Lesson 22

In this warm-up, students are presented with tape diagrams with a shaded portion, and they identify the percentage that is shaded. 

Display the image in the task statement for all to see, and ask students to think of at least one thing they notice. Ask a few students to share something they notice. It is likely that students notice there are three tape diagrams of the same length, but the first is divided into 10 equal pieces, the second into 2 equal pieces, and the last into 4 equal pieces. Students may make claims about what fraction of each tape is shaded. Remind students that when we wonder, "What percent of something is shaded?" it is understood that the whole thing is 100%. 

Arrange students in groups of 2. Give 1–2 minutes of quiet think time, followed by partner- and whole-class discussions.

Invite students to share how they reasoned about the percentage of each tape diagram that is shaded. Record and display their explanations for all to see. Highlight alternative ways of naming the size of the shaded portion, for example, “\(\frac34\) of the diagram” and “75% of the diagram.”

In this activity, students calculate three different benchmark percentages—50%, 10%, and 75%—given three different values that correspond to 100%. The repetition of the benchmark percentages allows students to notice regularity and engage in MP8. They generalize the patterns in their calculations to determine how to find those percentages when the 100% value is \(x\).

Ask students to complete the first three sub-questions of each problem mentally. If necessary, clarify that “using mental math” means working out an answer without writing down their calculations and just recording the answer. For the last sub-question, ask them to write a sentence or two to explain their approach. Give students quiet think time to complete the activity and then time to share their explanation with a partner.

If students struggle to get started with mental math, offer some scaffolding. For example, ask, “How much of 100% is 50%?” Suggest to students that if they know that 50% of a number is the same as \(\frac12\) of that number, then they can think about what \(\frac12\) of the number is.

Highlight the following:

50% of a quantity is \(\frac12\) of that quantity. We can calculate it by dividing the quantity by 2 or multiplying the quantity by \(\frac12\). If no students bring up the multiplication method, ask what fraction 50% reminds them of or what number they could multiply by to get the same answer; either \(\frac12\) or 0.5 is fine.

10% of a quantity is \(\frac{1}{10}\) of that quantity. You can calculate it by dividing the quantity by 10, or multiplying the quantity by \(\frac{1}{10}\).

75% of a quantity is \(\frac34\) of that quantity. You can calculate it by dividing the quantity by four and then multiplying by 3, or by multiplying the quantity by \(\frac34\).

In this activity students find the values for 100% given different benchmark percentages. Students are likely to calculate the answers quickly. They are to spend the majority of the task time discussing how they reason about the questions.

Give students quiet think time to complete the activity and then time to share their explanations with a partner.

Students may think 9 is 100% and try to find the percentage of the number given. You can use the diagram in the task statement to help them make sense of the first question and then encourage them to use mental math, or draw additional diagrams, to solve the other four questions.

Invite students to share different strategies for answering the questions, with or without using fractions. For example, some may multiply the given 9 by 2, 4, and 10, respectively, to find values of 100% given the values of 50%, 25%, and 10% in the first three problems. Highlight any fractions students used to make sense of and solve the problems.

In this activity, students calculate benchmark percentages. Students are likely to calculate the answers quickly They are to spend the majority of the task time discussing how they reason about the questions.

Give students quiet think time to complete the activity and then time to share their explanation with a partner.

Because 5 goes into 20 four times, students may answer that 5 is 4% of 20. If this happens, explain that 5 is really \(\frac14\) of 20 and ask them what percentage represents one quarter.

Invite students to share their mental math strategies. If needed, consider illustrating the relationship between of \(B\) and \(C\) (where \(A\%\) of \(B\) is \(C\)) using tape diagrams or double number line diagrams to help students visualize the association between the benchmark percentages and fractions.