Lesson 13

Expressions with Rational Numbers

13.1: True or False: Rational Numbers (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

  1. \((\text-38.76)(\text-15.6)\) is negative
  2. \(10,000 - 99,999 < 0\)
  3. \(\left( \frac34 \right)\left( \text- \frac43 \right) = 0\)
  4. \((30)(\text- 80) - 50 = 50 - (30)(\text- 80)\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their reasoning.

13.2: Card Sort: The Same But Different (10 minutes)

Activity

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.

Launch

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

Student Facing

Your teacher will give you a set of cards. Group them into pairs of expressions that have the same value.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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)?

Activity Synthesis

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.

Engagement: Develop Effort and Persistence. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Attention; Social-emotional skills
Speaking, Representing: MLR7 Compare and Connect. Use this routine to support whole-class discussion. Ask students to consider what is the same and what is different about the structure of each expression. Draw students’ attention to the association between quantities (e.g., adding the additive inverse is subtracting; multiplying by the multiplicative inverse is dividing). These exchanges strengthen students’ mathematical language use and reasoning to describe when comparing inverse-operation relationships involving rational numbers.
Design Principle(s): Support sense-making; Maximize meta-awareness

13.3: Near and Far From Zero (15 minutes)

Activity

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 \(5a\).

Launch

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

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.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention

Student Facing

\(a\) \(b\)     \(\text-a\)         \(\text-4b\)       \(\text-a+b\)     \(a\div \text-b\)       \(a^2\)         \(b^3\)    
\(\text-\frac12\) 6
\(\frac12\) -6
-6 \(\text-\frac12\)
  1. For each set of values for \(a\) and \(b\), evaluate the given expressions and record your answers in the table.

  2. When \(a= \text-\frac12\) and \(b= 6\), which expression:

    has the largest value?

    has the smallest value?

    is the closest to zero?

  3. When \(a= \frac12\) and \(b= \text-6\), which expression:

    has the largest value?

    has the smallest value?

    is the closest to zero?

  4. When \(a= \text-6\) and \(b= \text-\frac12\), which expression:

    has the largest value?

    has the smallest value?

    is the closest to zero?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Are there any values could you use for \(a\) and \(b\) that would make all of these expressions have the same value? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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 \(b\).
Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: "I agree because ….” or "I disagree because ….” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.
Design Principle(s): Support sense-making

13.4: Seagulls and Sharks Again (10 minutes)

Optional activity

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.

Launch

Display the diagram for all to see. Ask students to share anything they notice and wonder about the diagram. Some things to notice are that

  • The seagull is above the water and the shark is below the water.
  • \(a\) and \(b\) represent the vertical position of the seagull and shark, respectively.
  • \(a\) represents a positive value and \(b\) represents a negative value.
  • \(a\) is farther from 0 than \(b\), so \(|a|>|b|\).

To facilitate engagement with the task, consider demonstrating the placement of one animal before students start working. For example, ask students, “If there were a minnow with vertical position \(m\), and \(m=\frac13 b\), where is the minnow?” Help students interpret the equation. Explain that the value of \(m\) only gives the vertical position. The horizontal position of the minnow is unknown. Encourage students to plot all of the new points on the vertical axis (as specified in the task statement).

Keep students in the same groups as the previous activity. Encourage students to check in with their partner periodically, and work together to resolve any discrepancies.

Classes using the digital version have an applet to use. Students can choose their own vertical positions for \(a\) and \(b\). Teachers may allow students to turn on a grid to facilitate the placement of the other animals.

Conversing, Representing, Writing: MLR2 Collect and Display. As students work, circulate and listen to students' conversations as they decide where on the vertical axis to show the position of each new animal. Write down common or important phrases you hear students say as they reason about the relative value of each expression (e.g., twice as far, opposite direction, half the distance, etc.). Write the students’ words and expressions on a whole-class display of the task’s diagram. This will help students interpret and use mathematical language during their paired and whole-group discussions.

Design Principle(s): Support sense-making

Student Facing

A seagull has a vertical position \(a\), and a shark has a vertical position \(b\).

A vertical number line. Vertical position, meters. 

In the applet, you may choose to start by clicking on the open circles on the seagull and shark to drag them to new vertical positions. Once you have them in place, drag each of the other animals to the vertical axis to show its position, determined by the expression next to it.

