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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.
The best teaching happens when you make a lesson visual, spatial and auditory. That’s why I love teaching kids number sense with a math balance.
Utilizing a math balance in a whole-class setting of twenty-seven kids would be tricky, but at home with one child it’s easy. The balance we own came from Right Start and costs $25.
5 does not equal 8. It’s so easy to see.
Just like it’s easy to figure out that there are many number combinations that equal 5.
In fact, we spent a full ten minutes just figuring out the number 5!
In pedagogy, we call this “Constructivism”. It means learning a new concept through your own experimentation and discovery. Giving children the full Constructivist experience isn’t always possible, but a math balance makes it a lot easier.
One of the “I’m-a-mean-mom” Christmas presents I gave my son last year was The Giggly Guide to Grammar by Cathy Campbell. My eight-year-old would have much preferred another Lego kit, but I had my eye on the Common Core. I know Bruce’s teacher does a lot with grammar at school, and I’d like to support that at home.
I can see why The Giggly Guide to Grammar gets great reviews. It has fun drawings and even funnier sentences. Here’s an example from page 107: “Aunt Sylvia believes Elvis lives because she thinks that she saw him on a commercial for Levis.” (That’s a complex sentence with an anagram, btw.)
Unfortunately, I was hoping this book would be a good fit for Afterschooling, but it really isn’t. The Giggly Guide to Grammar would be great for public school, and it would be awesome for homeschoolers, but for an Afterschooling family it requires too much paper and pencil practice. That would be fine if we were using it during the summer, but for the school year it’s too much work. My goal with Afterschooling is not to load my kids up with extra duties, but rather to encourage them with fun enrichment.
A more passive approach to grammar would be the Royal Fireworks Press book Sentence Island by Michael Clay Thompson.
That being said, I keep finding The Giggly Guide to Grammar all over the house. On the kitchen table, laying in the hallway, in the bathroom (yuck); Bruce is clearly reading this book for enjoyment.
I’m not exactly sure how much Bruce is learning. I asked him about the book and he said he likes reading the funny sentences. I guess that’s why the full title is “The Giggly Guide to Grammar, Serious Grammar with a Sense of Humor”.