18.1: Number Talk: Additive Inverses (5 minutes)
The purpose of this Number Talk is to elicit strategies and understandings that students have for adding and subtracting signed numbers. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to rewrite subtraction as adding the opposite. While four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem, saving most of the time for the final question.
Display one problem at a time. Give students 30 seconds of 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.
Supports accessibility for: Memory; Organization
Find each sum or difference mentally.
\(\text-30 + \text-10\)
\(\text- 10 + \text-30\)
\(\text- 30 - 10\)
\(10 - \text- 30\)
When it comes up, emphasize that “subtract 10” can be rewritten “add negative 10.” Also that addition is commutative but subtraction is not. Mention these points even if students do not bring them up.
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 ___’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 _____’s strategy?”
- “Do you agree or disagree? Why?”
Design Principle(s): Optimize output (for explanation)
18.2: A Helpful Observation (10 minutes)
Students recall that subtracting a number (or expression) is the same as adding its additive inverse. This concept is applied to get students used to the idea that the subtraction sign has to stay with the term it is in front of. Making this concept explicit through a numeric example will help students see its usefulness and help them avoid common errors in working with expressions that involve subtraction.
Display the expression \(7 \frac34 + 3 \frac56 - 1 \frac34\) and ask students to evaluate. After they have had a chance to think about the expression, read through the task statement together before setting students to work.
Supports accessibility for: Language
Design Principle(s): Optimize output (for justification)
Lin and Kiran are trying to calculate \(7 \frac34 + 3 \frac56 - 1 \frac34\). Here is their conversation:
Lin: “I plan to first add \(7\frac34\) and \(3\frac56\), so I will have to start by finding equivalent fractions with a common denominator.”
Kiran: “It would be a lot easier if we could start by working with the \(1 \frac34\) and \(7 \frac34\). Can we rewrite it like \(7 \frac34 + 1 \frac34 - 3 \frac56\)?”
Lin: “You can’t switch the order of numbers in a subtraction problem like you can with addition; \(2-3\) is not equal to \(3-2\).”
Kiran: “That’s true, but do you remember what we learned about rewriting subtraction expressions using addition? \(2-3\) is equal to \(2+(\text-3)\).”
- Write an expression that is equivalent to \(7 \frac34 + 3 \frac56 - 1 \frac34\) that uses addition instead of subtraction.
- If you wrote the terms of your new expression in a different order, would it still be equivalent? Explain your reasoning.
Ensure everyone agrees that \(7\frac34 +3 \frac56 -1 \frac34\) is equivalent to \(7\frac34 +3 \frac56 + \left(\text- 1 \frac34\right)\) is equivalent to \(7\frac34 + \text- 1 \frac34 +3 \frac56 \). Use the language “commutative property of addition.”
18.3: Organizing Work (15 minutes)
Students learn that we can still organize our work with the distributive property in a familiar way, even with negative numbers where thinking in terms of area breaks down.
Display the image and ask students to write an expression for the area of the big rectangle in at least 3 different ways.
Collect responses. If students simply say “16,” ask them to explain how they calculated 16 and record these processes for all to see. Remind students that thinking about area gives us a way to understand the distributive property. This diagram can be used to show that \(2\boldcdot 5+2\boldcdot 3=2(5+3)\). Be sure that students see you write the partial products in the diagram, and that they see every piece of the associated identity \(2\boldcdot 5+2\boldcdot 3=2(5+3)\).
Tell students that when we are working with negative numbers, thinking about area doesn’t work so well, but the distributive property still holds when there are negative numbers. The expressions involved still have the same structure, and we can still organize our work the same way.
Supports accessibility for: Conceptual processing
Write two expressions for the area of the big rectangle.
Use the distributive property to write an expression that is equivalent to \(\frac12(8y + \text-x + \text-12)\). The boxes can help you organize your work.
Use the distributive property to write an expression that is equivalent to \(\frac12(8y - x - 12)\).
Are you ready for more?
Here is a calendar for April 2017.
Let's choose a date: the 10th. Look at the numbers above, below, and to either side of the 10th: 3, 17, 9, 11.
- Average these four numbers. What do you notice?
- Choose a different date that is in a location where it has a date above, below, and to either side. Average these four numbers. What do you notice?
- Explain why the same thing will happen for any date in a location where it has a date above, below, and to either side.
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Solicit responses to the second question and demonstrate thinking about one product at a time:
Then ask students to share how they approached the last question. Highlight responses where students noticed that \(\frac12(8y - x - 12)\) can be rewritten like \(\frac12(8y + -x + -12)\) (because of what they talked about in the warm-up). So the two questions have the same answer.
Design Principle(s): Maximize meta-awareness
Display two expressions like \(x + 2 - 3x - 10\) and \(x + 3x - 2 - 10.\) Ask students to think about why these expressions are not equivalent and explain to a partner. Two explanations should be highlighted:
- Subtraction isn't commutative. \(2-3x\) and \(3x-2\) are not equivalent; you can't just switch terms around a subtraction sign.
- Since \(-3x\) is the same as \(+ \text-3x\), the negative sign needs to stay with the \(3x\) when terms are rearranged.
Ask students how they could fix the second expression to make it equivalent to the first. Ensure that everyone agrees and understands why \(x + \text-3x + 2 + \text-10\) and \(x - 3x + 2 -10\) are equivalent to the first expression.
18.4: Cool-down - Equivalent to $4-x$ (5 minutes)
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Student Lesson Summary
Working with subtraction and signed numbers can sometimes get tricky. We can apply what we know about the relationship between addition and subtraction—that subtracting a number gives the same result as adding its opposite—to our work with expressions. Then, we can make use of the properties of addition that allow us to add and group in any order. This can make calculations simpler. For example:
\(\displaystyle \frac58 - \frac23 - \frac18\)
\(\displaystyle \frac58 + \text- \frac23 + \text-\frac18\)
\(\displaystyle \frac58 + \text-\frac18 + \text- \frac23 \)
\(\displaystyle \frac48 + \text-\frac23\)
We can also organize the work of multiplying signed numbers in expressions. The product \(\frac32(6y-2x-8)\) can be found by drawing a rectangle with the first factor, \(\frac32\), on one side, and the three terms inside the parentheses on the other side:
Multiply \(\frac32\) by each term across the top and perform the multiplications:
Reassemble the parts to get the expanded version of the original expression: \(\displaystyle \frac32(6y-2x-8)=9y-3x-12\)