Bicycle Budget Problem 6.EE - Reason about and solve one- variable equations and inequalities 5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. 8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Part A Ana is saving to buy a bicycle that costs $135. She has saved $98 and wants to know how much more money she needs to buy the bicycle. Teacher present this problem to students and ask them how they will solve the problem. Teacher can encorage students to write an equation instead of working the problem backwards.

MP6: Attend to Precision 1. Explain how the equation below models the following situation: “Ana is saving to buy a bicycle that costs $135. She has saved $98 and wants to know how much more money she needs to buy the bicycle.” 135 = x + 98 (6.EE.5) Students show a thorough understanding of equations in a contextual scenario, as well as a thorough understanding of substituting values accurately and efficiently into equations to verify whether or not they satisfy the equation. The student offers a correct and accurately interpretation of the equality in the context of the problem. Students state precisely the meaning of variables they use when setting up equations, this includes specifying whether the variable refers to a specific number, or to all numbers in some range x represents the additional amount of money Ana needs to buy the bicycle. x+ 98 represent the amount that Ana needs to buy the bicycle and the $98 she has saved. What does x represent? How could you test your solution to see if it answers the problem? What symbols or mathematical notations are important in this problem?

When substituting for x, which value(s), if any, from Task #1: When substituting for x, which value(s), if any, from {0, 37, 98, 135, 233} will make the equation true? Consider the following situation: When x = 0 When x = 37 When x = 98 135 = x + 98 135 = x + 98 135 = x + 98 135 = 0 + 98 135 = 37 + 98 135 = 98 + 98 135  98 135 = 135 135  196 False True False Students can accurately substitute and solve each equation and can state with precision that 37 is the only value in the range that makes the equation true. When x = 233 When x = 135 135 = x + 98 135 = x + 98 135 = 233 + 98 135 = 135 + 98 135  331 135  236 False False

How did you know your solution was reasonable? Task #2: Explain what this means in terms of the amount of money needed and the cost of the bicycle. Consider the following situation: When x = 0 When x = 37 When x = 98 135 = x + 98 135 = x + 98 135 = x + 98 135 = 0 + 98 135 = 37 + 98 135 = 98 + 98 135  98 135 = 135 135  196 False True False Students accurately substituted and solved each equation and can state with precision that 37 is the only value in the range that makes the equation true. This means that Ana will need exactly $37 more to buy the bicycle. How did you know your solution was reasonable? When x = 233 When x = 135 135 = x + 98 135 = x + 98 135 = 233 + 98 135 = 135 + 98 135  331 135  236 False False

Part B Ana considered buying the $135 bicycle, but then she decided to shop for a different bicycle. She knows the other bicycle she likes will cost at least $150.

MP6: Attend to precision 1. Explain how the inequality below models the following situation: “Ana considered buying the $135 bicycle, but then she decided to shop for a different bicycle. She knows the other bicycle she likes will cost more than $150.” X + 98 ≥ 150 What symbols or mathematical notations are important in this problem? (6.EE.5) Students show a thorough understanding of inequalities in a contextual scenario, as well as a thorough understanding of substituting values accurately and efficiently to verify whether or not they satisfy the inequality. The student offers a correct and accurately interpretation of the inequality in the context of the problem. x represents the additional amount of money Ana needs to buy the bicycle. x+ 98 > 150 represent Ana’s savings plus the amount needed to buy the other bicycle needs to be greater than $150. What does x + 98 > 150 represent? What does x represent?

What does this solution represent? Task #3: Ana earns her money by washing cars. She can potentially earn between $0 and $120 per month. Is it possible for Ana to afford her bicycle within a month? What is the least amount of money Ana needs to save to purchase the bicycle? Consider the following situation: x + 98 ≥ 150 Allow students to make conjectures about the solution of this problem before attempting to solve. The solution represent that Ana needs to save at least $52 in order to afford the bicycle that cost $150. Have a discussion with students about how much Ana needs to save if she would like to purchase a bicycle that cost more than $150. -98 -98 x ≥ 52 What does this solution represent?

What does the graph of the inequality represent? Task #3: Ana earns her money by washing cars. She can potentially earn between $0 and $120 per month. Is it possible for Ana to afford her bicycle within a month? What is the least amount of money Ana needs to save to purchase the bicycle? Graph the solution to the inequality: -98 -98 x ≥ 52 x + 98 ≥ 150 x ≥ 52 What does the graph of the inequality represent? Students offer a correct interpretation of the inequality in the context of the problem. Students will graph the inequality accurately. -250 250 -200 200 -150 150 -100 100 -50 50

The Paper Problem Joey had 26 sheets of papers in his desk. His teacher gave him some more and now he has 100 sheets. How many sheets of papers did his teacher give him?

MP6: Attend to Precision 1. Explain how the equation below models the following situation: “Joey had 26 sheets of papers in his desk. His teacher gave him some more and now he has 100 sheets. How many sheets of papers did his teacher give him? 26 + n = 100 n represent the number of papers the teacher gives Joey. What does n represent? How could you test your solution to see if it answers the problem?

Joey had 26 sheets of papers in his desk Joey had 26 sheets of papers in his desk. His teacher gave him some more and now he has 100 sheets. How many sheets of papers did his teacher give him? The equation 26 + n = 100 can be stated as “some number was added to 26 and the result was 100.” 100 26 n This situation can be represented by the equation 26 + n = 100 where n is the number of papers the teacher gives to Joey. Each bar represents one of the values in the problem. For example 26 states the number of paper Joy had, the 100 bar represent the total number of paper Joey has, and n represent the number of papers the teacher gives to Joey. Students use this visual representation to precisely demonstrate that 26 and the unknown value together make 100. What symbols or mathematical notations are important in this problem? What does each bar represent?

“Joey had 26 papers in his desk “Joey had 26 papers in his desk. His teacher gave him some more and now he has 100. How many papers did his teacher give him?” 26 + n = 100 -26 -26 n = 74 Student can precisely apply inverse operations to undo the equation and find the value of n. Since subtraction “undoes” addition then subtract 26 from 100 to get the numerical value of n.

Additional Practice The booklet of stamps costs 11 dollars and each stamp costs 44 cents. How many stamps are in the booklet? Explain the strategies used to determine the answer. Show that the solution is correct using substitution. Meagan spent $56.58 on three pairs of jeans. If each pair of jeans costs the same amount, write an algebraic equation that represents this situation and solve to determine how much one pair of jeans cost. The equation 0.44 s = 11 where s represents the number of stamps in a booklet. Sample Solution: There are 25 stamps in the booklet. I got my answer by dividing 11 by 0.44 to determine how many groups of 0.44 were in 11. By substituting 25 in for s and then multiplying, I get 11. 0.44(25) = 11 11 = 11 2)Sample Solution: Students might say: “I created the bar model to show the cost of the three pairs of jeans. Each bar labeled J is the same size because each pair of jeans costs the same amount of money. The bar model represents the equation 3J = $56.58. To solve the problem, I need to divide the total cost of 56.58 between the three pairs of jeans. I know that it will be more than $10 each because 10 x 3 is only 30 but less than $20 each because 20 x 3 is 60. If I start with $15 each, I am up to $45. I have $11.58 left. I then give each pair of jeans $3. That’s $9 more dollars. I only have $2.58 left. I continue until all the money is divided. I ended up giving each pair of jeans another $0.86. Each pair of jeans costs $18.86 (15+3+0.86). I double check that the jeans cost $18.86 each because $18.86 x 3 is $56.58.”