6.RP.A.1 Lesson 2
Students work with a set of cards showing different ways of expressing ratios, including both part-part statements and part-whole statements. They group the cards that show the same ratio of boys to girls, but without the explicit use of the term equivalent.
Ratios can be represented in a:b form, as fractions, as decimals, as factors, and in words; they can be expressed in part-part statements or in part-whole statements.
Goals and Learning Objectives
- Group cards showing ratios that are equivalent but expressed in different forms.
Warm Up - Number Talk: Dividing by 4 and Multiplying by 1/4
This number talk helps students recall that dividing by a number is the same as multiplying by its reciprocal. Four problems are given, however, they do not all require the same amount of time. Consider spending 6 minutes on the first three questions and 4 on the fourth question.
In grade 4, students multiplied a fraction by a whole number, using their understanding of multiplication as groups of a number as the basis for their reasoning. In grade 5, students multiply fractions by whole numbers, reasoning in terms of taking a part of a part, either by using division or partitioning a whole. In both grade levels, the context of the problem played a significant role in how students reasoned and notated the problem and solution. Two important ideas that follow from this work and that will be relevant to future work should be emphasized during discussions:
- Dividing by a number is the same as multiplying by its reciprocal.
- We can multiply numbers in any order if it makes it easier to find the answer.
Display one problem at a time. Give students 1 minute of quiet think time per problem and ask them to give a signal when they have an answer and a strategy. Allow students to share their answers with a partner and note any discrepancies. Pause after the third question and ask, “What do you notice about the first three questions? Do you notice the same thing if we divide 5 by 4? Why?”
Support for Students with Disabilities
Memory: Processing Time. Provide sticky notes or mini whiteboards to aid students with working memory challenges.
Support for English Language Learners
Heavier Support. Pair students heterogeneously by language fluency so that more fluent students can support less fluent students.
- 24 ÷ 4 = 6; Possible strategies: Divide 24 into 4 equal groups or know that 4⋅6=24.
- 1/4 ⋅ 24 = 6; Possible strategies: Divide 24 into 4 equal groups or know that 4⋅6=24.
- 24 ⋅ 1/4 = 6; Possible strategies: Divide 24 into 4 equal groups or know that 4⋅6=24 or Commutative Property from the second question.
- 5 ÷ 4 = 5/4 or equivalent; Possible strategies: Distributive Property (4+1)÷4=(4÷4)+(1÷4) or know that 5⋅14=54.
Ask students to share what they noticed about the first three problems. Record student explanations that connect dividing by a number with multiplying by its reciprocal. Revisit the meaning of “reciprocal” when the term comes up (or bring it up if it's not mentioned by students). Help students recall that the product of a number and its reciprocal is 1.
Discuss how students could use their observations on the first three questions to divide 5 by 4, and then any two whole numbers.
Find the value of each expression mentally.
24 ÷ 4
1/4 ⋅ 24
24 ⋅ 1/4
5 ÷ 4
Ratio of Boys and Girls
Read the prompt about boys and girls at the middle school aloud. Then have students work together to complete the given statements using each given ratio exactly once.
When the class is done, have students share how they knew which values went with which statements.
SWD: When students think aloud, it provides them with opportunities to practice the thinking process involved in solving a set of problems. Listen to the students’ thought processes, correct misconceptions, and fill in incomplete processes.
ELL: Read-aloud routines and procedures are important for ELLs because teachers explicitly model strategies and behaviors used by effective readers. These valuable strategies can be used to access texts, retrieve information, build comprehension, and organize information.
Note that some of the given ratio values represent part-whole relationships. Students will work with part-whole relationships in Lesson 13, when percents are introduced.
Point out that some of the statements describe the relationship between a “part” and “the whole” and some of the statements describe the relationship between one “part” and the other “part.”
- The ratio of boys to girls is 2:1.
- Of all the students, are girls.
- of all the students are boys.
- One out of every 3 students is a girl.
- For every 2 boys, there is 1 girl.
- There are half as many girls as boys.
- There are 2 times as many boys as girls.
Ratio of Boys and Girls
There are 200 boys and 100 girls at Thurgood Marshall Middle School.
Complete the following sentences using each of the given quantities shown below only once.
- The ratio of boys to girls is _____.
- Of all the students, _____ are girls.
- _____ of all the students are boys.
- One out of every _____ students is a girl.
- For every 2 boys, there is _____ girl.
- There are _____ as many girls as boys.
- There are _____ times as many boys as girls.
2:1 1 3 half 2
Discuss the Math Mission. Students will represent ratios in different ways.
Represent ratios in different ways.
Four Schools Card Sort
Have students work in pairs on the Card Sort Interactive. Have them take turns sorting the cards into the school with the same ratio of girls and boys. Encourage students to discuss their reasoning with their partner about their card placement
SWD: Students with disabilities may have difficulty with operations involving more complex numbers (decimals). Supporting students to practice skills related to ratios more independently may include:
- Allowing students access to a calculator
- Providing partially scaffolded answers
ELL: Encourage students to verbalize their explanations. Allow students to speak in small groups to gain confidence.
Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.
Listen for students who are challenging one another’s grouping choices and who are developing clearer, more coherent arguments to justify their choices.
Student doesn’t understand the task.
- Each of the cards shows a comparison of boys and girls; your job is to group them so all of the cards that express the same ratio are in a group together.
Student relates cards without reference to the data from each school.
- How can you tell this card should be grouped with these cards? Justify your choice to your partner. Can you think of another way to justify the choice?
- For which cards do you need to know the actual number of boys and girls at the school?
- Explain how you knew which group this card belonged to without having to check it against the number of boys and girls at the school.
Student has difficulty setting up appropriate calculations.
- Talk to your partner about why it is difficult to set up the calculation.
- Some of the statements are part-part statements and some are part-whole statements. What is the whole—the number of all students—at each school?
Four Schools Card Sort
Four high schools (A, B, C, and D) have different numbers of students and different ratios of boys to girls.
- Working with a partner, take turns matching cards that represent the same school.
- Explain to your partner how you know the cards match.
- Your partner should either agree with your explanation or challenge it if your explanation is not correct, clear, or complete.
To match the cards, find a strategy that will help you narrow down the choices. For example, you might start by choosing a school in which the ratio of boys to girls is easy for you to see. Then find all the cards that match that school.
- Answers will vary. Check students’ work. They should create one group of cards or statements, each with a different representation: ratio, decimal, fraction, factor, and part-part statement in words.
- Make a set of cards for another school using a ratio of girls to boys that is different from any of the ratios in the card sort.
Challenge—HANDOUT: Four Schools Card Sort: Create Your Own School
Cool - Down: A Collection of Animals
Support for Students with Disabilities
Heavier Support. Allow students to use their first language for their statements, and then translate with peers into English. Ask students to read at least one statement (in English) aloud to a partner, and if needed, revise/refine in writing.
Answers vary. Sample responses:
- The ratio of dogs to cats is 6:4.
- There are 3 dogs for every 2 cats.
- There is 1 mouse for every 2 cats.
- The ratio of cats to mice is 4:2.
Here is a collection of dogs, mice, and cats:
Write two sentences that describe a ratio of types of animals in this collection.