Identify Polygons - Lesson 12.3 ​

Describe Sides of Polygons - Lesson 12.4

​ Classify Quadrilaterals - Lesson 12.5

​Draw Quadrilaterals - Lesson 12.6

Describe Triangles - Lesson 12.7

Chapter 12 Performance Task Review For Test

Problem Solving - Compare Fractions - Lesson 9.1

Compare Fractions with the Same Denominator - Lesson 9.2

Compare Fractions with the Same Numerator - Lesson 9.3

Compare Fractions - Lesson 9.4

Compare and Order Fractions - Lesson 9.5

Model Equivalent Fractions - Lesson 9.6

Equivalent Fractions - Lesson 9.7

Divide by 2 - Lesson 7.1

Divide by 10 - Lesson 7.2

Divide by 5 - Lesson 7.3

Divide by 3 - Lesson 7.4

Divide by 4 - Lesson 7.5

Divide by 5 - Lesson 7.6

Mid-Chapter 7 Checkpoint on Division Facts and Strategies

Divide by 7 - Lesson 7.7

Divide by 8 - Lesson 7.8

Divide by 9 - Lesson 7.9

Problem Solving - Two-Step Problems - Lesson 7.10

Order of Operations - Lesson 7.11

Problem Solving - Model Division - Lesson 6.1

Size of Equal Groups - Lesson 6.2

Number of Equal Groups - Lesson 6.3

Model (Division) with Bar Model - Lesson 6.4

Relate Subtraction and Division - Lesson 6.5

Mid-Chapter 6 Checkpoint

Model (division) with Arrays - Lesson 6.6

Relate Multiplication and Division - Lesson 6.7

Write Related Facts - Lesson 6.8

Division Rules for 1 and 0 - Lesson 6.9

Chapter 6 Review for Test - Understanding Division

Multiply with 2 and 4 - Lesson 4.1

Multiply with 5 and 10 - Lesson 4.2

Multiply with 3 and 6 - Lesson 4.3

Distributive Property - Lesson 4.4

Multiply with 7 - Lesson 4.5

Associative Property of Multiplication - Lesson 4.6

Patterns on the Multiplication Table - Lesson 4.7

Multiply with 8 - Lesson 4.8

Multiply with 9 - Lesson 4.9

Review For Test on Chapter 4

Describe Patterns - Lesson 5.1

Find Unknown Factors - Lesson 5.2

Problem Solving: Using the Distributive Property - Lesson 5.3

Multiplication Strategies with Multiples of 10 - Lesson 5.4

Multiply Multiples of 10 by 1-Digit Numbers - Lesson 5.5

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Model Perimeter - Lesson 11.1

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Problem Solving - Area of Rectangles - Lesson 11.7

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Count Equal Groups - Lesson 3.1

Relate Addition and Multiplication - Lesson 3.2

Skip Count on a Number Line - Lesson 3.3

Problem Solving - Model Multiplication - Lesson 3.4

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Commutative Property of Multiplication - Lesson 3.6

Multiply with 1 and 0 - Lesson 3.7

Time to the Minute - Lesson 10.1

A.M. and P.M. - Lesson 10.2

Measure Time Intervals - Lesson 10.3

Use Time Intervals - Lesson 10.4

Problem Solving - Time Intervals - Lesson 10.5

Measure Length - Lesson 10.6

Estimate and Measure Liquid Volume - Lesson 10.7

Estimate and Measure Mass - Lesson 10.8

Equal Parts of a Whole - Lesson 8.1

Equal Shares - Lesson 8.2

Unit Fractions of a Whole - Lesson 8.3

Fractions of a Whole - Lesson 8.4

Fractions on a Number Line - Lesson 8.5

Relate Fractions and Whole Numbers - Lesson 8.6

Fractions of a Group - Lesson 8.7

Find Part of Group Using Unit Fractions - Lesson 8.8

Problem Solving: Find the Whole Using Unit Fractions - Lesson 8.9

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A: ( 5 , 1 ) ( 5 , 1 )  B: ( −2 , 4 ) ( −2 , 4 )  C: ( −5 , −1 ) ( −5 , −1 )  D: ( 3 , −2 ) ( 3 , −2 )  E: ( 0 , −5 ) ( 0 , −5 )  F: ( 4 , 0 ) ( 4 , 0 )

