problem solving in math 2

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Integrated math 2

Welcome to integrated math 2, unit 1: absolute value & piecewise functions, unit 2: quadratics: multiplying & factoring, unit 3: quadratic functions & equations, unit 4: irrational numbers, unit 5: complex numbers, unit 6: rational exponents and radicals, unit 7: exponential models, unit 8: similarity, unit 9: right triangles & trigonometry, unit 10: solid geometry, unit 11: circles, unit 12: conic sections, unit 13: probability.

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Find help from our team of math teachers who are here to help struggling students with their math skills. Find on-demand videos for every second-grade skill to teach students how to best tackle math problems. Teachers break down the concepts in an easily understandable format for younger students who need help with their math skills.

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Also known as "College Algebra"

OK. So what are you going to learn here?

You will learn about Numbers, Polynomials, Inequalities, Sequences and Sums, many types of Functions, and how to solve them.

You will also gain a deeper insight into Mathematics, get to practice using your new skills with lots of examples and questions, and generally improve your mind.

With your new skills you will be able to put together mathematical models so you can find good quality solutions to many tricky real world situations.

Near the end of most pages is a "Your Turn" section ... do these! You need to balance your reading with doing . Answering questions helps you sort things out in your mind. And don't guess the answer: use pen and paper and try your best before seeing the solution.

So what is this thing called Mathematics? And how do you go about learning it?

• Welcome to Mathematics
• Learning Mathematics
• The Language of Mathematics
• Symbols in Algebra

Next, we need to think about mathematics in terms of sets .

• Introduction to Sets

Now we know what a set is, let us look at different sets of numbers that are useful:

• The Evolution of Numbers
• Prime and Composite Numbers
• Fundamental Theorem of Arithmetic
• Whole Numbers and Integers
• Rational Numbers
• Using Rational Numbers
• Irrational Numbers
• 0.999... = 1
• Real Numbers
• Imaginary Numbers
• Complex Numbers
• Multiplying Complex Numbers
• The Complex Plane
• Common Number Sets

Inequalities

"Equal To" is nice but not always available. Maybe we only know that something is less than, or greater than. So let's learn about in equalities.

• Introduction to Inequalities
• Properties of Inequalities
• Solving Inequalities
• Solving Inequality Word Questions

We will be using exponents a lot, so let's get to know them well.

• Variables with Exponents
• Using Exponents in Algebra
• Squares and Square Roots
• Squares and Square Roots in Algebra
• Fractional Exponents
• Laws of Exponents
• Exponents of Negative Numbers

Polynomials

Polynomials were some of the first things ever studied in Algebra. They are simple, yet powerful in their ability to model real world situations.

• What is a Polynomial?
• Adding And Subtracting Polynomials
• Multiplying Polynomials
• Polynomials - Long Multiplication
• Dividing Polynomials
• Polynomials - Long Division
• Degree (of an Expression)
• Special Binomial Products
• Difference of Two Cubes
• Factoring in Algebra
• Solving Polynomials
• Roots of Polynomials: Sums and Products
• Rational Expressions
• Using Rational Expressions
• Fundamental Theorem of Algebra
• Remainder Theorem and Factor Theorem
• General Form of a Polynomial

Graphing Polynomials

• How Polynomials Behave
• Polynomials: The Rule of Signs
• Polynomials: Bounds on Zeros

And, of course, we need to know about equations ... and how to solve them.

• Equations and Formulas
• Solving Equations
• Solving Word Questions
• Zero Product Property
• Implication and Iff
• Theorems, Corollaries, Lemmas

Graphs can save us! They are a great way to see what is going on and can help us solve many things. But we need to be careful, as they sometimes don't give the full story.

• Cartesian Coordinates
• Pythagoras' Theorem
• Distance Between 2 Points
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• Finding Intercepts From an Equation
• Symmetry in Equations
• Linear Equations

They are just equations for lines. But they come in many forms.

• Equation of a Straight Line
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• General Form of Equation of a Line
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A function relates an input to an output. But from that simple foundation many useful things can be built.

• What is a Function?
• Domain, Range and Codomain
• Evaluating Functions
• Increasing and Decreasing Functions
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• Set-Builder Notation

Common Functions Reference :

• Square Function
• Square Root Function
• Cube Function
• Reciprocal Function
• Absolute Value Function
• Floor and Ceiling Function
• Function Transformations
• Equation Grapher
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• Composition of Functions
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Equations of Second Degree

"Second degree" just means the variable has an exponent of 2, like x 2 . It is the next major step after linear equations (where the exponent is 1, like x).

• Quadratic Equations
• Factoring Quadratics
• Completing the Square
• Derivation of Quadratic Formula
• Graphing Quadratic Equations
• Quadratic Equations in the Real World
• Circle Equations

We already have experience in solving, but now we can learn more!

• Mathematical Models and Mathematical Models 2
• Approximate Solutions
• Intermediate Value Theorem
• Solving Radical Equations
• Change of Variables
• Algebra Mistakes

We learned about inequalities above, now let's learn how to solve them.

• Graphing Linear Inequalities
• Inequality Graphing Tool
• Solving Quadratic Inequalities
• Solving Rational Inequalities
• Absolute Value in Algebra

Exponents and Logarithms

We already know about exponents ... well logarithms just go the other way. And together they can be very powerful.

• Introduction to Logarithms
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• Working with Exponents and Logarithms
• Exponential Function
• Logarithmic Function
• Exponential Growth and Decay
• Systems of Linear Equations

What happens when we have two or more linear equations that work together? They can often be solved! It isn't very hard but can take a lot of calculations.

