Seventh Grade Problem Solving Tasks Weekly Enrichment Teacher Materials. Summer Dreamers

Transcription

1 Seventh Grade Problem Solving Tasks Weekly Enrichment Teacher Materials Summer Dreamers

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3 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 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

4 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

5 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 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.

6 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 %

7 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.

8 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

9 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 % 20/32 1/8 1/5 20.4% ¼ 32.6% 5/

10 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.

11 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.

12 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) 1. Distribute Evaluate: Equivalent Rational Numbers to each student. 2. Prompt students to complete Evaluate: Equivalent Rational Numbers. 3. 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

13 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.

14 Activity Master: Number Line 1 0.5

15 Activity Master: Number Line 0

16 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 Set 2 Card 1 Set 2 Card 2 Set 2 Card 3

17 Activity Master: Fractions, Decimals, and Percents Set 3 Card 1 Set 3 Card 2 Set 1 Card 3

18 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 D. 8.35

19 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.

20 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%

21 3. Tyler estimated that 48.2% of the crystals in his sugar project developed correctly. Which number is NOT equivalent to 48.2%? A B 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

22 SOLVING MATH PROBLEMS KEY QUESTIONS WEEK 2 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

23 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.

24 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

25 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.

26 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: = =

27 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: = =

28 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 C A D B STUDENT WORKSHEETS FOLLOW!!!!!

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30 Name: Date:

31 Name: Date:

32 Name: Date:

33 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 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

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35 SOLVING MATH PROBLEMS KEY QUESTIONS WEEK 3 By the end of this lesson, students should be able to answer these key questions: What is a proportion? How can you represent a proportional relationship by using a ruler? What patterns are there in proportional relationships? How do you solve proportions? MATERIALS: For each student: Round Robin Warm Up Activity: Proportions School Project Paintbrushes Evaluate: Proportions

36 TEACHER TOOLS ENGAGE: The Engage portion of the lesson is designed to access students prior knowledge of comparing rates. This phase of the lesson is designed for groups of 4 students. (15 minutes) 1. Distribute Round Robin: Proportions warm up activity to each student. 2. Prompt students to complete Round Robin: Proportions warm up activity with their partners. 3. Actively monitor student work and ask facilitation questions when appropriate. Facilitating Questions How could you determine the number of 10 ounce bottles needed? Answers may vary. Possible answer: I could divide 60 ounces, the total amount of red paint needed, by 10 ounces. How could you determine the total cost of the 10 ounce bottles? Answers may vary. Possible answer: I could multiply $1.60 by the number of 10 ounce bottles needed. Does your answer seem reasonable? Why or why not? Answers may vary. Possible answer: Yes. If I round $1.60 up to $2.00 and multiply it by 6, the number of 10 ounce bottles needed, I get $12. My actual answer was $9.60. Since I rounded the price per bottle up, I know the actual answer will be less than my estimate. How could you determine the number of 15 ounce bottles needed? Answers may vary. Possible answer: I could divide 60 ounces, the total amount of red paint needed, by 15 ounces. How could you determine the total cost of the 15 ounce bottles? Answers may vary. Possible answer: I could multiply $2.20 by the number of 15 ounce bottles needed. How could you use the total cost of the 10 ounce bottles and the 15 ounce bottles to determine the better buy? Answers may vary. Possible answer: I could compare the total costs and see which one is less to determine the better buy.

37 EXPLORE: The Explore portion of the lesson provides the student with an opportunity to be actively involved in using a ruler to model proportional relationships. This phase of the lesson is designed for groups of 2 students. (25 minutes) 1. Distribute School Project to each student. 2. Prompt students to complete School Project with their partners. 3. Actively monitor students and ask facilitating questions when appropriate. Facilitating Questions Part 1 If 2 inches represents 15 feet, how many feet are represented by 4 inches? 30 feet How could you use the ruler to determine the number of feet on the mural? Answers may vary. Possible answer: Every 2 inches represents 15 feet, so I could mark an additional 15 feet every time I increase by 2 inches on the ruler. How could you use the fact that 3 inches is halfway between 2 inches and 4 inches on the ruler to determine the number of feet represented by 3 inches? Answers may vary. Possible answer: Since 3 inches is halfway between 2 inches and 4 inches, the number of feet must be halfway between 15 feet and 30 feet. I can find the average of 15 feet and 30 feet. What labels do you have on your ruler that could be used to determine the number of inches that represent 135 feet? Answers may vary. Possible answer: Since 8 inches represents 60 feet and 10 inches represents 75 feet, I can add those 2 values together to see that 18 inches are needed to represent 135 feet.