A: ( 4 , 2 ) ( 4 , 2 )  B: ( −2 , 3 ) ( −2 , 3 )  C: ( −4 , −4 ) ( −4 , −4 )  D: ( 3 , −5 ) ( 3 , −5 )  E: ( −3 , 0 ) ( −3 , 0 )  F: ( 0 , 2 ) ( 0 , 2 )

Answers will vary.

ⓐ yes, yes  ⓑ yes, yes

ⓐ no, no  ⓑ yes, yes

x - intercept: ( 2 , 0 ) ( 2 , 0 ) ; y - intercept: ( 0 , −2 ) ( 0 , −2 )

x - intercept: ( 3 , 0 ) ( 3 , 0 ) , y - intercept: ( 0 , 2 ) ( 0 , 2 )

x - intercept: ( 4 , 0 ) ( 4 , 0 ) , y - intercept: ( 0 , 12 ) ( 0 , 12 )

x - intercept: ( 8 , 0 ) ( 8 , 0 ) , y - intercept: ( 0 , 2 ) ( 0 , 2 )

x - intercept: ( 4 , 0 ) ( 4 , 0 ) , y - intercept: ( 0 , −3 ) ( 0 , −3 )

x - intercept: ( 4 , 0 ) ( 4 , 0 ) , y - intercept: ( 0 , −2 ) ( 0 , −2 )

− 2 3 − 2 3

− 4 3 − 4 3

− 3 5 − 3 5

− 1 36 − 1 36

− 1 48 − 1 48

slope m = 2 3 m = 2 3 and y -intercept ( 0 , −1 ) ( 0 , −1 )

slope m = 1 2 m = 1 2 and y -intercept ( 0 , 3 ) ( 0 , 3 )

2 5 ; ( 0 , −1 ) 2 5 ; ( 0 , −1 )

− 4 3 ; ( 0 , 1 ) − 4 3 ; ( 0 , 1 )

− 1 4 ; ( 0 , 2 ) − 1 4 ; ( 0 , 2 )

− 3 2 ; ( 0 , 6 ) − 3 2 ; ( 0 , 6 )

ⓐ intercepts  ⓑ horizontal line  ⓒ slope–intercept  ⓓ vertical line

ⓐ vertical line  ⓑ slope–intercept  ⓒ horizontal line  ⓓ intercepts

• ⓐ 50 inches
• ⓑ 66 inches
• ⓒ The slope, 2, means that the height, h , increases by 2 inches when the shoe size, s , increases by 1. The h -intercept means that when the shoe size is 0, the height is 50 inches.
• ⓐ 40 degrees
• ⓑ 65 degrees
• ⓒ The slope, 1 4 1 4 , means that the temperature Fahrenheit ( F ) increases 1 degree when the number of chirps, n , increases by 4. The T -intercept means that when the number of chirps is 0, the temperature is 40 ° 40 ° .
• ⓒ The slope, 0.5, means that the weekly cost, C , increases by $0.50 when the number of miles driven, n, increases by 1. The C -intercept means that when the number of miles driven is 0, the weekly cost is $60
• ⓒ The slope, 1.8, means that the weekly cost, C, increases by $1.80 when the number of invitations, n , increases by 1.80. The C -intercept means that when the number of invitations is 0, the weekly cost is $35.;