• Types of Matrix
• How to Multiply Matrices
• Determinant of a Matrix
• Inverse of a Matrix:
• Using Elementary Row Operations (Gauss-Jordan)
• Using Minors, Cofactors and Adjugate
• Scalar, Vector, Matrix and Vectors
• Matrix Calculator
• More at Matrix Index
• Solving Systems of Linear Equations Using Matrices
• Systems of Linear and Quadratic Equations
• Probability

Is it likely? You be the judge!

• The Basic Counting Principle
• Combinations and Permutations

Sequences, Series and Partial Sums

A Sequence is a set of things (usually numbers) that are in order. We can also sum up a series, where Sigma Notation is very useful.

• Sequences - Finding A Rule
• Sigma Notation
• Partial Sums
• Arithmetic Sequences and Sums
• Geometric Sequences and Sums

These last few subjects use what we have learned above.

• Partial Fractions
• Mathematical Induction
• Pascal's Triangle
• Binomial Theorem

And that is all!

But there are many other interesting algebra topics such as:

• Euler's Formula for Complex Numbers
• Taylor Series (needs a basic understanding of derivatives )

Home » Teacher & Parents Tips » Math Made Easy: Helping Grade 2 Students Thrive in Problem Solving

Math Made Easy: Helping Grade 2 Students Thrive in Problem Solving

Mathematics is an essential subject that students cannot afford to neglect. As early as grade 2, students are introduced to the basics of mathematical concepts such as numbers, addition, subtraction, multiplication, and division. An essential aspect of mathematics is problem-solving. While many students find it challenging, it can be made easy with the right approach. This article will discuss how parents and educators can turn problem-solving woes into victories and equip their grade 2 students with the skills needed to excel in math.

Table of Contents

The Importance of Problem Solving in Grade 2 Math

Problem-solving is an essential skill that students need to master to excel in math. It involves critical thinking, analysis, and decision-making. It also helps students develop their reasoning abilities and enables them to apply logic in real-life situations. In grade 2 math, problem-solving skills are essential as they help students understand mathematical concepts better. Students who can solve problems are more confident and enjoy math more than those who struggle.

Turning Problem-Solving Woes into Victories

Many students, especially in grade 2, find problem-solving challenging. However, with the right approach, it can be turned into a victory. One way to turn problem-solving woes into victories is by breaking down the problem into smaller parts. Encourage your child to read the problem carefully and identify the key elements. Once they have identified these elements, they can begin to solve the problem step by step. This approach helps students understand the problem better and reduces anxiety.

Another way to turn problem-solving woes into victories is by practicing regularly. The more students practice, the more comfortable they become with problem-solving. Encourage your child to practice regularly and provide them with different types of problems to solve. As they solve more problems, their confidence will increase, and they will become more efficient problem-solvers.

Strategies for Effective Problem Solving in Math

There are several strategies that students can use to solve problems effectively. One of the most effective strategies is the use of visual aids. Encourage your child to draw diagrams or pictures to help them understand the problem better. This approach helps students visualize the problem and enables them to make better decisions.

Another strategy is to use real-life situations to solve problems. This approach helps students understand how math can be applied in real-life situations. For example, if you want to teach your child about fractions, you can use pizza slices to help them understand the concept better.

Lastly, encourage your child to work with a partner or in a group. Group work helps students learn from one another and can be an effective way to solve problems. It also helps students develop their social skills and enables them to work collaboratively.

Fun and Engaging Math Activities for Grade 2 Students

Learning math can be fun and engaging. There are several math activities that parents and educators can use to help grade 2 students develop their problem-solving skills. One such activity is math games. Games such as Sudoku, Math Bingo, and Math Jeopardy can be used to teach students math concepts while making learning fun.

Another activity is math puzzles. Puzzles such as crosswords, word searches, and logic puzzles can be used to develop critical thinking and problem-solving skills. Additionally, math stories can be used to teach math concepts while making learning fun. Math stories can be found in storybooks or online, and they provide an interactive way to teach math concepts.

Developing Critical Thinking Skills through Math

Mathematics is an excellent way to develop critical thinking skills. Critical thinking involves analyzing, interpreting, and evaluating information to make informed decisions. In math, critical thinking is essential as it helps students understand mathematical concepts and apply them in real-life situations.

Encourage your child to think critically when solving problems. Teach them to ask questions and to consider different solutions to a problem. Additionally, encourage them to explain their reasoning and to justify their solutions. These skills are essential in problem-solving and can be applied in other areas of their lives.

Providing Support and Encouragement for your Child

Providing support and encouragement is essential when helping your grade 2 child excel in math. Encourage your child to ask questions and to seek help when they need it. Additionally, provide them with a positive learning environment and praise their efforts and progress. Celebrate their successes and encourage them to keep learning and practicing.

Equipping Your Child for Math Success in Grade 2

Mathematics is an essential subject that requires problem-solving skills. By turning problem-solving woes into victories, providing support and encouragement, and using effective strategies and fun activities, parents and educators can help grade 2 students excel in math. With the right approach, learning math can be fun and engaging while developing critical thinking skills that are essential in all areas of life.

