Week Of: April 27- May 1 Grade: 6 Content: Math Learning Objective
Week of: April 27- May 1 Grade: 6 Content: Math
Learning Objective:
Greetings 6th graders! We hope you were successful with last week’s work on ratios! We are all so proud of you for continuing your learning.
This week we will be reviewing some work that you’ve done on ratios and using those concepts to determine equivalent ratios. Some essential topics that you will be reviewing:
• A ratio is a comparison of two quantities. • Ratios can be expressed using words, with a colon, or in fractional form • A ratio can represent part-to-whole or part-to-part relationships. • Fractions and percents are special types of part-to-whole ratios.
Please review the Khan Academy video links below. This will assist you with your printable resources.
Video Links:
Khan Academy: https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra- ratio-word-problems/v/ratios-as-fractions
On-Line
All students now have access to an on-line program called Mathia!
Mathia- If you are already in Mathia, please continue to work in the program.
If you are new to Mathia: Please log-in to through Clever
Printable Resources:
Skills Practice: see the attached practice pages
Oh, Yes, 3 I Am the Muffin Man Determining Equivalent Ratios
WARM UP LEARNING GOALS Choose the correct statement to complete • Use drawings to model and determine each sentence and explain your reasoning. equivalent ratios. 1. When the manager at Sweets-a-Plenty • Reason about tape diagrams to model Bakery decides how many bakers are and determine equivalent ratios. needed to bake muffins for a given day, • Define and use rates and rate reasoning she needs to consider the total number to solve ratio problems. of muffins needed for the day. • Use scaling up and scaling down to determine equivalent ratios. a. Making fewer muffins with more • Use double number lines to solve bakers will take: real-world problems involving ratios. • less time. • an equal amount of time. KEY TERMS • more time. • equivalent ratios • scaling up • tape diagram • scaling down b. Making more muffins in a shorter • rate • double number amount of time requires: • proportion line • fewer workers. • an equal amount of workers. • more workers.
Informally comparing ratios, or qualitatively comparing ratios, is important. However, there are many instances when you need to make more specific comparisons. How can you use equivalent ratios in order to compare ratios more precisely?
LESSON 3: Oh, Yes, I Am the Muffin Man • M2-37
CC01_SE_M02_T01_L03.indd01_SE_M02_T01_L03.indd 3377 007/04/177/04/17 4:574:57 AMAM Getting Started
Which Has More?
Consider the given representations to answer each question. Explain your reasoning.
1. Which dinner order has more pizza?
Order 1 Order 2
2. Which pattern has more stars?
Pattern 1 Pattern 2
3. Which pile of laundry has more shirts? Pile 1 Pile 2
4. Which type of reasoning did you use for each question— additive or multiplicative? Explain why.
M2-38 • TOPIC 1: Ratios
CC01_SE_M02_T01_L03.indd01_SE_M02_T01_L03.indd 3388 007/04/177/04/17 4:574:57 AMAM ACTIVITY Using Drawings to Model 3.1 Equivalent Ratios
Kerri and her friends are going hiking. Kerri invites her friends to meet at her house for a quick breakfast before heading out on their hike. Kerri wants to offer muffins to her friends.
1. She knows that one muffin combo has four muffins that can feed four people.
a. Draw a model showing the relationship between the muffin combo and the number of people it will feed.
How do your models show a relationship b. If Kerri invites 6 friends, how many muffin combos will she between two need? Draw a model to show how many muffin combo(s) quantities? she will need, and explain your answer.
3 c. If Kerri has 2 __ muffin combos, how many friends can she 4 feed? Draw a model to show how many friends she can feed, and explain your answer.
LESSON 3: Oh, Yes, I Am the Muffin Man • M2-39
CC01_SE_M02_T01_L03.indd01_SE_M02_T01_L03.indd 3399 007/04/177/04/17 4:574:57 AMAM Let's consider a different variety pack. In one muffin variety pack, two out of every five muffins are blueberry. blueberry muffin
2. Draw a model to answer each question. Explain your reasoning.
a. How many muffins are blueberry muffins if there are a total of 25 muffins?
