# Lesson 15

Multiplying Rational Numbers

## 15.1: Which One Doesn’t Belong: Expressions (5 minutes)

### Warm-up

This warm-up prompts students to compare four expressions. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about the expressions in comparison to one another. To allow all students to access the activity, each expression has one obvious reason it does not belong.

### Launch

Arrange students in groups of 2–4. Display the expressions for all to see. Ask students to indicate when they have noticed one expression that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their reasoning why a particular expression does not belong and together find at least one reason each expression doesn’t belong.

### Student Facing

Which expression doesn’t belong?

$$7.9x$$

$$7.9\boldcdot (\text- 10)$$

$$7.9 + x$$

$$\text-79$$

### Activity Synthesis

Ask each group to share one reason why a particular expression does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given make sense.

## 15.2: Rational Numbers Multiplication Grid (10 minutes)

### Optional activity

In this optional activity, students revisit the representation of a multiplication chart, which may be familiar from previous grades; however, in this activity, the multiplication chart is extended to include negative numbers. Students identify and continue patterns (MP8) to complete the chart and see that it fits the patterns in the chart for the product of two negative numbers to be a positive number.

The blackline master has a multiplication chart that also includes the factors 1.5, -1.5, 3.2, and -3.2, so that students can see how the patterns extend to rational numbers that are not integers. Encourage students to complete the rows and columns for the integers first and then come back to 1.5, -1.5, 3.2, and -3.2 later. Directions are included on the blackline master for a way that students can fold their papers to hide the non-integers while they fill in the integers. If you want students to do this, it would be good to demonstrate and walk them through the process of folding their paper.

### Launch

Arrange students in groups of 3. If desired, distribute 1 copy of the blackline master to every student and instruct students to ignore the chart printed in their books or devices. (Also if desired, instruct students to fold their papers according to the directions on the top and right sides of the chart, so that the decimal rows and columns are temporarily hidden.) Give students 30 seconds of quiet think time. Have them share what patterns they notice about the numbers that are already filled in. Give the groups 5 minutes of work time followed by whole-class discussion.

If students have access to the digital materials, students can use the applet to complete the chart. The applet helps students focus on fewer of the numbers and patterns at a time, similar to the purpose of folding the blackline master. Also, the applet gives students immediate feedback on whether their answers are correct which helps them test their theories about ramifications of multiplying by a negative number.

Representation: Internalize Comprehension. Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

Look at the patterns along the rows and columns and continue those patterns to complete the table. When you have filled in all the boxes you can see, click on the "More Boxes" button.

What does this tell you about multiplication by a negative?

### Launch

Arrange students in groups of 3. If desired, distribute 1 copy of the blackline master to every student and instruct students to ignore the chart printed in their books or devices. (Also if desired, instruct students to fold their papers according to the directions on the top and right sides of the chart, so that the decimal rows and columns are temporarily hidden.) Give students 30 seconds of quiet think time. Have them share what patterns they notice about the numbers that are already filled in. Give the groups 5 minutes of work time followed by whole-class discussion.

If students have access to the digital materials, students can use the applet to complete the chart. The applet helps students focus on fewer of the numbers and patterns at a time, similar to the purpose of folding the blackline master. Also, the applet gives students immediate feedback on whether their answers are correct which helps them test their theories about ramifications of multiplying by a negative number.

Representation: Internalize Comprehension. Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

1. Complete the shaded boxes in the multiplication square.
2. Look at the patterns along the rows and columns. Continue those patterns into the unshaded boxes.
3. Complete the whole table.
4. What does this tell you about multiplication with negative numbers?

### Anticipated Misconceptions

Some students may need a reminder of how a mutliplication chart works: the factors are listed at the end of the rows and columns, and their products go in the boxes.