38 Part 2 Since 12 centimeters on the ruler is divided into 3 equal sections, how many cups of white paint does each section represent? How do you know? 3 = What patterns do you notice on the ruler? Part 3 Answers may vary. Possible answer: Every increase in 4 centimeters (blue paint) corresponds to an increase in cup of white paint. How could you label the ruler to help you solve this problem? Answers may vary. Possible answer: I could label the top of the ruler as silver stars and the bottom of the ruler as gold stars. Since there are 8 silver stars for every 3 gold stars, I could put an 8 above the 3 inch mark on the ruler. What other value(s) could you find on the ruler? How? Answers may vary. Possible answer: I could find the number of silver stars represented by 6, 9, and 12 gold stars by using the 6 inch, 9 inch, and 12 inch marks on the ruler. What patterns do you notice on the ruler? Answers may vary. Possible answer: Every time I increase 3 inches (the number of gold stars), I increase by 8 silver stars. EXPLAIN: The Explain portion of the lesson provides students with an opportunity to express their understanding of proportional relationships. The teacher will use this opportunity to clarify vocabulary terms and connect student experiences in the Explore phase with relevant procedures and concepts. (15 minutes) 1. Debrief School Project. 2. Use the facilitating questions to lead the discussion.

39 Facilitating Questions Part 1 How did you use the ruler to determine the number of feet on the mural? Answers may vary. Possible answer: Every 2 inches represented 15 feet, so I increased by an additional 15 feet every time I increased by 2 inches on the ruler. How did you use the ruler to determine the number of feet that 3 inches on the scale drawing would represent? Answers may vary. Possible answer: Since 3 inches is halfway between 2 inches and 4 inches, the number of feet must be halfway between 15 feet and 30 feet. I found the average of 15 feet and 30 feet which is 22.5 feet. What proportion could you use to determine the number of feet represented by 3 inches? Answers may vary. Possible answer: = How do you see this proportion on the ruler? Answers may vary. Possible answer: The 15 is above the 2 on the ruler, just like it is in the proportion. The unknown is above the 3 on the ruler, just like it is in the proportion. How could you solve the proportion = to verify the answer you found with the ruler? Part 2 Answers may vary. Possible answer: Since the scale factor is to verify the answer I found with the ruler., I could multiply 15 by How did you use the ruler to determine the number of inches on the scale drawing that would represent 135 feet on the mural? Answers may vary. Possible answer: Since 8 inches represents 60 feet and 10 inches represents 75 feet, I added those 2 values together to see that 18 inches would represent 135 feet. How did you use the ruler to fill in the missing blanks? Answers may vary. Possible answer: I noticed that 12 centimeters on the ruler was divided into 3 equal sections, so I divided of a cup into 3 equal parts.

40 How did you determine the number of cups of white paint needed for 16 cups of blue paint? Answers may vary. Possible answer: Since every 4 centimeters represents cup of white paint, I know that cup of white is used for 12 cups of blue paint. Therefore, I added cup to cup to determine that 1 cup of white paint would be needed. How did you use the ruler to determine the number of cups of blue paint that are needed for 3 cups of white paint? Answers may vary. Possible answer: Since 1 cup of white paint is used for 16 cups of blue paint, I know that 3 cups of white paint are used for 48 cups of blue paint. I know that cup of white paint is used for 12 cups of blue paint. Therefore, I added 48 cups and 12 cups to find that I would need 60 cups of blue paint. Part 3 How did you use the ruler to represent the situation? Answers may vary. Possible answer: I labeled the top of the ruler as silver stars and the bottom of the ruler as gold stars. Since there are 8 silver stars for every 3 gold stars, I put an 8 above the 3 inch mark on the ruler. How did you determine the number of silver stars needed for 9 gold stars? Answers may vary. Possible answer: Every time I increased by 3 inches (3 gold stars), I also increased by 8 silver stars. So 6 inches represented 16 silver stars and 9 inches represented 24 silver stars. What proportion could you use to determine the number of silver stars needed for 9 gold stars? Answers may vary. Possible answer: = How do you see this proportion on the ruler? Answers may vary. Possible answer: The 8 is above the 3 on the ruler, just like it is in the proportion. The unknown is above the 9 on the ruler, just like it is on the proportion. How could you solve the proportion = to verify the answer you found with the ruler?

