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Notice and Wonder Warm-Up Routine That Works in Any Math Lesson

  · @MrPurdyMath

Display a picture, like of a roll of cardboard on the board and ask, "What do you notice?" Within a minute, hands go up all over the room, including from students who rarely talk in math class. That's the power of Notice and Wonder: every student has something to say, and no answer is wrong.



This routine comes from the National Council of Teachers of Mathematics (NCTM), whose Notice and Wonder lesson plans pair a striking photo with real math. Their cardboard lesson (grades 6–8) is a great example. Its first part works as a five-minute warm-up in almost any lesson. When you have more time, the second part turns it into a full lesson on circles and cylinders.

The 4 steps

Here's the whole warm-up, using NCTM's photo of a rolled-up sheet of corrugated cardboard. The sample responses come straight from the lesson.

  1. Look silently for one minute. Show the image and say, "Take a minute to carefully look over the image." No hands yet. Everyone gets time to really see it before anyone talks.
  2. Ask: What do you notice? Every observation counts, so write them all down. Students might say:
    • It's cardboard.
    • It's wavy or grooved.
    • It's rolled up.
    • It's one large sheet.
    • There are 10 layers.
  3. Ask: What do you wonder? You're collecting questions here, not answers, so keep them for later. Students might wonder:
    • How many grooves are there?
    • How is cardboard made?
    • How long is the sheet?
    • What is it used for?
    • Why is it rolled up?
  4. Ask: What math questions could we ask? This is the step that turns curiosity into math. Help students reshape their wonders into questions that numbers can answer:
    • How thick is the cardboard?
    • How long is the roll of cardboard?
    • What percent of the cardboard circle can we see?

That's it. For a warm-up, you can stop here.

Making it work as a warm-up

  • Swap the image to fit your unit. The four steps don't depend on cardboard. A photo, a graph or a pattern connected to today's topic works just as well.
  • Protect the silent minute. It feels long, but it's what gives every student something to say, not just the fastest thinkers.
  • Write everything down. Recording every notice and wonder, word for word, tells students their thinking counts.
  • Don't answer the wonders yet. Keep the list posted. It makes a great hook for the lesson that follows, and students love seeing their own questions come back.
  • Use think, pair, share. Thirty seconds alone, thirty seconds with a partner, then share out keeps the pace up and the whole class involved.

Want more than a warm-up? Use Part 2

If you have a full class period, Part 2 of the lesson provides the questions, assessment and extension to use this as a full lesson. It picks up right where the warm-up ends. Students learn that this is corrugated cardboard, a wavy sheet of paper sandwiched between flat sheets, and that its thickness is measured with a tool called a caliper.

The scenario: the cardboard is 1/4 inch thick, it's rolled up 10 times with no extra space, and the photo shows exactly one-quarter of the circle.

Guiding questions. Students work through a chain of problems that build on each other:

  • Find the radius and diameter of the cardboard circle.
  • Find the area of the whole circle, then the area of the quarter shown in the photo.
  • If the sheet is 6 feet tall, name the shape the roll makes (a cylinder) and find its surface area and lateral surface area.
  • Design two different nets for a box made from a 6-foot by 6-foot sheet of cardboard.

Assessment. A student handout covers every question, and the teacher version includes an answer key. The box-net task is open-ended, so it's a nice window into how students think about surface area.

Extension. Students notice and wonder about a graph of cardboard recycling rates from the American Forest and Paper Association, then predict what the rate will be in future years and defend their choice.

Along the way, students practice the area of circles, surface area of cylinders and nets, and they apply formulas to a real-world object.

Get the lesson

The cardboard lesson is part of NCTM's Notice and Wonder Lesson Plans, a PK–12 series built around images that spark curiosity. NCTM members get the full instructional plans, activity sheets and presentation slides for each lesson.

Get the free lesson plan here: National Council of Teachers of Mathematics (NCTM), Notice and Wonder: Cardboard, grades 6–8

Want the warm-up steps in your pocket? Swipe through my Notice and Wonder carousel on Instagram at @MrPurdyMath, and save it for your next warm-up.

What would your students notice first? Tell me in the comments.

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