EQUATIONS. Main Overarching Questions: 1. What is a variable and what does it represent?

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1 EQUATIONS Introduction to Variables, Algebraic Expressions, and Equations (2 days) Overview of Objectives, students should be able to: Main Overarching Questions: 1. Evaluate algebraic expressions given replacement values 2. Identify solution of equations 3. Translate phrases into variable expressions 4. Multiply and divide given fractional replacement values 5. Add or subtract given fractional replacement values 1. What is a variable and what does it represent? 2. What is the difference between an equation and an expression? 3. Explain the difference between the terms simplify, evaluate, and solve. 4. What does the solution of an equation represent? 6. Simplifying Algebraic Expressions Use properties of numbers to combine like terms Use properties of numbers to multiply expressions Simplify expressions by multiplying and then combining like terms Objectives: Activities and Questions to ask students: Translate phrases into variable expressions Working in small groups or pairs, have students generate lists of words that mean add, subtract, multiply, divide, and equal. Review list as a class, and compile lists into a master list, having individual students add words to their own lists. Discuss the term variable and contrast it with the word constant. What makes a variable different than a constant? What does a variable represent? Model some basic algebraic expressions for students, like a number plus seven. Make sure to do at least one example that uses the words less than, pointing out that this requires the

2 Evaluate algebraic expressions given replacement values Multiply and divide given fractional replacement values student to switch the order of the terms. For example, a number less than seven, would be written as 7 n rather than n 7. Assign a short practice set, allowing students to work individually or in pairs using their list of operator words for assistance. Review the meaning of the word variable with students. Prompt them to generate a definition in their own words, and to provide examples of variables. Using one of the algebraic expressions from the previous lesson, ask students to determine the value of the expression without providing a value for the variable. Do we know what this expression is equal to? Assist students in concluding that we cannot determine the value of the expression without knowing the value of the variable. Provide a value for the variable in the expression, then ask students to determine the value of the entire expression (this may require a review of the order of operations). Provide several examples as guided practice problems, constantly checking for student understanding. Ask students to use their expressions from the previous lesson, and to assign values to each of the variables. Have them trade papers with a partner and evaluate each other s expressions. Then, have them check each other s work. Review the rules for multiplying with fractions. Must you have common denominators in order to multiply? Why not? Is it always ok to turn fractions into decimals in order to evaluate? Review reducing fractions using division of common factors. What are we really doing when we reduce a fraction? Model a few problems where you have to evaluate an expression by multiplying with fractions. Provide a brief problem set for independent practice. Review the rules for dividing with fractions. Model a few problems where you have to evaluate an expression by dividing with fractions. Provide a brief problem set for independent practice. Add or subtract given fractional replacement values Review the rules for adding/subtracting with fractions. Why do you have to have common denominators before you can add or subtract fractions? How do you determine the common denominator, and how do you get it? Model a few problems where you have to evaluate an expression by adding or subtracting with fractions. Provide a brief problem set for independent practice. Identify solution of equations What is an equation? What makes an equation different from an expression? Display a basic one-step equation, and allow students to work in pairs to make a conjecture

3 Simplifying Algebraic Expressions o o o Use properties of numbers to combine like terms Use properties of numbers to multiply expressions Simplify expressions by multiplying and then combining like terms about the solution to the equation. Ask them to share how they reached that solution. What does it mean to solve an equation? What are we finding the value of? Is solving the same as evaluating? Provide a few more one-step equations for students to work in their pairs or small groups. Have them check each other s work, then check solutions as a class. Review the differences between the terms evaluate and solve. What does it mean to simplify? Can I simplify the expression ? What is the result? Could I simplify the expression 5x + 2x 4x? What is the result? Plug in a value for x and verify that the simplified expression is equivalent to the expanded expression. Could I simplify the expression 5x + 2y 4z? Why or why not? Assign values to x, y, and z and substitute to test conjectures. Prompt students to determine that x, y, and z all represent different values. Why can t we combine these different values using addition and subtraction? 2 Could I simplify the expression 5x + 2x? Why or why not? Plug in a value for x to determine whether or not the students conjectures are correct. 2 2 Explain that x does not represent the same number as x, so 5 sets of x s wouldn t be the same as 5 sets of x s. If I want to combine terms, the sets have to be comparable. Go over the rules for combining like terms (same variables, same exponents). As a class, create several examples and non-examples of like terms. Model several examples of simplifying expressions by combining like terms. How could I simplify the expression 2(5x)? Illustrate that 2(5x) is equivalent to 5x + 5x. What is the sum of these two terms? How could I simplify the expression 2(5x + 2x 4x)? Have students work in pairs to determine how to simplify the expression. Ask them to share their methods with the class. (Combine the like terms first, then multiply; write out each term twice, then simplify; distribute the two, then multiply). Do two terms have to be like terms in order to multiply them? Are 2 and 5x like terms? Were we able to multiply them? Provide several examples of multiplying with unlike terms, guiding students through the process.