2nd Grade Fast Math Success Workbook

2nd Grade Fast Math Success Workbook: Math Worksheets Grade 2: Numeration, Addition, Subtraction, Telling Time and More with Answers

• Comparing Numbers Within 200
• Ordering Numbers Within 200
• Skip Counting: Count By 1s and 2s
• Place Value: Ones, Tens, and Hundreds
• Addition: 1 through 100
• Subtraction: 1 through 100
• Addition: Missing Number – 1 through 100
• Subtraction: Missing Number – 1 through 100
• Ordering Numbers – 1 through 1000
• Comparing Numbers – 1 through 1000
• Counting: Count by 4 to 6
• Addition and Subtraction: Double Digit
• Addition and Subtraction: Missing Number – Double Digit
• Addition: Triple Addend – 1 through 100
• Mixed Operations: 1 through 100
• Write the Numbers Before, After, and Between
• Addition and Subtraction: 1 through 1000
• Telling Time
• Time Passages
• Addition and Subtraction Games
• Final Review

MathBear: Math Workbook Grade 2

MathBear: Math Workbook Grade 2: 2nd Grade Math Workbook: Addition, Subtraction, Multiplication, and Numeration with Answers

Math Practice Workbook Grade 2

Math Practice Workbook Grade 2: 3051 Questions to Master Essential Math Skills (Numeration, Addition, Subtraction, Multiplication, Telling Time and More) With Answer Key

• Circle the Numbers
• Comparing Numbers
• Addition: Double Digit
• Subtraction: Double Digit
• Addition: Missing Number
• Subtraction: Missing Number
• Number Before, After and Between
• Addition: Triple Digit
• Subtraction: Triple Digit
• Basic Multiplication

MathBear: Homeschool Math Workbook Grade 2

MathBear: Homeschool Math Workbook Grade 2: 2nd Grade Homeschool Math Practice Workbook: Addition, Subtraction, Multiplication, Place Value with Answers

• Place Value
• Commutative Property
• Addition Games
• Subtraction Games

MathBear: Math Curriculum Workbook Grade 2

MathBear: Math Curriculum Workbook Grade 2: 2nd Grade Math Curriculum: Numeration, Place Value, Addition and Subtraction, Telling Time with Answers

• Ordering Numbers
• Number Before, After, or Between
• Number Lines
• Place Value: Ones, Tens, Hundreds
• Subtraction
• Addition: 3 Addend
• Introduction to Multiplication
• Measure the Rectangles
• Match the Answers

MathBear: Math Practice Grade 2

MathBear: Math Practice Grade 2: 2nd Grade Math Practice Workbook: Addition, Subtraction, Multiplication, Place Value, Telling Time, Commutative Property with Answers

MathBear: Math Skills Workbook Grade 2

MathBear: Math Skills Workbook Grade 2: 2nd Grade Math Skills Practice Workbook: Addition, Subtraction, Basic Multiplication, Place Value, Math Games, and More With Answers

All Seasons Math Tests Grade 2

All Seasons Math Tests Grade 2 (Student's Edition): 100 Math Practice Pages Grade 2: Timed Math Tests: For Classroom and Homeschool

Kids Math Book Ages 6-8

Kids Math Book Ages 6-8: Math Practice workbook Grade 1-3: Addition, Subtraction, Place Value, Telling Time

• Addition Target
• Subtraction Target
• Addition Square
• Ordering Numbers: 1 to 100
• Addition Table
• Subtraction Table
• Addition: Triple Addend
• Numbers Before, After, and Between
• What time was and will it be?

Math Workbook Grade 2: Addition and Subtraction

MathBear: Math Workbook Grade 2: Addition and Subtraction: 2nd Grade Double Digit Addition and Subtraction Workbook with Answers

• Addition: within 100
• Subtraction: within 100
• Basic Addition with Regrouping
• Basic Subtraction with Regrouping

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2nd Grade Math Word Problems Worksheets

Math word problem worksheets for grade 2.

These word problem worksheets place 2nd grade math concepts in contexts that grade 2 students can relate to.  We provide math word problems for addition, subtraction, multiplication, time, money and fractions.

We encourage students to read and think about the problems carefully, and not just recognize an answer pattern.  We facilitate this by:

• providing a number of mixed word problem worksheets (e.g. subtraction questions mixed in with addition questions)
• sometimes including irrelevant data within word problems so students must understand the context before applying a solution

Addition word problems for grade 2

Simple addition (1-2 digits)

Slightly harder addition (1-3 digits)

Subtraction word problems for grade 2

Simple subtraction (1 and 2 digit numbers)

Subtracting 1-3 digit numbers

Mixed addition and subtraction word problems

Mixed addition and subtraction within 20

Mixed addition and subtraction (two digits)

More mixed addition and subtraction word problems

Multiplication word problems

Multiplication within 25

Measurement word problems

Length word problems

Time word problems

Time and elapsed time (1/2 hour intervals)

Time and elapsed time (5 minute intervals)

Money word problems

Counting money (coins and bills)

Fraction word problems

Understanding fractions

Write and compare fractions from a story

Mixed word problems

Mix of all above types of grade 2 word problems

Mix of just addition / subtraction / multiplication word problems  

Sample Grade 2 Word Problem Worksheet

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• The basics of algebra are the commutative, associative, and distributive laws.
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• The basic rules of algebra are the commutative, associative, and distributive laws.
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• The golden rule of algebra states Do unto one side of the equation what you do to others. Meaning, whatever operation is being used on one side of equation, the same will be used on the other side too.
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• The basic laws of algebra are the Commutative Law For Addition, Commutative Law For Multiplication, Associative Law For Addition, Associative Law For Multiplication, and the Distributive Law.

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1.2: Problem Solving and Estimating

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• Page ID 74279

• Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia
• Rio Hondo College

1.2 Learning Objectives

• Apply the problems solving process to real life problems
• Use dimensional analysis to solve problems
• Use estimates to solve real life problems
• Round decimals

In many problems, it is tempting to take the given information, plug it into whatever formulas you have handy, and hope that the result is what you were supposed to find. Chances are, this approach has served you well in other math classes.

This approach does not work well with real life problems. Instead, problem solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backwards, asking “what information and procedures will I need to find this?” Very few interesting questions can be answered in one mathematical step; often times you will need to chain together a solution pathway, a series of steps that will allow you to answer the question.

