Training a Mathematical Mind Even If You Hated Math Series – Addition
The only way to flex your “relationship seeing” muscle is to pull out some blocks and look for relationships. In the following image, we have an orange block (o). We will be using that block as our reference. We are describing the other blocks as they are related to the o block. There are other relationships to be seen here, but we are not concerning ourselves with those today. To make this easier, we will stick with just addition. Immediately we notice that there are a lot of possible combinations of blocks that will “make same” with the orange block. This is called combinatorics and is part of set theory.
We can play around with (analyze) the image and ask questions like:
- How many combinations would only contain 2 blocks?
- How many combinations of 3 blocks could you make?
- What is the most number of blocks you could use?
- The fewest?
If we have several children at the table, we can ask each child to make as many combinations as they can find in a set amount of time. We then ask:
- “Who made a combination with a red and a white?
- “Who made a combination with all white?”
- “How about with no white?”
Now let’s say what you see starting at the top and working our way down. I am going to use letters that represent the colors. They are as follows: 0range = o, blue = b, brown = n, black = k, dark green = d, yellow = y, purple = p, light green = l, red = r and white = w. I am writing all of it as an exercise in equivalent addition. Each line represents one equivalent. If I were doing this with my son he would say each line and I would write it down for him exactly like I have entered it below. Right now, for the + sign he will say the word “and”. He uses the phrase “is the same as” for the equal sign. We will transition out of that when he starts writing on his own.
How we say the above image: o is the same as y and y is the same as r and r and r and r and r is the same as w and w and w and w and w and w and w and w and w and w is the same as b and r is the same as p and p and w and w is the same as l and l and w and l is the same as p and d is the same as b and l is the same as r and b.
We write it as follows: o = y + y = r + r + r + r+ r = w + w + w + w + w + w + w + w + w + w = b + r = p + p + w + w = l + l + w + l = p + d = b + l = w + b = r + b
What did we learn from this exercise?
- Numbers are made up of other numbers that can be arranged in various ways, yet all of them are still the same as (or equal to) the o block. This is the Commutative Property of Addition. It comes from the word commute, which means “to move around.”
- If we look above we can visually see that a b can be replaced with 2 p‘s and 2 w‘s can replace an r.
- Saying w + w + w + w . . . gets tedious quickly. Children want a faster way to say it. Next time we will write 10w and KNOW that it means w + w + w + w . . .
- If my student is older, the writing part of this exercise is done without the blocks. The child must recall by either memory or by visualization the possibilities for the orange block. This develops spatial memory and the ability to mentally manipulate objects and later numbers.