Representing Percentages in Different Ways
10 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is to elicit two ideas: that we can reason about percentages of a number in terms of fractions of a whole, and that tape diagrams can support this reasoning.
The given tape diagram represents the relationship between the baby weight and adult weight of Jada’s puppy, which students previously represented with a double number line diagram. This is done to encourage students to think about the structure of each representation and how the same relationship and quantities are presented differently in the two diagrams. The thinking here will support students in choosing representations to make sense of percentage problems.
While students may notice and wonder many things about the diagram, the important discussion points are what the number of parts, the size of the parts, and the entire diagram might tell us about the situation represented.
Launch
Arrange students in groups of 2. Display the tape diagram for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder about. Give students 1 minute of quiet think time, and then 1 minute to discuss the things that they notice and wonder about with their partner.
Student Task
What do you notice? What do you wonder?
Sample Response
Students may notice:
- The diagram looks like the ones used to show fractions.
- The diagram is partitioned into 5 parts and one part is shaded.
- All the parts are labeled with the number 9.
- The shaded part is 20% of something.
- The entire tape represents 5⋅9.
Students may wonder:
- Why is only one part shaded?
- What situation does the diagram represent?
- Does the diagram represent Jada’s puppy weight?
Activity Synthesis (Teacher Notes)
Ask students to share the things that they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If no students observed that (or wondered if) the tape diagram could represent the situation about Jada’s puppy in a recent activity, ask students: “Could this diagram represent the weight of Jada’s puppy and its adult weight? How do you know?”
Then ask students:
- “How can we see 100% in the diagram?” (It’s the entire length of the diagram.) “What does it represent in this situation?” (the adult weight of Jada’s puppy)
- “The adult weight of the puppy will be 45 pounds. How can you see that in the diagram?” (There are 5 groups of 9 in the diagram, and 5⋅9=45.)
- [Display the double number line diagram from “Puppies Grow Up.”] “Compare the way the tape diagram and the double number line diagram represent the same situation. How are they alike?” (They both show that 9 pounds is 20% of something. They both use the length of a diagram to represent weight and percentage.)
- “How are the diagrams different?” (The 100% is not shown on the double number line diagram. It doesn’t show how many groups of 20% are in 100%. In the tape diagram, we see that there are 5 parts that are the same size. Each part represents 20% or a fifth of the tape, so the entire length of the diagram is 100%.)
Point out to students that a percentage can sometimes be thought of as a fraction of a whole. While both representations we’ve used so far can help us solve problems, a tape diagram can show a number as a fraction of 100%, which can help us make sense of a situation.
Standards
- 6.RP.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
- 6.RP.A.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
15 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Representing Percentages in Different Ways
10 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is to elicit two ideas: that we can reason about percentages of a number in terms of fractions of a whole, and that tape diagrams can support this reasoning.
The given tape diagram represents the relationship between the baby weight and adult weight of Jada’s puppy, which students previously represented with a double number line diagram. This is done to encourage students to think about the structure of each representation and how the same relationship and quantities are presented differently in the two diagrams. The thinking here will support students in choosing representations to make sense of percentage problems.
While students may notice and wonder many things about the diagram, the important discussion points are what the number of parts, the size of the parts, and the entire diagram might tell us about the situation represented.
Launch
Arrange students in groups of 2. Display the tape diagram for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder about. Give students 1 minute of quiet think time, and then 1 minute to discuss the things that they notice and wonder about with their partner.
Student Task
What do you notice? What do you wonder?
Sample Response
Students may notice:
- The diagram looks like the ones used to show fractions.
- The diagram is partitioned into 5 parts and one part is shaded.
- All the parts are labeled with the number 9.
- The shaded part is 20% of something.
- The entire tape represents 5⋅9.
Students may wonder:
- Why is only one part shaded?
- What situation does the diagram represent?
- Does the diagram represent Jada’s puppy weight?
Activity Synthesis (Teacher Notes)
Ask students to share the things that they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If no students observed that (or wondered if) the tape diagram could represent the situation about Jada’s puppy in a recent activity, ask students: “Could this diagram represent the weight of Jada’s puppy and its adult weight? How do you know?”
Then ask students:
- “How can we see 100% in the diagram?” (It’s the entire length of the diagram.) “What does it represent in this situation?” (the adult weight of Jada’s puppy)
- “The adult weight of the puppy will be 45 pounds. How can you see that in the diagram?” (There are 5 groups of 9 in the diagram, and 5⋅9=45.)
- [Display the double number line diagram from “Puppies Grow Up.”] “Compare the way the tape diagram and the double number line diagram represent the same situation. How are they alike?” (They both show that 9 pounds is 20% of something. They both use the length of a diagram to represent weight and percentage.)
- “How are the diagrams different?” (The 100% is not shown on the double number line diagram. It doesn’t show how many groups of 20% are in 100%. In the tape diagram, we see that there are 5 parts that are the same size. Each part represents 20% or a fifth of the tape, so the entire length of the diagram is 100%.)
Point out to students that a percentage can sometimes be thought of as a fraction of a whole. While both representations we’ve used so far can help us solve problems, a tape diagram can show a number as a fraction of 100%, which can help us make sense of a situation.
Standards
- 6.RP.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
- 6.RP.A.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.