Using Gary Rubinstein’s “Russian Peasant” video with kids

Saw a neat tweet from Gary Rubinstein yesterday:

This morning I thought it would be fun to look at the “Russian Peasant” multiplication video with the boys. Here’s Rubenstein’s video:

I had the boys watch the video twice and then we talked through an example. My older son went first. He had a fun description of the process: “It is like multiplying, but you aren’t actually multiplying the numbers.”

Next my older son worked through a problem. This problem was the same as the first one but the numbers were reversed. It isn’t at all obvious that the “Russian Peasant” process is commutative when you see it for the first time, so I thought it would be nice to check one example:

Next we moved to discussing why the process produces the correct answer. My older son had a nice idea -> let’s see what happens with powers of 2.

The last video looking at multiplication with a power of 2 gave the kids a glimpse of why the algorithm worked. In this video they looked at an example not involving powers of 2 (24 x 9) and figured out the main idea of the “Russian Peasant” multiplication process:

This was a really great project with the boys. It’ll be fun to work through Rubinstein’s videos over the next few months. I’m grateful that he’s shared the entire collection of ideas.

A neat counting problem from the 1995 AJHME

Stumbled on a neat counting problem tonight on an old math contest. Here’s a link to the 1995 AJHME hosted by Art of Problem Solving:

Problem 23 from the 1995 AJHSME

and here’s the problem:

How many four-digit whole numbers are there such that the leftmost digit is odd, the second digit is even, and all four digits are different?

I thought it would interesting to have both kids talk through the problem just to see if they would take different approaches. It turned out the way they approached the problem was similar, and both were left with the same multiplication problem as the last step. They way the multiplied out the final product was drastically different.

here’s my older son:

and my younger son:

Definitely a nice counting problem, and the difference in multiplying strategies was fascinating to me.

Going through “Count like an Egyptian” with the boys

Earlier in the week I read Evelyn Lamb’s review of David Reimer’s Count like an Egyptian and thought it would have many fun projects for the boys. Lamb’s review is here:

Evelyn Lamb’s review of Count like an Egyptian

Link to the book is here:

Count Like and Egyptian by David Reimer

The book arrived a few days ago, and I was super excited to do our first project today!

The first chapter is about multiplication and division. The book explains a procedure for multiplication which is (i) so interesting and (ii) so simple that I was stunned that I’d never seen it before.

We started with a simple example -> 9 x 7

 

Next we more on to see how this simple procedure could be applied to a slightly more complicated multiplication problem -> 34 x 51. After we calculated this product two different ways, I showed the boys how this Egyptian multiplication process was connected to representing the numbers in binary.

 

With these two examples out of the way, I wanted to see if the boys understood the procedure, so I had each of them work through an example on their own. First up, my younger son working through 13 x 19:

 

and the my older son working through 27 x 36:

 

I was happy to see that they understood the procedure and were able to work through examples on their own after seeing only two examples from me. That sort of gets back to my surprise at never having seen this idea previously. It really is a nice way to multiply, but also a nice way to sneak in a little arithmetic practice while learning some new math. Seems like a great project for kids.

I didn’t want to go too long this morning but I did want to show them that the procedure was also pretty easy to reverse. We’ll probably revisit the division algorithm next weekend, but here’s a quick look:

 

So, a fun project from the ideas in the first chapter of Count like an Egyptian. Happy to have seen Evelyn Lamb’s review of the book and really looking forward to several more projects based on the ideas in this book.

Trying out Jo Boaler’s Number Talks idea

Yesterday (August 1, 2014) I saw this video on Jo Boaler’s YouCubed site:

I hadn’t come up with any ideas for our Family Math talks for this weekend, but after seeing this video I thought it would be fun to try out this idea with the boys this morning.

5×18 first

Younger son (will be in 3rd grade):

Older son (will be in 5th grade):

It was interesting to me that both of the kids approached the problem essentially the same way the first and second times. The subtle difference, which I think is somewhat interesting, is that 5*2*9 = 10*9 = 90, is a slightly easier computation than 5*9*2 = 45*2 = 90. At the time it didn’t occur to me to ask either kid why they chose to multiply in the order that they did.

Next comes the more difficult problem of 12 * 16. Rewatching Boaler’s video as I was writing this up, I see that I didn’t remember the second problem correctly – she used 12*15. Hopefully the difference between these two problems is not a big deal.

Older son first this time:

Younger son:

Again interesting to see how similar their approaches were. The first approach involved factoring, though the multiplication after that was different. I was surprised to see the way my younger son multiplied out 3*64. The second approach from both of them involved the distributive property, and the next two videos show some fun geometric ways to understand multiplication.

First, using our snap cubes we take a closer look at the picture I drew for my older son that connects the distributive property to squares and rectangles. We also talk about how thinking about multiplication this way helps understand multiplying polynomials. Naturally, I miss an easy opportunity to ask what happens in my algebraic example when x = 10. Oh well . . .

Finally I use the same geometric idea to show one way to understand why an negative number times a negative number is a positive number. Instead of viewing 12 * 16 as (10 + 2) * (10 + 6) we look at it as (20 – 8)*(20 – 4) in this video.

So, after going through this, I’m not sure what I’ll be doing differently going forward. I certainly agree completely that developing a good sense of numbers and arithmetic is extremely important and also see that this exercise is a good way to do that. I tend to focus on developing that number sense when we are working through problems, though, and personally prefer to work on it that way. That said, as reasonably quick and easy way to help kids develop number sense, the approach outlined in Boaler’s video seems pretty good.