# Algebraic Structure Part A: Comparing Operations (10 minutes)

## Session 9, Part A

Students begin to move from arithmetic to algebra (in the structural sense) when they start thinking about properties of operations rather than properties of numbers. This happens quite early. For example, “missing addend” problems (such as 4 + ? = 9) can be solved one at a time, each as a special case, by any number of techniques (counting up, counting back, even subtracting). But when your students start saying things like “subtraction is the opposite of addition” or “subtraction undoes addition,” they are starting to realize a structural relationship between two operations rather than a collection of relationships between pairs of numbers. Note 2

We took an initial look at algebra from a structural approach when we examined the concept of doing and undoing in Session 3. At that point, we were looking at relationships between operations with a focus on undoing, or inverting, operations. Another hallmark of a move to algebra as structure is a focus on comparing algorithms. For example, consider the following two algorithms:

Algorithm A:

Algorithm B:

### Problems

Problem A1

Explain why algorithms A and B will always give the same output if you give them the same input.

Problem A2

In what sense are algorithms A and B “the same?” In what sense are they different? Do you think algebra students would classify them as the same or different? Would you classify them as equivalent?

### Notes

Note 2

In this session, we’ll be exploring a structural approach to algebra. When we begin to think about properties of operations rather than of numbers, we’re moving from arithmetic to algebra, in a structural sense. You may recall that in Session 3 we looked at algorithms through the lens of “doing and undoing.” A focus on “what undoes what” often marks the beginning of reasoning about operations. In this session, we’ll continue to examine algorithms, as well as other systems that represent a structural approach to algebra.

Spend some time thinking through the following exercise.

Groups: You may want to put this on an overhead.

One hallmark of a move to algebra as structure is a focus on undoing, or inverting, processes; another is comparing them.

A simple case of this is the following:

Is adding 3 to 2 the same as adding 2 to 3?

Is subtracting 3 from 2 the same as subtracting 2 from 3?

Thinking about properties rather than individual numbers indicates a move to a structural point of view. Keep this in mind as you work on Problems A1 and A2.

### Solutions

Problem A1

Follow both algorithms using N as the input number:

Algorithm A: N -> N + 1 -> 2(N + 1) = 2N + 2
Algorithm B: N -> 2N -> 2N + 2

Each algorithm produces the same output, as long as you’re willing to trust that the distributive property is always true! If you don’t trust or remember the distributive property, remember that 2(N + 1) is the same as (N + 1) + (N + 1).

Problem A2

Most algebra students would classify them as identical, since they determine identical expressions. But the steps are different — it’s like giving two sets of directions to the same place. Certain problems have many different algorithms of solution, some much more difficult than others.