G8 U1 Constructed Response
Name: ________________________________________ Date:_______________
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers TASK 1: Decimal Representations/Expansions: Converting Fractions to Decimals and Terminating/Non-Terminating Decimals into Fractions (CCSS:8.NS.1 [a/s]; SMP: 1, 2, 3, 6, 7, 8)
1. Use the equation below to answer the following questions: −13 − 𝑦 = 10 a. Solve the equation. Show detailed and organized steps in your work.
Answer: _____________________ b. Analyze your solution. Under which categories of numbers does your solution fall? Explain your reasoning using the mathematical vocabulary that you have learned.
c. Give three examples of numbers that are the opposite of your solution.
Answer: ___________________ Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 1
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers 2. Examine the statement below. 24 cups of sugar out of 64 a. Write the fraction and decimal expansion that best represents this statement.
Answer: Decimal Expansion: ____________ Fraction: ____________ b. After showing the fractional and decimal representations of the statement, write an explanation in which you describe the characteristics of the decimal form you wrote for the fraction. What do you notice about this number?
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Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers 3. a. Convert the following decimal into a fraction. In the table below, show and explain the process necessary to find your solution. 2.3939… Mathematical Process
Explanation of Steps
Answer: b. Using strategies you have already learned, explain how you could show that that the fraction in your answer from part A is correct.
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Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers TASK 2: Estimating Irrational Numbers (CCSS: 8.NS.1 [a/s]; SMP: 1, 2, 3, 7)
4. Between which two integers is the value of √125? a. Before you begin solving the problem you read above, give a written explanation of the meaning of the problem and describe a starting point for solving the problem.
b. Solve the problem using the strategy you described above
Answer: _______________________ c. How would your process and your answer change if the square root were changed to 5√125?
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Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers 5. a. Estimate the value of √89 to the nearest hundredth. Show detailed and organized steps in your work.
Answer: ______________________ b. Write a discussion in which you explain how other concepts you have learned before where helpful when finding your solution. (How was this concept useful or important in this type of problem?)
c. After working with the process of estimation, describe the possible pros and cons for estimating the value of a number to more decimal points. In a laboratory or medical setting, would this be effective?
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Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers 6. Analyze the statement below: 2
6 9
is slightly larger than √10
a. After analyzing the data in the statement, write a critique in which you identify and explain any flaws in the statement. Correct any errors you may find and give the correct answer.
b. Demonstrate that your answer is correct by visually representing the comparison of the numbers on a number line.
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Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers TASK 3: Approximating Other Irrational Numbers and Their Expressions (CCSS: 8.NS.2 [a/s]; SMP: 1, 2, 6, 7)
7. a. Simplify the following expression: 2𝜋 + 5𝜋 2 − 7𝜋 2 + 𝜋 3
Answer: _______________________ b. Analyze your answer from above and describe a strategy you could use to calculate the value of your simplified expression. How would you use this strategy to solve this expression?
c. Using the strategy you described above, calculate the value of the expression to the nearest hundredth. Show detailed and organized steps for your work.
Answer: __________________________ Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 7
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers 8. a. Simplify the following expression: 8√51 + 3√15 − 2√51 − 𝜋 + 3√15
Answer: _______________________ b. What mathematical concepts were useful when simplifying the expression? How did you use these mathematical concepts to find your solution?
c. Calculate the value of the expression you simplified. Show your work in detailed and organized steps.
Answer: _____________________ Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 8
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers d. After simplifying and calculating the value of the expression, explain why it is a good strategy to first simplify this type of expression and then try to solve it. Why not solve it without simplifying? Justify your reasoning.
9. a. Calculate the value of the expression below to the nearest hundredth. Show your work in detailed and organized steps: 2
3
(√125) + 3√7 + (√33) + 2𝜋
Answer: _________________________
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Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers b. After calculating the value of the expression, explain how the process for solving this expression was similar and different to other expressions you have calculated. Were there any extra steps that were necessary to find your answer?
