Sixth Grade Problem Solving Tasks Weekly Enrichments Teacher Materials Summer Dreamers 2013
SOLVING MATH PROBLEMS KEY QUESTIONS WEEK 1 By the end of this lesson, students should be able to answer these key questions: How do you add or subtract fractions and mixed numbers? How do you add or subtract decimals? How do you determine whether a solution to a problem is reasonable? MATERIALS: For each student: Who s Correct? Pizza Anyone? 5 Mile Run Evaluate: Operations with Rational Numbers For each group of 2 students: Activity Master: Pizza Anyone Problems Activity Master: Vocabulary Conversation Starters
TEACHER TOOLS ENGAGE: The Engage portion of the lesson is designed to access students prior knowledge of adding and subtracting fractions. This phase of the lesson is designed for groups of 2 students. (10 minutes) 1. Distribute Who s Correct? to each student. 2. Prompt students to complete Who s Correct? with their partners. 3. Actively monitor student work and ask facilitation questions when appropriate. Facilitating Questions: What is the problem asking you to do? Answers may vary. Possible answer: Determine if Juan, Liz, or Michael is correct and justify why or why not. What is Kim asking the 3 students to do with the denominators in the problem? Answers may vary. Possible answer: Find the least common multiple of 2, 4, and 7. What is a multiple? Answers may vary. Possible answer: A multiple of a number is the product of that number and any whole number. What are the first 4 multiples of 2? Answers may vary. Possible answer: 2, 4, 6, 8 What strategy could you use to find the multiples of 2, 4, and 7? Answers may vary. Possible answer: I could make a list of the multiples of each number. Could you work backwards to solve this problem? If so, how? Answers may vary. Possible answer: Yes, I could look at each student s solution and determine which solution is a multiple of all three denominators. I could also look at each solution and determine which solution is divisible by all three denominators. Why were the students determining the least common multiple to add or subtract these fractions? Answers may vary. Possible answer: There were determining the least common multiple so that they could rewrite the fractions with a common denominator. They can then easily add or subtract the fractions.
EXPLORE: The Explore portion of the lesson provides the student with an opportunity to be actively involved in investigating addition and subtraction of fractions and decimals. This phase of the lesson is designed for groups of 2 students. (20 minutes) 1. Distribute Activity Master: Pizza Anyone Problems and Activity Master: Vocabulary Conversation Starters to each group of students. 2. Distribute Pizza Anyone? to each student. 3. Prompt students to complete Pizza Anyone? problems and Activity Master: Conversation Starters with their partners. 4. Actively monitor student work and ask facilitation questions when appropriate. Facilitating Questions: Problem A What information is given? Answers may vary. Possible answer: Carla ate pizza for three days in a row. She ate 1 medium pizzas on Saturday, 1 medium pizzas on Sunday, and 2 medium pizzas on Monday. What is the problem asking you to do? Answers may vary. Possible answer: Find out how much more pizza she ate on Monday than she ate on Sunday. Is there any extra information you do not need? Answers may vary. Possible answer: Yes; I do not need the amount she ate on Saturday. What strategy could you use to solve the problem? Answers may vary. Possible answer: I could draw a picture. What operation could you use to find out how much more pizza she ate on Monday than on Sunday? Explain. Answers may vary. Possible answer: I could use subtraction because I am finding the difference of the amounts she ate on Monday and Sunday. Is the order in which you subtract these fractions important? Why? Answers may vary. Possible answer: Yes, I should subtract 1 from 2 because I want to subtract the smaller amount of pizza from the larger amount of pizza to find the difference.
How could you estimate the solution? Answers may vary. Possible answer: I could round the amounts of pizza eaten on Monday and Sunday to estimate the solution. 2 pizzas 2 pizzas and 1 pizzas 1 pizza, 2 pizzas 1 pizza = 1 pizza, so I would estimate that Carla ate about 1 more pizza on Monday than on Sunday. What process could you use to subtract these fractions? Answers may vary. Possible answer: I could express both fractions as equivalent fractions with a common denominator, subtract the fractions, and then subtract the whole numbers. Can you subtract 1 from 2? Explain. Answers may vary. Possible answer: Yes, I can subtract if I regroup the whole numbers (whole pizzas) and fractional parts of the pizzas so that I can find the difference. Is your answer reasonable? How do you know? Answers may vary. Problem B What is the problem asking you to do? Answers may vary. Possible answer: Determine the number of miles Robert traveled from camp to the pizza store and back. What information is given? Answers may vary. Possible answer: I know the odometer readings before and after the trip. What is an odometer? Answers may vary. Possible answer: An odometer is a device used to indicate the distance an automobile has traveled. What does 2,182.4 represent in this problem? Answers may vary. Possible answer: It represents the distance traveled by the car before the trip to the pizza store. What does 2,279.0 represent in this problem? Answers may vary. Possible answer: It represents the distance traveled by the car after the trip to the pizza store. What operation could you use to find the number of miles traveled from the camp to the store and back? Why? Answers may vary. Possible answer: I could use subtraction because I am finding the difference between 2,182.4 and 2,279.0. Is the order in which you subtract important? Why? Answers may vary. Possible answer: Yes, I have to subtract the smaller amount of miles traveled from the larger amount of miles traveled to find the difference. How could you estimate the solution? Answers may vary. Possible answer: I could round the odometer readings to estimate the solution.
