Chalk, Animals, & Tile: Summertime Fun with Area & Perimeter

On my summer reading list: Jo Boaler's Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching.

Yesterday, I read her challenge to teach math content using a question rather than following a procedure. (p. 78) She offers this suggestion:

"Instead of asking students to find the area of a 12 by 4 rectangle, ask them how many rectangles they can find with an area of 24."

And voila...we have a summer afternoon activity for a slightly bored child whose siblings are all at camp!

I posed the situation this way...

A farmer has pens that each contain an area of 24 square units. The farmer wants to know how many different rectangular pens he can make.

Then we got out the chalk, tile, and animals.  The first pen he built, 4 x 6, was for the cows:

I asked what other pens fit the criteria of 24 square units. He thought for a bit. "6 x 4?" We agreed that since this had the same dimensions, we wouldn't built it. He soon thought of another: 3 x 8.

He asked if this farm could have penguins. Sure, why not? The next pen took a bit more thought, but after some think time he built a 2 x 12 for the pigs.

I asked if this included all the possibilities for pens with an area of 24. He wasn't sure, so we made a list. (Ideally, he would have played around with 24 tile, exploring how many different rectangles could be made, but the farmer was getting tired.)

Although he didn't want to make the 1 x 24, he talked about how LONGGGG that one would be.

Then I asked another question:

The farmer needs to buy fencing for each of the pens. One section of fence covers one side of a tile. Which pen has the cheapest fencing? Which has the most expensive?

At first he predicted that the "biggest" pen would have the most fence. (At this point, in his mind, the 4x6 pen was "biggest." After all, it did contain the cows! Later on, I asked about pen size and he was able to say that they are all the same.)

His findings:
4x6 area = 20 sections of fence
3x8 area = 22 sections of fence
2x12 area = 28 sections of fence
1x24 area = 50 sections of fence

His eyes got really big when he heard it would take 50 sections of fence. He remarked that the chunkier pens have less fence because more of the edges are in the middle. I asked if he knew another name for the "fence" or the distance around. He named it perimeter.

Thanks, Jo, for a great summertime exploration!

p.s. Try making your own farms with pens of 36, 100, or other areas!

Exploring Area with Pattern Blocks

Last week I mentioned how we used pattern blocks to explore area during our geometry class. Today, my student continued the lesson as he explored area using pattern blocks with puzzles.

I own several different pattern block books: Pattern Block Problems for Primary People, Patternables and Magnetic Pattern Blocks. I used the first two as it was easy to find puzzles that did not use the tan parallelogram and orange  square (not used in this lesson.) I took all the pages out of both books, slipped each page into a plastic sleeve, and put them all into a 3-ring binder. This allows multiple children to use materials at once; I just pull out a page or two at a time. Today, little student, age 4, put pattern blocks onto some of the easiest puzzles while big student, age 7, calculated area. [Side note: the magnetic books are great for younger kids who may have trouble keeping the blocks in place as they work on a puzzle.]

I again explained that "just for today" a green triangle has an area of 1. With that in mind, I asked him to figure the area for 6 puzzles. He first had to solve the puzzle itself, trying different pattern blocks until he found the ones that fit correctly. After the puzzle was finished, he figured the area for each figure if a triangle = 1. I put a small post-it note next to each puzzle where he could write the answer. When he finished, he explained to me how he got his answer. He calculated so quickly that I sometimes had to ask him repeat himself because "Mommy can't add that fast!" I videotaped one example...it's the first time he's sharing this puzzle with me. See if you can keep up! ;) I wouldn't expect students new to pattern blocks to be able to add so fast, but he's obviously familiar with the shapes and how many triangles fit in each. (You'll want to investigate how many triangles fit in each pattern block shape with your own student before figuring larger puzzles.)



If you keep the puzzles in plastic sleeves, you could ask your student to trace each shape using a dry erase marker, then write the value inside each outline. [For example, a trapezoid would have a "3" written inside it, a triangle a "1", etc.]

Before bedtime, we'll read Spaghetti and Meatballs for All!, a book that considers area and perimeter as tables and chairs are readied for a family reunion.

P.S. Pattern blocks are widely available for purchase. Amazon and The Math Learning Center are two options. Disclaimer: I make a few cents on purchases if you click from here to Amazon. I do not make money on pattern blocks through The Math Learning Center, but they are an awesome non-profit and worth supporting. :)