 
  1. A dragonfly at \(d\), where \(d=\text-b\)

  2. A jellyfish at \(j\), where \(j=2b\)

  3. An eagle at \(e\), where \(4e=a\).

  4. A clownfish at \(c\), where \(c=\frac{\text-a}{2}\)

  5. A vulture at \(v\), where \(v=a+b\)

  6. A goose at \(g\), where \(g=a-b\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Display the diagram for all to see. Ask students to share anything they notice and wonder about the diagram. Some things to notice are that

  • The seagull is above the water and the shark is below the water.
  • \(a\) and \(b\) represent the vertical position of the seagull and shark, respectively.
  • \(a\) represents a positive value and \(b\) represents a negative value.
  • \(a\) is farther from 0 than \(b\), so \(|a|>|b|\).

To facilitate engagement with the task, consider demonstrating the placement of one animal before students start working. For example, ask students, “If there were a minnow with vertical position \(m\), and \(m=\frac13 b\), where is the minnow?” Help students interpret the equation. Explain that the value of \(m\) only gives the vertical position. The horizontal position of the minnow is unknown. Encourage students to plot all of the new points on the vertical axis (as specified in the task statement).

Keep students in the same groups as the previous activity. Encourage students to check in with their partner periodically, and work together to resolve any discrepancies.

Classes using the digital version have an applet to use. Students can choose their own vertical positions for \(a\) and \(b\). Teachers may allow students to turn on a grid to facilitate the placement of the other animals.

Conversing, Representing, Writing: MLR2 Collect and Display. As students work, circulate and listen to students' conversations as they decide where on the vertical axis to show the position of each new animal. Write down common or important phrases you hear students say as they reason about the relative value of each expression (e.g., twice as far, opposite direction, half the distance, etc.). Write the students’ words and expressions on a whole-class display of the task’s diagram. This will help students interpret and use mathematical language during their paired and whole-group discussions.

Design Principle(s): Support sense-making

Student Facing

A vertical number line. Vertical position, meters. 

A seagull has a vertical position \(a\), and a shark has a vertical position \(b\). Draw and label a point on the vertical axis to show the vertical position of each new animal.

  1. A dragonfly at \(d\), where \(d=\text-b\)

  2. A jellyfish at \(j\), where \(j=2b\)

  3. An eagle at \(e\), where \(e=\frac14a\).

  4. A clownfish at \(c\), where \(c=\frac{\text-a}{2}\)

  5. A vulture at \(v\), where \(v=a+b\)

  6. A goose at \(g\), where \(g=a-b\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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 \(\frac{a}{2}\).

Activity Synthesis

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 \(\frac{\text- a}{\text -2}\).

Lesson Synthesis

Lesson Synthesis

Key takeaways:

  • Develop flexible thinking with all four operations across the rational numbers.
  • Recognize multiplicative and additive inverses and the difference between them.

Discussion questions:

  • Explain the rules for arithmetic with negative numbers.
  • Can you give an example of a number whose additive inverse is the same as its multiplicative inverse? Why not?

13.5: Cool-down - Make Them True (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We can represent sums, differences, products, and quotients of rational numbers, and combinations of these, with numerical and algebraic expressions. 

Sums:

\(\frac12 + \text-9\)

\(\text-8.5 + x\)

Differences:

\(\frac12 - \text-9\)

\(\text-8.5 - x\)

Products:

\((\frac12)(\text-9)\)

\(\text-8.5x\)

Quotients:

\(\frac12\div\text-9\)

\(\frac{\text-8.5}{x}\)

We can write the product of two numbers in different ways.

  • By putting a little dot between the factors, like this: \(\text-8.5\boldcdot x\).
  • By putting the factors next to each other without any symbol between them at all, like this: \(\text-8.5x\).

We can write the quotient of two numbers in different ways as well.

  • By writing the division symbol between the numbers, like this: \({\text-8.5}\div{x}\).
  • By writing a fraction bar between the numbers like this: \(\frac{\text-8.5}{x}\).

When we have an algebraic expression like \(\frac{\text-8.5}{x}\) and are given a value for the variable, we can find the value of the expression. For example, if \(x\) is 2, then the value of the expression is -4.25, because \(\text-8.5 \div 2 = \text-4.25\).