Getting Started |
| Misconception/Error The student does not demonstrate an understanding of subtraction as an unknown-addend problem. |
| Examples of Student Work at this Level The student uses a strategy such as Counting Down to solve the subtraction problem (10 – 7 = ____), and knows that the answer is three; however, the student cannot identify the corresponding addition equation, even with prompting. |
| Questions Eliciting Thinking You know that 10 – 7 = 3. So, what added to seven makes 10? (Write out both 10 – 7 = 3 and 7 + ____ = 10 for the student so that he or she can see the equations.)
If 9 – 3 = 6, what added to six makes nine? (Write out both 9 – 3 = 6 and 6 + ____ = 9 for the student so that he or she can see the equations.)
Can you see a relationship between these pairs of equations? |
| Instructional Implications Model related addition and subtraction problems with cubes. For example, start with 10 cubes and take away seven leaving three. Then, show how adding back the same seven cubes yields the original amount of 10. Relate the model to equations.
Use a number line to show the relationship between addition and subtraction (e.g., the difference between 7 and 10 is the same as the amount that must be added to seven to make 10.)
Have the student write related addition and subtraction equations (e.g., 6 – 2 = 4 and 4 + 2 = 6). |
Moving Forward |
| Misconception/Error The student can identify an addition problem related to a given subtraction problem, but only with significant prompting from the teacher, and even so, is not always successful. |
| Examples of Student Work at this Level Initially, the student says that 10 + 7 will help solve the problem 10 – 7.
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With prompting, the student determines that if 7 + 3 = 10, then 10 – 7 = 3. But, given the problem 9 – 3 = ____, the student says that the addition fact 9 + 3 would help find 9 – 3.Â
The student is unable to identify the addition equation that can be used to solve 9 – 3 = ____, even with prompting from the teacher.
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| Questions Eliciting Thinking What added to seven makes 10? How can we use that to help us know the addition problem that relates to the subtraction problem of 10 – 7?
What is 10 - 7? What addition facts go with that problem? Â
If 9 – 3 = 6, what added to six makes nine? (Write out both 9 – 3 = 6 and 6 + ____ = 9 for the student so that he or she can see the equations.)
Can you see a relationship between these pairs of equations?
What do you know about addition and subtraction? How are they related? |
| Instructional Implications Provide the student opportunities to see how addition and subtraction are related. Â
Use a number line to show the relationship between addition and subtraction (e.g., the difference between 7 and 10 is the same as the amount that must be added to seven to make 10).
Have the student solve related addition and subtraction problems and then discuss how the two sets of problems are alike.
Have the student write related addition and subtraction equations (e.g., 6 – 2 = 4 and 4 + 2 = 6).
Present the Counting On To strategy in the context of Compare (Result Unknown) subtraction problems as a way to use addition to solve subtraction problems. |
Almost There |
| Misconception/Error Some guidance from the teacher is needed in order for the student to identify an addition problem related to a given subtraction problem. |
| Examples of Student Work at this Level With guidance, the student can identify a corresponding addition equation, and can use it to solve a given subtraction equation.
The student is able to identify the addition equation that can be used to solve 10 - 7 = ____ with some guidance from the teacher
The student is able to identify the addition equation that can be used to solve 9 - 3 = ____ with some guidance from the teacher.
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| Questions Eliciting Thinking Why is 7 + 3 = ____ the addition problem that helps us find 10 - 7?
What do we know about the numbers 10, 7, and 3? Â
How are addition and subtraction related? Â
Will this always work with subtraction problems? Can you find an addition fact that goes with any subtraction problem? |
| Instructional Implications Provide the student with subtraction problems and ask him or her to write the related addition problem. Focus on the relationship between the problems rather than simply moving numbers to fit the equations.
Present the Counting On To strategy in the context of Compare (Result Unknown) subtraction problems as a way to use addition to solve subtraction problems.
Allow other students in the class to share their thinking about how to solve subtraction problems using addition strategies. |
Got It |
| Misconception/Error The student has no misconceptions or errors. |
| Examples of Student Work at this Level In response to the second problem, the student says 6 + 3 = 9 so 9 – 3 = 6 without any further prompting.
The student is able to describe, in general, how pairs of addition and subtraction problems are related.
The student can explain the inverse relationship between addition and subtraction. |
| Questions Eliciting Thinking How do you know that 7 + 3 = 10 is the addition problem that helps us solve 10 - 7?
Do you think we can always use addition to solve subtraction problems? Why? |
| Instructional Implications Allow the student opportunities to generalize this relationship. Encourage the student to try to do so without using numbers and rather to simply describe the way each operation works.
Provide the student with a subtraction problem that has larger numbers (e.g., 37 – 29 = ____) to determine if he or she can generate the addition equation that could be used to solve this problem.
Have the student create his or her own problems where inverse operations can be used as a strategy for solving. |
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