4 Solving Equations (3 days) Overview of Objectives, students should be able to: 1. Identify solutions of equations. 2. Use the addition property (If a = b, then a + c = b + c) of equality to solve equations. 3. Use the multiplication property (if a = b, then a*c = b*c) of equality to solve equations 4. Solve linear equations containing parentheses 5. Solve a formula or equation for one of its variables 6. Apply the general strategy for solving a linear equation Main Overarching Questions: 1. What is the difference between an equation and an expression? 2. When solving an equation, why don t you follow the order of operations? 3. Why is it sometimes useful to solve a literal equation for a specific variable? 4. What is the solution to an identity? 5. How do you know when an equation has no solutions? 6. If an equation contains fractions, what can you do as a first step to get rid of all the fractions? 7. How can you check the solution to an equation? 7. Recognize identities and equations with no solution 8. Solving Equations Containing Fractions a. Solve equations containing fractions b. Solve equations by multiplying by the LCD c. Solve proportions 9. Solving Equations Containing Decimals 10. Providing answers in set notation Objectives: Activities and Questions to ask students: Identify solutions of equations. What is an equation? What makes an equation different from an expression? Display a basic one-step equation, and allow students to work in pairs to make a conjecture about the solution to the equation. Ask them to share how they reached that solution. What does it mean to solve an equation? What are we finding the value of? Is solving the same as evaluating?

5 Provide a few more one-step equations for students to work in their pairs or small groups. Have them check each other s work, then check solutions as a class. Use the addition/subtraction properties (If a = b, then Review the different methods that students used to identify solutions of equations. Discuss a + c = b + c) of equality to solve equations. the idea of undoing the order of operations, or working backward to perform the inverse operation. What is the inverse operation of addition? Of subtraction? Model a couple examples of one-step equations that require the use of the addition/subtraction properties. How do you know whether to use addition or subtraction to solve an equation? Emphasize the importance of checking solutions by substituting them back into the original equations. Use the multiplication/division properties (if a = b, Provide a one-step equation that requires multiplication or division to solve. Have students then a*c = b*c) of equality to solve equations work in pairs to determine how they would solve the problem. Ask students to share methods and results. Review the idea of undoing the order of operations by using inverse operations. What is the inverse operation of multiplication? Of division? Model a few examples of one-step equations that require the use of the multiplication/division properties. How do you know when to multiply and when to divide? Emphasize the importance of checking solutions. When equations contain more than one operation, like multiplication and addition, what do you undo first? Model a few examples of two-step equations, then provide a problem set for independent practice. Solve linear equations containing parentheses What purpose do parentheses serve in mathematics? What does the expression 2(x + 3) really mean? What if I know what that expression equals? Using a basic equation, like 2(x + 3) = 12, ask students to work in pairs to determine the solution. Have them share their methods and results. Discuss the different methods for solving equations with parentheses, including distributing first, and dividing first. Guide students through a few examples, then provide a problem set for independent practice. Emphasize the importance of checking solutions. Solve a formula or equation for one of its variables Use the following investigation to introduce students to solving literal equations: A track coach is calculating the average speed of each team member. Using the formula d = rt, have students complete the following table:

6 Distance Rate Time 1500 m 4.98 min 800 m 3.08 min 800 m 2.93 min 400 m 1.18 min Now ask students to use the formula r = d/t to find the rates. What do you notice about the two values? (they are the same) Which formula was easier to use? Discuss the idea that sometimes it is easier to isolate a certain value in a formula one time, rather than having to isolate that value every time you use it to solve an equation. Ask students to recall the formula for the area of a triangle. If we had several triangles where the area and bases were known, but we wanted to find the heights, what could we do to the formula to make it easier to solve for each height? Guide students through the required steps to solve for h. Have students work in pairs and compare their answers. Apply the general strategy for solving a linear equation Review simplifying expressions from the previous section. How do you know when two terms can be combined? Is it possible to have like terms in an equation? If there are like terms in an equation, should they be combined? Provide students with the following example: 3x+ 4x 2 = 12. What should we do before we begin solving this problem? (combine the 3x and 4x). Do we use inverse operations when we combine like terms? Why not? Emphasize that you only use inverse operations when you move terms across the equal sign. Provide an example that requires students to move terms across the equal sign, like 5a 14 = a. Which terms are like terms, and how can we get them together? Model several additional examples, guiding students through the steps required to simplify, then solve. Recognize identities and equations with no solution Divide the class in half. Provide half of the students with the equation 2(x 4) = 2x + 3. Provide the other half of the class with the equation 2(x 4) = 2x -8. Ask each group of students to use prior knowledge to solve their equations. Have them share their answers with their groups and discuss their results. Allow one student from each group to present their results (along with work) to the class. What does it mean when I end up with 0 = 0, or -8 = -8, or any number equal to itself in an equation? These statements are always true, which leads me to believe that no matter what I plug in for the variable, I m always going to get a true statement in the end. Test the

7 Solving Equations Containing Fractions o Solve equations containing fractions o Solve equations by multiplying by the LCD o Solve proportions hypothesis by plugging in several different values for x into the second equation. Discuss the term identity. Why is this a good name for an equation where all real numbers are solutions? What does it mean when I end up with a statement that is never true, like -8 = 3? Is there any number that I could plug in for x that would make -8 equal to 3? What does this mean for the solution of our equation? **After learning about identities and no solutions, students sometimes tend to think that every equation can be labeled as an identity or a no solution equation. Explain that these are special cases, and that most of the equations that they solve will have numerical solutions. Provide several examples of equations, and ask students to identify which ones have numerical solutions, which ones have no solutions, and which ones are identities. Check answers as a class. Review the rules for performing operations with fractions. What operation does the fraction bar represent? Write the equation, 3 x 1 = 2 on the board. In the first term, what is the (3/4) doing to 4 4 the x-value? (it is multiplying x by ¾) If the fraction bar means division, how is it possible that x is being multiplied by ¾? Guide students to the conclusion that x is being multiplied by 3 and divided by four. Would it be possible to undo that division by four as my first step instead of waiting until the end? How can I undo division? (multiplication) Review the multiplication property, reminding students that whatever you multiply to one side of an equation, you must multiply to the other. Guide students through the problem, showing how multiplication by four cancels out divisions by four. Provide an additional example with common denominators and allow students to work through the problem independently, checking their work as they go. Provide an example where the denominators are different, and ask students to determine whether or not you could use a similar process to get rid of those denominators. What would you have to multiply by to cancel out both denominators in the same step? Guide students through a few additional examples. x 4 Provide an example of a simple proportion, like =. Ask students what the 3 is doing to 3 12 the x (it is dividing into x). What could I do to undo that division? Guide students through solving the equation, and checking the solution. Provide an example where x is the denominator. Can I solve this proportion in the same way?

8 Why or why not? Guide students through the process for solving a proportion by crossmultiplication. Model several examples, then provide a problem set for independent practice. Solving Equations Containing Decimals What does the decimal point in a number represent? If I want to move the decimal point one place to the right, by what number would I multiply? What if I wanted to move it two places to the right? One place to the left? Discuss the idea that the process for solving equations with decimals is no different than for any other equation. Tell students that if they are uncomfortable working with decimals, that we can use multiplication to move the decimal, similar to multiplying by the common denominator to rid an equation of fractions. Provide an example, like 0.5x = By what number would I have to multiply to get rid of all the decimals in the equation? Assist students in moving the decimal, then allow students to solve the resulting equation independently. Check results as a class. Model several additional examples, but remind students that moving the decimal is not a necessary step, and does not have to be used unless the student is uncomfortable working with decimals. Work some examples without moving the decimal. Providing answers in set notation Formulas and Problem Solving (3 days) Overview of Objectives, students should be able to: 1. Use problem-solving steps to solve problems 2. Translate a problem to an equations, then use the equation to solve the problem 3. Use formulas to solve problems 4. Solve problems modeled by proportions 5. Solving Percent Problems with Equations Main Overarching Questions: 1. What methods/steps should be used to approach and solve a word problem? 2. How can a percent problem be written as an equation/proportion? 3. When solving a percent problem, how do you know when/where to move the decimal? 4. What is the formula for percent increase/decrease? How do you know whether it is an increase or decrease? 5. What is the formula for simple interest and how is it used? a. Write percent problems as equations b. Solve percent problems