Problem Solving Process

• Identify the question you’re trying to answer.
• Work backwards, identifying the information you will need and the relationships you will use to answer that question.
• Continue working backwards, creating a solution pathway.
• If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
• Solve the problem, following your solution pathway.
• Check your answer. Does the answer make sense?

In most problems we work, we will be estimating a solution, because we will not have perfect information. We will begin with a few examples where we will be able to estimate the solution using basic knowledge from our lives.

Definition: Estimation

Estimation is the process of arriving at an approximate answer to a question. Usually by rounding answers to a place value that is easier to express in the situation.

Depending on the situation, you may be asked to round your answers to a specific place value. The tables below list more common values, but there are many more. 

Place values of a whole number:

Place values of a decimal number:

Name and Write Decimals

Notice that “ten thousand” is a number larger than one, but “one ten-thousand th ” is a number smaller than one. The “th” at the end of the name tells you that the number is smaller than one.

When we name a whole number, the name corresponds to the place value based on the powers of ten. We read 10,000 as “ten thousand” and 10,000,000 as “ten million.” Likewise, the names of the decimal places correspond to their fraction values.

We summarize the steps needed to name a decimal below.

• Name the number to the left of the decimal point.
• Write “and” for the decimal point.
• Name the “number” part to the right of the decimal point as if it were a whole number.
• Name the decimal place of the last digit.

Name the decimal \(4.3\).

Try It Now 1

Name the decimal \(6.7\).

six and seven tenths

Try It Now 2

Name the decimal: \(−15.571\).

When we write or read a check we interpret both the numerals and the name of the number. Let’s see how to write the decimal from the name.

We summarize the steps to writing a decimal .

• Place a decimal point under the word “and.” Translate the words before “and” into the whole number and place it to the left of the decimal point.
• If there is no “and,” write a “0” with a decimal point to its right.
• Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
• Translate the words after “and” into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
• Fill in zeros for place holders as needed.

Write “fourteen and twenty-four thousandths” as a decimal.

Try It Now 3

Write as a decimal: thirteen and sixty-eight thousandths.

Round Decimals

Rounding decimals is very much like rounding whole numbers. We will round decimals with a method based on the one we used to round whole numbers.

We summarize the steps for rounding a decimal here.

• Locate the given place value and mark it with an arrow.
• Underline the digit to the right of the place value.
• Yes—add 1 to the digit in the given place value.
• No—do not change the digit in the given place value.
• Rewrite the number, deleting all digits to the right of the rounding digit.

Round 18.379 to the nearest hundredth.

Try It Now 4

Round 6.582 to the nearest

• whole number.

We can use these estimation techniques to answer questions that involve some problem solving. Bringing the two concepts from above together, we apply them to several examples below.

How many times does your heart beat in a year?

This question is asking for the rate of heart beats per year. Since a year is a long time to measure heart beats for, if we knew the rate of heart beats per minute, we could scale that quantity up to a year. So the information we need to answer this question is heart beats per minute. This is something you can easily measure by counting your pulse while watching a clock for a minute.

Suppose you count 80 beats in a minute. To convert this beats per year:

\(\dfrac{80 \text { beats }}{1 \cancel {\text { minute }}} \cdot \dfrac{60 \cancel {\text { minutes }}}{1 \cancel {\text { hour }}} \cdot \dfrac{24 \cancel {\text { hours }}}{1 \cancel {\text { day }}} \cdot \dfrac{365 \cancel{ \text { days }}}{1 \text { year }}=42,048,000\) beats per year

How thick is a single sheet of paper? How much does it weigh?

While you might have a sheet of paper handy, trying to measure it would be tricky. Instead we might imagine a stack of paper, and then scale the thickness and weight to a single sheet. If you’ve ever bought paper for a printer or copier, you probably bought a ream, which contains 500 sheets. We could estimate that a ream of paper is about 2 inches thick and weighs about 5 pounds. Scaling these down,

\(\dfrac{2 \text { inches }}{ \cancel{\text { ream }}} \cdot \dfrac{1 \cancel{ \text { ream }}}{500 \text { pages }}=0.004\) inches per sheet

\(\dfrac{5 \text { pounds }}{ \cancel {\text { ream }}} \cdot \dfrac{1 \cancel {\text { ream }}}{500 \text { pages }}=0.01\) pounds per sheet, or 0.16 ounces per sheet.

A recipe for zucchini muffins states that it yields 12 muffins, with 250 calories per muffin. You instead decide to make mini-muffins, and the recipe yields 20 muffins. If you eat 4, how many calories will you consume?

There are several possible solution pathways to answer this question. We will explore one.

To answer the question of how many calories 4 mini-muffins will contain, we would want to know the number of calories in each mini-muffin. To find the calories in each mini-muffin, we could first find the total calories for the entire recipe, then divide it by the number of mini-muffins produced. To find the total calories for the recipe, we could multiply the calories per standard muffin by the number per muffin. Notice that this produces a multi-step solution pathway. It is often easier to solve a problem in small steps, rather than trying to find a way to jump directly from the given information to the solution.

We can now execute our plan:

\(12 \cancel{\text { muffins}} \cdot \dfrac{250 \text { calories }}{ \cancel {\text { muffin }}}=3000\) calories for the whole recipe

\(\dfrac{3000 \text { calories }}{20 \text { mini-muffins }}\) gives 150 calories per mini-muffin

\(4 \cancel {\text { mini-muffins }} \cdot \dfrac{150 \text { calories }}{ \cancel {\text { mini-muffin }}}\) totals 600 calories consumed.

You need to replace the boards on your deck. About how much will the materials cost?

There are two approaches we could take to this problem: 1) estimate the number of boards we will need and find the cost per board, or 2) estimate the area of the deck and find the approximate cost per square foot for deck boards. We will take the latter approach.