41 Answers may vary. Possible answer: Since the scale factor is 3, I could multiply 8 by 3 to verify the answer I found with the ruler. How did you find the number of gold stars represented by 64 silver stars? Summary Answers may vary. Possible answer: The ruler only went up to 12 inches so the most I could represent was 12 gold stars and 32 silver stars. Since I needed to find the number of gold stars for 64 silver stars, I just pretended there were 2 rulers. There would be a total of 24 inches which would represent 24 gold stars and 64 silver stars. How does a ruler help you visualize a proportional relationship? Answers may vary. Possible answer: It helps me organize my information and look for patterns. It is similar to creating a table to represent a proportional relationship. How is using a ruler similar to writing a proportion? Answers may vary. Possible answer: The information is organized in a similar manner. I can see the proportion when I set up the ruler. ELABORATE: The Elaborate portion of the lesson affords students the opportunity to extend or solidify their knowledge of proportional relationships. This phase of the lesson is designed for individual investigation. (10 minutes) 1. Distribute Paintbrushes to each student. 2. Prompt students to complete Paintbrushes. 3. Actively monitor student work and ask facilitating questions when appropriate. Facilitating Questions How could you determine the cost of 1 paintbrush in a 3 pack of paintbrushes? I could divide $2.25 by 3. How could you determine the cost of 1 paintbrush in a 5 pack of paintbrushes? Answers may vary. Possible answer: Once I know the price of 1 paintbrush in the 3 pack, I could subtract 5 from that price to find the price of 1 paintbrush in the 5 pack.

42 How could you use the price of 1 paintbrush in a 5 pack to determine the cost of the 5 pack of paintbrushes? Answers may vary. Possible answer: I could multiply the price of 1 paintbrush by 5. 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. 1. Distribute Evaluate: Proportions to each student. 2. Prompt students to complete Evaluate: Proportions. 3. Upon completion of Evaluate: Proportions, the teacher should use the error analysis 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 C A B D (Guess) 3 D A B C 4 B A C D STUDENT WORKSHEETS FOLLOW!!!!

43 Name: Date:

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47 Name: Date: Evaluate: Proportions 1. A 12 ounce bottle of prescription lotion to treat poison ivy costs $18. If the price per ounce is the same, how many ounces would be in a bottle that costs $27? Show all work and explain your answer below. 2. George is on the track team and can run 50 meters in 8 seconds. If he runs at the same rate, about how many seconds will it take him to reach the 375 meter mark? A. 7.5 seconds B. 47 seconds C. 60 seconds D. 68 seconds 3. A 1 liter bottle of soda costs 99 and a 2 liter bottle of soda costs $1.59. Which of these statements will help a shopper decide which bottle is the better buy? A. The 1 liter bottle of soda is the better buy because it is less expensive per liter. B. The 1 liter bottle of soda is the better buy because 99 is about $1, and $1 goes into $1.59 about 1.5 times. C. The 2 liter bottle of soda is the better buy because it is more expensive. D. The 2 liter bottle of soda is the better buy because it is less expensive than two of the 1 liter bottles of soda.

48 4. The prices of 4 different bottles of lotion are given in the table. Bottle Size (ounces) Lotion Prices Price (dollars) 25 $ $ $ $0.86 Which size bottle of lotion has the lowest price per ounce? A. The 25 ounce bottle B. The 15 ounce bottle C. The 10 ounce bottle D. The 5 ounce bottle

49 SOLVING MATH PROBLEMS KEY QUESTIONS WEEK 4 By the end of this lesson, students should be able to answer these key questions: What is a rate? What is a unit rate? How can knowing the unit rate help you to complete a pattern? How can you determine if a problem situation can be solved using a unit rate? MATERIALS: For whole class demonstration: Timer or watch with a second hand For each student: Rulers (for straightedge) Warm Up Activity: The Name Game Word Power Worksheet Independent Practice: Rush Hour Rates to Tables, to Graphs Evaluate: Rates