I think I see a pattern? Do you see it? b. How many muffins are blueberry muffins if there are a total of 35 muffins?
c. How many total muffins are there if 8 muffins are blueberry?
As you solved these problems, you determined equivalent ratios. Equivalent ratios are ratios that represent the same part-to-part or part-to-whole relationship.
M2-40 • TOPIC 1: Ratios
CC01_SE_M02_T01_L03.indd01_SE_M02_T01_L03.indd 4400 007/04/177/04/17 4:574:57 AMAM ACTIVITY 3.2 Tape Diagrams
The local bakery sells muffins in variety packs of blueberry, pumpkin, and bran muffins. They always sell the muffins in the ratio of 3 blueberry muffins : 2 pumpkin muffins : 1 bran muffin. Don't forget to label each 1. Write the ratio that expresses each relationship. Identify each quantity with as a part-to-part or a part-to-whole ratio. the unit of measure! a. blueberry muffins to total muffins
b. pumpkin muffins to total muffins
c. bran muffins to total muffins
d. blueberry muffins to pumpkin muffins
e. bran muffins to pumpkin muffins
f. blueberry muffins to bran muffins
A ratio can be represented by drawing the objects themselves, Blueberry but they also can be represented using a tape diagram. A tape diagram illustrates number relationships by using rectangles to Pumpkin represent ratio parts. A tape diagram representing the ratio of each type of muffin is shown. Bran
2. What does each small rectangle represent in the given tape diagram?
LESSON 3: Oh, Yes, I Am the Muffin Man • M2-41
CC01_SE_M02_T01_L03.indd01_SE_M02_T01_L03.indd 4411 007/04/177/04/17 4:574:57 AMAM Tape diagrams provide a visual representation of ratios, but they also Remember, in this can be used to solve problems. scenario the ratio of muffins in each WORKED EXAMPLE variety pack is always 3 blueberry muffins : Suppose you purchase an 18-pack of muffins. How many 2 pumpkin muffins : 1 blueberry, pumpkin, and bran muffins will you purchase? bran muffin. There are 6 muffins represented in the tape diagram, and you want 18 total muffins that are in the same ratio.
Therefore, to determine how many muffins you need to maintain the same ratio, you can divide 18 by 6. 18 4 6 5 3
Therefore, each rectangle will represent 3 muffins.
Blueberry 333
Pumpkin 3 3
Bran 3
From the tape diagram, you can see that there are 9 blueberry muffins, 6 pumpkin muffins, and 3 bran muffins.
3. Is the ratio 9 : 6 : 3 equivalent to 3 : 2 : 1? Explain how you know.
M2-42 • TOPIC 1: Ratios
CC01_SE_M02_T01_L03.indd01_SE_M02_T01_L03.indd 4422 007/04/177/04/17 4:574:57 AMAM 4. Suppose you purchase a 36-pack of muffins. Use the tape diagram to illustrate how many blueberry, pumpkin, and bran NOTES muffins you will receive.
Blueberry
Pumpkin
Bran
5. Suppose you wanted 20 pumpkin muffins in your variety pack. How many total muffins will be in your variety pack? Complete the tape diagram to determine the answer.
Blueberry
Pumpkin
Bran
LESSON 3: Oh, Yes, I Am the Muffin Man • M2-43
CC01_SE_M02_T01_L03.indd01_SE_M02_T01_L03.indd 4433 007/04/177/04/17 4:574:57 AMAM 6. The table shows the number of muffins in specific sized variety packs. Complete just the missing cells in the columns for the 6-pack and 36-pack of muffins.
Total Number of Muffins 612182436 Number of Blueberry Muffins 9 Number of Pumpkin Muffins 6 Number of Bran Muffins 3
7. Analyze the completed columns in the table.
a. What do you notice about the numbers?
b. How could you have determined the number of each type of muffin in the 18-pack without using the tape diagram?
c. How could you have determined the number of each type of muffin in the 36-pack without using the tape diagram?
d. Use what you noticed about the numbers in the table to complete the remaining columns for the number of each type of muffin in a 12-pack and in a 24-pack of muffins. Explain your strategy.