not parallel; same line

perpendicular

not perpendicular

y = 2 5 x + 4 y = 2 5 x + 4

y = − x − 3 y = − x − 3

y = 3 5 x + 1 y = 3 5 x + 1

y = 4 3 x − 5 y = 4 3 x − 5

y = 5 6 x − 2 y = 5 6 x − 2

y = 2 3 x − 4 y = 2 3 x − 4

y = − 2 5 x − 1 y = − 2 5 x − 1

y = − 3 4 x − 4 y = − 3 4 x − 4

y = 8 y = 8

y = 4 y = 4

y = 5 2 x − 13 2 y = 5 2 x − 13 2

y = − 2 5 x + 22 5 y = − 2 5 x + 22 5

y = 1 3 x − 10 3 y = 1 3 x − 10 3

y = − 2 5 x − 23 5 y = − 2 5 x − 23 5

x = 5 x = 5

x = −4 x = −4

y = 3 x − 10 y = 3 x − 10

y = 1 2 x + 1 y = 1 2 x + 1

y = − 1 3 x + 10 3 y = − 1 3 x + 10 3

y = −2 x + 16 y = −2 x + 16

y = −5 y = −5

y = −1 y = −1

x = −5 x = −5

ⓐ yes  ⓑ yes  ⓒ yes  ⓓ yes  ⓔ no

ⓐ yes  ⓑ yes  ⓒ no  ⓓ no  ⓔ yes

y ≥ −2 x + 3 y ≥ −2 x + 3

y < 1 2 x − 4 y < 1 2 x − 4

x − 4 y ≤ 8 x − 4 y ≤ 8

3 x − y ≤ 6 3 x − y ≤ 6

Section 4.1 Exercises

A: ( −4 , 1 ) ( −4 , 1 )  B: ( −3 , −4 ) ( −3 , −4 )  C: ( 1 , −3 ) ( 1 , −3 )  D: ( 4 , 3 ) ( 4 , 3 )

A: ( 0 , −2 ) ( 0 , −2 )  B: ( −2 , 0 ) ( −2 , 0 )  C: ( 0 , 5 ) ( 0 , 5 )  D: ( 5 , 0 ) ( 5 , 0 )

ⓑ Age and weight are only positive.

Section 4.2 Exercises

ⓐ yes; no  ⓑ no; no  ⓒ yes; yes  ⓓ yes; yes

ⓐ yes; yes  ⓑ yes; yes  ⓒ yes; yes  ⓓ no; no

$722, $850, $978

Section 4.3 Exercises

( 3 , 0 ) , ( 0 , 3 ) ( 3 , 0 ) , ( 0 , 3 )

( 5 , 0 ) , ( 0 , −5 ) ( 5 , 0 ) , ( 0 , −5 )

( −2 , 0 ) , ( 0 , −2 ) ( −2 , 0 ) , ( 0 , −2 )

( −1 , 0 ) , ( 0 , 1 ) ( −1 , 0 ) , ( 0 , 1 )

( 6 , 0 ) , ( 0 , 3 ) ( 6 , 0 ) , ( 0 , 3 )

( 0 , 0 ) ( 0 , 0 )

( 4 , 0 ) , ( 0 , 4 ) ( 4 , 0 ) , ( 0 , 4 )

( −3 , 0 ) , ( 0 , 3 ) ( −3 , 0 ) , ( 0 , 3 )

( 8 , 0 ) , ( 0 , 4 ) ( 8 , 0 ) , ( 0 , 4 )

( 2 , 0 ) , ( 0 , 6 ) ( 2 , 0 ) , ( 0 , 6 )

( 12 , 0 ) , ( 0 , −4 ) ( 12 , 0 ) , ( 0 , −4 )

( 2 , 0 ) , ( 0 , −8 ) ( 2 , 0 ) , ( 0 , −8 )

( 5 , 0 ) , ( 0 , 2 ) ( 5 , 0 ) , ( 0 , 2 )

( 4 , 0 ) , ( 0 , −6 ) ( 4 , 0 ) , ( 0 , −6 )

( 3 , 0 ) , ( 0 , 1 ) ( 3 , 0 ) , ( 0 , 1 )

( −10 , 0 ) , ( 0 , 2 ) ( −10 , 0 ) , ( 0 , 2 )

ⓐ ( 0 , 1000 ) , ( 15 , 0 ) ( 0 , 1000 ) , ( 15 , 0 ) ⓑ At ( 0 , 1000 ) ( 0 , 1000 ) , he has been gone 0 hours and has 1000 miles left. At ( 15 , 0 ) ( 15 , 0 ) , he has been gone 15 hours and has 0 miles left to go.

