At this point I have the choice of continuing to cut strips or simply marking the measurements on the strips I already have. So I might hold strip B to one end of strip A and put a mark on it, then move it to the other end of strip A and put a mark there. (The space between those two marks would correspond to what in my picture is the length of strip D.) I keep moving the strips around always measuring from one of my previous marks until the strips are totally covered in marks. I have a ruler.
In the example I drew out it eventually becomes clear that B is eleven times the length of E, and that C is 23 times E. A is 34 times E. I have the ratios! At this point I get giddy with excitement and my children look at me bored so I assign them the task of measuring out my strips on a long, long strip of paper in the hallway. I make them use A and B and we talk about where they line up together. I show on a calculator how many units of E it is by the time they line up together and why. Ours don’t line up perfectly because we’re careless but it works well enough. My children tell me it is magic and I say no, it isn’t, it is math.
I have a few follow-up activities I want to do with the kids. I want my seven year old to find the ratio between piece of paper eight inches and twelve inches and I want to try to encourage him to think of it in terms of the greatest common multiple.
I want him to find the ratio between the sizes of common objects and draw pictures that maintains the ratio.
I want him to find the ratio between amounts of ingredients in baked goods. In baking people often have to double all the ingredients to create a double batch. In those cases they don’t have to know the ratios between the ingredients, just the ratio of the original recipe to the expanded one, but what if a person was given just a random amount of flour? Then the ratio between the original and expanded recipe is not there, but the recipe could be rebuilt from the ratios between the ingredients.
I wonder if I could fix up a balance scale so that he could find the difference in ratio of the weights of two different piles of beans, where he first has to find what will be the common unit by which he can compare the weights, subtracting piles of beans like we subtract the lengths of paper.
We have a few books about ratios, the favorite being Ann McCallum’s book The Measure of a Giant. In it the friendly giant Ray is five times Jack’s height. The children figure out how to make basketball nets sized right for each of them. I want to give my seven year old pieces of paper representing two different giants and let him figure out how high they are compared to each other, perhaps ones I will have cut specifically so the ratio between them is 3:7.
I also want to play pirates with the children and divide up the treasure according to a set formula giving the captain more than the first mate who gets more than the plain crew member. I want to play with math as frequently and in as many ways as possible.