# Thoughts on linear equations

March 8, 2016

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
🙂

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?

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