### Activity Synthesis

The most important takeaway is that it makes sense for the product of two negative numbers to be a positive number, whether or not the numbers are integers. This fits in with the patterns in the extended multiplication chart. Those patterns depend on the distributive property. For example, the reason the numbers in the top row go up by 5s is that $$5(n+1) = 5n + 5$$. So when students extend the pattern to negative numbers, they are extending the distributive property.

Display a complete chart for all to see, and ask students to explain the ways in which the chart shows that the product of a negative and a negative is a positive. The general argument involves assuming that a pattern observed in a row or column will continue on the other side of 0.

## 15.3: Card Sort: Matching Expressions (10 minutes)

### Optional activity

This activity reminds students of the links between positive fractions and multiplication and prepares them to think about division as multiplication by the reciprocal; this will be important for dividing negative numbers. Students will use earlier work from grade 6 and their work in previous lessons in this unit to extend what they know about division of positive rationals to all rational numbers (MP7).

Ask students as they are working if there is an easy way to tell if two expressions are not equivalent, making note of students who reason about how many negative numbers are multiplied, and what the outcome will be. For example, they may have first gone through and marked whether each product would be positive or negative before doing any arithmetic.

Teacher Notes for IM 6–8 Accelerated
This activity is not optional for this course. It is an important opportunity for students to practice multiplying rational numbers, including unit fractions. This helps prepare students for dividing rational numbers in the next lesson.

### Launch

Ask students to recall the rules they have previously used about multiplication of signed numbers.

This is the first encounter with an expression where three integers are multiplied, so students might need to see an example of evaluating an expression like this one step at a time. Display an expression like this for all to see, and ask students how they might go about evaluating it: $$\displaystyle (\text-2)\boldcdot (\text-3)\boldcdot (\text-4)$$ The key insight is that you can consider only one product, and replace a pair of numbers with the product. In this example, you can replace $$(\text-2)\boldcdot (\text-3)$$ with 6. Then, you are just looking at $$(6)\boldcdot (\text-4)$$, which we already know how to evaluate.

Arrange students in groups of 2. Distribute sets of cards.

Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Conceptual processing; Organization
Conversing: MLR8 Discussion Supports. Arrange students in groups of 2. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “ ____ and ____ are equal because . . .”, and “I noticed ___ , so I matched . . .” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning about multiplication of signed numbers.
Design Principle(s): Support sense-making; Maximize meta-awareness

### Student Facing

Your teacher will give you cards with multiplication expressions on them. Match the expressions that are equal to each other. There will be 3 cards in each group.

### Activity Synthesis

Ask the previously identified students to share their rationale for identifying those that do not match.

Consider highlighting the link between multiplying by a fraction and dividing by a whole number. If desired, ask students to predict the values of some division expressions with signed numbers. For example, students could use the expression $$\text-64 \boldcdot \frac{1}{8}$$ to predict the value of $$\text-64 \div 8$$. However, it is not necessary for students to learn rules for dividing signed numbers at this point. That will be the focus of future lessons.

## 15.4: Row Game: Multiplying Rational Numbers (10 minutes)

### Optional activity

This optional activity gives students an opportunity to practice multiplying signed numbers. The solutions to the problems in each row are the same, so students can check their work with a partner.

### Launch

Arrange students in groups of 2. Make sure students know how to play a row game. Give students 5–6 minutes of partner work time followed by whole-class discussion.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts. For example, after students have completed the first 2-3 rows of the table, check-in with either select groups of students or the whole class. Invite students to share how they have applied generalizations about multiplying signed numbers from the previous activity so far.
Supports accessibility for: Conceptual processing; Organization; Memory

### Student Facing

Evaluate the expressions in one of the columns. Your partner will work on the other column. Check in with your partner after you finish each row. Your answers in each row should be the same. If your answers aren’t the same, work together to find the error.

