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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.
Here’s my “I Brake for Moms” column about Common Core Math in today’s Daily Herald:
If you’re looking for ways to support your children’s math development at home, here are some of my favorite activities:
If you’ve read the recent article My Daughter’s Homework is Killing Me, then you know that parents all over our country are scratching their heads, wondering when homework got so dang hard.
I think that part of the reason is that complex concepts (like algebra) are being introduced in earlier grades.
In an ideal world, an early introduction to algebra would help prepare students to master advanced math in middle school and high school. It’s scaffolding for the future.
In the meantime, parents look at their kids’ homework and go “Whoa.”
Here’s a trick that might make homework easier. Add candy to the equation!
To show how this can work, I’m using an example similar to what you would find in the 5th grade Houghton Mifflin Math Expressions textbook, which the Edmonds School District uses.
Problem: Mrs. Garcia’s neighborhood has 28 pets. There are twice as many cats as hamsters and four times as many dogs as hamsters. How many of each pet are there?
You could use guess and check to figure this out, which would take forever. Or you could use algebra. Or you could use algebra and candy…even better!
One piece of candy corn equals the number of hamsters. Two pieces of candy corn equals the number of cats, which is twice the number of hamsters. Four pieces of candy corn equals the number of dogs, which is four times the number of hamsters. In all, the total number of pets is 28. That would mean 4 hamsters, 8 cats, and 16 dogs in the neighborhood.
Once you introduce candy into the equation, math homework becomes more fun. Just don’t forget to have toothbrushes on the ready.