9 6. Solving Percent Problems with Proportions a. Write percent problems as proportions b. Solve percent problems 7. Applications of Percents a. Solve applications involving percent b. Find percent increase and percent decrease c. Calculate sales tax and total price d. Calculate discount and sale price e. Calculate simple interest Objectives: Activities and Questions to ask students: Use problem-solving steps to solve problems Ask students what approaches/methods they use to read and interpret word problems. For example, draw a diagram, underline important information, eliminate irrelevant information, assign variables to unknowns, guess and check, look for a pattern, use a formula, write an equation, etc. Discuss the general 4-step problem solving approach: 1) Read (understand) the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check the solution. Ask students, Does this mean that every word problem will be solved in exactly four steps? Translate a problem to an equations, then use the equation to solve the problem Present a problem similar to the following: A rectangle has a perimeter of 20 inches. The length of the rectangle is 4 more than the width. What are the dimensions of the rectangle? Allow students to work in pairs or small groups and follow the problem solving plan to complete the problem. Ask individual group members to share their problem solving plan with the class, along with their solutions. Discuss how different approaches can lead to the same correct solution. If no group used an equation to solve the problem, guide students through setting up the equation 2x + 2(x + 4) = 20 and solve the equation to arrive at the same answer. Use formulas to solve problems Ask students how they knew to add all the sides in the previous example. What is the formula for perimeter of a rectangle?

10 Solicit other common formulas from students. How can formulas assist us in solving word problems? Present a problem similar to the following: The speed limit on highway 47 is 60 mph. Sally traveled from Trenton to Lake City (50 miles) in 40 minutes. Was she speeding? Ask students to determine what formula could assist us in solving this problem. If they do not remember, provide them with the formula d = rt. Ask students to work in pairs to determine whether or not Sally was speeding on her trip. You may have to review converting between hours and minutes. Ask individual group members to share their solutions with the class, along with their problem-solving approaches. Solve problems modeled by proportions Using the previous example as a guide, ask students, If Sally could travel 50 miles in 40 minutes, then how many miles would she have traveled in 30 minutes traveling at the same speed? Allow students to work in pairs to determine the solution. Ask students to share their problem solving methods with the class, and compare solutions. Remind students that a proportion is when two ratios are set equal to one another. Ask students how the formula d = rt could be written as a ratio (r = d/t). If Sally s rate (speed) remains constant, then wouldn t d/t = d/t? How could we solve this proportion? Provide several additional examples (not necessarily using distance, rate, and time). Guide students through setting up the proportions and then solving using cross-multiplication. Solving Percent Problems with Proportions What does the word percent mean? (Part of 100) Could we write a percent as a ratio? What would the denominator be? o Write percent problems as proportions How could I write 10% as a ratio? If I know that 10% is equal to 10/100 (10 parts out of every 100), then how could I write 10% of 300 as a proportion? o Solve percent problems Guide students in setting up the problem as a proportion, then solving using cross products. Provide the formula: %/100 = is/of OR %/100 = part/whole. Provide several additional examples of percent problems, and assist students in setting up the proportions correctly. Allow the students to solve independently then check their work with a partner. Solving Percent Problems with Equations What s nice about using proportions to solve percent problems is that you do not have to worry about moving any decimals. However, knowing that 10% is equal to 10/100, what o Write percent problems as equations would 10% be as a decimal? Is there a shortcut to writing percents as decimals? Provide an example similar to the following: What is 10% of 300? We already know the o Solve percent problems solution, and how to find it using proportions. Is there a way that we could solve this by using