M2-44 • TOPIC 1: Ratios
CC01_SE_M02_T01_L03.indd01_SE_M02_T01_L03.indd 4444 007/04/177/04/17 4:574:57 AMAM ACTIVITY 3.3 Rates and Proportions
One of the rounds at the Math Quiz Bowl tournament is a speed round. A team of four students will represent Stewart Middle School in the speed round of the Math Quiz Bowl. One student of the team will be chosen to solve as many problems as possible in 20 minutes. The results from this week’s practice are recorded in the table.
Number of Correctly Solved Problems Student in a Specified Time
Kaye 4 problems correct in 5 minutes
Susan 7 problems correct in 10 minutes
Doug 1 problem correct in 2 minutes
Mako 3 problems correct in 4 minutes
1. Explain how Tia's reasoning and Lisa's reasoning about who should compete in the speed round are incorrect.
Tia
Susan should definitely compete in the speed round because she correctly solved the most problems.
Lisa
It took Susan the longest time to complete her problems. She should not compete in the speed round.
LESSON 3: Oh, Yes, I Am the Muffin Man • M2-45
CC01_SE_M02_T01_L03.indd01_SE_M02_T01_L03.indd 4455 007/04/177/04/17 4:574:57 AMAM Each quantity in the table is a rate. A rate is a ratio that compares two quantities that are measured in different units. The rate for each student in this situation is the number of problems solved per amount of time.
WORKED EXAMPLE Kaye’s rate is 4 problems correct per 5 minutes. This rate can be written as: It's important to line up the 4 problems correct ______. units when 5 minutes writing equal ratios.
2. Write the rates for the other three team members.
a. Susan b. Doug c. Mako
When two ratios or rates are equivalent to each other, you can write them as a proportion. A proportion is an equation that states that two ratios are equal. In a proportion, the quantities composing each part of the ratio have the same multiplicative relationship between them.
WORKED EXAMPLE
For example, you know that Kaye got four problems correct per 5 minutes. So, you can predict how many problems she could answer correctly in 20 minutes. 34
problems correct 16 4 5 minutes 5 20
34
Kaye can probably answer 16 problems correctly in 20 minutes.
M2-46 • TOPIC 1: Ratios
CC01_SE_M02_T01_L03.indd01_SE_M02_T01_L03.indd 4466 007/04/177/04/17 4:574:57 AMAM When you change one ratio to an equivalent ratio with larger numbers, you are scaling up the ratio. Scaling up means you multiply both parts of the ratio by the same factor greater than 1.
4 16 3. Use the definition of a ratio to verify that __ is equivalent to ___ . 5 20
Remember, one way to represent a ratio is in fractional form. It doesn’t matter which quantity is in the numerator or denominator; it matters that the unit of measure is consistent among the ratios.
WORKED EXAMPLE You can write the proportion in a different way. 34 minutes 5 5 20 problems correct 4 16
34 This is the same strategy you used in 4. Determine the number of problems each student can probably elementary solve in 20 minutes. Explain the scaling up you used to school to write determine the equivalent ratio. equivalent fractions. Susan Doug Mako
5. Which team member is the fastest? Who would you pick to compete? Explain your reasoning.
LESSON 3: Oh, Yes, I Am the Muffin Man • M2-47
CC01_SE_M02_T01_L03.indd01_SE_M02_T01_L03.indd 4477 007/04/177/04/17 4:574:57 AMAM Warm Up Answers 6 8 1. __ , __ 7 9 7 5 2. ___ . ___ 13 11 4 4 3. __ , __ Going 2 5 3 Strong Comparing Ratios to Solve Problems
WARM UP LEARNING GOALS Use reasoning to compare • Apply qualitative ratio reasoning to compare ratios in each pair of fractions. real-world and mathematical problems. • Apply quantitative ratio reasoning to compare ratios in 6 8 __ __ 1. 7 and 9 real-world and mathematical problems. • Compare and order part-to-part and part-to-whole ratios 7 5 ______2. 13 and 11 represented verbally, pictorially, and numerically.
