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What is the area of this concave polygon?
Maye there is a formula out there to figure it out, but we are going to use our brains instead. It’s going to be hard, but it’s going to be fun!
First let’s make a rectangle around the whole polygon. This tells us that the polygon is going to be less than 12 square inches.
Now let’s splice and dice.
The blue triangle in that yellow rectangle is 1.5 square inches.
The blue triangle in this yellow square equals 2 square inches. (Click here for how we figure out areas of triangles.)
The area of this blue triangle is 2 square inches. Plus I need to add the 1 square inch from that square in the upper right corner.
Now I have 1.5 + 2 + 2 + 1 = 6.5 square inches.
Hard, but fun, right?
Sorry for the lame picture. Things have been a bit busy here.
What you’re looking at is the Melissa and Doug 11 x 11 geoboard kit I bought for Jenna(3) so she wouldn’t be left out of the geoboard action her big brother Bruce is doing.
This is a real geoboard that you could do some serious math with. Since it is 11 x 11, there is the potential to do bigger problems than on a 7 x7 board. But!!! The square inch tiles won’t work. (See here for more info.)
What this kit has going for it, is the picture cards that slide into the back. Jenna is still too young to be able to actually create the pictures, but she thinks she is is creating the pictures. There is also a lot of fine motor activity going on which will help her handwriting muscles.
P.S. I am adding this to my Grandma Please Buy This page.
How many rectangles do you see?
This was a problem from Bruce’s homework last week. He answered the question correctly by labeling each rectangle and listing them out, all by himself.
After I checked his answer, I thought “Wow! This would have been so much easier on a geoboard.”
Then I forced him to watch me solve the problem and endured listening to him say “Moooooom!”
And yet you listen to me without the dram. You gotta love blogging….
So here it goes on the geoboard. In that first picture you can see the five squares right? Since squares are also rectangles, those are numbers 1, 2, 3, 4, and 5. Then we’ve got:
Check out this blue Isosceles triangle.
To find out the area of this blue triangle we could (and will eventually) use the standard formula. But first let’s do some cool stuff!
Let’s figure out the area of the rectangle that triangle is in. Easy, right? The area of the rectangle is 8.
Or maybe I might want to look at it this way. Now I’ve spliced the blue triangle in half. I can see that I’ve ended up creating four congruent right triangles.
Now I have a lot of information.
If I wanted to, I could even turn this into an algebra problem.
Y + Y + B = 8
B = Y +Y
Y + Y + Y + Y = 8
2 = 8
B = 2 + 2
Or, I could go back to the traditional formula.
area of the blue triangle = 1/2(2 X 4) = 4
Geometry, multiplication, fractions and algebra all in one lesson?
Yup! Geoboards are awesome.
Here’s a great starter lesson with geoboards: figuring out the area of squares and rectangles.
In the above picture I threw in some square inch tiles so that it is really easy to figure out the area of the rubber band square, just by counting. This also teaches early multiplication skills because we are looking at a 4 x 4 array.
Here’s another example, this time with a rectangle. To find the area of this rectangle kids can count the squares or…
…They can transition into just multiplying 2 x 4.
So now we’ve got an activity that teaches shapes, counting, area and multiplication all in the same lesson. You could do this with your four year old, or your nine year old, and they would both get something out of it.
I finally broke down an ordered a 7 x 7 geoboard. It was only $6 and now I’m thinking “Why didn’t I buy one of these sooner?” There are soooooo many cool things you can do with geoboards, that I thought I might create a new Pinterest board, all about them.
It’s really hard for classroom teachers to use math manipulatives “enough”.
Even when teachers have training in using things like geoboards, they might not have enough materials for all 30 students in their room. Then, even if they do have 30 geoboards, they are under enormous pressure by the state to makes sure kids learn computation skills like adding, subtracting, multiplication and division. Sometimes math manipulatives like geoboards sit on the shelf of a classroom, unused.
The sad thing is that geoboards can actually make learning easy.
They are kick-butt awesome at teaching geometry, fractions, division, multiplication, and logic. But to be fair to teachers (I was one of them so I know), managing geoboards, little boys and rubber bands, is really hard. The rubber band part especially!
That’s why using geoboards one-0ne-one with your child at home “for fun” is a great Afterschooling idea.
This month I’m going to be looking at some cool things you can do with geoboards at home:
- Area of squares and rectangles
- Area of triangles
- More area of triangles
- How many rectangles?
- Square numbers
- Area of circles
P.S. If you have never bought a geoboard before, be warned that there are two different types out there, 7 x 7 and 11 x 11. We happen to own both. But the 7 x 7 frame is what I’ll be using for these activities.