Elementary School Grade 3 Multiplication Operations and Algebraic Thinking

Counting Fifty Dollar Bills

This yielded a few really interesting mistakes, all pointing me to the same conclusion: my 3rd Graders need to learn how large numbers work. Though, as always, the student thinking itself was interesting.

Sorry that this kid used the “highlighter.” But they got 6000 for those 30 fifty dollar bills.

Their count: 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000, 5000, 6000. Thousands are stupid, we should just eliminate them entirely and count “11 hundred, 12 hundred, etc.” It would solve a lot of problems! We’ll have to wait until I’m King of Math to do that one.

This same student made another mistake while working on this problem. They asked me if 1000 was made of three 50s. I asked, “wait what?” They clarified that since two 50s make 100, maybe three 50s make 1000?


Area Measurement & Data Multiplication Numbers & Operations in Base 10

Finding the Area of a Rectangle [Desmathmistakes Activity]

Here’s the work of a 4th Grader named Jaden. He has a lot of interesting ideas for finding the area of a rectangle. What do you notice in his work? What do you wonder?


When I asked teachers this question as part of a Desmathmistakes activity, there were a lot of interesting responses. While all sorts of observations about student work are valuable, it can be especially valuable to transform our observations about student thinking into some next step. (Researchers look at work as an end in itself. When teachers look at student work it’s almost always to evaluate it or to figure out what to do next in class. We’re doing the latter here.)

Here were three of my favorite responses to the activity, with thanks to (in order) Mary, K, and Cindy.





In case you’re curious, here is everybody’s rectangles:



Finally, on twitter Kristin Gray is thinking in a different direction:


Kristin’s idea is for a string of area calculation problems that all total to to the same area, but are partitioned in different ways:

Some meta-questions: What were people thinking about during this activity? What were they doing? Were they learning something? Could they be learning something?

Jump into the comments if you have some thoughts about Desmathmistakes Experiment #2.

Multiplication Numbers & Operations in Base 10

What Else Could She Know? [on 12×14=280]



1. Isn’t this an interesting multiplication mistake?

2. I used to ask “how could we help this student?” or “why do you think this student made this mistake?” I still think that these questions are valuable to ask when looking deeply at student thinking. But, when teaching, the better question seems to be not “what mistake did this student make?” but instead “what could this student know that might help her?”

In this case, I’d say that this student could use more versatile ways of breaking numbers apart more than any sort of reflection on the errors of her ways.

Every once in a while people get in touch with me because they don’t like that this site is focused on mistakes. I think this is probably what they’re getting at.

Multiplication Numbers & Operations in Base 10

“Write a story problem for 13 x 2.”

In this case, the mistake (or whatever we call it) isn’t about what the student wrote, but what he said.

At the end of class, I asked my 3rd Graders to write a story problem for 13 x 2 and hand it in. As he was leaving, a boy handed me this slip and apologized for it.



“Why are you apologizing?”

“Because my story is for 2 x 13, not for 13 x 2.”


The big lesson here is that the order matters in multiplication, as it does with addition (for most young kids 9+2 is much easier than 2+9) and as it does for algebra (4 + 2x = 10 is not the same as 10 = 2x + 4). Each of these problems has a different flavor for people who are beginning to get comfortable with these types of problems. Saying that two problems are “the same” is a substantive mathematical claim, and it needs to be taken with the seriousness that all mathematical claims require.

Multiplication Numbers & Operations in Base 10

Getting Better At Multiplying Two Digit Numbers

Previously, I shared my 4th Graders strategies for multiplying two-digit numbers. That work was taken at the beginning of our unit, and it’s interesting to me to follow up that post with an update of how their multiplication strategies developed over the course of the past two weeks.


Multiplication Numbers & Operations in Base 10

13,234 times 187



What’s this kids next multiplication strategy? How would you help him get there?

Multiplication Numbers & Operations in Base 10

Multiplication Strategies My Students Are Starting With

Here’s the breakdown of student thinking about double-digit multiplication that I’m seeing as we begin our unit in my 4th Grade class.

Direct Modeling:

Directly Modeling


Direct Modeling With Composition Into Groups:



Breaking The Numbers Apart With Addition:

Breaking The Numbers Apart


Breaking The Numbers Apart 2


Breaking The Numbers Apart With Arrays:

Arrays Arrays - Breaking Numbers Apart


Use of Standard Algorithm:

Standard Algorithm Standard Algorithm - Wrong


No Real Strategy, But Knowledge Of Multiplication by Multiples of 10:

Not Sure

Multiplication Numbers & Operations in Base 10

15 x 12 = 20



This comes via submission. Thoughts?

(“Algorithms unteach place value.”)

Multiplication Numbers & Operations in Base 10

34 x 68 = 98


Multiplication Numbers & Operations in Base 10

6 x __ = 240



You can’t say that the kid is incapable of understanding what the box means here. Still, in the space of one line, it slipped through her fingers.

Is this connected to the way kids inconsistently treat exponents? I’m struggling to articulate a general principle, but it goes something like “Operations defined in terms of others are strongly associated with their parent operation, to the point that students often perform the parent in place of the derivative operation. As a result, students should always be introduced to a new operation in its own context, not in terms of other operations, whenever possible.”