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# Thoughts on linear equations

I recently received this email from a teacher I work with:

“Dan, I have a question for you. I just introduced my [pre-algebra students] to slope and then to slope-intercept form of linear equations and wanted to explore with them some word problems which could be written in that form. (Ex: . For babysitting, Anna charges a flat fee of \$10, plus \$5 per hour. Write an equation for the cost, C, after h hours of babysitting. What do you think the slope and the y-intercept represent? How much money will she make if she baby-sits 5 hours?)

Do you have any ideas how to make this kind of lesson more fun, hands-on and exploratory for students?”

Here’s my response:

“This is a great question. Fortunately, there’s a large community online that’s working to solve it. I’ve got some ideas too ðŸ™‚

Strategies for making linear equations more relevant, more interesting, more exploratory:

1. Same problems, slicker delivery
An example might be the 100×100 cheeseburger problem. Same idea as the babysitting problem, but real life, involves a menu, and a compelling premise.
2. Slick delivery, unanticipated result
You can sometimes grab students attention with a problem that seems easy, but has a twist, like this stacking cups activity.
3. Pared down delivery
For example, visual patterns has the same info and the same question each time: here’s steps 1-4, and you can get step 43 as a hint. Write the equation. Purely visual, so students can begin immediately, and there’s actually more thinking work for them to do. It’s harder for them to just use a recipe approach.
4. Give two options to compare.
For example, a Would You Rather structure, as in, “Would you rather charge \$5 base rate plus \$7 per hour, or \$15 base rate plus \$5 per hour to babysit? Defend with algebra.” This is a harder question, and involves actually having to make an argument, which is a more compelling, more social reason to do something, and usually generates a deeper understanding.
5. Use Desmos.
I know a number of teachers who really like what they can do with this tool in the classroom. It basically allows kids to explore and solve problems with their computers or tablets, in some creative ways. There are some clever lessons in the teacher portal.
6. Raise the ceiling.
If you were to show the sequence of dominoes in a growth pattern below, there are natural questions that you (or the kids!) might ask:
-How many columns will there be in the nth stage?
-How many dominoes will there be in the nth stage (double n dominoes)?
-How many dominoes will be in the tallest column of this organization in the nth stage?
-how many dots total on all the dominoes in the nth stage?

Some of these questions go beyond linear equations, but provide a natural stretch questions, and can actually help kids understand how to model with equations even better.”

The teacher who wrote me closed by say “I have a few ideas, but would love to hear yours.” I feel exactly the same way. What else goes on this list?

1. Robert Kaplinsky

This was a fun reflection to read. Thanks for sharing. I’m biased, but I think Open Middle goes well with what you said, especially raising the ceiling (and lowering the floor).

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Dan

I’d never actually checked out http://www.openmiddle.com/. Some great stuff there! I’m a huge fan of low floor/high ceiling questions as well.
It seems like there’s starting to be quite a wealth of places to find good problems online, if you know where to look.

2. Anusha

Very cool problems.
Another set of very practical problems would be to use Excel or other software to fit a line to a bunch of data points (price of milk vs price of oil, height vs age for each kid in the class, baseball batting average vs years of experience of player, etc.). Then have the kids think about what happens if they extrapolate the linear equation. Does it make sense to even extrapolate some data? This is a context in which linear equations are used pretty much in every quantitative field. And can be super open ended.

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