Can you subtract 0.4 from 0.0 in this problem? Explain. Answers may vary. Possible answer: Yes, if I regroup I can subtract. Problem C What is the problem asking you to do? Answers may vary. Possible answer: Determine the amount of pizza that was eaten. What does the shaded region represent in the pizzas? Answers may vary. Possible answer: It represents the amount of pizza that was not eaten. What does the unshaded region represent in the pizzas? Answers may vary. Possible answer: It represents the amount of pizza that was eaten. What operation could you use to find the total amount of pizza that was eaten? Why? Answers may vary. Possible answer: I could use addition because I am finding the total amount of pizza eaten. What procedure could you use to find the total amount of pizza that was eaten? Answers may vary. Possible answer: I could find the sum of the fractional parts of all 3 pizzas that are not shaded. How could you estimate the solution? Answers may vary. Possible answer: I could use the pictures of leftover pizza to estimate the solution. What does it mean if the numerator is larger than the denominator in your answer? Answers may vary. Possible answer: I have an improper fraction that I need to simplify. How could you rewrite an improper fraction as a mixed number? Answers may vary. Possible answer: The numerator of the fraction tells me how many parts I have. The denominator tells me how many of the parts it takes to make 1 whole. I could think about regrouping the parts that I have into wholes to see how many wholes I can make. This value will be the whole number part of my mixed number. The parts that are leftover will represent the numerator of the fraction part of my mixed number. The denominator will remain the same because the number of parts in 1 whole has not changed.
EXPLAIN: The Explain portion of the lesson provides students with an opportunity to express their understanding of adding and subtracting fractions and decimals. The teacher will use this opportunity to clarify key vocabulary terms and connect student experiences in the Explore phase with relevant procedures and concepts. (15 minutes) 1. Debrief Pizza Anyone? 2. Use the facilitating questions to lead the discussion. Facilitating Questions: Problem A What math vocabulary did you and your partner use in your discussion? Answers may vary. Possible answer: difference, subtraction, least common multiple, least common denominator, equivalent fraction, regroup, etc. What is the problem asking you to do? Answers may vary. Possible answer: Find out how much more pizza Carla ate on Monday than she ate on Sunday. What discussion did you and your partner have to help you solve the problem? Answers may vary. Possible answer: We discussed that we had to find the difference of the amount she ate on Monday and on Sunday. What information did you need to solve this problem? Answers may vary. Possible answer: We needed the amount she ate on Monday, 2, and the amount she ate on Sunday, 1. How did you estimate the solution to this problem? Answers may vary. Possible answer: I rounded to 2 and 1. I then subtracted 1 from 2 and got 1. She ate about 1 more pizza on Monday than she ate on Sunday. What procedures (steps) did you use to solve the problem? Answers may vary. Possible answer: I expressed both fractions as equivalent fractions with a common denominator. I then had to regroup in order to find the difference of the fractions, and finally I found the difference of the whole numbers. How much more pizza did Carla eat on Monday than Sunday? Answers may vary. Possible answer: Carla ate 7/8 of a pizza more on Monday than Sunday. Is your answer reasonable? How do you know? Answers may vary. Possible answer: Yes, I estimated my answer to be about 1 pizza, and 7/8 is almost 1 whole.
Problem B What math vocabulary did you and your partner use in your discussion? Answers may vary. Possible answer: difference, regroup, etc. What is the problem asking you to do? Answers may vary. Possible answer: Find out how many miles Robert traveled. What discussion did you and your partner have to help you solve the problem? Answers may vary. Possible answer: We discussed that we had to determine the difference in miles that was recorded before and after the trip. We also discussed that we had to line up our decimals before we subtracted. We discussed that we had to subtract the smaller number from the larger number. What information did you need to solve this problem? Answers may vary. Possible answer: We needed the odometer reading before and after the trip. How did you estimate the solution to this problem? Answers may vary. Possible answer: I rounded 2,182.4 to 2,200 and 2,279.4 to 2,300, and then I subtracted 2,200 from 2,300 and estimated about 100 miles. What procedures (steps) did you use to solve the problem? Answers may vary. Possible answer: I first placed the larger number on top and the smaller number on bottom making sure that I lined up the decimals. I then found the difference of the number of miles before and after the trip. How many miles did it take Robert to travel to the store and back? Robert drove a total of 96.6 miles. Is your answer reasonable? How do you know? Answers may vary. Possible answer: Yes, my answer is 96.6 miles which is close to my estimation of 100 miles. Problem C What math vocabulary did you and your partner use in your discussion? Answers may vary. Possible answer: sum, improper fraction, proper fraction, simplify, etc. What is the problem asking you to do? Answers may vary. Possible answer: Find out how much pizza was eaten. What discussion did you and your partner have to help you solve the problem? Answers may vary. Possible answer: We discussed that we had to find the sum of the 3 fractional amounts of pizzas eaten (unshaded portion of the pizzas). We discussed that we had to find equivalent fractions with common denominators.
