homeschooling,  mathematics

A few more math books and activities….

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Our regular math curriculum has one again been interrupted. Instead of following it, we’ve been reading some math books and doing some mathematical play.

The Number Devil is a delightfully illustrated chapter book by Hans Magnus Enzensberger telling the story of a boy’s encounter with a helpful though temperamental number devil. Through dream-visits with the number devil the boy learns to think of math as something more challenging and more exciting than the arithmetic problems he had been exposed to at school. What I disliked most about the book is the mentions of normal “school math” being boring. My children don’t talk like that yet, and I don’t like books telling them they should think that. What I really liked about the book is it helped remind my oldest of things we had already learnt about like prime numbers, triangular numbers and Pascal’s triangle. His eyes glowed as he recognized ideas he was familiar with already and they have continued to sparkle since when he recognizes in other stories and situations the ideas he was introduced to within The Number Devil.

One of the things The Number Devil talks about are Fibonacci numbers. In The Number Devil they are visualized partly through the life cycles of magical rabbits but also through the divisions of branches within a tree. (At one unit off the ground there is one branch, at two units there is one branch, at three units there are two branches, at four units there are five branches, at five units there are eight branches, and so on.) I took the time to draw out a picture of a tree similar to the one in the book and show it in comparison to a separate tree where the branches double at every unit. I want my kids to think about how different patterns of numbers can be visualized differently. We also watched the YouTube video Doodling in Math, which shows Fibonacci numbers appearing within nature in different ways.
One problem with The Number Devil is that it uses its own names for things. For example, square roots are called rutabagas. I suspect this was meant to keep the story light hearted and playful but I would have preferred if the normal terminology could have been used. Still the book is worth reading.

The Lion’s Share is a fun storybook by Matthew McElligott.  The book tells of an ant attending the animal king’s dinner party, where each guest takes half of what is left of the cake until the ant has only crumbs to try to share with the king. Appalled she offers to bake the king a cake, and the animals then take turns trying to top each other’s offers, each offering to bake double the number of cakes of the previous animal. It is a good way to review fractions and to introduce exponents. The author’s webpage (linked above) includes lesson plan ideas though the most obvious possibility is to simply have a little tea party dividing up a cake or paper-cake the same way as demonstrated in the book.
A pretend paper-cake wasn’t enough for the kids, so after we did that they wanted to prepare a full party. Baking a cake is an easy opportunity to practice math and cutting the pound of butter into the right amount (half a pound) was particularly satisfying given the story we were talking about. Just for fun we then cut that part into smaller and smaller pieces. It also gave me a chance to make sure my oldest at least understood the idea that when you are cutting something in half the size depends on what you started with and the same size piece can be described different ways. Half a cup of butter is a quarter of the block of butter. 

 The kids set about decorating napkins and place mats. I’m not going to post pictures of all of them, but here is a picture of a math napkin my oldest son drew. Included in it was a square root, the start of Pascal’s Triangle, and a little picture of a balance beam demonstrating the equation 9 = 3x. Math becomes art and play. The strange weird aspects of math that I have introduced the children to becomes their own.

While they did that I tidied the kitchen, and explored some math myself. I used graph paper to draw out the image of the cake being divided up over and over and then another set of graph paper to draw a Fibonacci spiral. The kids had learned about Fibonacci spirals already from Doodling in Math and because the figures look so similar I wanted to make sure the kids notice the difference. In one you alternate drawing rectangles and squares. In the other you draw just squares (which joined with the previous squares makes a rectangle.)
I enjoyed pointing out to the children the way in which the cut cake keeps switching back and forth between squares and rectangles. My next goal is to challenge the children to think of other ways in which the cake could be cut apart. Cutting the cake diagonally, the pieces could be arranged as a sort of spiral. What other designs are possible? What would it look like if a circular cake was divided into slices, each person taking half of what was left? Could you color a circle in the colors of the rainbow, with the first color taking up half the space, and each color after taking up half of the space left? This is another way to introduce Zeno’s paradox (which I had already introduced to my kids here.)
When I cut our cake up I divided it first with each piece being half the size of the last cut, but without dishing them out of the pan. We talked about what was fair or not and then I showed how I could extend the lines that I had drawn and divide the cake into 16 equal pieces. I asked my seven year old to tell me how many pieces half of the cake would be and I made a quick little chart for him showing:
1 cake = 16 pieces
1/2 cake = 8 pieces
1/4 cake = 4 pieces
1/8 cake = 2 pieces
1/16 cake = 1 piece.
Then I drew a little picture of another cake, divided just into four pieces. He could tell me easily that each piece equalled a quarter of the cake so I asked, if I wanted to give his baby sister just half of a piece of cake from the cake cut into quarters, how much of the cake would she get to eat? When he answered that I drew his attention back to the cake with the 16 pieces and asked how much of the cake half a piece of it would be.

Another book about doubling numbers is One Grain of Rice: A Mathematical Folktale by Demi. In it a clever heroine outwits the miserly ruler and saves the people from starvation for asking that for 30 days the Raja gives her every day double the amount of rice he gave her the day before, starting with one day. One reviewer on the website suggests imitating the story by giving a child pennies instead of rice and limiting the number of pennies to a week so that one doesn’t have to break the bank to do it. I might try doing that. I’m also thinking about having my seven year old graph it out, possibly together alongside graphing out the Fibonacci numbers, triangular numbers and perhaps multiples of six.
My son brought up another way to talk about doubling numbers. He had made a small family tree at his beaver’s group, showing him, his parents and grandparents. A simplified family tree like that pointed out that everyone has two (biological) parents, four grandparents, eight great-grandparents, and so on. He asked how many generations back it would be that we have 1000 relatives. My husband answered quickly: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. If each generation takes about 20 years, then you are a direct descendant of approximately 1000 people who lived 200 years ago and a million people who lived 400 years ago. Unless of course one of your ancestors married his cousin, in which case, that cuts down the number of ancestors.
The math curriculum we have been using emphasises learning to skip count by all different numbers. I thought it strange at first counting by sixes, sevens and eights, because although I had in elementary school memorized the multiplication tables I had never had to skip count it. Now having seen how the skip counting helps my son and I’m thinking it is also useful for him to know the beginning numbers of other patterns such as what the first ten or so powers of 2 are. I’m also discovering that when we surround ourselves with plenty of mathematical stories and games it isn’t that hard to memorize the more frequently reoccurring patterns.
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