Section 4.4 Exercises

−3 2 = − 3 2 −3 2 = − 3 2

− 1 3 − 1 3

− 3 4 − 3 4

− 5 2 − 5 2

− 8 7 − 8 7

ⓐ 1 3 1 3   ⓑ 4 12 pitch or 4-in-12 pitch

3 50 3 50 ; rise = 3, run = 50

ⓐ 288 inches (24 feet)  ⓑ Models will vary.

When the slope is a positive number the line goes up from left to right. When the slope is a negative number the line goes down from left to right.

A vertical line has 0 run and since division by 0 is undefined the slope is undefined.

Section 4.5 Exercises

slope m = 4 m = 4 and y -intercept ( 0 , −2 ) ( 0 , −2 )

slope m = −3 m = −3 and y -intercept ( 0 , 1 ) ( 0 , 1 )

slope m = − 2 5 m = − 2 5 and y -intercept ( 0 , 3 ) ( 0 , 3 )

−9 ; ( 0 , 7 ) −9 ; ( 0 , 7 )

4 ; ( 0 , −10 ) 4 ; ( 0 , −10 )

−4 ; ( 0 , 8 ) −4 ; ( 0 , 8 )

− 8 3 ; ( 0 , 4 ) − 8 3 ; ( 0 , 4 )

7 3 ; ( 0 , −3 ) 7 3 ; ( 0 , −3 )

horizontal line

vertical line

slope–intercept

• ⓒ The slope, 2.54, means that Randy’s payment, P , increases by $2.54 when the number of units of water he used, w, increases by 1. The P –intercept means that if the number units of water Randy used was 0, the payment would be $28.
• ⓒ The slope, 0.32, means that the cost, C , increases by $0.32 when the number of miles driven, m, increases by 1. The C -intercept means that if Janelle drives 0 miles one day, the cost would be $15.
• ⓒ The slope, 0.09, means that Patel’s salary, S , increases by $0.09 for every $1 increase in his sales. The S -intercept means that when his sales are $0, his salary is $750.
• ⓒ The slope, 42, means that the cost, C , increases by $42 for when the number of guests increases by 1. The C -intercept means that when the number of guests is 0, the cost would be $750.

not parallel

• ⓐ For every increase of one degree Fahrenheit, the number of chirps increases by four.
• ⓑ There would be −160 −160 chirps when the Fahrenheit temperature is 0 ° 0 ° . (Notice that this does not make sense; this model cannot be used for all possible temperatures.)

