Lesson 2

The purpose of this Math Talk is to elicit strategies and understandings students have for calculating percentages in their head. These understandings help students develop fluency and will be helpful later when students will need to be able to reason about sales tax and write expressions for situations involving percent calculations in an associated Algebra 1 lesson. Math Talks build fluency by encouraging students to think about expressions or diagrams and rely on what they know about properties of operations to mentally solve a problem. In this activity, students have an opportunity to notice and make use of structure (MP7) as well as repeated reasoning (MP8), because they can exploit the proportional structure of percents to use previous answers to help them answer more difficult questions—e.g., using the value of 75% of a number to find the value of 7.5% of the same number.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with 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:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

Monitor for students who lack conceptual understanding of percent as a rate per 100, or students who lack strategies for evaluating percentages mentally. Check in with these students individually to see if they can use a calculator to evaluate percentages. If many students are struggling to understand percent as a rate per 100, (and therefore why 50% is half of a number, for example), consider illustrating each of these examples using a double number line or tape diagram.

The goal of this activity is to help students practice identifying and representing pertinent information in a word problem. In the associated Algebra 1 lesson, students create equations that represent relationships between two or more variables in contexts. This activity prepares students to be successful in the Algebra 1 lesson by helping them interpret word problems before they write equations to represent the relationships.

Demonstrate how to identify important values and variables using the example: At a department store, t-shirts are on sale for $5 each and shorts are on sale for $8 each. A customer plans to purchase several t-shirts and shorts.

Ask students questions about the example that will guide their thinking about what is important in the problem and how to represent the important quantities. Here are sample questions to help guide students' thinking:

• "What information would be important for customers to know?" (The cost of each item and how many of each item they can get.)

• "What values might be important for purchasing purposes?" ($5 and $8)

• "If you were the customer, what things would you make sure to keep in mind as you shop?" (The cost of each item, the total of everything I want to purchase, and the amount of money I have to spend on the items.) 

Tell students that they are not solving a problem here, but identifying pertinent information and describing what each thing represents. Through a class discussion, students should identify some information from the word problem:

• $5 is the cost of a t-shirt.
• $8 is the cost of a pair of shorts.
• \(t\) represents the number of t-shirts purchased.
• \(s\) represents the number of shorts purchased.
• \(C\) represents total cost of the purchase.

Remind students that the letters they choose are not as important as knowing what quantities the variables represent. As demonstrated in the solution to the example, it is common, but not required, to use the initial letter of the thing they wish to represent.

The goal of this discussion is for students to understand how to represent different quantities from a situation. Here are sample questions to promote class discussion:

• "How do you decide when to use a letter to represent a quantity?" (Letters are used to represent unknown quantities. Known quantities are given specific values that may not change in the situation.)

• "How do we represent unknown quantities?" (We use variables to represent them.)

• "How did you determine which quantities are important for each situation?" (I thought about what quantities would be useful in solving a problem. For example, if we know the price per shirt, then we would also like to know the number of shirts and the total price.)

In this activity, students take turns with a partner, matching expressions to situations. Students trade roles, explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).

Monitor for students who may match the two blank cards, which will lead to the remaining two unmatched cards not going together. Ask students, “Is it possible to match all of them by using the blanks differently?”

Monitor for students drawing pictures or other representations to help them figure out the relationships.

Arrange students in groups of 2. Demonstrate how to set up and find matches. Choose a student to be your partner. Mix up the cards and place them face-up. Point out that the cards contain either an expression or a situation. Select one of each style of card and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree, for example, by explaining your mathematical thinking or asking clarifying questions. Give each group a set of cut-up cards for matching. After groups have matched the cards, tell them to sort the pairs into groups based on mathematical operation used: addition, subtraction, multiplication, and division. 

Tell students to work with their partner to sort the cards based on the mathematical operation used: addition, subtraction, multiplication, or division. The goal is for students to recognize patterns in choosing specific operations to represent relationships between quantities. Here is a sample question to promote class discussion:

• "What pictures could you draw for the situations that would help connect the situation to the expression?" (A teapot with 2 tea bags in it and stick people to the side with a tea bag under each one.)

If any students created pictures or other representations to help them figure out the relationship, ask them to share with the class and invite them to explain how the picture helped them figure out the relationship.