What information did you need to solve this problem? Answers may vary. Possible answer: We needed the fractional part (unshaded region) eaten from each pizza: 7/8, ¾, and 6/8. How did you estimate the solution to this problem? Answers may vary. Possible answer: I looked at the amount not eaten on the pictures and determined a little more than ½ of a pizza was left; therefore, a little less than 2 ½ pizzas were eaten. What procedures (steps) did you use to solve the problem? Answers may vary. Possible answer: I found the sum of the fractions. Did you simplify your answer? Explain. Answers may vary. Possible answer: Yes, my answer was an improper fraction so I changed it to a proper fraction. How much pizza was eaten at the party? 2 pizzas were eaten at the party. Is your answer reasonable? How do you know? Answers may vary. Possible answer: I know my answer is reasonable because 2 is a little less than 2 or 2 ½ which is close to my estimation. ELABORATE: The Elaborate portion of the lesson affords students the opportunity to extend or solidify their knowledge of adding fractions, decimals, and whole numbers. This phase of the lesson is designed for individual investigation. (15 minutes) 1. Distribute 5 Mile Run to each student. 2. Prompt students to complete 5 Mile Run. 3. Actively monitor student work and ask facilitating questions when appropriate. Facilitating Questions: What is the problem asking you to do? Answers may vary. Possible answer: Determine the total number of miles Alex ran in 4 days. What information do you need? Answers may vary. Possible answer: We need the number of miles he ran each of the 4 days. What do you notice about the numbers? Answers may vary. Possible answer: They are in different forms.
How could you make the problem easier to solve? Answers may vary. Possible answer: I could convert my decimal to a mixed number or convert my 2 mixed numbers and whole number to decimals so they all are the same form. Could you estimate your answer? How? Answers may vary. Possible answer: Yes, I could round each of my mixed numbers and my decimal to the nearest whole number. I would then add all the whole numbers together to determine an estimation. How could you convert the decimal to a fraction? Answers may vary. Possible answer: I could first read my decimal, two and seventy five hundredths, and then write it as a mixed number. I could then simplify my mixed number. How could you convert the fractions to decimals? Answers may vary. Possible answer: First, I would determine if I could find an equivalent fraction with a denominator of 10 or 100. I would then use place value to write it as a decimal. When adding the fractions with unlike denominators, what could you do first to help solve the problem? Answers may vary. Possible answer: I could express the mixed numbers as equivalent mixed numbers with common denominators. Is your answer reasonable? How do you know? Answers may vary. EVALUATE: During the Evaluate portion of the lesson, the teacher will assess student learning about the concepts and procedures that the class investigated and developed during the lesson. (10 minutes) 1. Distribute Evaluate: Operations with Rational Numbers to each student. 2. Prompt students to complete Evaluate: Operations with Rational Numbers. 3. Upon completion of Evaluate: Operations with Rational Numbers, the teacher should discuss error analysis (shown below)to assess student understanding of the concepts and procedures the class addressed in the lesson.
Answers and Error Analysis for Evaluate: Operations with Rational Numbers Question Number Correct Answer Conceptual Error Procedural Error 1 D A B C 2 308.05 3 A B C D 4 C B A D STUDENT WORKSHEETS FOLLOW!!!!!
Activity Master: Pizza Anyone Problems Read each problem below. Discuss with your partner how you could solve the problem. Include as much math vocabulary as possible in your conversation. Use the words on Activity Master: Vocabulary Conversation Starters in your discussion. Solve each problem on Pizza Anyone? Problem A Carla loves pizza. In fact she loves pizza so much that she ate pizza for 3 days. She ate 1 2 medium pizzas on Saturday, 1 1 medium pizzas on Sunday, and 3 4 2 1 medium pizzas on Monday. How much more pizza did she eat on Monday than she ate 8 on Sunday? Problem B Lou, Arturo, and Robert went camping for Spring Break. The boys were tired of roasting hotdogs so Robert agreed to drive to the pizza store. Robert was curious about the number of miles it would take to drive to the store and back, so he recorded the number of miles before and after the trip. The odometer read 2,182.4 before the trip and 2,279.0 after the trip. How many miles did Robert travel? Problem C Sherry and Belinda bought 3 medium square pizzas for the party. The pictures below show how much of the pizza was left. How much pizza was eaten at the party?
Activity Master: Vocabulary Conversation Starters DIFFERENCE SUM LEAST COMMON MULTIPLE EQUIVALENT FRACTION IMPROPER FRACTION PROPER FRACTION NUMERATOR DENOMINATOR
Name: Period: Date: Who s Correct? Juan, Liz, and Michael were absent when Mrs. Mateo taught her students how to add and subtract fractions with unlike denominators. Mrs. Mateo asked Kim to help the 3 students solve the problem: 1 + 3 + 1 = 2 4 7 Kim asked each student to find the least common multiple of the denominators 2, 4, and 7. To Kim s surprise, each student had a different answer! Who is correct? Juan s Solution 8 Liz s Solution 14 Michael s Solution 28 Is Juan correct? Why or why not? Work: Is Liz correct? Why or why not? Is Michael correct? Why or why not?
