Training a Mathematical Mind Even If You Hated Math – Subtraction

This is my third post in the series How to Train a Mathematical Mind. The whole purpose is to train our minds to see the relationships that are “visually obvious” in base ten blocks so that we can help our kids discover them as well.  I put visually obvious in quotes because I honestly don’t think it is visually obvious to most people. And I certainly don’t think most people see how all of it is really the same. Not in the way Ben Rogers from Crewton Ramone’s House of Math means same. You will find the introduction with addition here and multiplication here. I specifically chose to use Cuisenaire rods for showing the relationships because I think they are easier to see. You don’t get lost in the numbers on the blocks. We are just talking about relationships. As others have said, “Once you figure that out, all you can do is change the numbers.” We will get to the numbers soon enough. I am convinced that this kind of training with the blocks translates into being able to see interrelationships in all kind of areas. Not just with base ten blocks and not just with math. As a homeschooling mom, I am always looking to cross-teach as much as I can.

0range matWe have been using the same image this whole series. That is important. There is a lot to uncover in this image. Once we are done, we will be able to relate our knowledge to any image. And we will be able to insert numbers as we wish. While we are working on subtraction, we will also bring into our work addition and multiplication. Since you guys have all been keeping a handy dandy notebook,  it is all an old hat by now.  If you are on a cell phone I am sorry but this is going to be hard for you. Bookmark it and come back later.

We have at the top our Orange rod which we labeled o. The first time going through this exercise with a student, specifically a pre-k/k student, I would use 3-4 equivalencies. Unlike in addition, we are not going to read each line as is. We are not adding we are subtracting. Instead of building blocks that are the same as o, we will start with o and see what happens when you remove one or more blocks. In other words:

  • If I take a y from an o what do I have left?
  • If I take a k from an o what do I have left?
  • If I take a p and another p from an o what do I have left?
  • If I take 2w and an r what do I have left?

If the child is small s/he will probably want to physically line up the blocks and remove them. An older child should be able to look and see what is going on. We call the above image a ‘mat’. The more equivalencies on the mat the more varied and complex the answers become. Young children need to grow into a mat this size. They do that by playing blocks and talking about blocks and by being given challenges to solve. We start with one line and add another. If that goes well, we add another.

If we want to get creative we can say 0 – (y-p) = 4r + w   and we can learn that the previous sentence is very different from 0 – y – p = w   If your student decided to use this as a sentence, now would be a good time to introduce the use of parenthesis. If we look at lines 10 and 11 we can see that b + w = 0. If we take a b from o we will get a w. What happens if we take (b – w) from 0? If we pull that w and put it on top of the b we can see we have an r sitting there.

My kindergartner would say something along these line for me to write down.

0 – b = 2w

o – 2l = 2r

o – y = 3w + r

0 – l = k

o – l = p + (d – 3w)

This kind of work makes the student flexible in his/her thinking. I am sure there are many children would find this kind of work boring, but we like puzzles at our house. We delight in finding the most complex way available to say the same thing. Why would that be useful? It develops an intimate knowledge of the relationships presented to us.

“One cannot understand… the universality of laws of nature, the relationship of things, without an understanding of mathematics. There is no other way to do it.”
– Richard P. Feynman


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