11 Applications of Percents o Solve applications involving percent o Find percent increase and percent decrease o Calculate sales tax and total price o Calculate discount and sale price o Calculate simple interest an equation? In the example, the word what could be represented with a variable. What mathematical symbol represents the word is? What mathematical operation is related to the word of? Recognizing that what is our variable, is means equals, 10% means.10, and of means multiply, our equation becomes x =.10(300). Guide students toward the solution. Model several additional examples, making sure to provide several different types of percent problems (solving for the percent, solving for the is, and solving for the of ). What real-life examples can you think of that would involve percents? If I went to a department store and saw a pair of jeans that was marked down 50%, then another 25%, what is the total percent discount that I would receive? Many people would say 75%, and the department stores would like for us to think that this is the case, but let s find out if it really is. Let s say the jeans had an original price of $50. If we were to mark them down by 50%, what would the new price be? ($25). Then, we would take that $25 and mark it down 25%. What would the new price be? ($18.75). Let s return to our original price and determine whether or not the final price is marked down 75% from the original price of $50. Ask students to find out what a 75% discount from $50 would have been, and what the resulting price would be. Did we receive a 75% discount on our jeans? Let s see if we can find out the actual percent discount that we received. We started with a $50 pair of jeans, and we paid $18.75 for them. That s a price cut of $ What percent of $50 is $31.25? Ask students to draw upon previous lessons to determine the percent. (62.5%). Although this may be a good deal, it s not the 75% discount that we thought we were getting. When we found the total percent discount in the last problem, we were actually finding a percent of change. Percent of change can refer to an increase or decrease. In our example, did we see a percent increase or decrease? How can you tell? The formula that we use to find a percent of change is (new old)/old, where new refers to the new price or quantity, and old refers to the original price or quantity. Here s another example for us to try. If I were to increase the price of a $100 item by 10%, then decrease that new price by 10%, would the item return to its original $100 price? Assist students in using the percent of change formula to determine the solution. Another common real-world use of percents is in calculating interest. The formula for simple interest is I = prt, or Interest earned = principal * rate * time. The principal represents the initial investment. The rate is usually expressed as an annual percent, but must be converted to a decimal for the purposes of the formula. The unit of time must match the unit of time

12 used for the rate. For example, if the rate is 5% per year, then time must be expressed in years or as a fraction of a year. Model a few examples, making sure to provide problems that require you to solve for different variables. Provide several examples of interest problems, and allow students to work in pairs or small groups to determine solutions. Solving Linear Inequalities and Problem Solving (2 days) Overview of Objectives, students should be able to: 1. Graph inequalities on a number line 2. Use the addition property of inequality to solve inequalities Main Overarching Questions: 1. What is the difference between an equation and an inequality? 2. How many solutions can an inequality have? 3. How is solving an inequality similar to solving an equation? a. a < b and a + c < b + c are equivalent 3. Use the multiplication property of inequality to solve inequalities a. If c positive then a < b and ac<bc are equivalent 4. When multiplying or dividing by a negative number in an inequality, what must you do? 5. How do you represent the solution to an inequality on a number line? 6. How do you represent the solution to an inequality in set notation? b. If c negative, then a < b and ac>bc are equivalent 4. Use both properties to solve inequalities 5. Solve problems modeled by inequalities 6. Provide answers in set notation Objectives: Activities and Questions to ask students: Graph inequalities on a number line Ask students to compare the solutions to x = 2 and x > 2. How many solutions does the equation have? How many solutions does the inequality have? Draw a number line and ask