Name: Period: Date: Pizza Anyone? SEE: PLAN: Sum Find the Difference of and_. Product Quotient DO: (Solve) A Estimation: REFLECT: SEE: PLAN: Sum Find the Difference of and. Product Quotient DO: (Solve) B Estimation: REFLECT: SEE: PLAN: Find the Sum Difference Product Quotient of,_, and _. DO: (Solve) C Estimation: REFLECT:
Name: 5 Mile Run Period: Date: Alex trained 4 days for a 5 mile race. He ran 2.75 miles on Tuesday, 3 2 5 miles on Wednesday, 3 1 miles on Thursday, and 5 miles on Friday. How far did he run while training? 4 Describe the procedure (steps) you would use to solve the problem. Use as much math vocabulary as possible. Solve the problem. Is your answer reasonable? How do you know?
Name: Period: Date: Evaluate: Operations with Rational Numbers 1 Mr. Zavala has $20.00 to spend at the candy store for the Valentine s Party. Chocolate heart candies cost $2.75 per pound, and strawberry lollipops costs $2.25 per pound, tax included. If Mr. Zavala buys 5 pounds of lollipops, how can he determine how much money he has left to spend on chocolate heart candies? 3 Janet needed 1 2 3 gallons of paint for the living room, 2 1 gallons for 5 1 the game room, and 1 gallons for 8 the breakfast room. Which procedure can Janet use to find the total number of gallons she needs for the 3 rooms? A Divide 5 by $2.25 B Multiply $2.75 and 5 C Add $2.25 and $2.75 D Subtract the product of 5 and $2.25 from $20.00 2 Noah had $285.66 in his bank account. He took out $15.88 to buy a book. The next week he earned $38.27 and put that money in his bank account. How much money, in dollars and cents is in Noah s bank account? A Add the sum of the whole numbers to the sum of the fractions. B Find the product of the sum of the whole numbers and the sum of the fractions. C Subtract the sum of the fractions from the sum of the whole numbers. D Find the quotient of the sum of the whole numbers and the sum of the fractions. Record your answer and fill in the bubbles. Be sure to use the correct place value.
4 Johnny has 1 meter of rope for a Boy Scout project. One part of the project requires 1 2 meter of rope, and the other part of the project requires 1 3 meter of rope. Each bar below represents 1 meter of rope. Which bar below is shaded to show the total amount of rope that is left after he finishes the project? A C B D
SOLVING MATH PROBLEMS KEY QUESTIONS WEEK 3 By the end of this lesson, students should be able to answer these key questions: How do you generate equivalent rational numbers? How do you compare rational numbers? MATERIALS: Warm Up: Who is Correct? Activity Master: Number Line assembled and posted on the wall For each student: Fractions, Decimals, and Percents, Oh My! Race Car Stat Evaluate: Equivalent Rational Numbers For each group of 2 students: Activity Master: Fractions, Decimals, and Percents cut apart, 1 set of cards per group
TEACHER TOOLS ENGAGE: The Engage portion of the lesson is designed to access students prior knowledge of percent models. This phase of the lesson is designed for groups of 2 students. (10 minutes) 1. Distribute Who is Correct? warm up. 2. Prompt students to individually complete the warm up Who is Correct? 3. Upon completion of the warm up, prompt students to share and justify their solutions with a partner. 4. Actively monitor student work and ask facilitating questions when appropriate. Facilitating Questions: What is the question asking you to do? Answers may vary. Possible answer: Determine who is correct by determining the percent of the flag that is shaded. What do you know? Answers may vary. Possible answer: I know the answer given by each person, and I was given a picture of the flag. What do you need to know? Answers may vary. Possible answer: I need to know the percent of the flag that is shaded in order to determine who is correct. What strategy could you use to determine who is correct? Answers may vary. Possible answer: I could find the percent of the flag that is shaded then compare my answer to the answer of each person to determine who is correct. How many squares make up the flag? 32 How many squares are shaded? 12 How could you write a ratio that compares the number of shaded squares to the total number of squares on the flag? Answers may vary. Possible answer: 3 to 8, 3:8, 3 out of 8, 3/8 What do you know about percents? Answers may vary. Possible answer: I know that percents are how many out of a 100.
How could you use the ratio of the shaded squares to the total number of squares to help you determine what percent of the flag is shaded? Answers may vary. Possible answer: I know the ratio of shaded squares to total squares is 3/8; therefore, I could use a factor of change to rewrite the ratio as a fraction with a denominator that is a power of 10, such as 1000. What factor of change could you use to change 3/8 to thousandths? Multiply by 125 What is 3/8 written as thousandths? 375/100 How could you rewrite 375/1000 as a percent? Answers may vary. Possible answer: Multiply the numerator and the denominator by 1/10 in order to generate an equivalent fraction with a denominator of 100, 37.5/100. Then I could use the numerator as my percent, since percent means out of 100. EXPLORE: The Explore portion of the lesson provides the student with an opportunity to be actively involved in investigating equivalent rational numbers. This phase of the lesson is designed for groups of 2 students. (25 minutes) 1. Distribute 1 set of Activity Master: Fractions, Decimals, and Percents to each group of students and Fractions, Decimals, and Percents, Oh My! to each student. (NOTE: Have Activity Master cards pre cut for student use.) 2. Prompt students to complete Fractions, Decimals, and Percents, Oh My! 3. Actively monitor student work and ask facilitating questions when appropriate. Facilitating Questions: What information is found on the cards? Answers may vary. Possible answer: The cards contain rational numbers written in different forms. Rewriting Fractions as Decimals How could rewriting each fraction as hundredths help you write the decimal representation of the fraction? Answers may vary. Possible answer: Decimals are just fractions that have denominators that are 10, 100, 1000, etc. (powers of 10). So if I rewrite the fraction as hundredths, I could write the decimal by using place value.
