I truly think the hardest part about Algebra for students is that they have to look for a meaning *behind* the numbers and variables.

As I’ve thought a lot more about that this year for my students, I decided that I would do some research, and what I found was a Common Core performance assessment from Illustrativemathematics.com that **strictly asked for meaning**. I fell immediately in love with the idea of **no numbers, no vocabulary**, just strict meaning of the problems.

Granted, my students had never been exposed to multiple variables in one expression with only three weeks of Algebra under their belt. But I thought they were up for the challenge.

Below is the problem:

A company uses two different-sized trucks to deliver sand. The first truck can transport x cubic yards, and the second y cubic yards. The first truck makes S trips to a job site, while the second makes T trips. What quantities do the following expressions represent in terms of the problem’s context?

- S+T
- x+y
- xS+yT
- (xS+yT) / (S+T)

I used this problem to get students to understand that **numbers are not required to have an expression**. It also forced students to understand what each variable meant, and then what each variable meant when multiplied or divided by another.

I used it in my classroom as a group activity to introduce students to in-depth thinking about Algebra and unknowns. Students were simply required to describe *in words* what each expression meant. They then posted their thoughts on butcher paper throughout the room.

What was useful about the group setting is that each student had a piece of the information needed to make the whole. James, “Y is the number of cubic yards transported by the second truck.” Moneka, “T is the number of trips the second truck took.” Moneka + James, “yT must meant the total number of cubic yards transported from all trips of the second truck.” To prevent a stalling group, I preassigned groups so that there wouldn’t be any groups with all the super smart kids or any groups that seemed absolutely stuck.

Highly useful was the word formation for students. They weren’t used to putting the information in their own words and to do so was very rewarding.

I look forward to more abstract problems that don’t have concrete numbers or formally expressed meanings. Instead, students have the ability to decipher for themselves the *possible* meaning behind a word problem.

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Kaci McCoy

said:This is great! I find that my algebra students in the past (I teach IMP this year, not algebra 1) have great difficulty working strictly with variables. They can’t make the connection that expressions or equations with just variables are really like expressions or equations with numbers. I usually have to TELL them that and provide examples. I love the idea of them exploring that on their own. And what rich discussion you must have heard! LOVE IT!

algebrasfriend

said:I love the problem you chose! I work with advanced students … 9th and 10th graders taking Algebra 2. I dare some of them would struggle with explaining the variables! I plan to add this problem to my toolbox. I also checked out your source and found a problem for our current unit on systems – the Cash Box!

http://www.illustrativemathematics.org/illustrations/462

I am also participating in MTBoS … check out my blog at http://algebrasfriend.blogspot.com/.

Mary Dooms

said:My 7th graders would benefit from being regularly exposed to such algebraic thinking. Thanks for your insights!

wwndtd

said:This is somewhat like what my 12th-grade physics teacher called “the So What? factor”. That is, okay, you calculated a number… so what? What’s it mean?

I use this a lot in my chemistry classes too.

unmuddlemath

said:Very nice! This will certainly help my developmental algebra students with the transition into variables. Thank you for sharing!

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