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Here’s an especially tricky problem from 5th grade geometry. Everyone knows that the area of a triangle is 1/2 (b * h). But with this particular triangle, what qualifies as “the height” is difficult to see. At least it was for me the first time I looked at it.
I don’t know–maybe you’ll look at this problem and say “Duh, Jenny.” But for me, the scalene triangle was strange looking.
When I first looked at this I saw that it would be easy to solve with the Pythagorean Theorem. But Houghton Mifflin Math Expressions hadn’t covered that yet. So there was another even easier way to solve this problem that wasn’t jumping out to my me or my son.
Can you figure out what it is?
Figuring out the perimeter of the red triangle is easy. That’s 17 + 9 + 10 = 36 cm. But what about the area?
First, I’ll show the way that ends up being the most complicated: using the Pythagorean Theorem.
This is a perfect example of how being algorithm dependent can screw up your number sense. I was so sure the Pythagorean Theorem was the way to go, I initially missed seeing the easier solution.
Now after all of that, let’s look at the original problem and try a third method to solve this problem, using the formula 1/2 (b*h). This is arguably the easiest method.
1/2 (9 * 8) = 36 sq cm.
Okay, so why didn’t I use the formula to begin with? When my son first looked at this, why didn’t I say “Dude, plug in the formula 1/2 (b * h),”?
Because that’s not what good math teachers do. Math is more than memorizing and applying formulas. Math is about experimenting, visualizing, internalizing and sometimes struggling until you reach a higher level of understanding.
This is an example of a problem that is simple yet confusing. Those are the best types! I’ve gone through college level calculus and I still looked at the picture and couldn’t viscerally understand why 8 cm was the height of the triangle. Neither could anyone in my family. (My husband, btw, is a lot smarter in math than me!)
So we played with this problem. We turned it inside out. Now, it makes sense. Along the way, we got to do a lot of cool math.
Bruce had homework regarding rotational symmetry and it totally confused me because I’m really bad at visual-spatial things.
What is rotational symmetry? That means a shape that can be rotated less than 360 degrees and still look the same. More info right here.
For spatially challenged people like me (you should see me parallel park!), rotational symmetry can be hard to picture. Hands-on learning can help.
A long time ago, I blogged about using flour and cookie cutters to learn about flips and turns. Guess what? That idea also works for rotational symmetry too!
A word to the wise: this activity is messy! It’s the perfect example of something that would be really hard to do with thirty fifth graders in a classroom, but doable with your child at home.
Just be sure to have a vacuum ready!
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.
Area of a triangle= 1/2 bh
If you are a kid that formula can look like gibberish.
Let’s use geoboards to help that formula make sense!
The area of this square is 16. I can count those squares and that makes perfect sense.
Now I’m cutting the square in half and making a triangle.
What’s the area of that triangle? Half of 16 = 8
Booyah! I understand that.
1/2 of 4 X 4 = 8
1/2 of b x h
Let’s try this again:
I’m cutting up another triangle. Let’s find the area of that blue one.
Logic tells me that the blue triangle is half of the green one. The green triangle = 8. So the blue trianle should equal 4, right?
Let’s proove it!
If I use my imagination I can flip half of the triangle over and it becomes a complete square. Now I can “see” that the area of the blue triangle is four.
Or I can use the formula:
1/2 of b x h = 1/2 (4 x 2) = 4
Pretty cool, hunh?
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.
Bruce has a 3rd grade Houghton Mifflin Math Expressions geometry test tomorrow that includes the concepts: flip, turn, and slide. We did some at-home practice with this from a Constructivist perspective. Instead of worksheets, he built each type of movement with cookie cutters in a baking sheet full of flour.
This is a half turn, or half rotation.
Here’s the quarter turn.
…and the slide. Those bunnies get around!
At the end of this I had intended to use my omnium-gatherum of cookie cutters to do a pattern activity with Jenna(2.5) in the flour. But by that point, there was flour everywhere and I was starting to go a bit crazy! I’ll have to save that idea to do with Jenna another day.
