This One Really Takes the Cake! (Explaining Your Math Thinking)

 Hi All,

If you don't know Marilyn Burns yet, please check out her books and blog. She has been writing for over 50 years about teaching math. Here's a lesson from an excellent book she wrote, 50 Problem-Solving Lessons Grades 1-6 (Math Solutions, 1996). I put my little spin on it, but it was in Burns' book and attributed to Carolyn Felux of Converse, Texas.

Begin by talking with the class about cakes, a subject near and dear to many kids. What are some favorite flavors? What are some shapes cakes come in? Do people cut all cakes the same way? Write answers on the board and draw some of the cake's shapes and ways they can be cut.

Give students some 8 1/2" by 11" sheets of white copy paper. Say, "Let's pretend this is a cake. What shape is it?" (rectangle) "Right. Now use a pencil and divide the cake up into four equal pieces. So, divide it into fourths."

Allow students time to complete this. Encourage them to get more paper and do it as many ways as they can.

Here are some common students responses.

Now ask for volunteers to come to the board and share a diagram of one way to evenly divide the cake into four parts. 

Say, "All of these cakes have been cut up into four pieces. Now, take  scissors and cut one piece from each cake you divdided. Compare the pieces to see if each one gives you the same amount of cake as the others. Get a new piece of paper and write about your decision explaining your thinking."

This activity can be very revealing and is a great assessment. Many children will write something like, "I don't think the pieces are the same size because they aren't the same shape. Some are skinny and some are fat."

Note that the word "size" works in this context as well as the word "amount."

There is often a debate as student thinking is shared on this. At this point, I like to take the investigation a step further and provide students with 8 1/2" x 11" one-inch-square (or 1 cm2) graph paper. I then ask them to draw their cakes again, divide them, cut them up with scissors if they want, and then compare them. 

 

Often the graph paper can help make it more apparent that different shapes can have the same area. Students can actually count squares and pieces of squares, to get an exact or almost exact count. This can be a huge discovery for many as they literally and physically prove to themselves that different shapes can have the same area.

You can follow up with more explorations of this idea by providing a set amount of square tiles, say 24 tiles, and letting students make different shape cakes with them and then split those into fourths. They then can literally and easily count the squares.

All the best,

Bob

PS - I'm having fun picking and choosing lessons and activities, but please feel free to let me know in the comment section if there is any particular kind of lesson you are looking for.