What factor of change could you use to rewrite this fraction as hundredths? Answers may vary. Rewriting Fractions as Percents How could rewriting each fraction as hundredths help you write the percent representation of the fraction? Answers may vary. Possible answer: Percents are just fractions that have denominators of 100. So if I rewrite the fraction as hundredths, I could write the percent by using the numerator. What factor of change could you use to rewrite this fraction as hundredths? Answers may vary. x 125. 62.5% Rewriting Decimals as Percents How could rewriting each decimal as a fraction help you write the decimal as a percent? Answers may vary. Possible answer: Place value is based on powers of 10: 10, 100, 1000, etc. So if I rewrite the decimal as a fraction, I could apply a factor of change to the fractions to rewrite the fractions as hundredths then I could write the percent by using the numerator. What factor of change could you use to rewrite this fraction as hundredths? Answers may vary.
Rewriting Decimals as Fractions How could place value help you write each decimal as a fraction in simplest form? Answers may vary. Possible answer: Place value is based on powers of 10: 10, 100, 1000, etc. So I could rewrite the decimal as a fraction with a denominator of tenths, hundredths, or thousandths then simplify. Rewriting Percents as Fractions What procedures could be used to write a percent as a fraction in simplest form? Answers may vary. Possible answer: Percents are just fractions that have denominators of 100. So if I rewrite the fraction as hundredths, then I could simplify. Rewriting Percents as Decimals How could rewriting a percent as a fraction help you write a percent as a decimal? Answers may vary. Possible answer: Percents are just fractions that have denominators of 100. So if I rewrite the percent as a fraction, I could write the decimal by using place value. Ordering from Least to Greatest Which representation is the easiest to use to help you determine which rational number represents the largest amount? Why? Answers may vary. Possible answer: To compare the rational numbers, I could use the fractions written as hundredths, the percent, or the decimal form to compare easily. Which representation is the easiest to use to help you determine which rational number represents the smallest amount? Why? Answers may vary. Possible answer: To compare the rational numbers, I could use the fractions written as hundredths, the percent, or the decimal form to compare easily. Comparing 83.5% How could you determine which of the numbers are equivalent to 83.5%? Answers may vary. Possible answer: I could rewrite 83.5% as a fraction and as a decimal and compare my values with the values of the answer choices. What process could you use to rewrite 83.5% as a fraction with a denominator of 100? Answers may vary. Possible answer: Percents are just fractions that have denominators of 100. So I could rewrite 83.5% as a fraction where 83.5 is the numerator and 100 is the denominator.
What process could you use to rewrite 83.5% as a fraction with a denominator of 1000? Answers may vary. Possible answer: I could rewrite 83.5% as a fraction where 83.5 is the numerator and 100 is the denominator then use a factor of change to rewrite the fraction as thousandths. What factor of change could be used to convert 83.5/100 to thousandths? 10 What process could you use to rewrite 83.5% as a decimal? Answers may vary. Possible answer: I could rewrite 83.5% as a fraction with a denominator of 1000 then write the decimal by using place value. 0.835 EXPLAIN: The explain portion of the lesson provides students with an opportunity to express their understanding of equivalent rational numbers. The teacher will use this opportunity to clarify vocabulary terms and connect student experiences in the Explore phase with relevant procedures and concepts. (20 minutes) 1. Display assembled Activity Master: Number Line on the wall in front of the room. 2. Prompt 1 group of students with Card Set 1 to post their cards in the appropriate place on Activity Master: Number Line. Students will need to approximate the placement. 3. Prompt 1 group of students with Card Set 2 and 1 group with Card Set 3 to add their cards to the number line. Note: There are repeated rational numbers throughout the 3 different sets of cards; however, the repeated rational numbers are in different forms. 4. Use the facilitating questions to lead a whole group discussion as students add their cards to the number line. Number Line Key 0.625 12.5% 20/32 1/8 1/5 20.4% ¼ 32.6% 5/8 0 0.5
Facilitating Questions Which representation is the easiest to use to help you determine which rational number represents the largest amount? Why? Answers may vary. Possible answer: To compare the rational numbers, I could use the fractions written as hundredths, the percent, or the decimal form to compare easily. Which representation would be the hardest to use to determine which rational number represents the largest amount? Why? Answers may vary. Possible answer: To compare the rational numbers, I would not use the fractions in the simplest form because these fractions are not easily compared without a common denominator. How could you use the placement of the cards on the number line to determine which rational number represents the largest amount? Answers may vary. Possible answer: The number that represents the largest amount would be the number that is farthest to the right on the number line. Which representation is the easiest to use to help you determine which rational number represents the smallest amount? Why? Answers may vary. Possible answer: To compare the rational numbers, I could use the fractions written as hundredths, the percent, or the decimal form to easily compare. Which representation would be the hardest to use to determine which rational number represents the smallest amount? Why? Answers may vary. Possible answer: To compare the rational numbers, I would not use the fractions in simplest form because they do not have a common denominator. How could you use the placement of the cards on the number line to determine which rational number represents the smallest amount? Answers may vary. Possible answer: The number that represents the smallest amount would be the number that is farthest to the left on the number line. Which representation is the easiest to use to help you determine where a rational number lies on the number line? Why? Answers may vary. Possible answer: Since the number line is in decimal form, it is easier to determine the placement of the decimal representations. 5. Debrief questions 3 on Fractions, Decimals, and Percents, Oh My! 6. Use the facilitating questions to lead the discussion.
