Unit 5, Lesson 5
Representing Subtraction
Grade 7
5.2
Activity
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Here is an unfinished number line diagram that represents a sum of 8.
A number line, 21 evenly spaced tick marks. Tick marks begin at negative 10 and count up by 1 to the final tick mark at 10. Above the number line is an arrow that begins at 0 and stops at 3. There is a dot on the tick mark labeled 8.- How long should the other arrow be?
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For an equation that goes with this diagram:
- Mai writes .
- Tyler writes .
Do you agree with either of them?
- What is the unknown number? How do you know?
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Here are two more unfinished diagrams that represent sums.
Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1's. An arrow, pointing to the left, goes from 0 to negative 3.
Number line. 21 evenly spaced tick marks. Scale negative 10 to 10, by 1's. There is a point at negative 8. An arrow, pointing to the right, goes from 0 to 3.
For each diagram:
- What equation would Mai write if she used the same reasoning as before?
- What equation would Tyler write if he used the same reasoning as before?
- How long should the other arrow be?
- What number would complete each equation? Be prepared to explain your reasoning.
- Draw a number line diagram for What is the unknown number? How do you know?
5.3
Activity
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Match each diagram to one of these expressions:
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Which expressions in the first question have the same value? What do you notice?
Pause here so your teacher can review your work.
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Which expression has the same value as ?
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Which expression has the same value as ?
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Which expression has the same value as ?
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Choose one of the three previous problems. Draw and label diagrams to show that the two expressions have the same value.
Student Lesson Summary
We can use the relationship between addition and subtraction to reason about subtracting signed numbers. For example, the equation is equivalent to . Here is a diagram that represents the addition equation.
A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 5, and is labeled "plus 5." A second arrow starts at 5, points to the right, ends at 7, and is labeled with a question mark. There is a solid dot indicated at 7.
To get to the sum of 7, the second arrow must be 2 units long, pointing to the right. This tells us that positive 2 is the number that completes each equation: and .
Notice that the addition expression also equals 2.
A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 7, and is labeled "plus 7". A second arrow starts at 7, points to the left, ends at 2, and is labeled "minus 5". There is a solid dot and a question mark labeled at 2.
So we can see that .
Here's another example. The equation is equivalent to .
A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 5, and is labeled "plus 5." A second arrow starts at 5, points to the left, ends at 3, and is labeled with a question mark. There is a solid dot indicated at 3.
To get the to the sum of 3, the second arrow must be 2 units long, pointing to the left. This tells us that -2 is the number that completes each equation: and .
Notice that the addition expression also equals -2.
A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, ends at 3, and is labeled "plus 3". A second arrow starts at 3, points to the left, ends at negative two, and is labeled "minus 5". There is a solid dot and a question mark labeled at 2.
So we can see that .
This pattern always works. In general:
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