4 4 __ __ 3. 5 and 3
You know how to write a ratio as a comparison of two quantities. How can you compare two ratios to make decisions in real-world situations?
LESSON 2: Going Strong • M2-25
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LESSON 2: Going Strong • M2-25
CC01_TIG_M02_T01_L02_025-036.indd01_TIG_M02_T01_L02_025-036.indd 2525 44/4/17/4/17 6:046:04 PMPM Answers 1. Jen’s glass of lemonade will have the stronger tasting Getting Started lemon fl avor. Initially, Jen’s glass of lemonade had the stronger tasting lemon Lemony-er Lemonade fl avor. Since both glasses Tammy’s glass of lemonade has a weaker tasting lemon fl avor than have the same volume of Jen’s glass of lemonade. The shaded portion in each glass represents lemonade and the same an amount of lemonade. amount of lemon mix was added to both glasses; Jen’s glass will still have the stronger tasting lemon fl avor. Tammy’s Glass Jen’s Glass
1. If one teaspoon of lemon mix is added to both Jen’s and Tammy’s glasses, which glass will contain the lemonade with the stronger lemon fl avor? Explain your reasoning.
M2-26 • TOPIC 1: Ratios
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M2-26 • TOPIC 1: Ratios
CC01_TIG_M02_T01_L02_025-036.indd01_TIG_M02_T01_L02_025-036.indd 2626 44/4/17/4/17 6:046:04 PMPM Answers 1a. John’s glass will contain the lemonade with the ACTIVITY Qualitative Comparisons stronger lemon fl avor. 2.1 1b. Jake’s glass will contain the lemonade with the stronger lemon fl avor. In this activity you will compare ratios without measuring or counting 1c. There is not enough quantities. When you reason like this, it is called qualitative reasoning. information to determine which glass will contain 1. The shaded portion in each glass represents an amount of the lemonade with the lemonade. Answer each question and explain your reasoning. stronger lemon fl avor. a. Beth’s glass of lemonade has a weaker tasting lemon fl avor than John’s glass of lemonade. If two ounces of water is added to Beth’s glass and one teaspoon of lemon mix is added to John’s glass, which glass will contain the
lemonade with the stronger lemon fl avor? Beth’s Glass John’s Glass
b. Jimmy and Jake have glasses of lemonade that taste the same. If one teaspoon of lemon mix is added to each glass, which glass will contain the lemonade with the stronger lemon fl avor? Jimmy’s Glass Jake’s Glass
c. Jack’s glass of lemonade has a stronger tasting lemon fl avor than Karen’s glass of lemonade. If one teaspoon of lemon mix is added to Karen’s glass and one ounce of water is added to Jack’s glass, which glass will contain the lemonade with the stronger lemon fl avor? Jack’s Glass Karen’s Glass
LESSON 2: Going Strong • M2-27
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ELL Tip Because of the quantity of words in these situations, English Language Learners may struggle to understand the situations. For the fi rst problem, have students engage in a Think-Pair-Share activity, but with three students per group. Each student should be assigned one of the three parts. Make sure that each ELL student is paired with native English speakers. Having them talk through their part will help them understand what is being asked and will also allow them to practice their spoken language skills.
LESSON 2: Going Strong • M2-27
CC01_TIG_M02_T01_L02_025-036.indd01_TIG_M02_T01_L02_025-036.indd 2727 44/4/17/4/17 6:046:04 PMPM Answers 2a. Luke’s orange drink today would have a stronger 2. Choose the correct statement to complete each sentence and tasting orange fl avor be- explain your reasoning. If the answer cannot be determined, cause the ratio of orange explain why not. mix to water increased. a. If Luke plans to use four more tablespoons of orange 2b. Cannot be determined. mix today than what he used yesterday to make the There is not enough infor- same amount of orange drink, his orange drink today mation given. would have: 2c. The speed of the race car • a stronger tasting orange flavor. would be faster today. • a weaker tasting orange flavor. • a mix that has the same strength of orange taste as yesterday.
b. Dave and Sandy each made a pitcher of orange drink. Sandy’s pitcher is larger than Dave’s pitcher. Sandy used more orange mix than Dave. Dave’s orange drink has:
• a stronger tasting orange flavor. • a weaker tasting orange flavor. • a mix that has the same strength of orange taste as Sandy’s drink.
c. If a race car travels more laps in less time than it did yesterday, its speed would be:
• slower. • exactly the same. • faster.
