Training a Math Mind Series Part 4

This is the 4th post in my series on Training a Math Mind. We are looking to uncover the relationships that are the most visually obvious in the blocks so that you can help your children to see them as well. Again, I chose to use Cuisenaire Rods for this series because the units are delineated on other base ten blocks. We actually use the Cuisenaire Rods for this kind of work in our homeschool. When we are looking to understand the mathematical relationships the numbers get in the way. Which is funny because we assume that numbers = math. Today I am covering division and fractions and we will also be using addition, subtraction, and multiplication to describe how blocks are related to the orange block, but also how some blocks are related to other blocksmultiplication mat.

When we examine the image labeled with a 2 in the left-hand corner. We notice that the orange block (o) is the same as 2 of the yellow blocks (y). We can say that o = 2y.  We can also see that o ÷ 2 = y. If o ÷ 2 = y  then o ÷ y must be the same as 2.  If we are discussing fractions  we write that as 1/2 o = y. We read that as: “One-half of o  is the same as y.”     1/2 0 could be rewritten as o/2.  That is just another way of saying o ÷ 2 . We know that it is the same as y.

We wrote this entire image before as multiplication equivalencies. o = 2y = 5r = 10w.  If we look at the image we can say that

  • 2y/r = 5. The fraction 2y over r is the same as saying 2y ÷ r.
  • o ÷ 10 = w
  • o/w = 10.

Math is the study of numbers but it is also the study of relationships. What we say and write has a lot to do with what relationships we are describing and what we want to know about those relationships.

Exhausting the Math Relationshipsmath with cuisenaire rods

This larger image provides us with enough math work for a 1st grader for a year. It would get boring, but it is unlikely that even an 8-year- old could exhaust the relationships found in that image. Our goal is not to bore children with math but to spark their imaginations. If your child doesn’t think of this work as a fun math game, we need to rethink our approach or skip it and find something that your child does enjoy doing. Base ten blocks are a geometric model of math. These images can represent many ideas, as we have seen. The question becomes does the child understand the model? And does it make sense to the child? If not, we should back up or try a different model.

Let’s look at 6 lines we can write from this image. We do this by way of substitution. In the second line below we substitute a w for what is clearly 1/2 of an r. In the 3rd line, we substitute 5r for the o. In line 4 there is a lot of substituting and it might take a child 3-4 months of working with the blocks to get to that point. But we are still dealing in sameness when we substitute. We are still dealing in equivalencies.

  • o  ÷ r = 5
  • y  ÷ r = 2 1/2 
  • 5r = 2p + 2w
  • 1/2 n + 2r + 2w = 2y
  • (k-p) + 1/2 d + 2r = y + 2 1/2r
  • 3l – (r +w) + p = b + w


For an excellent series on considering relationships in math and training your mathematical intuition, I suggest a series of posts by Kalid over at Better Explained.


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