For this solution pathway, we will be able to answer the question if we know the cost per square foot for decking boards and the square footage of the deck. To find the cost per square foot for decking boards, we could compute the area of a single board, and divide it into the cost for that board. We can compute the square footage of the deck using geometric formulas. So first we need information: the dimensions of the deck, and the cost and dimensions of a single deck board.

Suppose that measuring the deck, it is rectangular, measuring 16 ft by 24 ft, for a total area of \(384 \mathrm{ft}^{2}\).

From a visit to the local home store, you find that an 8 foot by 4 inch cedar deck board costs about $7.50. The area of this board, doing the necessary conversion from inches to feet, is:

\(8 \text { feet } \cdot 4 \cancel { \text{inches } }\cdot \dfrac{1 \text { foot }}{12 \cancel {\text{ inches} }}=2.667 \mathrm{ft}^{2}\).

The cost per square foot is then

\(\dfrac{\$ 7.50}{2.667 \mathrm{ft}^{2}}=\$ 2.8125 \text { per } \mathrm{ ft}^{2}\).

This will allow us to estimate the material cost for the whole \(384 \mathrm{ ft}^{2}\) deck

\(\ 384 \cancel {\mathrm{ ft}^{2} }\cdot \dfrac{\$ 2.8125}{\cancel{\mathrm{ ft}^{2}}}=\$ 1080\) total cost.

Of course, this cost estimate assumes that there is no waste, which is rarely the case. It is common to add at least 10% to the cost estimate to account for waste.

Is it worth buying a Hyundai Sonata hybrid instead the regular Hyundai Sonata?

To make this decision, we must first decide what our basis for comparison will be. For the purposes of this example, we’ll focus on fuel and purchase costs, but environmental impacts and maintenance costs are other factors a buyer might consider.

It might be interesting to compare the cost of gas to run both cars for a year. To determine this, we will need to know the miles per gallon both cars get, as well as the number of miles we expect to drive in a year. From that information, we can find the number of gallons required from a year. Using the price of gas per gallon, we can find the running cost.

From Hyundai’s website, the 2013 Sonata will get 24 miles per gallon (mpg) in the city, and 35 mpg on the highway. The hybrid will get 35 mpg in the city, and 40 mpg on the highway.

An average driver drives about 12,000 miles a year. Suppose that you expect to drive about 75% of that in the city, so 9,000 city miles a year, and 3,000 highway miles a year.

We can then find the number of gallons each car would require for the year.

\(9000 \cancel {\text { city miles } }\cdot \dfrac{1 \text { gallon }}{24 \cancel{ \text { city miles }}}+3000 \cancel{ \text { hightway miles}}. \dfrac{1 \text { gallon }}{35 \cancel{ \text { highway miles }}}=460.7\text{ gallons}\)

\(9000 \cancel {\text { city miles }}\cdot \dfrac{1 \text { gallon }}{35 \cancel{ \text { city miles }}}+3000 \cancel {\text { highway miles}}. \dfrac{1 \text { gallon }}{40 \cancel {\text { highway miles }}}=332.1\text{ gallons}\)

If gas in your area averages about $3.50 per gallon, we can use that to find the running cost:

Sonata: \(460.7 \text { gallons } \cdot \dfrac{\$ 3.50}{\text { gallon }}=\$ 1612.45\)

Hybrid: \(\text { 332.1 gallons } \cdot \dfrac{\$ 3.50}{\text { gallon }}=\$ 1162.35\)

The hybrid will save $450.10 a year. The gas costs for the hybrid are about \(\dfrac{\$ 450.10}{\$ 1612.45} = 0.279 = 27.9\%\) lower than the costs for the standard Sonata.

While both the absolute and relative comparisons are useful here, they still make it hard to answer the original question, since “is it worth it” implies there is some tradeoff for the gas savings. Indeed, the hybrid Sonata costs about $25,850, compared to the base model for the regular Sonata, at $20,895.

To better answer the “is it worth it” question, we might explore how long it will take the gas savings to make up for the additional initial cost. The hybrid costs $4965 more. With gas savings of $451.10 a year, it will take about 11 years for the gas savings to make up for the higher initial costs.

We can conclude that if you expect to own the car 11 years, the hybrid is indeed worth it. If you plan to own the car for less than 11 years, it may still be worth it, since the resale value of the hybrid may be higher, or for other non-monetary reasons. This is a case where math can help guide your decision, but it can’t make it for you.

Try it Now 5

If traveling from Seattle, WA to Spokane WA for a three-day conference, does it make more sense to drive or fly?

There is not enough information provided to answer the question, so we will have to make some assumptions, and look up some values.

Assumptions:

a) We own a car. Suppose it gets 24 miles to the gallon. We will only consider gas cost.

b) We will not need to rent a car in Spokane, but will need to get a taxi from the airport to the conference hotel downtown and back.

c) We can get someone to drop us off at the airport, so we don’t need to consider airport parking.

d) We will not consider whether we will lose money by having to take time off work to drive.

Values looked up (your values may be different)

a) Flight cost: \(\$184\)

b) Taxi cost: \(\$25\) each way (estimate, according to hotel website)

c) Driving distance: \(280\) miles each way

d) Gas cost: \(\$3.79\) a gallon

Cost for flying: \(\$184\text{ flight cost }+ \$50\text{ in taxi fares }= \$234\).

Cost for driving: \(560\) miles round trip will require 23.3 gallons of gas, costing \(\$88.31\).

Based on these assumptions, driving is cheaper. However, our assumption that we only include gas cost may not be a good one. Tax law allows you deduct \(\$0.55\) (in 2012) for each mile driven, a value that accounts for gas as well as a portion of the car cost, insurance, maintenance, etc. Based on this number, the cost of driving would be \(\$319\).

