T60
Mathematics Success – Grade 6
LESSON 4: Ratios and Unit Rates
[OBJECTIVE] The student will understand and apply the concepts of ratios and unit rates and use ratio and unit rate language to describe the relationship between two quantities. [PREREQUISITE SKILLS] interpreting data from a table, equivalent fractions [MATERIALS] Student pages S31 – S40 Two-color counters (12 per student pair) Toothpicks (9 per student pair) [ESSENTIAL QUESTIONS] 1. How can I describe a relationship between two quantities? 2. How can I write a ratio using numbers and words? using numbers and symbols? 3. How can I describe and determine a unit rate using a ratio? [WORD WALL WORDS] ratio, quantity, compare, relationship, unit rate [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) *For Cooperative Pairs (CP) activities, assign the roles of Partner A and Partner B to students. This allows each student to be responsible for designated tasks within the lesson. [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Verbal Description, Graphic Organizer, Pictorial Representation, Concrete Representation [WARM-UP] (IP, I, WG) S31 (ANSWERS ARE ON T72.) • Have students turn to S31 in their books to begin the Warm-Up. Students will answer questions using data displayed in a table. Monitor students to see if any of them need help during the Warm-Up. Have students complete the problems and then review the answers as a class. {Verbal Description, Graphic Organizer} [HOMEWORK] Take time to go over the homework from the previous night. [LESSON] [2 days (1 day = 80 minutes) – M, GP, IP, WG, CP]
Mathematics Success – Grade 6
T61
LESSON 4: Ratios and Unit Rates
SOLVE Problem
(WG, GP) S32 (Answers on T73.)
Have students turn to S32 in their books. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to determine and use ratios and unit rates. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Graphic Organizer, Verbal Description}
Ratios - Concrete M, WG, GP, CP:
(CP, WG, M, GP, IP) S32 (Answers on T73.)
Distribute counters and toothpicks to partners. Make sure students know their designation as Partner A or Partner B. Have students use the workspace below the directions for the concrete representations. {Verbal Description, Concrete Representation}
T62
Mathematics Success – Grade 6
LESSON 4: Ratios and Unit Rates
MODELING Ratios - Concrete Step 1: Have Partner A, place 1 counter on the workspace. • Partner B, place 2 toothpicks beside the counter on the workspace. • With your partner, compare the number of counters to toothpicks in the set you just created. (1 counter and 2 toothpicks) • Partner A, if the counters represent the first quantity on the workspace, identify the first quantity. (1) • Partner B, if the toothpicks represent the second quantity on the workspace, identify the second quantity. (2) Step 2: Have student pairs discuss how they could describe the relationship between the counter and toothpicks and then share answers with the whole group. • Partner A, explain the relationship between the counters and the toothpicks in the set. (1 counter to 2 toothpicks) • Partner B, explain the relationship between the toothpicks and the counters. (2 toothpicks to 1 counter)
*Teacher Note: Explain to students that the number of objects to be compared will be referred to as quantities. A quantity is how much there is of something. It has a number value.
Step 3: Have students place a set of 3 counters and 5 toothpicks on the workspace. • Partner A, if the counters represent the first quantity on the workspace, identify the first quantity. (3) • Partner B, if the toothpicks represent the second quantity on the workspace, identify the second quantity. (5) Step 4: Have student pairs discuss how they could describe the relationship between the counter and toothpicks and then share their answers with the whole group. • Partner A, explain the relationship between the counters and the toothpicks in the set. (3 counters to 5 toothpicks) • Partner B, explain the relationship between the toothpicks and the counters. (5 toothpicks to 3 counters)
Mathematics Success – Grade 6
T63
LESSON 4: Ratios and Unit Rates
IP, CP, WG:
Have students work with their partners to create different relationships with the counters and toothpicks. Partners should take turns creating and identifying relationships. Monitor closely to make sure students are using the appropriate vocabulary. Have students come back together as a class and share their results. {Verbal Description, Concrete Representation}
Ratios - Pictorial M, WG, GP, CP:
(CP, WG, M, GP, IP) S32, S33 (Answers on T73, T74.)