Here’s a geometry-themed adaption of an idea I originally saw on the blog One Mouthful. In the original example, Lia had written numbers on the bottom of an egg carton, and let her preschooler smash the numbers with a toy hammer. Since Jenna(2.5) is still working on visulalization and quantities, instead of numeral recognition, I decided to make a geometry smasher instead. I called out each colored shape, and Jenna smashed away.
The unexpected consequence of this activity, was that Jenna’s older brother Bruce(6.5) was very jealous. He wanted a geometry smasher too! So I thought, why not? I emptied out some eggs in the fridge, and made a geometry smasher for Bruce that concentrated on concepts he has already learned in his third grade Houghton Mifflin Math Expressions book at school.
Here’s the before picture.
Here’s the after picture. Apparently, six and a half year olds like to smash stuff too!
This weekend Bruce(6.5) and I had fun building real life triangles. I was inspired to try this activity after reading Keith Devlin’s thought provoking book, The Language of Mathematics. Unlike The Math Instinct, The Language of Mathematics focuses more on the history of mathematics rather than best practices in teaching mathematics.
Dr. Devlin wrote The Language of Mathematics for the general reader, but I found it quite challenging even though I was an A- student in mathematics all the way through college level Calculus. I wish I had read this book back in high school, because it might have helped me knock off those minuses! On page 44, he discusses 3-4-5 triangles and ancient builders.
A 3-4-5 triangle is also mentioned in a children’s book Bruce and I read recently by Avi called The Barn. That too, was an excellent book although I would not recommend it for children under ten (oops) because it dealt with a really depressing storyline. In The Barn, the children use 3-4-5 triangles to create the right angle of the barn they are building for their father who is dying after suffering from a stroke. Here is how we built our own 3-4-5 triangle:
First, Bruce measured out 12 segments on masking tape that was folded on itself so it wasn’t sticky. He used Legos as his nonstandard unit of measurement. It would have been better to tie knots in rope or yarn at 12 equal lengths, but I knew that tying knots with a six year old would have added about ten minutes of utter frustration to the activity, and fine motor skill improvement was not my learning objective. Plus, I couldn’t find any rope. 🙂
At the 3 and 7 marks, we pulled the triangle tight. This picture doesn’t do the triangle justice, because I was also trying to hold the camera, but we did end up forming a right triangle with sides that were equal to 3-4-5 Legos. This triangle was also scalene because all three sides were different lengths.
After building our right triangle, we built a giant Isosolese triangle on the ground for fun. We also got out our protractor and measured angles, but as you can probably tell from the picture, since we had built our triangle with masking tape instead of rope, the angles were off by a few degrees. Bruce had previously had trouble remembering what an Isosclese triangle was, so hopefully this helps him remember. You can fit an “I” in an Isosceles triangle. Hopefully he also remembers what a right triangle is too.
A while back I posted about how my son Bruce (6) has been working through the Right Start Level D workbook this summer, sans scripted lessons in the Teacher’s Guide. I wouldn’t recommend this for everyone, but we have all of the materials and the geometry booklet, plus I’ve had a ton of training in Constructivism, since I use to be a teacher. In the future, I won’t attempt flying solo for Level E because I have never taught higher than fourth grade!
Summer is nearing completion and Bruce has completed 35 pages. That’s not as far as I hoped, but isn’t bad if you look at a one page a day M-F average. We’ve jumped around a lot, and now he is doing the geometry section. Are we doing this right? Hmmm… That’s a very good question. Bruce is definitely learning a lot, but I’m not sure I’m teaching this exactly how Joan Cotter would like. Please feel free to tell me what you think (or give me a geometry slap-down as the case might be). 🙂
Here’s how I’ve been teaching segmenting circles and drawing octagons etc. These next pictures are with my own handwriting:
That was all using the 45 degree triangle. I’m thinking maybe I should write the angles on the triangles with a Sharpie. Why didn’t I do that before? With older kids you could probably be more heavy handed about angles, and go through and label them all, match up the congruent ones etc. If you have a compass, and triangles you can do all of these activities at home. Of course, the caveat is that I might be teaching you the wrong way to do this!