### Reinforcing Ratio Reasoning

I’m excited to be presenting a workshop at the North Carolina Council for Teachers of Mathematics (NCCTM) Convention in this week. My session is **Reinforcing Ratio Reasoning – Throughout the Middle School Math Curriculum. **If you’re attending the convention, please stop by! My session is at 8:30 AM on Friday in the Turnberry room.

That’s awfully early for a Friday morning, but I hope to have a good turnout because I think developing ratio reasoning in middle school students one of the most critical things teachers can do to ensure success in higher level math. Unfortunately, I think some teachers approach ratio and proportion by teaching students to set up proportions, cross multiply, and solve for the unknown value. The End. But in my experience, very few of their students — even the ones who can pretty consistently get the right answer this way — have much (if any!) understanding of how the quantities relate to each other.

Even after teaching ratio concepts (6th and 7th grade) in what I believe to be a very conceptual way, without ever even *mentioning* cross multiplication (much less teaching students how to use this “trick”), many of my students still really struggle to see the connections between ratio quantities. This is especially true when those quantities aren’t “nice” numbers that are obvious factors and multiples of each other. When large numbers, fractions, or decimal values are thrown into the mix, their ratio reasoning breaks down — they start adding and subtracting rather than multiplying and dividing. They guess at the operation they should use, or at which number should be divided into (or divided by) the other. That’s why I wasn’t at all surprised by the results of a 1986 study, in which researchers asked 12- and 13-year-old students to answer these questions (with blanks where the numbers would go):

**The Price of Paint **

#### Paint is priced at **☐** per gallon. How much will ☐gallons cost?

#### It costs ☐ for ☐ gallons of paint. How much is the cost of the paint per gallon?

#### How will you solve each of these problems? (which operations will you use, on what part)

After students had determined how they would answer each question, the researchers gave students the same questions again, this time with numbers where the blanks had been.

### The Price of Paint

#### Paint is priced at **$16** per gallon. How much will **0.85** gallons cost?

#### It costs **$16** for **0.85** gallons of paint. How much is the cost of the paint per gallon?

#### How will you solve each of these problems? (which operations will you use, on what part)

The researchers reported that about 50% of the students changed their operations/strategies when the numbers were revealed. This suggests that the students “may rely more on the quantities in a context than they do the context itself.” I was curious to see how my 7th grade students would respond. It turns out that *over *50% of them changed their strategies once the numbers were revealed (some to the correct operation and others to the *wrong* operation), including about 1/3 of them who said they thought they needed to use a different strategy, but they weren’t sure what to change it to.

Keep in mind that this was about 7 weeks into the school year, and 7.RP is the only domain we had addressed in those 7 weeks. And these are 7th graders who covered the 6.RP standards in 6th grade. Clearly, these students need more work on ratio and proportional reasoning. But we can’t keep working on 7.RP at the expense of the other CCSS domains, so I need to find ways to incorporate ratio and proportion into other content standards as often as possible.

So rather than offering resources that primarily address the CCSS Ratio and Proportional Reasoning standards (6.RP, 7.RP, leading into 8.F), my workshop will focus on ideas for reinforcing ratio reasoning while addressing *other* content standards. This is exactly what the CCSS Critical Areas are intended for:

**Critical Areas should be used as a lens through which to view the Content Standards at a particular grade level.****Critical Areas are meant to help teachers****plan meaningful learning opportunities for their students that connect throughout the school year****and form a firm foundation on which to build concepts and procedures in later years.**

I wanted to share my materials and handouts with my workshop participants, and I’ve uploaded them to Google Drive and decided to share them here as well, in case anyone else is looking for ideas. So below you’ll find files and links to materials used in my session. First is the main handout, which outlines all of the content areas I linked to ratio and proportional reasoning. The file includes hyperlinks to web resources I used. Below that is a link to the Google Drive file with copies of all the files I used.

**If you’ve read this far and have ideas for other ways to incorporate ratio and proportional reasoning into other parts of the middle school curriculum, please share in the comments! **

Link to Google Drive folder containing all materials used in the workshop. Note: for ease in sharing between operating systems, I’ve saved all files as PDFs. But I’m happy to share the original files in Pages format (or exported into Word, which might cause the file to lose some of its formatting) if you find something you’d like to adapt for your own use.