Students will continue to work on S32 using the toothpicks and counters. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Concrete
Representation, Pictorial Representation, Graphic Organizer}
MODELING Ratios - Pictorial Step 1: Have students create a set of 1 counter and 2 toothpicks on the workspace. • Partner A, remove the counter and draw a picture of it. • Partner B, remove the toothpicks and draw a picture of them. Step 2: • Partner A, identify the number of counters in the set. (1) • Partner B, identify the number of toothpicks in the set. (2) • With your partner, discuss the relationship of the counter to the toothpicks. • Partner A, describe the relationship of the counter to the toothpicks. (1 counter to 2 toothpicks) Record. Step 3: Have students look at the graphic organizer below Question 1. • Partner A, identify the First Quantity from Question 1. (counters) What is the value of the first quantity? (1) Record. • Partner B, identify the Second Quantity from Question 1. (toothpicks) What is the second quantity? (2) Record. • Have student pairs discuss the relationship between the quantities. • Partner B, explain the relationship. (There is 1 counter to 2 toothpicks.) Record. Step 4: • Partner A, identify the number of toothpicks in the set. (2) • Partner B, identify the number of counters in the set. (1) • With your partner, discuss the relationship of the toothpicks to the counters. • Partner A, describe the relationship of the toothpicks to the counter. (2 toothpicks to 1 counter) Record.
T64
Mathematics Success – Grade 6
LESSON 4: Ratios and Unit Rates
Step 5: Have students look at the graphic organizer below Question 2. • Partner A, if the first quantity is the number of toothpicks, record the number of toothpicks in the graphic organizer in the First Quantity box. What is the value of the first quantity? (2) Record. • Partner B, if the second quantity is the number of counters, what is the second quantity? (1) Record. • Have student pairs discuss the relationship between the quantities. • Partner B, explain the relationship. (There are 2 toothpicks to 1 counter.) Record. Step 6: Have student pairs discuss any observations about the two relationships they may have and then share with the whole group. (Answers may vary, but can include that the values are the same with different order, comparing the same two items, quantities do not change.) • Direct students to look at the third box in the graphic organizer for Question 1. • Have student pairs discuss possible strategies to shorten how they would describe and write the relationship between the two quantities. Share answers as a whole group. • What if we take the statement, “There is 1 counter to 2 toothpicks” and shorten it by using only the information of the quantities? • Partner A, how could we write the relationship using only the values of the two quantities? (1 to 2) Record. • We can also write the relationship using a colon. (1:2) Record. • The third way we can write the relationship is by using a fraction bar. ( 12 ) Record. Step 7: Have student pairs discuss what they noticed about the order of the quantities. • Partner A, what did you notice about the order of the quantities? (The number of counters was first, and the number of toothpicks was second.) • Partner B, explain why the relationship was written this way. (The relationship we were describing was counters to toothpicks.) Step 8: Have student pairs work together to fill in the blanks showing the relationship between the two quantities in the last box in the graphic organizer for Question 2. • Partner A, how could we write the relationship using only the values of the two quantities? (2 to 1) Record.
Mathematics Success – Grade 6
T65
LESSON 4: Ratios and Unit Rates
• Partner B, what was the second way we could write the relationship? (using a colon; 2:1) Record. • Partner A, what was the third way we could write the relationship? (using a fraction bar; 2 ) Record. 1 • Partner B, what did you notice about the order of the quantities? (The number of toothpicks was first, and the number of counters was second.) • Partner A explain why the relationship was written this way. (The relationship we were describing was toothpicks to counters.) Step 9: Partner B, what were the three ways we wrote the relationship between the two numbers? (used the word “to”, separated with a colon, or writing it as a fraction) • The word we can use to describe that relationship is ratio. Record in the graphic organizer for Questions 1 and 2. Step 10: Have students answer Questions 3 and 4 at the bottom of the page and review the answers as a whole group.