Facilitating Questions How did you determine which of the numbers are not equivalent to 83.5%? Answers may vary. Possible answer: I rewrote 83.5% as a fraction and as a decimal and compared my values with the values of the answer choices. Did you eliminate any of the answer choices? Why? Answers may vary. Possible answer: Yes, since 83.5% is less than 100% and 100% is equivalent to 1, I was able to eliminate 8.35 because it is greater than 1. What process did you use to rewrite 83.5% as a fraction with a denominator of 100? Answers may vary. Possible answer: Percents are just fractions that have denominators of 100. So I rewrote 83.5% as a fraction where 83.5 is the numerator and 100 is the denominator. What process did you use to rewrite 83.5% as a fraction with a denominator of 1000? Answers may vary. Possible answer: I rewrote 83.5% as a fraction where 83.5 is the numerator and 100 is the denominator then used a factor of change of 10 to rewrite as thousandths. (835/1000) What process did you use to rewrite 83.5% as a decimal? Answers may vary. Possible answer: I rewrote 83.5% as a fraction with a denominator of 1000 then wrote the decimal by using place value. Which number is not equivalent to 83.5%? Why? 8.35 ELABORATE: The Elaborate portion of the lesson affords students the opportunity to extend or solidify their knowledge of equivalent rational numbers. This phase of the lesson is designed for individual investigation. (10 minutes) 1. Distribute Race Car Stat to each student. 2. Prompt students to complete Race Car Stat. (Answer A) 3. Actively monitor student work and ask facilitating questions when appropriate. Facilitating Questions What is the question asking you to do? Answers may vary. Possible answer: Determine between which 2 fractions 3/8 lies on a number line. What do you know? Answers may vary. Possible answer: I know 4 different possible sets of fractions that 3/8 may fall between.
What do you need to know? Answers may vary. Possible answer: I need to know a common denominator so that I can make comparisons. What procedure could you use to determine which pair of fractions 3/8 may fall between? Answers may vary. Possible answer: I could find a common denominator, simplify, or use a factor of change to rewrite each fraction using the common denominator then compare numerators. EVALUATE: During the Evaluate portion of the lesson, the teacher will assess student learning about the concepts and procedures that the class investigated and developed during the lesson. (20 minutes) 4. Distribute Evaluate: Equivalent Rational Numbers to each student. 5. Prompt students to complete Evaluate: Equivalent Rational Numbers. 6. Upon completion of Evaluate: Equivalent Rational Numbers, the teacher should discuss error analysis (shown below)to assess student understanding of the concepts and procedures the class addressed in the lesson. Answers and Error Analysis for Evaluate: Equivalent Rational Numbers Question Number Correct Answer Conceptual Error 1 B A C D 2 B A C D 3 A B C D 4 A B C D STUDENT WORKSHEETS FOLLOW!!!!! Procedural Error
Warm Up: Who is Correct? Kobie shaded 3/8 of the flag black, as shown below. Cassie stated that 12% of the flag was shaded, and Kobie said that 37.5% of the flag was shaded. Who is correct? Explain your answer.
Activity Master: Number Line 1 0.5
Activity Master: Number Line 0
Activity Master: Fractions, Decimals, and Percents Cut each card out along lines. 1 set per group of students 3 cards per set. 20.4% Set 1 Card 1 Set 1 Card 2 Set 1 Card 3 0.625 Set 2 Card 1 Set 2 Card 2 Set 2 Card 3
Activity Master: Fractions, Decimals, and Percents Set 3 Card 1 Set 3 Card 2 Set 1 Card 3
Name: Date: Complete the table below. Fractions, Decimals, and Percents, Oh My! CARD Card 1 Fraction (in simplest form) Decimal Percent Card 2 Card 3 1. List the cards in order from least to greatest. 2. Which representation is the easiest to use to help you determine the order from least to greatest? Why? 3. It is estimated that Jimmy Johnson completed 83.5% of the laps in the 2004 Talladego Race. Which number is NOT equivalent to 83.5%? A. B.. C. 0.835 D. 8.35
Name: Date: Race Car Stat Tony Stewart was either the winner or the runner up in 3 out of the last 8 races in the series. The fraction 3/8 is found between which pair of fractions on a number line? A. and B. and C. and D. and Justify your answer choice and state why the other answer choices are incorrect.