50 TEACHER TOOL ENGAGE: The Engage portion of the lesson is designed to create student interest in the concepts addressed. This part of the lesson is designed for whole group instruction. (10 minutes) 1. Distribute The Name Game warm up activity to each student. 2. Using a timer or a watch with a second hand, prompt students to begin printing their first name as many times as possible in 15 seconds. 3. Prompt students to record this information in the table on The Name Game in the appropriate row. (NOTE: Students should record fractional parts of their name if they were unable to completely write their full name when time is called.) 4. Prompt students to use their recorded data to complete The Name Game. 5. Actively monitor student work and ask facilitating questions when appropriate. Facilitating Questions How would you find the fractional part of your name? Answers may vary. Possible answer: I can count how many letters are in my name that will be the denominator of my fraction. The number of letters I was able to write will be the numerator. How do you multiply fractions? Answers may vary. Possible answer: Change any mixed numbers to improper fractions. Then multiply the numerators together and multiply the denominators together. How can you find the number of names you would be able to write in 5 seconds? Answers may vary. Possible answer: I can divide the number of names I wrote in 15 seconds by 3. How do you divide fractions? Answers may vary. Possible answer: I can multiply the first fraction by the reciprocal of the second fraction.

51 How could you use the first row of your table to find the number of times you could write your name in 60 seconds? Answers may vary. Possible answer: Since 60 seconds is four times 15 seconds, I can multiply the number of names in the first row by 4. What is a strategy you can use to find the number of times you could write your name in 105 seconds? Answers may vary. Possible answer: I can add together the answers I got for 45 seconds and 60 seconds. EXPLORE: The Explore portion of the lesson provides the student with an opportunity to be actively involved in the exploration of the mathematical concepts addressed. This part of the lesson is designed for groups of 2 students. (20 minutes) 1. Distribute Word Power to each student. 2. Prompt students to complete Word Power. 3. Actively monitor student work and ask facilitating questions when appropriate. Facilitating Questions What is different about the two rates? Answers may vary. Possible answer: Amy s rate is expressed in words per minutes and Yvonne s rate is expressed in words per second. How are the tables different? Answers may vary. Possible answer: Yvonne s table uses seconds and Amy s table uses minutes. What values can you put in your table for the time periods? Answers may vary. Possible answer: I can put any time period in the table and then determine the number of words that can be typed in that amount of time. Since Yvonne s rate was 11 words in 15 seconds, I thought it would be easiest to use time periods that are multiples of 15 seconds. I put 11 words in 15 seconds as the first row of the table and then extended the table from there.

52 How many seconds are in 1 minute? 60 seconds What fraction of a minute is 15 seconds? of a minute How can you determine the number of words Yvonne can type in 2 minutes? Answers may vary. Possible answer: I can multiply 11 words by 4 to find the number of words Yvonne can type in 1 minute. Then I can multiply 44 words by 2.5 to find the number of words Yvonne can type in 2 minutes. How can you determine the number of words Amy can type in 1 minute? In 15 seconds? Answers may vary. Possible answer: I can divide 130 words by 2 to find how many words Amy can type in 1 minute. Since 2 minutes equals 150 seconds, I can divide 130 words by 10 to find how many words Amy can type in 15 seconds. EXPLAIN: The Explain portion of the lesson is directed by the teacher to allow the students to formalize their understanding of the concepts addressed in the lesson. (15 minutes) 1. Debrief Word Power. 2. Use facilitating questions to lead the discussion. Facilitating Questions What is a rate? Answers may vary. Possible answer: A rate is a ratio that compares 2 different units. What 2 units were compared on Word Power? Words and time (seconds or minutes) How did you determine the values to place in your table? Answers may vary. Possible answer: I used an increment of time that was a multiple or factor of the time period I was given. I could have used any time periods, but it was easiest to use factors or multiples.