IP, CP, WG:
Have students work with their partners to complete Questions 1-5 on S33. Monitor closely to make sure students are using the appropriate vocabulary. Have students come back together as a class and share their results. {Verbal Description, Pictorial Representation, Graphic Organizer}
Ratios - Abstract M, WG, GP, CP:
(CP, WG, M, GP, IP) S34 (Answers on T75.)
Have students turn to S34 in their books. Make sure students know their designation as Partner A or Partner B. Students will be working with part-to-part ratios and part-to-whole ratios. {Verbal Description, Graphic Organizer}
T66
Mathematics Success – Grade 6
LESSON 4: Ratios and Unit Rates
MODELING Ratios - Abstract Step 1: Partner A, what is different about how the information for the relationship between quantities is presented on S34 and S33? (There is no picture. The information is given in a chart.) Step 2: Have student pairs review the information in the chart and complete Question 1. • Partner A, what does the First Quantity represent? (books Elisa read) Explain why this is the first quantity. (It is listed first in the question.) • Partner B, how many books did Elisa read? (3) Record. • Partner A, what does the Second Quantity represent? (books Tomeka read) Explain why this is the second quantity. (It is listed second in the question.) • Partner B, how many books did Tomeka read? (1) Record. Step 3: • Partner A, what is the relationship between the quantities? (For every 3 books Elisa read, Tomeka read 1 book.) Record. • Partner B, explain to Partner A how this relationship is written as a ratio. (3 to 1, 3:1, or 3 ) Record. 1 Step 4: Have student pairs read Question 2 and complete the answers. • Partner A, what is the relationship between the quantities? (For every 2 books Andy read, Elisa read 3 books.) • Partner B, explain to Partner A how this relationship is written as a ratio. (2 to 3, 2:3, or 2 .) Record. 3 Step 5: Have student pairs read Question 3 and discuss how it is different from Question 2. • Partner A, explain how the two questions are different. (Question 2 shows the ratio between two of the parts, and Question 3 shows the ratio between a part and the whole group.) • Partner B, how many books did Mario read? (2) • Partner A, what was the total number of books read? (12) • Partner B, explain how to write the ratio for that relationship. (2 to 12, 2 2:12, or 12 ) Record.
• Partner A, why are the numbers written in that order? (That is the order of the quantities in the question.)
Mathematics Success – Grade 6
T67
LESSON 4: Ratios and Unit Rates
IP, CP, WG:
Have students work with their partners to complete Questions 4 and 5 using the table on S34. Monitor closely to make sure students are using the appropriate vocabulary. Have students come back together as a class and share their results. {Verbal Description, Graphic
Organizer}
Unit Rates – Concrete and Pictorial M, GP, WG, CP:
(CP, WG, M, GP, IP) S35, S36 (Answers on T76, T77.)
Have students turn to S35 in their books. Distribute 9 counters and 3 toothpicks to partners. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Concrete Representation, Pictorial Representation}
MODELING Unit Rates – Concrete and Pictorial Step 1: Have student pairs look at Question 1 on S35. • Partner A, place nine toothpicks on the workspace. • Partner B, place three counters on the workspace. • Partner A, how many toothpicks are in the set? (9) Record. • Partner B, how many counters are in the set? (3) Record. • Partner A, what is the ratio of the set of toothpicks and counters? (9 to 3, 9:3, or 9 ) Record. 3 Step 2: Have students discuss how they could make groups with only 1 counter in each group. • Model for students how to separate the counters. Then model how to move the toothpicks one at a time so that they are evenly divided between the counters. • Partner B, how many equal groups can be created with 1 counter in each group? (3) Record. • Partner A, how many toothpicks are in each group? (3) Record. • Partner B, how many counters are in each group? (1) Record. • Have students replace the counters and toothpicks one at a time by drawing them as they remove each counter and toothpick. • With your partner, create 3 groups with 1 counter and 3 toothpicks in each by drawing a circle around each group.