Name: Date: Evaluate: Equivalent Rational Numbers 1. The fraction is found between which pair of fractions on the number line? A. and B. and C. D. and and 2. A specialty paint shop had 4 different race cars to complete. The shop completed,,, and of the work on each car. Which list shows the percent of the work completed on each car in order from greatest to least? A. 50%, 62.5%, 75%, 20% B. 75%, 62.5%, 50%, 20% C. 0.75%, 0.625%, 0.5%, 0.2% D. 20%, 50%, 62.5%, 75%
3. Tyler estimated that 48.2% of the crystals in his sugar project developed correctly. Which number is NOT equivalent to 48.2%? A. 4.82 B. 0.482 C. D.. 4. The table shows the driver and the portion of allowable gas each driver used in the race. Gas Usage Driver Portion of Allowable Gas Used Busch Johnson Burton 48.3% Earnhardt Harvick 48.2% Which of the following lists the racers in order from least to greatest portion of allowable gas used? A. Busch, Earnhardt, Harvick, Burton, Johnson B. Busch, Earnhardt, Burton, Harvick, Johnson C. Johnson, Burton, Harvick, Busch, Earnhardt D. Johnson, Burton, Harvick, Earnhardt, Busch
SOLVING MATH PROBLEMS KEY QUESTIONS WEEK 5 By the end of this lesson, students should be able to answer these key questions: What is a ratio? What are the different ways to write a ratio? How can you determine if 2 ratios are equivalent? What is a proportion? How can you determine if a problem situation can be solved using a proportion? What are the different ways to write a proportion given a problem situation? MATERIALS: For each student: Warm Up Activity: Find Someone Who Heartbeats Speed Racer Evaluate: Ratios and Proportions For each group of 2 to 3 students: Activity Master: Spinners cut apart 1 set per group Paperclip 1 per group TEACHER TOOLS ENGAGE: The Engage portion of the lesson is designed to access students prior knowledge about ratios. This phase of the lesson is designed for whole group instruction. (15 minutes)
1. Distribute a Formula Chart (optional) and Find Someone Who to each student. 2. Prompt students to complete and Find Someone Who 3. Actively monitor student work and ask facilitating questions when appropriate. Facilitating Questions What is a ratio? Answers may vary. Possible answer: A ratio is a comparison of two or more numbers. Does order matter when writing a ratio? Why? Answers may vary. Possible answer: Yes, the relationship must stay the same; however, if I label my values, then order does not matter. What are the different ways to write a ratio? Answers may vary. Possible answer: Different ways to write a ratio include: using a colon, as a fraction, verbally using the word to or using the words out of. How could you simplify a ratio? Answers may vary. Possible answer: You can simplify a ratio by dividing each of the numbers in the ratio by the same factor. Where could you find the relationship between seconds and minutes? Answers may vary. Possible answer: Formula Chart What is the relationship between seconds and minutes? 60 seconds = 1 minute How could you use this relationship when finding the number of heartbeats in a minute? Answers may vary. Possible answer: Since I know there are 60 seconds in a minute and 10 seconds goes into 60 seconds 6 times, I could multiply the number of heartbeats in 10 seconds by 6. EXPLORE: The Explore portion of the lesson provides the student with an opportunity to be actively involved in investigating ratios and proportions. This phase of the lesson is designed for groups of 2 to 3 students. (20 minutes) 1. Distribute Heartbeats to each student. 2. Distribute 1 set of Activity Master: Spinners and a paperclip to each group of students. (Use a pencil with the paperclip to spin the paperclip.) 3. Prompt students to complete Heartbeats. 4. Actively monitor student work and ask facilitating questions.
Facilitating Questions What ratio of seconds to heartbeats did you get when you used the 2 spinners? Answers may vary. Is it possible to simplify this ratio? Answers may vary. Which time value(s) are multiples of the time interval you spun? Answers may vary. Possible answer: Since I spun 30 seconds, 60 seconds is a multiple of 30 seconds. Which time value(s) are factors of the time interval you spun? Answers may vary. Possible answer: Since I spun 30 seconds, 15 seconds is a factor of 30 seconds. What factor could be used to scale your time interval up or down? Answers may vary. Possible answer: Since I spun 30 seconds, I could use a factor of ½ to scale down to 15 seconds, a factor of 1 ½ to scale up to 45 seconds, and a factor of 2 to scale up to 60 seconds. How could you use this factor to determine the number of heartbeats? Answers may vary. Possible answer: Since I spun 30 seconds and 8 heartbeats, I could use a factor of ½ to scale down the number of seconds to 15 and the number of heartbeats to 4 heartbeats. What is the relationship between seconds and minutes? 60 seconds = 1 minute If you know the number of heartbeats in 30 seconds, how would you determine the number of heartbeats in 1 minute? Answers may vary. Possible answer: Since I could multiply 30 seconds by a factor of 2 to get 60, I could multiply the number of heartbeats by 2. EXPLAIN: The Explain portion of the lesson provides students with an opportunity to express their understanding of ratios and proportions. The teacher will use this opportunity to clarify vocabulary and connect student experiences in the Explore phase with relevant procedures and concepts. (15 minutes) 1. Debrief Heartbeats. 2. Use the facilitating questions to lead the discussion. Facilitating Questions What is a ratio? Answers may vary. Possible answer: A ratio is a comparison of two values. What are some was ratios can be recorded? Answers may vary. Possible answer: Ratios can be recorded as a fraction, with a colon, or with the words to or out of.
