How would you use mental math to solve 36×25?

Last week I asked #MTBoS followers how they would use mental math to solve 36×25.

Below are a few responses that I received:


Having students use mental math to solve multiplication problems such as 36×25 can provide them with opportunities to informally use mathematical properties that are fundamental to algebraic reasoning.

For instance,


When students talk about their mathematical thinking, they have opportunities to clarify their ideas and to make meaning from symbol sentences. Tina, Annie, and Max all mentioned how they interpreted the symbol sentences they wrote. I think it is especially interesting how Tina said that the first two sentences were how she “thought it through.”


When I investigate students’ reasoning in my own research, I have learned that students may use the same written representation, but think about it very differently. The selection of Tweets that I included here is only a sampling of the interesting ways that #MTBoS followers thought about 36×25.


I wonder what might have changed if I had asked #MTBoS followers how they would use mental math to solve 25×36.


This semester, I am teaching a fully online class – Expanding conceptions of algebra— #MTED5622. In #MTED5622 we are investigating how students in grades K-12 engage in Algebraic Reasoning. One of the resources we are using is from NCTM’s Essential Understanding Series –Developing Essential Understanding of Algebraic Thinking Grades 3-5. This fall, I’ll be working to include blog posts focused on my work in teaching this course.


Why is it so hard for students to make sense of rate?

Earlier this year, I searched for some newspaper headlines that dealt with rates. Here are a few that I came across:

  • “Colorado unemployment rate among 10 lowest in the country”
  • “Unintended pregnancy rate in U.S. is high, but falling”
  • “Oil ends sharply higher. Logs 10% weekly gain as output draws focus”

Many of the headlines talked about rates, and not just rates, but varying rates. When I think about these headlines, I wonder how students make sense of varying rates. For example, what might a student think it means for a rate to be affected, for a rate to be low, or for a rate to be high, but falling? Furthermore, what does it mean for something to end “sharply” higher? Are there other kinds of ways to end higher? For example, what might ending “gradually” higher be like?

In my research, I investigate how students make sense of change, and more specifically, I study how students make sense of variation in change. Put another way, I want to know how—from a student’s perspective—an “increase” (or “decrease”) can be a thing that can vary. When I interact with students, I work to design learning experiences that can provide them opportunities to investigate different kinds of increases (and decreases).

When it comes to rate, I have found that it matters how students form and interpret relationships between quantities

If we want students to think about rate as something that is capable of varying, we should help students coordinate change in two different quantities, such as the height and volume of liquid in a filling bottle. Specifically, we should provide opportunities for students to think about one quantity as continuing to change while another quantity is changing along with it. For example, students could think about how the height of the liquid in a filling bottle continues to change while the volume of liquid in the filling bottle changes along with it.

I developed a framework that explains how students’ different ways of forming relationships between quantities can impact how they think about rate. I wrote about this framework in a 2015 article published in the journal Mathematical Thinking and Learning. The published version of the article is available here:

Here is the full citation for the article (APA 6th)

Johnson, H. L. (2015). Secondary students’ quantification of ratio and rate: A framework for reasoning about change in covarying quantities. Mathematical Thinking and Learning, 17(1), 64-90.

Here is the accepted manuscript of the article that you can download: