Lesson 13

The purpose of this warm-up is for students to reason about numeric expressions using what they know about operations with negative and positive numbers without actually computing anything.

Explain to students that we sometimes leave out the small dot that tells us that two numbers are being multiplied. So when a number, or a number and a variable, are next to each other without a symbol between them, that means we are finding their product. For example, \((\text-5)\boldcdot (\text-10)\) can be written \((\text-5) (\text-10)\) and \(\text-5 \boldcdot x\) can be written \(\text-5x\).

Display one problem at a time. Give students 1 minute of quiet think time per problem and ask them to give a signal when they have an answer. Follow with a whole-class discussion.

Ask students to share their reasoning.

In this activity students continue to build fluency operating with signed numbers as they match different expressions that have the same value. Students look for and use the relationship between inverse operations (MP7).

As students work, identify groups that make connections between the operations, for example they notice that subtracting a number is the same as adding its opposite, or that dividing is the same as multiplying by the multiplicative inverse.

Arrange students in groups of 2. Distribute pre-cut slips from the blackline master.

If students struggle to find matches, encourage them to think of the operations in different ways. You might ask:

• How else can you think of ____ (subtraction, addition, multiplication, division)?

Select students to share their strategies. Highlight stategies that compared the structure of the expressions instead of just the final answers.

Discuss:

• Subtracting a number is equivalent to adding the additive inverse.
• A number and its additive inverse have oppostie signs (but the same magnitude).
• Dividing by a number is equivalent to multiplying by the multiplicative inverse.
• A number and its multiplicative inverse have the same sign (but different magnitudes).

If desired, have students order their pairs of equivalent expressions from least to greatest to review ordering rational numbers.

In the previous activity, students interpreted the meaning of -\(x\) when \(x\) represented a positive value and when \(x\) represented a negative value. The purpose of this activity is to understand that variables can have negative values, but if we compare two expressions containing the same variable, it is not possible to know which expression is larger or smaller (without knowing the values of the variables). For example, if we know that \(a\) is positive, then we know that \(5a\) is greater than \(4a\); however, if \(a\) can be any rational number, then it is possible for \(4a\) to be greater than \(5a\), or equal to \(5a\).

\(\displaystyle a, b, \text-a, \text-4b, \text-a+b, a\div \text-b, a^2, b^3\)

Display the list of expressions for all to see. Ask students: Which expression do you think has the largest value? Which has the smallest? Which is closest to zero? It is reasonable to guess that \(\text-4b\) is the smallest and \(b^3\) is the largest. (This guess depends on supposing that \(b\) is positive. But this activity is designed to elicit the understanding that it is necessary to know the value of variables to be able to compare expressions. For the launch, you are just looking for students to register their initial suspicions.)

If you know from students’ previous work that they struggle with basic operations on rational numbers, it might be helpful to demonstrate a few of the computations that will come up as they work on this activity. Tell students, “Say that \(a\) represents 10, and \(b\) represents -2. What would be the value of . . .

• \(\text-b\)
• \(b^3\)
• \(a \boldcdot \frac{1}{b}\)
• \(\frac{a}{b} \div a\)
• \(a+\frac{1}{b}\)
• \(\left(\frac{1}{b}\right)^2\)

Arrange students in groups of 2. Encourage them to check in with their partner as they evaluate each expression, and work together to resolve any discrepancies.

If students struggle to find the largest value, smallest value, or value closest to zero in the set, encourage them to create a number line to help them reason about the positions of different candidates.

Display the completed table for all to see. Ask selected students to share their values that are largest, smallest, and closest to zero from each set and explain their reasoning.

Ask students if any of these results were surprising? What caused the surprising result? Some possible observations are:

• \(b^3\) was both the largest and smallest value at different times. 6 and -6 are both relatively far from 0, so \(b^3\) is a large number when \(b\) is positive and a small number when \(b\) is negative.
• \(a \div \text-b\) was often the closest to zero, because it had the same absolute value no matter the sign of \(a\) and \(b\).

The purpose of this activity is for students to interpret an expression in terms of the position it represents on a number line and to interpret the meaning of an associated equation. Expressions are equal when they represent the same position on a number line. This activity uses a familiar context that students have encountered before, so that they can more quickly engage with the meaning of the expressions. In this activity, students use the structure of the number line to reason about the relative values of expressions (MP7).

In grade 6 and earlier in this unit, students have seen fractions with a negative sign in front of the entire fraction. This activity is the first time students see a fraction with a negative sign in the numerator, in the equation \(c = \frac{\text-a}{2}\). Students can apply what they know about dividing signed numbers to make sense of expressions like this.

For students who are struggling to measure out a length of \(a\) or \(b\) or a sum, difference, or multiple of them, suggest that they measure and cut strips of paper for the lengths of \(a\) and \(b\) to help guide them. Ask how they could use the strips to find other distances such as \(a-b\) and \(\frac{a}{2}\).

Display the diagram from the task statement. Ask selected students to share their responses and the reasoning behind them, and record them on the diagram. As the discussion proceeds, illustrate the meaning of the equal sign by saying, for example, “We could either label this point with \(v\), or we could label it with \(a+b\), since we know \(v=a+b\). Since these expressions are equal, they represent the same position on the number line.”

If not brought up by students, consider drawing attention to the equation \(c = \frac{\text- a}{2}\) and discuss how this relates to what they have learned about dividing signed numbers. It is important for students to understand that \(\frac{\text- a}{2} = \text- \frac{a}{2}\) as well as \(\frac{a}{\text-2}\), but these are not equal to \(\frac{\text- a}{\text -2}\).