T68
Mathematics Success – Grade 6
LESSON 4: Ratios and Unit Rates
Step 3: Have students discuss other words they may know that can be used to describe 1 of something. (For example, another word for a chapter or section in their math book is known as a unit.) • Partner A, if we only have 1 counter, what can we call it? (a unit) • Partner B, what is the ratio of toothpicks to 1 counter? (3:1) • Partner A, what is unique or special about the ratio we just identified? (the second value is 1 or a unit) • Explain to students that when the second value is a 1, or a unit, we call that ratio a unit rate. • Partner B, what is the unit rate for each group of counters and toothpicks? (The unit rate is 3 toothpicks to 1 counter.) Record.
*Teacher Note: Explain to students that when determining a unit rate, the denominator will always be 1.
Step 4: Have students look at Problem 2. • Partner A, how many stars are shown? (8) Record. • Partner B, how many moons are shown? (4) Record. With your partner, determine the ratio of stars to moons. (8 to 4, 8:4, or 8 4 .) Record. Step 5: Model for students how to group each set of stars and moons equally so that there is 1 moon in each equal group. • Partner A, how many equal groups can be created with 1 moon in each group? (4) Record. • Partner B, how many stars are in each group? (2) Record. • Partner A, how many moons are in each group? (1) Record. • What is the unit rate of stars to moons? (2 stars to every moon) Record.
Mathematics Success – Grade 6
T69
LESSON 4: Ratios and Unit Rates
IP, CP, WG:
Have students work as partners to complete Problems 3 – 4 on S36. Explain to students that for Problem 4 they will be creating their own model using the shapes given in the problem. Monitor closely to make sure students are using the appropriate vocabulary. Have students come back together as a class and share their results. {Verbal Description, Pictorial Representation}
Unit Rates - Abstract M, GP, WG, CP:
(CP, WG, M, GP, IP) S37 (Answers on T78.)
Have students turn to S37 in their books. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Graphic Organizer} MODELING Unit Rates - Abstract
Step 1: Direct students’ attention to Problem 1 on S37 and use the following instructions to complete this step. • With your partner, determine what the first problem is asking you to find. (The unit rate of water to 1 cup of juice) Step 2: Look at the graphic organizer below Question 1. • Partner A, what is the first quantity in the ratio of water to juice? (9) Record. • Partner B, what is the second quantity in the ratio of water to juice? (3) • Partner A, what is the ratio of the water to juice? ( 9 ) Record. 3
• Partner B, explain why the ratio is written in that order. (The water is mentioned first in the ratio statement.) • Partner A, in a unit rate what must the denominator be? (1) • Partner B, explain how we can change the fraction that represents the ratio of water to juice into a fraction with a denominator of 1. (Simplify by dividing both the numerator and denominator by 3.) • Partner A, what is the equivalent fraction for 9 with a denominator of 3 1? ( 3 ) Record. 1 • With your partner, identify the unit rate in the problem. (There are 3 cups of water to 1 cup of juice.) Record. Step 3: Have students complete Problem 2 using the graphic organizer.
T70
Mathematics Success – Grade 6
LESSON 4: Ratios and Unit Rates
Step 4: Compare Problem 3 to the two previous problems. (There is no graphic organizer.) • Partner A, what is the problem asking you to find? (The unit rate of the price of the pencils) • Partner B, how can we write the ratio relationship using the items price from the problem? ( pencil ) Record. • Partner A, what values can we substitute into the ratio? ( 80 ) Record. 8
• Partner B, explain how to find the unit rate. (Divide both the numerator and denominator by 8, so that we have a denominator of 1 for the unit rate.)
80 8 10 ÷ 8 = 1 The unit rate of the price of the pencils is 10 8
cents per pencil. Record.