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Are ‘Forever Chemicals’ a Forever Problem?

The environmental protection agency says “forever chemicals” must be removed from tap water. but they lurk in much more of what we eat, drink and use..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Sabrina Tavernise. And this is “The Daily.”

[THEME MUSIC]

This month for the first time, the Environmental Protection Agency began to regulate a class of synthetic chemicals, known as forever chemicals, in America’s drinking water. But the chemicals, which have been linked to liver disease and other serious health problems, are in far more than just our water supply. Today, my colleague Kim Tingley explains.

It’s Wednesday, April 17.

So Kim, any time the EPA announces a regulation, I think we all sort of take notice because implicit in it is this idea that we have been exposed to something — something bad, potentially, lead or asbestos. And recently, the EPA is regulating a type of chemical known as PFAS So for those who don’t know, what are PFAS chemicals

Yeah, so PFAS stands for per and polyfluoroalkyl substances. They’re often called forever chemicals just because they persist so long in the environment and they don’t easily break down. And for that reason, we also use them in a ton of consumer products. They’re in makeup. They’re in carpet. They’re in nonstick cookware. They’re in food packaging, all sorts of things.

Yeah, I feel like I’ve been hearing about these chemicals actually for a very long time. I mean, nonstick pans, Teflon — that’s the thing that’s in my mind when I think PFAS.

Absolutely. Yeah, this class of chemicals has been around for decades. And what’s really important about this is that the EPA has decided, for the first time, to regulate them in drinking water. And that’s a ruling that stands to affect tens of millions of people.

So, help me understand where these things came from and how it’s taken so long to get to the point where we’re actually regulating them.

So, they really actually came about a long time ago. In 1938, DuPont, the people who eventually got us to Teflon, they were actually looking for a more stable kind of refrigerant. And they came upon this kind of chemical, PFAS. The thing that all PFAS chemicals have is a really strong bond between carbon atoms and fluorine atoms. This particular pairing is super strong and super durable.

They have water repellent properties. They’re stain resistant. They’re grease resistant. And they found a lot of uses for them initially in World War II. They were using them as part of their uranium enrichment process to do all these kinds of things. And then —

Well, good thing it’s Teflon.

In the 1950s is when they really started to come out as commercial products.

Even burned food won’t stick to Teflon. So it’s always easy to clean.

So, DuPont started using it in Teflon pans.

Cookware never needs scouring if it has DuPont Teflon.

And then another company, 3M also started using a kind of PFAS —

Scotchgard fabric protector. It keeps ordinary spills from becoming extraordinary stains.

— in one of their big products, Scotchgard. So you probably remember spraying that on your shoes if you want to make your shoes waterproof.

Use Scotchgard fabric protector and let your cup runneth over.

Right — miracle product, Scotchgard, Teflon. But of course, we’re talking about these chemicals because they’ve been found to pose health threats. When does that risk start to surface?

Yeah, so it’s pretty early on that DuPont and 3M start finding effects in animals in studies that they’re running in house.

Around the mid ‘60s, they start seeing that PFAS has an effect on rats. It’s increasing the liver and kidney weights of the rats. And so that seems problematic. And they keep running tests over the next decade and a half. And they try different things with different animals.

In one study, they gave monkeys really, really high levels of PFAS. And those monkeys died. And so they have a pretty strong sense that these chemicals could be dangerous. And then in 1979, they start to see that the workers that are in the plants manufacturing, working with these chemicals, that they’re starting to have higher rates of abnormal liver function. And in a Teflon plant, they had some pregnant workers that were working with these chemicals. And one of those workers in 1981 gave birth to a child who had some pretty severe birth defects.

And then by the mid 1980s, DuPont figures out that it’s not just their workers who are being exposed to these chemicals, but communities that are living in areas surrounding their Teflon plant, particularly the one in Parkersburg, West Virginia, that those communities have PFAS in their tap water.

Wow, so based on its own studies, DuPont knows its chemicals are making animals sick. They seem to be making workers sick. And now they found out that the chemicals have made their way into the water supply. What do they do with that information?

As far as we know, they didn’t do much. They certainly didn’t tell the residents of Parkersburg who were drinking that water that there was anything that they needed to be worried about.

How is that possible? I mean, setting aside the fact that DuPont is the one actually studying the health effects of its own chemicals, presumably to make sure they’re safe, we’ve seen these big, regulating agencies like the EPA and the FDA that exist in order to watch out for something exactly like this, a company that is producing something that may be harming Americans. Why weren’t they keeping a closer watch?

Yeah, so it goes kind of back to the way that we regulate chemicals in the US. It goes through an act called the Toxic Substances Control Act that’s administered by the EPA. And basically, it gives companies a lot of room to regulate themselves, in a sense. Under this act they have a responsibility to report to the EPA if they find these kinds of potential issues with a chemical. They have a responsibility to do their due diligence when they’re putting a chemical out into the environment.

But there’s really not a ton of oversight. The enforcement mechanism is that the EPA can find them. But this kind of thing can happen pretty easily where DuPont keeps going with something that they think might really be a problem and then the fine, by the time it plays out, is just a tiny fraction of what DuPont has earned from producing these chemicals. And so really, the incentive is for them to take the punishment at the end, rather than pull it out early.

So it seems like it’s just self-reporting, which is basically self-regulation in a way.

Yeah, I think that is the way a lot of advocacy groups and experts have characterized it to me, is that chemical companies are essentially regulating themselves.

So how did this danger eventually come to light? I mean, if this is in some kind of DuPont vault, what happened?

Well, there’s a couple different things that started to happen in the late ‘90s.

