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Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Keep all previous problems and work displayed throughout the talk.
Find the value of each expression mentally.
To involve more students in the conversation, consider asking:
Highlight two points: that dividing a number by 100 gives the same result as multiplying the number by , and that a fraction can be interpreted as division ( can be understood as ).
In this activity, students encounter percentage problems that are inconvenient to solve by drawing double number line diagrams or by using benchmark percentages, encouraging them to reason differently and look for a more practical approach.
To find 30% of a value in the first question, students can use a familiar percentage, 10%, as a stepping stone. For instance, they may find 10% or of the value and then multiply the result by 3. The next two questions ask students to find 44% and 102% of some values, respectively, prompting students to find different intermediate steps or use alternative strategies.
Some students may opt to use 4% and 2% as stepping stones to answer the two questions. Others may choose to find the value for 1% in each situation and multiply the result by the targeted percentage. Monitor for students who take this path and select them to share later.
Ask students if they have attended evening activities at their school or at another school. Invite them to share some evening activities they know about or have attended.
Display the first paragraph of the task statement for all to see, and read it aloud. Tell students that the activity has to do with the number of people attending different evening events. Invite students to ask clarifying questions about any unfamiliar event.
Arrange students in groups of 2. Give students 5 minutes of quiet work time and then time to share their explanation with a partner.
A school held several evening activities last month—a music concert, a basketball game, literacy night, and a drama play. The music concert was attended by 250 people.
How many people came to each of these activities? Show your reasoning.
The aim of the discussion is to draw attention to the benefits of using 1% as an intermediate step for finding percentages of a number.
Invite a couple of students to share their work on the first question. Then focus the discussion on the last two questions. Select several students who effectively found 44% and 102% of 250 to share their strategies, saving the strategy involving 1% for last. If no one used this method, ask whether finding 1% of 250 would be a helpful stepping stone for finding each of the three percentages.
Highlight that this method of first finding 1% of a quantity can be applied regardless of the percentage involved, much like finding a unit rate is an effective way to solve any ratio problem.
Consider using a table to illustrate this point. Start with the first two rows showing 100% and 1% of 250. Add a new row to show how each percentage can be found using the value for 1%, ending with a table such as shown:
Students may note that it is not possible to have 2.5 people or a non-whole number of people at an event, which is the case here if the percentage is an odd number. Acknowledge that this is true. Clarify that finding 1% of a quantity is still a valid intermediate step even in situations where only even-numbered percentages make sense.
Suppose a business donates 1% of its profits to charity each year. How much would it donate if it made \$7,500 in profits?
To find 1% of 7,500, we can multiply 7,500 by or 0.01.
, so the business would donate \$75.
What if the business donates 6% of its profits to charity? Because 6% of 7,500 is 6 times as much as 1% of 7,500, we can calculate or .
, so the business would donate \$450.
The same reasoning can help us find 1%, 6%, and other percentages of another number:
In general, to find of any number, , we can calculate: .
Three music artists plan to donate a percentage of the money they make from selling merchandise.
Complete the tables to show how much each artist would donate for different amounts of merchandise sales.
Artist A plans to donate 1% of the amount of merchandise sales.
| sales (\$) | 1 | 40 | 100 | 3,200 | |
|---|---|---|---|---|---|
| donation (\$) |
Artist B plans to donate 15% of the amount of merchandise sales.
| sales (\$) | 1 | 40 | 100 | 3,200 | |
|---|---|---|---|---|---|
| donation (\$) |
Artist C plans to donate 67% of the amount of merchandise sales.
| sales (\$) | 1 | 40 | 100 | 3,200 | |
|---|---|---|---|---|---|
| donation (\$) |
Another artist plans to donate of the money made from selling merchandise. In the table, write an expression for the amount of donation for each sales amount.
| sales (\$) | 1 | 40 | 100 | 3,200 | |
|---|---|---|---|---|---|
| donation (\$) |
Students who use 0.01, 0.15, and 0.67 to describe how to find 1%, 15%, and 67% of a number, respectively, may be unsure how to express of a number in decimal form. Encourage students to look at the values in the first column of each table. Consider asking:
Then ask students how that expression could help them write an expression for finding of any amount, .