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Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers TASK 1: Decimal Representations/Expansions: Converting Fractions to Decimals and Terminating/Non-Terminating Decimals into Fractions (CCSS:8.NS.1 [a/s]; SMP: 1, 2, 3, 6, 7, 8)
1. Use the equation below to answer the following questions: −13 − 𝑦 = 10 a. Solve the equation. Show detailed and organized steps in your work.
Answer: _____________________ b. Analyze your solution. Under which categories of numbers does your solution fall? Explain your reasoning using the mathematical vocabulary that you have learned.
c. Give three examples of numbers that are the opposite of your solution.
Answer: ___________________ Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 1
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers 2. Examine the statement below. 24 cups of sugar out of 64 a. Write the fraction and decimal expansion that best represents this statement.
Answer: Decimal Expansion: ____________ Fraction: ____________ b. After showing the fractional and decimal representations of the statement, write an explanation in which you describe the characteristics of the decimal form you wrote for the fraction. What do you notice about this number?
Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 2
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers 3. a. Convert the following decimal into a fraction. In the table below, show and explain the process necessary to find your solution. 2.3939… Mathematical Process
Explanation of Steps
Answer: b. Using strategies you have already learned, explain how you could show that that the fraction in your answer from part A is correct.
Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 3
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers TASK 2: Estimating Irrational Numbers (CCSS: 8.NS.1 [a/s]; SMP: 1, 2, 3, 7)
4. Between which two integers is the value of √125? a. Before you begin solving the problem you read above, give a written explanation of the meaning of the problem and describe a starting point for solving the problem.
b. Solve the problem using the strategy you described above
Answer: _______________________ c. How would your process and your answer change if the square root were changed to 5√125?
Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 4
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers 5. a. Estimate the value of √89 to the nearest hundredth. Show detailed and organized steps in your work.
Answer: ______________________ b. Write a discussion in which you explain how other concepts you have learned before where helpful when finding your solution. (How was this concept useful or important in this type of problem?)
c. After working with the process of estimation, describe the possible pros and cons for estimating the value of a number to more decimal points. In a laboratory or medical setting, would this be effective?
Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 5
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers 6. Analyze the statement below: 2
6 9
is slightly larger than √10
a. After analyzing the data in the statement, write a critique in which you identify and explain any flaws in the statement. Correct any errors you may find and give the correct answer.
b. Demonstrate that your answer is correct by visually representing the comparison of the numbers on a number line.
Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 6
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers TASK 3: Approximating Other Irrational Numbers and Their Expressions (CCSS: 8.NS.2 [a/s]; SMP: 1, 2, 6, 7)
7. a. Simplify the following expression: 2𝜋 + 5𝜋 2 − 7𝜋 2 + 𝜋 3
Answer: _______________________ b. Analyze your answer from above and describe a strategy you could use to calculate the value of your simplified expression. How would you use this strategy to solve this expression?
c. Using the strategy you described above, calculate the value of the expression to the nearest hundredth. Show detailed and organized steps for your work.
Answer: __________________________ Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 7
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers 8. a. Simplify the following expression: 8√51 + 3√15 − 2√51 − 𝜋 + 3√15
Answer: _______________________ b. What mathematical concepts were useful when simplifying the expression? How did you use these mathematical concepts to find your solution?
c. Calculate the value of the expression you simplified. Show your work in detailed and organized steps.
Answer: _____________________ Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 8
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers d. After simplifying and calculating the value of the expression, explain why it is a good strategy to first simplify this type of expression and then try to solve it. Why not solve it without simplifying? Justify your reasoning.
9. a. Calculate the value of the expression below to the nearest hundredth. Show your work in detailed and organized steps: 2
3
(√125) + 3√7 + (√33) + 2𝜋
Answer: _________________________
Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 9
Grade 8 Unit 1 Constructed Response Rational and Irrational Numbers b. After calculating the value of the expression, explain how the process for solving this expression was similar and different to other expressions you have calculated. Were there any extra steps that were necessary to find your answer?
Copyright © Swun Math Grade 8 Unit 1 Constructed Response, Page 10