53 What could be done to find Yvonne s typing rate in words per minute? Answers may vary. Possible answer: Since Yvonne s typing rate is given in words and seconds, I can find how many words she can type in 60 seconds. Thhat would be the same as how many words she can type in 1 minute. = = 44 words per minute What could be done to find Amy s typing rate in words per minute? Answers may vary. Possible answer: I can divide the numerator and denominator by 2 to find the unit rate. = = 52 words per minute How can you use a unit rate to determine the fastest typist? Answers may vary. Possible answer: I can see who types the most words in 1 minute. What was Kathy s reading rate? or = 6 words per second How could you find Kathy s reading rate in words per minute? Answers may vary. Possible answer: Since Kathy s reading rate is given in words and seconds, I can find how many words she can read in 60 seconds. That would be the same as how many words she can read in 1 minute. = = 360 words per minute What was Jamie s reading rate? = = words per second How could you find Jamie s reading rate in words per minute? Answers may vary. Possible answer: Since 15 is a factor of 60, I could find how many words Jamie reads in 15 seconds and use that information to find how many words she can read in 60 seconds.

54 = What was Betsy s readying rate? = = 280 words per minute How could you find Betsy s reading rate in words per minute? Answers may vary. Possible answer: Find the unit rate by dividing the numerator and denominator by. = = 320 words per minute How could you find Betsy s reading rate in words per second? Answers may vary. Possible answer: Find the unit rate by dividing the numerator and denominator by 90 since there are 90 seconds in minutes. = = or words per second How could you use the graph to determine the fastest typist/reader? Answers may vary. Possible answer: Look at one time period and read the graph to see who typed (read) the most words for that time period. ELABORATE: The Elaborate portion of the lesson provides an opportunity for the student to apply the concepts of the concept within a new situation. This part of the lesson is designed for individual investigation. (20 minutes) 1. Distribute Independent Practice: Rush Hour to each student. 2. Prompt student to complete Independent Practice: Rush Hour. The teacher may choose to use several of these problems as guided practice to check for student understanding of the concepts and procedures addressed in the lesson. 3. Actively monitor student work and ask facilitating questions when appropriate.

55 Facilitating Questions What ratio can be written to represent the relationship? Answers may vary. What is the unknown in this problem? Answers may vary. Does your answer seem reasonable? Answers may vary. Could you solve this problem another way? How? Answers may vary. Does the answer have a unit of measure? If so, what is it? 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. (15 minutes) 1. Distribute Evaluate: Rates to each student. 2. Prompt students to complete Evaluate: Rates. 3. Upon completion of Evaluate: Rates, the teacher should use the error analysis to assess student understanding of the concepts and procedures the class addressed in the lesson. Question Number Correct Answer Answers and Error Analysis for Evaluate: Rates Conceptual Error Procedural Error 1 D A B C 2 A B C D (Guess) 3 D A B C 4 B C D A (Guess) STUDENT WORKSHEETS FOLLOW!!!!

56 Name: Date: The Name Game 1. Your teacher is going to time you for 15 seconds. Your job is to print your first name as many times as you can during those 15 seconds. Record the total number of times you were able to write your name and any fractional parts of your name in the appropriate row of the table below. 2. Use the data you recorded to complete the rest of the table. Time (in seconds) Number of names you wrote 3. What patterns do you see in the table? 4. How many times do you think you could print your name in 5 minutes? Justify your answer. 5. How many names could you have written in 5 seconds?

57 Name: Date: Word Power 1. Yvonne types 11 words in 15 seconds. Amy types 130 words in 2 minutes. Use the tables and graph below to determine who types at the faster rate of speed. Justify your answers. Yvonne s Typing Rate Amy s Typing Rate Seconds Words Minutes Words Typing Rates

58 2. Kathy, Jamie, and Betsy took a speed reading course. Kathy read 180 words in 30 seconds. Jamie read 210 words in 45 seconds. Betsy read 480 words in 1 minutes. Use the tables and graph below to explore their reading rates. Then arrange the readers in order from the slowest reader to the fastest reader. Justify your answer. Kathy s Rate Jamie s Rate Betsy s Rate Seconds Words Seconds Words Seconds Words Reading Rates

59 Name: Date: Independent Practice: Rush Hour A rate is a ratio that compares 2 different units. Examples of rates include feet per second, miles per gallon, and words per minute. You can determine equivalent rates and create graphs and tables in order to make predictions and comparisons. Practice: During morning rush hour, Paul drove 5 miles in 15 minutes. How many miles per hour was Paul driving at that time? 1. What is Paul s driving rate? 2. How many miles per hour was Paul driving at that time? 3. Create a table to show Paul s driving rate up to 75 minutes.