What patterns did you see in the table? Answers may vary. Possible answer: The number of heartbeats is always a multiple of 16. How did you determine the number of heartbeats in 1 minute? Answers may vary. Possible answer: Since I know that 1 minutes is 60 seconds, I continued the pattern in the table until I found the number of heartbeats in 60 seconds. I set up and solved a proportion. What is a proportion? Answers may vary. Possible answer: A proportion is an equation showing that two ratios are equivalent. How do you know if a proportion could be used to solve these problems? Answers may vary. Possible answer: A proportion may be used if the problem contains a ratio and the situation requires the ratio to be scaled up or down. How could you set up a proportion to solve this problem? Answers may vary. Possible answer: = = What process could you use to find the missing value in your proportion? Answers may vary. Possible answer: Since 60 is a multiple of 15 and 4 times 15 equals 60, then I could multiply 16 times 4. How many times would the heart beat in 1 minute? Answers may vary. Possible answer: 64 times. How do you determine the number of heartbeats in 3 minutes? Answers may vary. Possible answer: I multiplied the number of heartbeats for 1 minute by 3. How many times would the heart beat in 3 minutes? Answers may vary. Possible answer: 192 times. How did you determine the number of seconds for 240 heartbeats? Answers may vary. Possible answer: I set up and solved a proportion. How could you set up a proportion to determine the number of seconds for 240 heartbeats? Answers may vary. Possible answer: = = What process could you use to find the missing value in your proportion? Answers may vary. Possible answer: Since 16 times 15 equals 240, I could multiply 15 times 15. How many seconds will pass for the heart to beat 240 times? Answers may vary. Possible answer: 225 seconds. How could you find the number of minutes for 240 heartbeats? Answers may vary. Possible answer: Since I know that 60 seconds is 1 minute, I could divide 225 by 60.
What process did you use to determine how many times Carol s heart beat while walking for 4 minutes? Answers may vary. Possible answer: Since her heart beats 18 times in 10 seconds, I found how many times her heart would beat in 1 minute then multiplied that number by 4 to find the number of heartbeats in 4 minutes. How could you set up a proportion to solve this problem? Answers may vary. Possible answer: = = What factor could you use to find the missing value in your proportion? Answers may vary. Possible answer: Since 24 times 10 equals 240, I multiplied 18 times 24. About how many times would Carol s heart beat during 4 minutes of walking? 432 times. ELABORATE: The Elaborate portion of the lesson affords students the opportunity to extend or solidify their knowledge of ratios and proportions. This phase of the lesson is designed for individual investigation. (15 minutes) 1. Distribute Speed Racer to each student. 2. Prompt students to complete Speed Racer. 3. Actively monitor student work and ask facilitating questions when appropriate. Facilitating Questions What is the question asking you to do? Answers may vary. Possible answer: Determine which girl answered the question correctly. What information is given to you? Answers may vary. Possible answer: We know each girl s answer choice and the problem they solved. What strategy could be used to determine which girl is correct? Answers may vary. Possible answer: I could solve the problem and then compare my answer to the answers of Maria and Louisa. What ratio is described in thee problem situation? Answers may vary. Possible answer: 85 miles/1 hour What proportion describes the situation? Answers may vary. Possible answer: = =
Which verbal description matches the process you could use to find the missing value in the proportion? Answers may vary. Possible answer: Since I would need to determine the factor to use to scale up 85 to 255, I could divide 255 by 85. Therefore, answer choice A has the correct verbal description. EVALUATE: During the Evaluate portion of the lesson, the teacher will assess student learning about the concepts and procedures that the class investigated and developed during the lesson.(20 minutes) 1. Distribute Evaluate: Ratios and Proportions to each student. 2. Prompt students to complete Evaluate: Ratios and Proportions. 3. Upon completion of Evaluate: Ratios and Proportions, the teacher should use the error analysis, provided below, to assess student understanding of the concepts and procedures the class addressed in the lesson. Question Number Answers and Error Analysis for Evaluate: Equivalent Rational Numbers Correct Answer Conceptual Error Procedural Error 1 B A C D 2 C A B D 3 24 4 C A D B STUDENT WORKSHEETS FOLLOW!!!!!
Name: Date:
Name: Date:
Name: Date:
Name: Date: Evaluate: Ratios and Proportions 1. If the ratio of cats to dogs in the veterinarian clinic is 2 to 3, which ratio does NOT show a possible number of cats to dogs in the clinic? A. 36 cats, 54 dogs B. 34 cats, 21 dogs C. 24 cats, 36 dogs D. 30 cats, 45 dogs 2. Claire was making necklaces for the craft show. She completed 9 necklaces in 30 minutes. If Claire continued making necklaces at this rate, how many necklaces would she make in 2 hours? A. 4 B. 9 C. 36 D. 45 3. The ratio of butterflies to bees in Jane s insect collection is 3 to 4. If there were 32 bees, how many butterflies would be there? Show all work and explain your reasoning. 4. There were 16 box cars and 24 students registered for the box car tournament. Which ratio accurately compares the number of students to the number of box cars? A. 2: 12 B. 3: 1 C. 3: 2 D. 16: 24