Section 4.6 Exercises

y = 4 x + 1 y = 4 x + 1

y = 8 x − 6 y = 8 x − 6

y = − x + 7 y = − x + 7

y = −3 x − 1 y = −3 x − 1

y = 1 5 x − 5 y = 1 5 x − 5

y = − 2 3 x − 3 y = − 2 3 x − 3

y = 2 y = 2

y = −4 x y = −4 x

y = −2 x + 4 y = −2 x + 4

y = 3 4 x + 2 y = 3 4 x + 2

y = − 3 2 x − 1 y = − 3 2 x − 1

y = 6 y = 6

y = 3 8 x − 1 y = 3 8 x − 1

y = 5 6 x + 2 y = 5 6 x + 2

y = − 3 5 x + 1 y = − 3 5 x + 1

y = − 1 3 x − 11 y = − 1 3 x − 11

y = −7 y = −7

y = − 5 2 x − 22 y = − 5 2 x − 22

y = −4 x − 11 y = −4 x − 11

y = −8 y = −8

y = −4 x + 13 y = −4 x + 13

y = x + 5 y = x + 5

y = − 1 3 x − 14 3 y = − 1 3 x − 14 3

y = 7 x + 22 y = 7 x + 22

y = − 6 7 x + 4 7 y = − 6 7 x + 4 7

y = 1 5 x − 2 y = 1 5 x − 2

x = 4 x = 4

x = −2 x = −2

y = −3 y = −3

y = 4 x y = 4 x

y = 1 2 x + 3 2 y = 1 2 x + 3 2

y = 5 y = 5

y = 3 x − 1 y = 3 x − 1

y = −3 x + 3 y = −3 x + 3

y = 2 x − 6 y = 2 x − 6

y = − 2 3 x + 5 y = − 2 3 x + 5

x = −3 x = −3

y = −4 y = −4

y = x y = x

y = − 3 4 x − 1 4 y = − 3 4 x − 1 4

y = 5 4 x y = 5 4 x

y = 1 y = 1

y = x + 2 y = x + 2

y = 3 4 x y = 3 4 x

y = 1.2 x + 5.2 y = 1.2 x + 5.2

Section 4.7 Exercises

ⓐ yes  ⓑ no  ⓒ no  ⓓ yes  ⓔ no

ⓐ yes  ⓑ no  ⓒ no  ⓓ yes  ⓔ yes

ⓐ no  ⓑ no  ⓒ no  ⓓ yes  ⓔ yes

y < 2 x − 4 y < 2 x − 4

y ≤ − 1 3 x − 2 y ≤ − 1 3 x − 2

x + y ≥ 3 x + y ≥ 3

x + 2 y ≥ −2 x + 2 y ≥ −2

2 x − y < 4 2 x − y < 4

4 x − 3 y > 12 4 x − 3 y > 12

• ⓑ Answers will vary.

Review Exercises

ⓐ ( 2 , 0 ) ( 2 , 0 )   ⓑ ( 0 , −5 ) ( 0 , −5 )   ⓒ ( −4.0 ) ( −4.0 )   ⓓ ( 0 , 3 ) ( 0 , 3 )

ⓐ yes; yes  ⓑ yes; no

( 6 , 0 ) , ( 0 , 4 ) ( 6 , 0 ) , ( 0 , 4 )

− 1 2 − 1 2

slope m = − 2 3 m = − 2 3 and y -intercept ( 0 , 4 ) ( 0 , 4 )

5 3 ; ( 0 , −6 ) 5 3 ; ( 0 , −6 )

4 5 ; ( 0 , − 8 5 ) 4 5 ; ( 0 , − 8 5 )

plotting points

ⓐ −$250  ⓑ $450  ⓒ The slope, 35, means that Marjorie’s weekly profit, P , increases by $35 for each additional student lesson she teaches. The P –intercept means that when the number of lessons is 0, Marjorie loses $250.  ⓓ

y = −5 x − 3 y = −5 x − 3

y = −2 x y = −2 x

y = −3 x + 5 y = −3 x + 5

y = 3 5 x y = 3 5 x

y = −2 x − 5 y = −2 x − 5

y = 1 2 x − 5 2 y = 1 2 x − 5 2

y = − 2 5 x + 8 y = − 2 5 x + 8

y = 3 y = 3

y = − 3 2 x − 6 y = − 3 2 x − 6

ⓐ yes  ⓑ no  ⓒ yes  ⓓ yes  ⓔ no

y > 2 3 x − 3 y > 2 3 x − 3

x − 2 y ≥ 6 x − 2 y ≥ 6

Practice Test

ⓐ yes  ⓑ yes  ⓒ no

( 3 , 0 ) , ( 0 , −4 ) ( 3 , 0 ) , ( 0 , −4 )

y = − 3 4 x − 2 y = − 3 4 x − 2

y = 1 2 x − 4 y = 1 2 x − 4

y = − 4 5 x − 5 y = − 4 5 x − 5

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Access for free at https://openstax.org/books/elementary-algebra/pages/1-introduction

• Authors: Lynn Marecek, MaryAnne Anthony-Smith
• Publisher/website: OpenStax
• Book title: Elementary Algebra
• Publication date: Feb 22, 2017
• Location: Houston, Texas
• Book URL: https://openstax.org/books/elementary-algebra/pages/1-introduction
• Section URL: https://openstax.org/books/elementary-algebra/pages/chapter-4

© Feb 9, 2022 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.