M2-28 • TOPIC 1: Ratios
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M2-28 • TOPIC 1: Ratios
CC01_TIG_M02_T01_L02_025-036.indd01_TIG_M02_T01_L02_025-036.indd 2828 44/4/17/4/17 6:046:04 PMPM Answers 1a. HR 6D has the strongest chocolate taste. HR 6C ACTIVITY Comparing Comparisons has the weakest chocolate 2.2 taste. 1b. Answers will vary 1c. Answers will vary
The 6th grade students are making hot chocolate to sell at the Winter Carnival. Each homeroom suggested a different recipe.
The “T” in each HR 6A HR 6B recipe stands for Tablespoon! 2 cups milk 5 cups milk 3 T cocoa powder 8 T cocoa powder
HR 6C HR 6D 3 cups milk 4 cups milk 4 T cocoa powder 7 T cocoa powder
1. Consider the given recipes to answer each question.
a. Use reasoning to determine which recipe has the stron- gest chocolate taste and which recipe has the weakest chocolate taste. b. Show how you used ratio reasoning to order the recipes. Identify the ratios that you used as part-to-part or part-to- whole. c. Create a poster to explain your answer and strategies to the class. Prepare to share!
LESSON 2: Going Strong • M2-29
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ELL Tip Even though the margin note points out that T stands for tablespoon, it is possible that English Language Learners may not be familiar with standard units of measurement.
LESSON 2: Going Strong • M2-29
CC01_TIG_M02_T01_L02_025-036.indd01_TIG_M02_T01_L02_025-036.indd 2929 44/4/17/4/17 6:046:04 PMPM Answers 1. See table below ACTIVITY Ordering Part-to-Part and 2.3 Part-to-Whole Ratios
Suppose your class is in charge of providing punch at the upcoming open house. The Parent-Teacher Association bought lemon-lime soda and pineapple juice to combine for the punch, but they did not tell your class how much of each to use. Your classmates submitted suggestions for how to make the tastiest punch.
Cut out the punch ratio cards at the end of the lesson. Order the cards from the least lemon-lime concentration to the most lemon-lime con- centration. If you think more than one card describes the same ratio of The shading, or lack lemon-lime soda and pineapple juice, group those cards together. of shading, of each cup represents the difference in the type of concentration. lemon-lime soda pineapple juice
1. Describe the strategies you used to sort and order the cards.
M2-30 • TOPIC 1: Ratios
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1. Order from Least Pictorial Part-to-Part Part-to-Whole to Greatest Representation 1 : 3 K C 1 : 2 I B 2 : 3 J L F 4 : 5 G D 1 : 1 A E H
M2-30 • TOPIC 1: Ratios
CC01_TIG_M02_T01_L02_025-036.indd01_TIG_M02_T01_L02_025-036.indd 3030 44/4/17/4/17 6:046:04 PMPM Answers 1. Amber should take the penalty shot because NOTES TALK the TALK she has the greatest ratio of shots made to shots Put Me In, Coach attempted.
A soccer team has been awarded a penalty shot at the end of a tie game. If they make the penalty shot, they will win the league championship. The coach is considering three players to take the penalty. Amber has taken 4 penalty shots this season and has made 3 of them. Lindsay has taken 6 penalty shots and made 4. Li has taken 3 penalty shots and made 2.