13 Use the addition property of inequality to solve inequalities a < b and a + c < b + c are equivalent Use the multiplication property of inequality to solve inequalities If c positive then a < b and ac<bc are equivalent If c negative, then a < b and ac>bc are equivalent students to identify the solution to x = 2 on the number line. Draw a second number line and ask students to identify the solutions to the inequality. Would 2 be a solution to the inequality? How do we show that 2 is not included in our solution set on the number line? Ask students to compare the solutions to x > 2 and x 2. What number is in the solution set to the 2 nd inequality that was not included in the first? How could we represent this on the graph? Ask students to compare the solutions of x > 2 and 2 < x. Many students get confused when the variable is not on the left side of the inequality. Make sure that students understand that these are equivalent inequalities, and that in either situation, x is larger than 2. Ask students to graph 2 < x, and make sure that they understand that the graph should be identical to the graph of x > 2. Model additional examples, making sure to include some where the variable is on the right side of the inequality sign. Solicit a student volunteer to solve a one-step equation, like x + 2 = 8, or x 7 = 9. Encourage students to show their steps as they solve. Using the same examples, change the = to inequality signs. Would this change the way that we solved the problem? Would it change the solutions? Guide students to the realization that the methods are the same, but the solutions are different. How many solutions did our equations have? How many solutions do the inequalities have? Ask student volunteers to graph the solutions on the board, then have students choose numbers from the solution set to substitute back into the inequality to check. Provide a few additional examples for guided practice, then provide a problem set for independent practice. Solicit a student volunteer to solve a one-step equation, like x/2 =4, or 2x = 10. Encourage students to show their steps as they solve. Using the same examples, change the = sign to inequality signs. Would this change the way that we solved the problem? Would it change the solutions? Guide students to the realization that the methods are the same, but the solutions are different. How many solutions did our equations have? How many solutions do the inequalities have? Ask student volunteers to graph the solutions on the board, then have students choose numbers from the solution set to substitute back into the inequality to check. Using an example that requires multiplication or division by a negative number (ex. -4x > 16), ask students to work with a partner at their seats to solve the inequality. Once they arrive at a solution, direct them to choose a number from the solution set to plug back into the inequality and check. Was their solution correct? Why or why not?

14 Ask a few students to share their methods and solutions with the class. Try to choose both students who came to correct and incorrect solutions. What did some students do to get the correct answer? Assist students in recognizing that multiplication or division by a negative in an inequality requires them to flip the inequality sign. Model a few examples for students, using both multiplication and division. Make sure to include some examples that require division into a negative number, but not by a negative number. This can be confusing for some students. For example, 2x > -4 would not require you to flip the sign, but -2x > -4 would. Use both properties to solve inequalities Provide a two-step equation for students to work on in pairs, like 2x 6 = 12. Ask groups to compare their solutions, then ask for a student volunteer to display their work to the class. Using the same example, change the = sign to an inequality. How would this change our steps? Would it change the solutions? Ask students to work in pairs to solve the inequality, and graph the solutions. Provide several additional examples for guided practice, making sure to include a few problems that require more than 2 steps, and problems that require multiplication or division by a negative. Solve problems modeled by inequalities Ask students to write the solutions to the following situation as an inequality: A bus can seat at most 48 students. Many students will likely write the solutions as x > 48. Ask students, If this is my solution set, then doesn t that mean that I could seat 50 students on the bus? Is this the case? Guide students to understand that at most means that 48 is the maximum acceptable value, so all feasible solutions must lie at or below 48. Model the correct inequality for students. Provide several additional examples for students to work in pairs. Examples: a) You must be at least 16 years old to obtain a driver s license, b) It is not safe to use a light bulb of more than 60 watts in this fixture, c) At least 350 students attended the football game Friday night, d) The Navy s flying squad, the Blue Angels, makes more than 75 appearances each year. After students have written their inequalities, ask them to choose a value from each solution set and test it in the original problem for feasibility. Did they change any answers? Provide a problem similar to the following, and ask students to work in pairs to write and solve an inequality: Your brother has $2000 saved for a vacation. His airplane ticket is $637. Write and solve an inequality to find how much he can spend for everything else. Ask several students to share their inequalities and methods for solving. Test values to determine the correctness of their results. Provide a problem similar to the following, and have students work in pairs: The science club

15 charges $4.50 per car at their car wash. Write and solve an inequality to find out how many cars they have to wash to earn at least $300. Ask several students to share their inequalities and methods for solving. Test values to determine the correctness of their results. Remember that some inequalities require more than one step to solve. Have students work in pairs to solve the following problem: The perimeter of an isosceles triangle is at most 27 cm. One side is 8 cm long. Find the possible lengths of the two congruent sides. Ask several students to share their inequalities and methods for solving. Test values to determine the correctness of their results. Provide answers in set notation Another way to represent the solutions to an inequality is by using set notation. For the solutions to the equation x = 3, we could write {3}. But for most inequalities, we have an infinite number of solutions, so we couldn t possibly list all of them. For the inequality x > 3, the set notation would be {x x > 3}. This may seem like an extra step now, but this idea will be expanded upon in higher level math classes, so it is good to learn it now. Have students go back to a previous assignment from this section, and rewrite some of their answers using set notation.