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I have seen many lessons in which the distinction between how to model or represent a story problem and how to solve the story problem get confused. This happens especially with story problems in which either the starting value or the change is unknown, a type of problem which the Common Core expects students to learn in grade 1. For example :

Some birds are sitting on the telephone wire. 14 birds fly down and join them. Now there are 21 birds on the telephone wire. How many birds flew down?

Teachers sometimes argue that this should be represented with a subtraction sentence. I disagree. This is an add-to situation, and therefore it should be represented with an addition sentence, like __ + 14 = 21. Of course, this is not a number sentence for which the missing number can be directly calculated, and so it is not helpful for finding the answer to the story problem. The best number sentence to solve this problem is 21 – 14 = __. But it is not obvious to students why subtraction makes sense here, because subtraction heretofore has been associated only with take-away situations.

Before I get into how one might help students understand why subtraction is appropriate, I want to note that the Common Core itself seems confused about the difference between modeling and solving:

K.OA.A.1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

In fact, this standard seems to confuse four things: problem situations, ways to model those situations, mathematical operations, and ways to calculate. Although these things are interrelated, it is valuable to understand how they are distinct.

Consider this story: Five children are sitting at a table. Two children leave.

This situation can be modeled, or represented, in various ways:

• Create a group of 5 blocks to represent the 5 children. Drag 2 blocks away to represent the children who leave.
• Draw a picture or diagram, e.g. of five circles representing the 5 children, with 2 circles crossed out to represent the idea that 2 children left.
• Write a subtraction number sentence: 5 – 2.

Each of these ways has its own way to determine the answer: count the remaining blocks; count the circles not crossed out; and know that 5 – 2 = 3.

Note, however, that the subtraction number sentence represents the situation, not the other way around. So why does the standard say to “Represent… subtraction with objects…”? If students solve a bare-number calculation like 5 – 2 using blocks or a diagram, are they representing the subtraction with the blocks? No. They are using the blocks to get the answer. They know that 5 – 2 represents take-from situations (5 things remove 2), and blocks or a diagram can also represent those same situations. So if you get the answer using blocks, that’s the same as the answer to the calculation.

Addition and subtraction are mathematical operations (done with numbers) that can be used to model add-to, put-together, take-from, and compare situations. Blocks, drawings, and fingers can be used model those same situations. For that reason, they can be used to calculate the answers to addition and subtraction sentences if students cannot calculate them in their heads.

Back to grade 1: The Standards for grade 1 are more clear about the difference between representing and solving:

1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Note “Use addition and subtraction… to solve” and “using objects, drawings, and equations… to represent the problem.” An equation can be used to represent and can be used to solve. Diagrams can be used to represent – but in grade 1 they should not be used to solve. No counting circles or objects! By grade 1 students should be calculating.*

Here’s the grade 1 story problem again:

Some birds are sitting on the telephone wire. 14 birds fly down and join them. Now there are 21 birds on the telephone wire. How many birds flew down?

How should grade 1 students represent this problem with a number sentence? How should they solve it? As I argued above, this story must be represented by an addition number sentence like __ + 14 = 21. Despite my admonition above, we know that some students will solve this problem by counting up from 14 to 21. (Having the start unknown makes this less likely.) But we need students to know that they can solve missing-addends problems using subtraction, and they deserve to understand why subtraction is appropriate.

A tape diagram can help students recognize this as a part-part-whole situation with a missing part. The very same missing-part diagram previously represented take-away situations. Since they solved those previous problems with a missing part using subtraction, they can see why subtraction is appropriate here, too.

*Yes I know that some grade 1 students will need to count, using pictures or their fingers. But we should be clear with students that we expect them to calculate, and then do everything we can to help students get there.