1. Which player would you recommend take the penalty shot? Why?
LESSON 2: Going Strong • M2-31
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LESSON 2: Going Strong • M2-31
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M2-32 • TOPIC 1: Ratios
CC01_TIG_M02_T01_L02_025-036.indd01_TIG_M02_T01_L02_025-036.indd 3232 44/4/17/4/17 6:046:04 PMPM Punch Ratio Cards A B
For every lemon-lime soda, there is a pineapple juice.
C D
One-fourth of the punch is lemon-lime soda.
E F
Half of the mixture is pineapple juice.
G H
Lemon-lime soda : Pineapple juice 5 4 : 5
I J
For every lemon-lime soda, there are For every lemon-lime soda, there are 1 __ two pineapple juices. 1 2 pineapple juices.
K L
Three-fi fths of the punch is pineapple Pineapple juice : lemon-lime soda 5 3 : 1 juice.
LESSON 2: Going Strong • M2-33
LESSON 2: Going Strong • M2-33 C01_SE_M02_T01_L02.indd 34 3/29/17 10:00 AM
M2-34 • TOPIC 1: Ratios
CC01_TIG_M02_T01_L02_025-036.indd01_TIG_M02_T01_L02_025-036.indd 3434 44/4/17/4/17 6:046:04 PMPM Assignment Answers Write Assignment Answers will vary.
Practice Write Remember 1. Answers will vary. Write two recipes for hot One ratio can be less than, greater than, or equal to another ratio. chocolate, each with a different 2. Answers will vary. ratio of chocolate mix to water 3. Answers will vary. or milk. Describe how the two recipes are similar and different. 4. Answers will vary. 5. Answers will vary. A ratio of 9 cups of fruit juice and 4 cups of ginger ale might not be too fruity or too Practice weak, based on the fi rst 4 Megan is making fruit punch using fruit juice and ginger ale. She tries different combinations to get the mixture just right. If the ratio of fruit juice to ginger ale is too high, the punch is too fruity; if the ratio is too attempts. low, the punch is too gingery. Stretch For each attempt, write a ratio Megan can try next time. 1. She tried 16 cups of fruit juice and 4 cups of ginger ale. That was too fruity. Recipe 2, Recipe 3, Recipe 1 2. She tried 10 cups of fruit juice and 8 cups of ginger ale. That was too gingery. 3. She tried 10 cups of fruit juice and 1 cup of ginger ale. That was too fruity. 4. She tried 8 cups of fruit juice and 4 cups of ginger ale. That was a little too gingery. 5. Based on Megan’s attempts in Questions 1-4, what might be a good ratio of fruit punch to ginger ale? Explain your thinking.
Stretch Which of the given recipes will make cookies with the most chocolate chips per cookie? Order the recipes from the least chocolate chips per cookie to the most chocolate chips per cookies. Explain your answer. 3 __ Recipe 1: 1 4 cups of chips for a batch of 2 dozen cookies Recipe 2: 1 cup of chips for a batch of 18 cookies 3 __ Recipe 3: 4 cup of chips for a batch of 12 cookies
LESSON 2: Going Strong • M2-35
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LESSON 2: Going Strong • M2-35
CC01_TIG_M02_T01_L02_025-036.indd01_TIG_M02_T01_L02_025-036.indd 3535 44/4/17/4/17 6:046:04 PMPM Assignment Answers Review 1a. 30 : 25, part-to-part Assignment 1b. 30 : 75, part-to-whole 2a. 51.1 square meters Review 2b. 75 square inches 1. During the spring sports season, students at Hillbrook Middle School have the opportunity to either play 5 baseball, run outdoor track, or play lacrosse. Of the 75 students at Hillbrook who play a spring sport, 3a. __ 6 30 run track, 25 play baseball, and 20 play lacrosse. Write the ratios and determine whether a part-to-part 3b. 1 __ 1 or part-to-whole relationship exists. 8 a. track runners to baseball players b. track runners to total number of athletes
2. Determine the area of each face of a cube with the given surface area. a. 306.6 m2 b. 450 in.2
3. Determine each sum. 1 2 5 1 __ 1 __ __ 1 __ a. 6 3 b. 8 2
M2-36 • TOPIC 1: Ratios
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M2-36 • TOPIC 1: Ratios