Lesson 16

The purpose of this warm-up is to encourage students to reason about properties of operations in equivalent expressions. While students may evaluate each expression to determine if the statement is true or false, encourage students to think about the properties of arithmetic operations in their reasoning.

Display one problem at a time. Tell students to give a signal when they have decided if the equation is true or false. Give students 1 minute of quiet think time followed by a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• Do you agree or disagree? Why?
• Who can restate ___’s reasoning in a different way?
• Does anyone want to add on to _____’s reasoning?

After each true equation, ask students if they could rely on the same reasoning to determine if other similar problems are equivalent. After each false equation, ask students how the problem could be changed to make the equation true.

The purpose of this activity is for students to see that to find what percentage one number is of another, divide them and then multiply by 100. First they find what percentage of 20 various numbers are, then they organize everything into a table. Using the table, students then describe the relationship they see between dividing the numbers by 20 and finding the percentage (MP8).

Depending on students' strengths in grade 5 work, it may be beneficial to offer some strategies for writing a fraction in an equivalent decimal form before students start working. Display the fraction \(\frac{15}{25}\) and ask students how they might think about writing it in decimal form. Possible strategies are writing the equivalent fraction \(\frac{60}{100}\) and knowing that "60 hundredths" is written 0.60 or 0.6. Another possibility is to write the equivalent fraction \(\frac35\), which is equal to \(\frac{6}{10}\) and knowing that "6 tenths" can be written as 0.6.

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

Have students share their strategies for the first three problems. Then ask what they noticed about the last two columns. There are many ways to formulate the relationship. Make sure everyone sees that to find what percentage a number is of 20, divide it by 20 and then multiply by 100. A table showing the percentage for 1 minute might also help:

Note that to find the percentage for 1 minute, we divide 100 by 20, and to find the percentage for any number of minutes, we multiply the result by that number of minutes. That is the same as dividing the number of minutes by 20 and multiplying by 100.

This activity gives students a chance to practice their new-found insight about how to find percentages. It also gives them an opportunity to practice dividing whole numbers in preparation for the next unit on base-ten numbers. 

Note that it is expected and perfectly okay for students to revert to less-efficient methods that they trust. Monitor for students with more- and less-efficient methods. A less-efficient representation can be used to make sense of a more-efficient method, and these connections can be made in the discussion that follows the task.

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

Invite selected students to share various approaches to the three problems. Sequence less-efficient methods before more-efficient ones. Make sure all students have a chance to see the approach that is the focus of this lesson. For example, to find "9 is what percent of 75?" we can divide 9 by 75 and then multiply by 100.