In this activity, students determine what percentage one number is of another number. They answer questions of the form “ is what percentage of ?” and match each question with an answer in a list of percentages.
Each pair of values is related by a factor that is a benchmark fraction, so students are likely to identify the matches quickly. They should spend some time discussing how they reason about each question.
Although students are likely to make the matches mentally, some may find it helpful to draw a tape diagram to visualize the relationship between the numbers.
Launch
Display this tape diagram for all to see.
Ask students to think about what question it could represent and to give a signal if they have a response. Select a couple of students to share their responses. Record questions along the lines of “7 is what percentage of 14?” and display the question(s) for all to see. Invite students to share the answer to the question and to explain how they know.
Tell students that they will now answer similar questions and think about what percentage one number is of another number.
Arrange students in groups of 2. Give students 2–3 minutes of quiet think time and another minute to share their responses and reasoning with a partner.
Action and Expression: Provide Access for Physical Action. Activate or supply background knowledge. Provide students with access to blank tape diagrams to support information processing. Supports accessibility for: Visual-Spatial Processing, Organization
Activity
None
Match each question in the left column with a percentage in the right column. One percentage will be left over. Be prepared to explain your reasoning.
5 is what percentage of 20?
3 is what percentage of 30?
6 is what percentage of 8?
20 is what percentage of 5?
4%
10%
25%
75%
400%
Student Response
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Building on Student Thinking
Because 5 goes into 20 four times, students might rush to say that 5 is 4% of 20. If this happens, encourage students to check their thinking by asking questions such as:
“10 is what percentage of 20?”
“If 10 is 50% of 20, what percentage of 20 is 5?”
“If 5 is 4% of 20, then does that mean 10 is 8% of 20?”
Activity Synthesis
The goal of the discussion is to help students see that a number can be expressed as a percentage of another number by comparing the numbers multiplicatively and using fractions.
Invite students to briefly share their responses and reasoning. Highlight that when finding what percentage one number is of another number, it can be helpful to think about what fraction one number is of the other number, for instance:
5 is what fraction of 20?
3 is what fraction of 30?
6 is what fraction of 8?
Once we know that 5 is of 20, 3 is of 30, and 6 is (or ) of 8, we can relate the fractions to the percentages 25% of 20, 10% of 30, and 75% of 8, respectively.
Then discuss how students reasoned about the last question. If not mentioned by students, emphasize that unlike in the first question, the 5 in this question is the value that corresponds to 100%, so the percentage of 20 of 5 must be greater than 100. Because 20 is , the percentages must also be related by a factor of 4. Consider using a table or a tape diagram to illustrate these relationships.
24.2
Activity
Standards Alignment
Building On
Addressing
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.
In this activity, students find both and where of is but do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.
The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).
In each question on the problem cards, the value that corresponds to 100% is different. This means that students who opt to use a diagram or a table need to create a separate representation for each question. Monitor for this awareness.
Also monitor for the strategies that students use when reasoning about \$24 as a percentage of \$25. For instance, students may:
Create a double number line diagram, draw and label the first few tick marks, see that each tick mark corresponds to \$1 and 4%, bypass drawing the remaining tick marks, and multiply 24 by 4.
Notice that \$24 is \$1 away from \$25 and subtract 4% from 100%.
Use a table to find the percentage corresponding to \$1 and then multiply that percentage by 24.
Consider selecting students who use efficient strategies to share during discussion.
This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.
Launch
Math Community
Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms or invite a student to read them out loud. Tell students that during this activity they are going to practice looking for their classmates putting the norms into action. At the end of the activity, students can share what norms they saw and how the norm supported the mathematical community during the activity.
Tell students that they will continue to solve problems involving percentages. Display the Info Gap graphic that illustrates a framework for the routine for all to see.
Remind students of the structure of the Info Gap routine, and consider demonstrating the protocol if students are unfamiliar with it. There is an extra set of cards available for demonstration purposes.
Arrange students in groups of 2. In each group, give a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give students the cards for a second problem and instruct them to switch roles.
Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of the Info Gap graphic visible throughout the activity or provide students with a physical copy. Supports accessibility for: Memory, Organization
Activity
None
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the problem card:
Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me ?”
Explain to your partner how you are using the information to solve the problem. “I need to know because . . . .”
Continue to ask questions until you have enough information to solve the problem.
Once you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.
If your teacher gives you the data card:
Silently read your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know ?”
Listen to your partner’s reasoning and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner!
These steps may be repeated.
Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.
Activity Synthesis
After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Consider inviting previously selected students to share their strategies, ending with methods that are more efficient.
Here are some questions for discussion:
“To answer the first question on Problem Card 1, we needed to find 60% of the cost of Device C. In that case, what amount corresponds to 100%?” (\$40, the cost of Device C)
“In the second question, what amount corresponds to 100%?” (\$25, the cost of Device B)
“In the question on Problem Card 2, what amount corresponds to 100%?” (\$60, the cost of Device D)
Then display the following questions for all to see. Explain that they represent the questions on the cards but are phrased differently:
“What is 60% of \$40?”
“\$24 is what percentage of \$25?”
“If \$24 is 40% of a number, what is that number?”
Highlight for students that in each question, we were looking for different pieces of information. In the first question, it was the dollar amount for 60% that was unknown. In the second question, it was the percentage. In the last one, it was the dollar amount for 100%. We can reason about the answer to each question using known strategies and representations.
Math Community
Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:
“I noticed our norm ‘’ in action today and it really helped me/my group because .”
Lesson Synthesis
In situations that can be described with percentages, there are generally three different questions we can ask. To illustrate this point, present a scenario and display a partially labeled double number line diagram. Annotate the diagram with the known and unknown values as each question is discussed.
Suppose a rain barrel has collected some rainwater.
If we know . . .
we can ask . . .
The barrel is 20% filled and can hold 400 liters when full.
What is 20% of 400 liters?
The barrel is 20% filled and has 80 liters of water.
How many liters are in a full barrel if 20% of a full barrel is 80 liters?
The barrel has 80 liters of water and can hold 400 liters when full.
80 liters is what percentage of 400 liters?
To solve the third problem, we can start by asking: “What fraction of 400 is 80?” , so we can partition into five equal parts the intervals between 0 and 400 and between 0% and 100% on a double number line diagram, as shown here:
Since of 100 is 20, that percentage is 20. We can also use a table:
Highlight that regardless of the question asked, it is important to think about which quantity corresponds to 100%.
Student Lesson Summary
In a situation that involves percentages, there are often three questions we are interested in answering.
Suppose a tank is filled with some water.
If we know that the tank is 25% filled and can hold 36 liters, we can ask: What is 25% of 36 liters?
If we know that the tank has 9 liters and is 25% filled, we can ask: How many liters are in a full tank?
If we know that the tank has 9 liters but can hold 36 liters when full, we can ask: What percentage of 36 liters is 9 liters?
We can use a double number line diagram, a table, or a tape diagram to help us reason about each question.
We can also use our knowledge of fractions or relationships between numbers. For instance, we know that 9 is of 36, or , so it is of 100% or 25%.
In general, in a situation where of is , we can find the value of , , or if we know the other two values.
Standards Alignment
Building On
5.NF.B
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
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.
If students draw a complete double number line diagram with all 25 tick marks on each line and take the time to label each tick mark by counting by 4s, ask them to explain their strategy. Then urge them to look for a pattern in their process to help them answer the question more efficiently. For example, consider asking:
“How many times would you need to count by 4s?”
“What might be a quicker way to find the percentage for \$24 other than adding 4 twenty-four times?”