The community around Parkersburg, West Virginia, people had reported seeing really strange symptoms in their animals. Cows were losing their hair. They had lesions. They were behaving strangely. Some of their calves were dying. And a lot of people in the community felt like they were having health problems that just didn’t really have a good answer, mysterious sicknesses, and some cases of cancers.

And so they initiate a class action lawsuit against DuPont. As part of that class action lawsuit, DuPont, at a certain point, is forced to turn over all of their internal documentation. And so what was in the files was all of that research that we mentioned all of the studies about — animals, and workers, the birth defects. It was really the first time that the public saw what DuPont and 3M had already seen, which is the potential health harms of these chemicals.

So that seems pretty damning. I mean, what happened to the company?

So, DuPont and 3M are still able to say these were just a few workers. And they were working with high levels of the chemicals, more than a person would get drinking it in the water. And so there’s still an opportunity for this to be kind of correlation, but not causation. There’s not really a way to use that data to prove for sure that it was PFAS that caused these health problems.

In other words, the company is arguing, look, yes, these two things exist at the same time. But it doesn’t mean that one caused the other.

Exactly. And so one of the things that this class action lawsuit demands in the settlement that they eventually reach with DuPont is they want DuPont to fund a formal independent health study of the communities that are affected by this PFAS in their drinking water. And so they want DuPont to pay to figure out for sure, using the best available science, how many of these health problems are potentially related to their chemicals.

And so they ask them to pay for it. And they get together an independent group of researchers to undertake this study. And it ends up being the first — and it still might be the biggest — epidemiological study of PFAS in a community. They’ve got about 69,000 participants in this study.

Wow, that’s big.

It’s big, yeah. And what they ended up deciding was that they could confidently say that there was what they ended up calling a probable link. And so they were really confident that the chemical exposure that the study participants had experienced was linked to high cholesterol, ulcerative colitis, thyroid disease, testicular cancer, kidney cancer, and pregnancy induced hypertension.

And so those were the conditions that they were able to say, with a good degree of certainty, were related to their chemical exposure. There were others that they just didn’t have the evidence to reach a strong conclusion.

So overall, pretty substantial health effects, and kind of vindicates the communities in West Virginia that were claiming that these chemicals were really affecting their health.

Absolutely. And as the years have gone on, that was sort of just the beginning of researchers starting to understand all the different kinds of health problems that these chemicals could potentially be causing. And so since the big DuPont class action study, there’s really just been like this building and building and building of different researchers coming out with these different pieces of evidence that have accumulated to a pretty alarming picture of what some of the potential health outcomes could be.

OK, so that really kind of brings us to the present moment, when, at last, it seems the EPA is saying enough is enough. We need to regulate these things.

Yeah, it seems like the EPA has been watching this preponderance of evidence accumulate. And they’re sort of deciding that it’s a real health problem, potentially, that they need to regulate.

So the EPA has identified six of these PFAS chemicals that it’s going to regulate. But the concern that I think a lot of experts have is that this particular regulation is not going to keep PFAS out of our bodies.

We’ll be right back.

So, Kim, you just said that these regulations probably won’t keep PFAS chemicals out of our bodies. What did you mean?

Well, the EPA is talking about regulating these six kinds of PFAS. But there are actually more than 10,000 different kinds of PFAS that are already being produced and out there in the environment.

And why those six, exactly? I mean, is it because those are the ones responsible for most of the harm?

Those are the ones that the EPA has seen enough evidence about that they are confident that they are probably causing harm. But it doesn’t mean that the other ones are not also doing something similar. It’s just sort of impossible for researchers to be able to test each individual chemical compound and try to link it to a health outcome.

I talked to a lot of researchers who were involved in this area and they said that they haven’t really seen a PFAS that doesn’t have a harm, but they just don’t have information on the vast majority of these compounds.

So in other words, we just haven’t studied the rest of them enough yet to even know how harmful they actually are, which is kind of alarming.

Yeah, that’s right. And there’s just new ones coming out all the time.

Right. OK, so of the six that the EPA is actually intending to regulate, though, are those new regulations strict enough to keep these chemicals out of our bodies?

So the regulations for those six chemicals really only cover getting them out of the drinking water. And drinking water only really accounts for about 20 percent of a person’s overall PFAS exposure.

So only a fifth of the total exposure.

Yeah. There are lots of other ways that you can come into contact with PFAS. We eat PFAS, we inhale PFAS. We rub it on our skin. It’s in so many different products. And sometimes those products are not ones that you would necessarily think of. They’re in carpets. They’re in furniture. They’re in dental floss, raincoats, vinyl flooring, artificial turf. All kinds of products that you want to be either waterproof or stain resistant or both have these chemicals in them.

So, the cities and towns are going to have to figure out how to test for and monitor for these six kinds of PFAS. And then they’re also going to have to figure out how to filter them out of the water supply. I think a lot of people are concerned that this is going to be just a really expensive endeavor, and it’s also not really going to take care of the entire problem.

Right. And if you step back and really look at the bigger problem, the companies are still making these things, right? I mean, we’re running around trying to regulate this stuff at the end stage. But these things are still being dumped into the environment.

Yeah. I think it’s a huge criticism of our regulatory policy. There’s a lot of onus put on the EPA to prove that a harm has happened once the chemicals are already out there and then to regulate the chemicals. And I think that there’s a criticism that we should do things the other way around, so tougher regulations on the front end before it goes out into the environment.

And that’s what the European Union has been doing. The European Chemicals Agency puts more of the burden on companies to prove that their products and their chemicals are safe. And the European Chemicals Agency is also, right now, considering just a ban on all PFAS products.

So is that a kind of model, perhaps, of what a tough regulation could look like in the US?

