Getting Started |
| Misconception/Error The student does not understand that opposite quantities have a sum of zero. |
| Examples of Student Work at this Level In response to question two, the student writes:
- AÂ subtraction problem that does or does not have a difference of zero (e.g., -6 - 6 or 6 - 6).
- An addition problem in which three or more quantities have a sum of zero [e.g., -2 + (-2) + 4].
- AÂ multiplication problem with a product of zero.Â
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| Questions Eliciting Thinking What does it mean for numbers to be called opposites?
What are examples of numbers that are opposites? Why are they opposite?
What is a sum? Did you write your problem as a sum?
What happens when you add a number to its opposite? Can you show that in a numerical expression? |
| Instructional Implications Initially approach the concept of opposites from the perspective of net gain and loss, (e.g., using a context such as yards gained or lost in football or deposits and withdrawals from a bank account).
Provide scenarios in which integer quantities can be added to a given quantity, (e.g., a bank account contains $100. The owner withdraws $60 from the account and then adds $40 to the account). Guide the student to record the changes to the account on a number line and by writing numerical expressions that involve addition of integers.
Be sure to include scenarios in which opposite quantities are added, (e.g., the bank account contains $10 and then $10 is withdrawn; or, the bank account contains $50 and $10 is withdrawn and then $10 is added).
Ask students to compose a net gain/loss situation that can be represented by a numerical expression such as -8 + 8, to represent the sum on a number line, and to interpret the resulting value in the context of the situation. Provide feedback as needed. |
Moving Forward |
| Misconception/Error The student is unable to illustrate opposite quantities combining to make zero on a number line. |
| Examples of Student Work at this Level The student plots the selected quantities in question two as points on a number line. The student may draw arrows, arcs, or “hops” from each point to zero.
The student plots a different numerical expression than he or she gave in question two.
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| Questions Eliciting Thinking Can you tell me how your number line shows the sum of opposites?
What addition or subtraction problem does your number line show? Is it the same as the problem you wrote for question two? |
| Instructional Implications Model the process of completing addition and subtraction problems on a number line. Be sure the student understands that each integer can be represented by a specific location on the number line. Relate the operations of addition and subtraction to movement either to the left or to the right on the number line. Provide additional opportunities to illustrate integer addition on the number line including sums of opposites. |
Making Progress |
| Misconception/Error The student is unable to describe a situation in which opposite quantities have a sum of zero. |
| Examples of Student Work at this Level The student describes:
- A “take away” situation.
- A situation that cannot be reasonably modeled by a negative number.
- A multiplication or division situation.
- A general situation without providing specific numerical values.
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| Questions Eliciting Thinking Can you come up with a word problem that uses the numbers you wrote for number two?
Does your word problem involve a sum of opposites?
What are the opposites being added together?
Does it make sense for a person to have a negative number of something? |
| Instructional Implications Expose the student to a variety of real-world situations in which integers are used to represent quantities such as gain or loss, increase or decrease, and above or below sea level. Guide the student to represent integer quantities in the context of problems.
Ask the student to use one of these contexts to describe a situation that can be modeled by a sum of opposites, [e.g., 6 + 6 or 6 + (-6)]. |
Almost There |
| Misconception/Error The student is unable to formulate a general explanation of why two opposite values have a sum of zero. |
| Examples of Student Work at this Level The student states that the sum of any two opposite numbers is always zero without an explanation or because:
- Zero is in the middle of the number line.
- A positive number and a negative number make zero.
- Opposites “cancel” each other out.
The student declares that the sum of any two opposite numbers is always zero and cites vocabulary terms such as zero pairs or additive inverse but does not provide an explanation.
The student provides an example rather than an explanation.
The student states that the sum of any two opposite numbers is not always zero because:
- The numbers are sometimes negative.
- The numbers might not be opposites.
- The numbers might be different.
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| Questions Eliciting Thinking Can you show me how you would add -4 + 4 on the number line? Can you show me how you would add 6 + (-6) on the number line? How would you add -3 + 3 on the number line? What is happening in each case?
What term do you use to describe the distance from zero on a number line?
Can you use that term to describe why any two opposite numbers have a sum of zero?
Would x and –x have a sum of zero? Why? |
| Instructional Implications Ask the student to construct a list of key terms and concepts, (e.g., zero pairs, inverse, additive inverse, sum, opposite, and absolute value). Ask the student to explain the relevance of each term to the sum of a pair of opposites.
Ask the student to demonstrate adding several pairs of opposites on a number line. Then ask the student to consider a general way to describe what is happening in each case.
Model explaining why x + (–x) = 0. |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level The student generates a word problem and expression in which the two given values are clearly opposites and uses vectors to illustrate the expression on the number line.
The student states that the sum of two opposites is always zero and explains using a number line. The student says, “When you add a pair of opposites you start at one of the numbers on the number line and travel as many units toward zero as the number you started with. You always move toward zero because the numbers are opposite in sign.”
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| Questions Eliciting Thinking Does it matter which number you start with? What property tells you that you can add two numbers in either order and will get the same sum?
Think about this expression: x + (–x). What if x were negative? Then what would –x mean? Would the sum still be zero?
Do you know what absolute value means? What can you say about the absolute values of numbers and their opposites? |
| Instructional Implications Be sure the student understands the concept of absolute value and models an explanation of why opposites sum to zero using absolute value.
Ask students to rewrite a sum of a positive and negative number by rewriting one of the addends in terms of the opposite of the other, {e.g., -9 + 4 = [-5 + (-4)] + 4 = -5 + (-4 + 4) = -5}. Challenge the student to find a second way to rewrite the sum, {e.g., -9 + 4 = -9 + [9 + (-5)] = (-9 + 9) + (-5) = -5}.
Engage the student in a discussion of the different ways that the minus or negative symbol is used in mathematics. Encourage the student to interpret expressions such as –x as meaning “the opposite of x.” |
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