Solving More Ratio Problems
10 min
Teacher Prep
Setup
Narrative
This Warm-up reminds students of previous work with tape diagrams and encourages a different way to reason with them. Students are given only a tape diagram and are asked to generate a concrete context to go with the representation.
Students’ stories should have the following components:
- The same unit for both quantities in the ratio
- A ratio of 7:3
- Scaling by 3, or 3 units per part
The stories may also have a quantity of 30 that represents the sum of the two quantities in the ratio.
As students work, identify a few different students whose stories are clearly described and are consistent with the diagram so that they can share later.
Launch
Ask students to share a few things they remember about tape diagrams from the previous lesson. Students may recall that:
- We draw one tape for every quantity in the ratio. Each tape has parts that are the same size.
- We draw as many parts as the numbers in the ratio show. (For instance, for a 2:3 ratio, we draw 2 parts in a tape and 3 parts in another tape).
- Each part represents the same value.
- Tape diagrams can be used to think about a ratio of parts and the total amount.
Tell students their job is to come up with a valid situation to match a given tape diagram. Give students some quiet think time and then time to share their response with a partner.
Student Task
Describe a situation with two quantities that this tape diagram could represent.
Sample Response
Sample responses:
- One batch of purple paint is made by mixing 7 cups of blue with 3 cups of red. In three batches, there are 30 cups of purple paint, which is made of 21 cups of blue paint and 9 cups of red paint.
- There are 30 fish in an aquarium. The ratio of blue fish to red fish is 7:3. There are 21 blue fish and 9 red fish.
Activity Synthesis (Teacher Notes)
Invite a few students to share their stories with the class. As they share, consider recording key details about each story for all students to see. Then, ask students to notice similarities in the different scenarios.
Guide students to see that they all involve the same units for both quantities of the ratio, a ratio of 7 to 3, and either 3 units per part or scaling by 3. They may also involve an amount of 30 units, representing the sum of the two quantities.
Anticipated Misconceptions
Students may misunderstand the meaning of the phrase “with two quantities” and simply come up with a situation involving ten identical groups of three. Point out that the phrase means that the row of seven groups of three should represent something different than the row of three groups of three.
Students may also come up with a situation involving different units, for example, quantity purchased and cost, or distance traveled and time elapsed. Remind them that the parts of tape are meant to represent the same value, so we need a situation that uses the same units for both parts of the ratio.
Standards
- 6.RP.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
- 6.RP.A.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
20 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Solving More Ratio Problems
10 min
Teacher Prep
Setup
Narrative
This Warm-up reminds students of previous work with tape diagrams and encourages a different way to reason with them. Students are given only a tape diagram and are asked to generate a concrete context to go with the representation.
Students’ stories should have the following components:
- The same unit for both quantities in the ratio
- A ratio of 7:3
- Scaling by 3, or 3 units per part
The stories may also have a quantity of 30 that represents the sum of the two quantities in the ratio.
As students work, identify a few different students whose stories are clearly described and are consistent with the diagram so that they can share later.
Launch
Ask students to share a few things they remember about tape diagrams from the previous lesson. Students may recall that:
- We draw one tape for every quantity in the ratio. Each tape has parts that are the same size.
- We draw as many parts as the numbers in the ratio show. (For instance, for a 2:3 ratio, we draw 2 parts in a tape and 3 parts in another tape).
- Each part represents the same value.
- Tape diagrams can be used to think about a ratio of parts and the total amount.
Tell students their job is to come up with a valid situation to match a given tape diagram. Give students some quiet think time and then time to share their response with a partner.
Student Task
Describe a situation with two quantities that this tape diagram could represent.
Sample Response
Sample responses:
- One batch of purple paint is made by mixing 7 cups of blue with 3 cups of red. In three batches, there are 30 cups of purple paint, which is made of 21 cups of blue paint and 9 cups of red paint.
- There are 30 fish in an aquarium. The ratio of blue fish to red fish is 7:3. There are 21 blue fish and 9 red fish.
Activity Synthesis (Teacher Notes)
Invite a few students to share their stories with the class. As they share, consider recording key details about each story for all students to see. Then, ask students to notice similarities in the different scenarios.
Guide students to see that they all involve the same units for both quantities of the ratio, a ratio of 7 to 3, and either 3 units per part or scaling by 3. They may also involve an amount of 30 units, representing the sum of the two quantities.
Anticipated Misconceptions
Students may misunderstand the meaning of the phrase “with two quantities” and simply come up with a situation involving ten identical groups of three. Point out that the phrase means that the row of seven groups of three should represent something different than the row of three groups of three.
Students may also come up with a situation involving different units, for example, quantity purchased and cost, or distance traveled and time elapsed. Remind them that the parts of tape are meant to represent the same value, so we need a situation that uses the same units for both parts of the ratio.
Standards
- 6.RP.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
- 6.RP.A.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.