There’s two sides to that question. And the first side is that a lot of people feel like it would be better if these chemical companies had to meet a higher standard of proof in terms of demonstrating that their products or their chemicals are going to be safe once they’ve been put out in the environment.

The other side is that doing that kind of upfront research can be really expensive and could potentially limit companies who are trying to innovate in that space. In terms of PFAS, specifically, this is a really important chemical for us. And a lot of the things that we use it in, there’s not necessarily a great placement at the ready that we can just swap in. And so it’s used in all sorts of really important medical devices or renewable energy industries or firefighting foam.

And in some cases, there are alternatives that might be safer that companies can use. But in other cases, they just don’t have that yet. And so PFAS is still really important to our daily lives.

Right. And that kind of leaves us in a pickle because we know these things might be harming us. Yet, we’re kind of stuck with them, at least for now. So, let me just ask you this question, Kim, which I’ve been wanting to ask you since the beginning of this episode, which is, if you’re a person who is concerned about your exposure to PFAS, what do you do?

Yeah. So this is really tricky and I asked everybody this question who I talked to. And everybody has a little bit of a different answer based on their circumstance. For me what I ended up doing was getting rid of the things that I could sort of spot and get rid of. And so I got rid of some carpeting and I checked, when I was buying my son a raincoat, that it was made by a company that didn’t use PFAS.

It’s also expensive. And so if you can afford to get a raincoat from a place that doesn’t manufacture PFAS, it’s going to cost more than if you buy the budget raincoat. And so it’s kind of unfair to put the onus on consumers in that way. And it’s also just not necessarily clear where exactly your exposure is coming from.

So I talk to people who said, well, it’s in dust, so I vacuum a lot. Or it’s in my cleaning products, so I use natural cleaning products. And so I think it’s really sort of a scattershot approach that consumers can take. But I don’t think that there is a magic approach that gets you a PFAS-free life.

So Kim, this is pretty dark, I have to say. And I think what’s frustrating is that it feels like we have these government agencies that are supposed to be protecting our health. But when you drill down here, the guidance is really more like you’re on your own. I mean, it’s hard not to just throw up your hands and say, I give up.

Yeah. I think it’s really tricky to try to know what you do with all of this information as an individual. As much as you can, you can try to limit your individual exposure. But it seems to me as though it’s at a regulatory level that meaningful change would happen, and not so much throwing out your pots and pans and getting new ones.

One thing about PFAS is just that we’re in this stage still of trying to understand exactly what it’s doing inside of us. And so there’s a certain amount of research that has to happen in order to both convince people that there’s a real problem that needs to be solved, and clean up what we’ve put out there. And so I think that we’re sort of in the middle of that arc. And I think that that’s the point at which people start looking for solutions.

Kim, thank you.

Here’s what else you should know today. On Tuesday, in day two of jury selection for the historic hush money case against Donald Trump, lawyers succeeded in selecting 7 jurors out of the 12 that are required for the criminal trial after failing to pick a single juror on Monday.

Lawyers for Trump repeatedly sought to remove potential jurors whom they argued were biased against the president. Among the reasons they cited were social media posts expressing negative views of the former President and, in one case, a video posted by a potential juror of New Yorkers celebrating Trump’s loss in the 2020 election. Once a full jury is seated, which could come as early as Friday, the criminal trial is expected to last about six weeks.

Today’s episode was produced by Clare Toeniskoetter, Shannon Lin, Summer Thomad, Stella Tan, and Jessica Cheung, with help from Sydney Harper. It was edited by Devon Taylor, fact checked by Susan Lee, contains original music by Dan Powell, Elisheba Ittoop, and Marion Lozano, and was engineered by Chris Wood.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for The Daily. I’m Sabrina Tavernise. See you tomorrow.

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• April 19, 2024   •   30:42 The Supreme Court Takes Up Homelessness
• April 18, 2024   •   30:07 The Opening Days of Trump’s First Criminal Trial
• April 17, 2024   •   24:52 Are ‘Forever Chemicals’ a Forever Problem?
• April 16, 2024   •   29:29 A.I.’s Original Sin
• April 15, 2024   •   24:07 Iran’s Unprecedented Attack on Israel
• April 14, 2024   •   46:17 The Sunday Read: ‘What I Saw Working at The National Enquirer During Donald Trump’s Rise’
• April 12, 2024   •   34:23 How One Family Lost $900,000 in a Timeshare Scam
• April 11, 2024   •   28:39 The Staggering Success of Trump’s Trial Delay Tactics
• April 10, 2024   •   22:49 Trump’s Abortion Dilemma
• April 9, 2024   •   30:48 How Tesla Planted the Seeds for Its Own Potential Downfall
• April 8, 2024   •   30:28 The Eclipse Chaser

Hosted by Sabrina Tavernise

Featuring Kim Tingley

Produced by Clare Toeniskoetter ,  Shannon M. Lin ,  Summer Thomad ,  Stella Tan and Jessica Cheung

With Sydney Harper

Edited by Devon Taylor

Original music by Dan Powell ,  Elisheba Ittoop and Marion Lozano

Engineered by Chris Wood

Listen and follow The Daily Apple Podcasts | Spotify | Amazon Music

The Environmental Protection Agency has begun for the first time to regulate a class of synthetic chemicals known as “forever chemicals” in America’s drinking water.

Kim Tingley, a contributing writer for The New York Times Magazine, explains how these chemicals, which have been linked to liver disease and other serious health problems, came to be in the water supply — and in many more places.

On today’s episode

Kim Tingley , a contributing writer for The New York Times Magazine.

Background reading

“Forever chemicals” are everywhere. What are they doing to us?

The E.P.A. issued its rule about “forever chemicals” last week.

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

Fact-checking by Susan Lee .

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson and Nina Lassam.

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