In an email to me after her trip to the supermarket, my friend Ann wrote:

*I asked for one-fourth of a pound of the already sliced turkey. The deli person pulled out a huge hunk of turkey and I saw .40 on the scale. As the deli person was about to wrap the hunk, I said, “That’s too much; I can’t use that much fast enough.”*

* She looked surprised and removed one slice.
*

*“It’s still too much—point forty is nearly half a pound,” I said and added, “One-fourth is really point twenty-five, but point thirty would be OK.”*

*“No,” the deli person insisted, “one-quarter is point twenty-five, but one-fourth is point forty.”*

*I couldn’t stop myself and said, with as much authority as I could, “A quarter of a pound and a fourth of a pound are the same thing.”*

*She looked amazed . . . and skeptical.*

*Marilyn, you have your work cut out for you.*

Ann’s message is a reminder that all of us as teachers have our work cut out for us.

I’ve had similar experiences myself. When shopping in Macy’s during a “25 percent discount taken at register” sale, a woman with a $24.95 sweatshirt in hand asked me if I knew how much the discounted price would be. I almost slipped into teacher mode, thinking about questions to ask her to help her figure it out for herself. But I caught myself. She didn’t want a math lesson— she just wanted the information. So, in my head, I figured that a fourth of $24 is $6, so she’d save a little more than $6. I subtracted $6 from $25, and said to her, “It will be just under $19.”

And one more. I treated myself to a pedicure a few days ago, a welcome break from my computer. I overheard the woman in the chair next to me ask, “Do you do head and neck massages?” The woman doing her nails answered, “Yes, it’s $15 for ten minutes.” The woman in the chair then asked, “How much would it cost for fifteen minutes?” There was a long silence. A really long silence. The woman doing her nails had stopped working and both of their foreheads were scrunched with that look I see on students who are thinking hard. My friend Ann was right—we do have our work cut out for ourselves.

As teachers, we have our work cut our for us when teaching math.

I’ve thought about the deli person’s confusion, the reason why the woman in Macy’s didn’t feel she could figure out the sale price, my experience at the nail salon, and situations like these that come up with students. When teaching in the classroom, it’s important that we probe to understand how students think and reason, and that we encourage them to persevere to find solutions for themselves. To that end, we need to know what students know, what they haven’t learned yet, and what misconceptions they have. In the classroom, this calls for having students explain their reasoning for answers they give, whether those answers are right or wrong. When teaching, I’m diligent about asking students, “How did you figure that out?” And I’ve found that after a while, students automatically explain their reasoning as part of their answers, without being prompted. Then my task shifts to listening carefully to what the students are saying and asking questions if I need to in order to understand their thinking. This helps me know whether my students are prepared to handle the everyday math problems they’ll encounter in real life with confidence or, if not, what I need to do to get them there.

## Classroom Suggestions

Below are two ideas for students who have studied about fractions and percents, and a third suggestion about how to adapt my Macy’s experience for younger students.

- If your students have experience with fractions and percents, read Ann’s email to them. Ask them how they might respond to the woman at the deli counter. Have them share their ideas with their partners. And then lead a class discussion for them to report their ideas. (This think-pair-share instructional strategy is a staple in my teaching.) I like ending lessons with writing assignments. Here, give a sentence frame for students to complete:
*To help the deli worker understand, I would . . .*

- A problem like the one I faced at Macy’s presents an opportunity for students to engage with mental math, which is important for helping them develop and hone their numerical reasoning skills. If your students have studied about percents, give them the Macy’s discount problem and ask them to figure out mentally about how much the discounted price would be. Then talk about their different strategies. Again, have them think alone first, then talk with their partners, and then report to the class.

- Simpler versions of the Macy’s problem can work for younger students, using a dollar amount for the discount instead of a percent. For example:
*If a sweatshirt costs $25 and is on sale for $19, how much would you save?*Choose numbers that students can deal with in their heads. I’ve found it helpful to set up the word problem by writing on the board:

*A sweatshirt costs $_____.*

*The sale price is $_____.*

*You save $____.*Then I can fill in numbers that are appropriate and also leave different amounts blank to vary where the unknown is. Following are three versions of the sweatshirt problem using different numbers, with the unknown in a different position in each:

*A sweatshirt costs $25.*

*The sale price is $19.*

*You save $____.*

*A sweatshirt costs $30.
*

*The sale price is $____.*

*You save $7.*

*A sweatshirt costs $____.
*

*The sale price is $15.*

*You save $4.*

Finally, for an individual assignment, ask students to copy the problem with three blanks, decide on numbers to use for two of the blanks, and solve the problem mentally. Ask them to write an explanation of how they reasoned. Then have students share their thinking.

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