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

Hi #MTBoS I’m gathering some information for my #MTED5622 class.

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

— Heather Johnson (@HthrLynnJ) September 2, 2016

Below are a few responses that I received:

@HthrLynnJ @mpershan 25×4 is 100. 4×9=36 so 900. I did 25×36=25x4x9=100×9 but I thought it through like the first two sentences.

— Tina Cardone (@crstn85) September 2, 2016

@HthrLynnJ 3*25=75, but it is really 30, not 3 so it’s 750. 6*25=150. 750+150=(700+100)+(50+50)=800+100=900

— Annie Forest (@mrsforest) September 2, 2016

@HthrLynnJ @mpershan

I notice these are square numbers.

= (5*6)^2

= 30^2

Hmm

= 10^2 * 3^2

= 100 * 9 = 900— Max Goldstein (@maxgoldst) September 2, 2016

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,

- Tina’s solution shows a way to use the associative property of multiplication to find the product: 25×36=25x(4×9)=(25×4)x9
- Annie’s solution shows a way to use the distributive property of multiplication over addition to find the product: 25×36=25*30+25*6
- Max’s solution shows a way to use the associative property of multiplication as well as different equivalent expressions to find the product: 25×36=(5×5)x(6×6)=(5×6)x(5×6)=(5×6)^2=30^2=10^2*3^2=100*9

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.*