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### Simple ways to encourage productive struggle in the math classroom

I had been considering making a meatloaf, so looked up a recipe online that called for 2 pounds of ground meat. Because it would be my first meatloaf, and because I only have 1 pound of ground beef in the refrigerator, I started mentally halving all of the ingredients to get a feel for the recipe.

The recipe called for 3/4 of a cup of breadcrumbs, so I found myself mentally halving this fraction, then thinking about how I mentally halved the fraction, then wondering how others may mentally halve the fraction. So I took to Facebook to ask how others would do this:

For such a seemingly straightforward problem, there was a wide variety of answers. Here are some of them:

- Change it to eighths

- I know that 1/2 of 1/4 is 1/8 and I need 3 of them, so 3/8.

- Convert to a decimal, .75 and then find half of that. 37 + 37 = 74, so half of .75 must be .375 - 3/4 means 3 out of 4 circles shaded. So split all 4 circles into 2 parts, so you have 6 shaded parts out of 8 parts. 1/2 of 6 is 3, so 3/8.

- Just double the denominator = 3/8

- Convert to decimal 1st.

- 4 * 2 = 8 3/8

- I split it into 1/4 and 1/4, then half of a fourth and add together as 8ths, so 2/8 + 1/8= 3/8

- Multiply denominator by 2 and simplify if necessary Funny story I did it today with pancake mix finding half of 2/3 🤦‍♂️, but hey it worked

- Ask Siri…or Alexa

- Equivalent fraction to make the numerator even, then halve it.

- Multiply 4x2

- I know I 3 parts out of four, it’s 3 quarter. It’s 0.75

- Convert it eighths first.

- Divide it in half

- Multiply by 1/2

- Half of .75

- I split 3/4ths in half using a bar model and count the pieces of the half left out of the total new pieces to get 3/8ths

- Double the denominator

- 3/4 x 1/2 = 3/8

- Picturing it as pizza slices.

- Not the shortest way, but my tendency with fourths is to think in terms of quarters. If I had 3 quarters, split in two groups, then I would have \$0.25 + \$0.125 in each group… then I realized keeping it as a fraction would have been more efficient! So 3/4 of a pan of brownies, each 1/4 could be cut in half. Then half the 3/4 would be 3/8. I love helping students think through problems by relating the scenario to money, food, or time!

- Half of 3 is 1.5. denominator stays same

- Half of 3/4 is 1.5/4, double to get the numerator a whole number

- 1/4 + 1/8 for 3/8 - 1/2 of 3/4 means 1/2 x 3/4 which is 3/8

- 1/2 of 1/2=1/4, 1/2 of 1/4=1/8, 1/4=2/8, 1/2 of 3/4=3/8. It seems like a long way around, but I can visualize it almost instantly.

- Convert to 8ths by multiplying both numerator and denominator by 2.

- Convert to an equivalent fraction with an even numerator (6/8). Half of 6 is 3, so 3/8. That's what my brain does.

- Keep the numerator then multiply the denominator by 2. Multiply 3/4 by 1/2 is the same thing.

- Multiply denominator by 2

- Double the bottom. “Half of” multiply by 1/2… So to find half of any fraction, just double the denominator.

- If you can't half the top you double the bottom

- In the real world, I don’t…I just take the 3/4 measuring cup and fill it half way.

- Three one-fourths cut in half is 1.5 one-fourths, which is equivalent to 3 one-eighths.

- .5 x .75 = .375 /2 of 3/4 is 1.5/4 so 3/8

- If I'm dealing with measuring cups/spoons....I split in to 1/2 and 1/4....1/2 of a 1/2 is a fourth and a half of a fourth is an eighth.

- I visualized 1/4 and asked what’s 1/2 of 1/4. Then multiplied by 3 or write 1/8 three times and added to get 3/8.

- 6/8, 3/8

I love all these answers because they highlight just how differently we all approach math in equally valid ways. Some methods may take longer, some may not work for other people, but the variety of the answers is what math is all about. Just maybe not the Siri one.

About a week ago, I saw an Instagram post from Brittany Hege from Mix and Math about the
book study she is spearheading this summer. The book is Productive Math Struggle, by John J. SanGiovanni, Susie Katt and Kevin J. Dykema. I'm a slow reader so was worried about keeping up with the book study, so I got the book to read on my own. Without too many spoilers in case you will be participating in the book study this summer, I want to highlight one sentence from the first chapter of the book:

"Each and every student is capable of doing math on their own and for their own reasons."

This sentence immediately made me think of the equally valid answers to "half of 3/4" from the Facebook post. It also made me think about how great Number Talks are.

Number Talks slow down student thinking and ask students to find answers in their own way. We used them to find percentages in our consumer math class. Each of my students who chose to share out their thinking had a completely different way to find the percentage.

Another simple way to incorporate productive struggle into class is with "See Think Wonder". Like Number Talks, see think wonder slows down the pace of class to focus on students' individual thinking. I used STW at the beginning of our vertex form functions unit in algebra 2. The first was our unit on graphing absolute value functions.

I shined a vertex form absolute value graph on the board along with its equation, and gave each student a STW sheet. Students got a couple minutes to write what they saw, a couple minutes to write what they thought, and a couple minutes to write what they wondered.

Students were able to make connections between the vertex of the function and the numbers in the equation before we even started the bulk of the unit. This independent discovery helped later when we graphed quadratic and radical functions. This could also be used with a linear function graph and its equation to get students making connections between slope, y-intercept and where these show up in graphs and equations.

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