column A column B
$$790\div 10$$  $$(7.9)\boldcdot 10$$
$$\text- \frac67 \boldcdot 7$$ $$(0.1) \boldcdot \text- 60$$
$$(2.1) \boldcdot \text- 2$$ $$(\text-8.4) \boldcdot\frac12$$
$$(2.5) \boldcdot (\text-3.25)$$ $$\text{-} \frac52 \boldcdot \frac{13}{4}$$
$$\text-10 \boldcdot (3.2) \boldcdot (\text-7.3)$$ $$5\boldcdot (\text-1.6) \boldcdot (\text-29.2)$$

### Student Facing

#### Are you ready for more?

A sequence of rational numbers is made by starting with 1, and from then on, each term is one more than the reciprocal of the previous term. Evaluate the first few expressions in the sequence. Can you find any patterns? Find the 10th term in this sequence.

$$\displaystyle 1\qquad\quad 1+\frac{1}{1}\qquad\quad 1+\frac{1}{1+1}\qquad\quad 1+\frac{1}{1+\frac{1}{1+1}} \qquad \quad 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+1}}}\qquad\quad \dots$$

### Activity Synthesis

Ask students, "Were there any rows that you and your partner did not get the same answer?" Invite students to share how they came to an agreement on the final answer for the problems in those rows.

Consider asking some of the following questions:

• "Did you and your partner use the same strategy for each row?"
• "What was the same and different about both of your strategies?"
• "Did you learn a new strategy from your partner?"
• "Did you try a new strategy while working on these questions?"

## Lesson Synthesis

### Lesson Synthesis

Display a number line with the numbers -1, 0, and 1 labeled. Ask students to give examples of multiplications problems with a product that is:

• greater than 1 (Sample responses: $$5 \boldcdot 3$$ or $$\text-5 \boldcdot \text-3$$)
• less than -1 (Sample responses: $$5 \boldcdot \text-3$$ or $$\text-5 \boldcdot \text-3 \boldcdot \text-1$$)
• between 0 and 1 (Sample responses: $$\frac15 \boldcdot \frac13$$ or $$\text-\frac15 \boldcdot \text-\frac13$$)
• between -1 and 0 (Sample responses: $$\frac15 \boldcdot \text-\frac13$$ or $$\text-\frac15 \boldcdot \text-\frac13 \boldcdot \text-1$$)

## Student Lesson Summary

### Student Facing

We can think of $$3\boldcdot 5$$ as $$5 + 5 + 5$$, which has a value of 15.

We can think of $$3\boldcdot (\text-5)$$ as $$\text-5 + \text-5 + \text-5$$, which has a value of -15.

We know we can multiply positive numbers in any order: $$3\boldcdot 5=5\boldcdot 3$$

If we can multiply signed numbers in any order, then $$(\text-5)\boldcdot 3$$ would also equal -15.

Now let’s think about multiplying two negatives.

We can find $$\text-5\boldcdot (3+\text-3)$$ in two ways:

• Applying the distributive property:

$$\text-5\boldcdot 3 + \text-5\boldcdot (\text-3)$$

• Adding the numbers in parentheses:

$$\text-5\boldcdot (0) = 0$$

This means that these expressions must be equal.

$$\text-5\boldcdot 3 + \text-5\boldcdot (\text-3) = 0$$

Multiplying the first two numbers gives

$$\text-15 + \text-5\boldcdot (\text-3) = 0$$

Which means that

$$\text-5\boldcdot (\text-3) = 15$$

There was nothing special about these particular numbers. This always works!

• A positive times a positive is always positive.

For example, $$\frac35 \boldcdot \frac78 = \frac{21}{40}$$.

• A negative times a negative is also positive.

For example, $$\text-\frac35 \boldcdot \text-\frac78 = \frac{21}{40}$$.

• A negative times a positive or a positive times a negative is always negative.

For example, $$\frac35 \boldcdot \text-\frac78 = \text-\frac35 \boldcdot \frac78 = \text-\frac{21}{40}$$.

• A negative times a negative times a negative is also negative.

For example, $$\text-3 \boldcdot \text-4 \boldcdot \text-5 = \text-60$$.
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