Using symbols to represent unknowns – Grade 3

[Duane]

The Standards 3.OA.3 and 3.OA.8 discuss using symbols to represent unknowns. In 3.OA.3 it seems as if it might be optional but for 3.OA.8 it reads as mandatory. The Progressions doc gives an example on page 28: a diagram and explanation that is quite complicated and abstract (for Grade 3) but with a comment that “students may be able to solve this problem without writing such equations”. The Grade 3 examples I’ve seen on the Illustrative Mathematics site use much simpler language and the only symbol is a question mark or an empty box.

Which examples (the Progressions or the IM) are most in keeping with the intent of those Standards? I know what seems most likely of being accomplished by Grade 3 students – the use of a question mark or an empty box. That’s not to say letters can’t be appropriate in some situations but the example in the Progressions seems a step too far.  Given that 3.OA.8 stipulates letters specifically (and not any old symbol) I’m concerned about how abstract it is meant to be at that Grade. I mean, if students can extract the relevant numbers from a two-step word problem (with or without distractors) and identify then perform the correct operations in the correct sequence I’m more than happy, regardless of whether they used letters for unknowns! Your advice, Bill, would be appreciated as always.

[Bill McCallum]

Just refreshing this topic as some advice would be appreciated.

[Kristin Umland]

Here is how I interpret the standard: Students can solve two-step word problems. They can also represent such problems with equations that use a letter to represent the unknown. That doesn’t mean they are required to represent them like this at all times (in fact, some times they might solve them simply be reasoning about the situation and not write an equation at all). I would imagine that teachers would work students up to such representations, possibly using question marks or boxes at first and then having them graduate to letters. So by the end of third grade, the ability to represent word problems with algebraic equations is one of the tools that students have in their tool belt, but like all tools, it doesn’t get used for every single construction project.

This doesn’t solve two issues related to the illustrations of this standard at the Illustrative Mathematics website that I think your post brings out. First, the set of tasks that illustrate this standard need to include some tasks that appropriately address the symbolic expectations of the standard–I don’t think the task that is there needs to have that representation, but we need some tasks that do. Second, in looking at this task, it has been moved around some and also it has been rated down a couple of times (I wish people would post comments explaining what they don’t like about a task so we could fix it), and to be honest, it isn’t my favorite task for illustrating this standard anyway (although I like the task fine). So for this reason also I think that having this task as the only task illustrating the standard is a mistake.

I can imagine a task that would have some story problems and some equations and might ask students to match them might be good. I’d like to see story problems that match to multiple equations (maybe, for example, a problem involving subtraction that can be represented by an equation involving addition and another involving subtraction to help emphasize the relationship between those operations). What do you think of that idea?

[Bill McCallum]

Thanks Kristin, I agree with you about the IM task not fully representing what is required. The example on page 28 of the OA Progressions is too far at the other end of the scale from the IM example to get a feel for what is required. Your idea about having stories match multiple equations to show what is possible is a good one and I agree that you wouldn’t expect students to use symbols all the time, only when necessary.

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