Staircase and Steepness

AHHHH, I love how this lesson has turned out so far!!

Today was our 3rd day of State testing, and because I have 8th graders for homeroom, we still have three more days of testing next week. Ick.

I didn’t want to cram/review again with my 6th graders today, so I thought of doing a lesson on “steepness” (adapted from Swan and Ridgway). I wanted to do this because we had a great lesson couple months ago on “squareness,” and I wrote a little bit about it in this post. Kate Nowak wrote two wonderful posts on a similar squareness lesson, here and here.

I began the lesson by showing a bunch of images, like these, and asked, What do you see?


Kids’ responses: going down, going up, all using their legs, exercise, at an angle.

I made this worksheet during testing. I’d be flattered if you wanted to download the lesson — in pdf steep_ness or docx steep_ness. I purposely wrote the lesson as if it were unfinished because I didn’t want it to end on my terms. I wanted the kids’ conversations and discoveries to guide me to closure, if any.



Question 1: Without measuring the staircases, put them in order of “steepness,” starting with the shape with the least “steepness.”


Question 2: Explain how you came up with your ranking in #1. Because you were asked NOT to measure, what “tools” or strategies did you use to make your decision?

You can see from these photos how they thought of “steepness.”


Question 3: Now discuss your ranking in #1 with another classmate. Are you going to change your ranking? If so, please indicate your new ranking.


Question 4: Now discuss your ranking in #3 with a different classmate. Are you going to change your ranking? If so, please indicate your new ranking.

Six students made no change to their original ranking after talking with two other classmates, 16 made one change, and 9 made two changes.

I was already so happy to see how the activity was coming along. I didn’t realize how much more I was about to learn from Question 5 and beyond.

Question 5: You may now measure the staircases with whatever tool(s) you need. Use the space below to keep track of your measurements, calculations, and notes.

A few kids asked for a protractor, most used a ruler. Most of those who used a ruler measured this length.


Me: Now that you have these measured, what do the numbers mean

Matthew: (Silent, mumbling…)

Me, pointing to staircase F: I see this has the longest length. Was this your steepest shape?

Matthew: Oh no. D was the steepest.

Me: Okay. Shape C has the shortest length. What does that mean in terms of steepness?

Matthew: I don’t think these numbers are right.

I went over to Troi’s desk, she too had measured the same lengths as Matthew did.

Troi: These numbers didn’t do anything for me.

Me: What makes you say that?

Troi: Well, the staircases are all different sizes, you’d have to make them all the same to compare them.

She then measured the rise of each step. I left her to do so.

Rapha: I’m measuring the height, but it depends on the width too.

She didn’t do any more with the two sets of numbers.

Zoe: Can you show me how to use a protractor?

Me: Sure. Which angle do you want to measure?

She pointed to the middle of the staircase. I worked with her for a little while.

I checked on a small group of boys who seemed to be using the protractors correctly.

Mike: We agree on the rest of the ranking. We’re just not sure about B and E. They’re like one degree apart.

Ryan, holding up two different brands of protractors: These aren’t even measuring the same.

Me: Hmmm. Tools aren’t perfect, are they?

I was watching the time; we had 20 minutes left of our 2-period class. I asked them for a “final answer” after whatever measurements they’d done, “Put this ranking at the bottom of your worksheet.”

By then the kids who used a ruler had abandoned the tool for the protractor. Make no mistake, these kids were pretty confident that if Ryan and Mike were measuring the angles, then they ought to be doing the same. More importantly, they noticed that the angle measurements correlated with their steepness rankings.


I asked Miles first — just because he sat front row, right side — for his ranking: D A B E C F. I asked if anyone else had the exact same as Miles’. Twenty hands went up. I asked for Moses’ ranking: D A E B C F. Ten hands went up for this ranking. That only left Sierra with a different ranking. (Sierra was one of six who never changed her original decision.) Because B and E do have the same steepness, all 30 of 31 kids were correct.

Now what? Somehow ending the lesson here seemed weird, even though we had a lot of good conversations. They used angles to figure out steepness. I hadn’t planned this! What about the ruler?!

So I said, What if you didn’t have a protractor? What if you only had a ruler? What would you measure instead?

I then defined lengths on the staircase that could be measured with a ruler so we could all speak the same language about what was measured.


I said, It was great that you figured out steepness using a protractor. But now I want you to figure out how to find steepness using a ruler. Which of these lengths would you measure? Do you need to measure more than one? And if you measured more than one, what would you do with the two/three numbers you have?

They began measuring ferociously, calculators in hands. (Yes, we use calculators all the time!) I didn’t see anyone measuring the slant. I got this much from them by walking around and asking:

Base minus height.

Base divided by height.

Base times height.

Height divided by base.

I ended with, For homework, please finish measuring and calculating for all six staircases. Do your calculations support the ranking?

Troi walked up to my desk when most students had already left, Can I change? I already knew… I did base times height. That didn’t work.


Staircases and Steepness, Continued

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  1. Kit
    Posted October 19, 2014 at 12:59 pm | Permalink

    I’m trying this lesson with my students right now. I’ve worked on it with them for about two days so far (we have 44 minute periods sometimes and 88 minute periods other days, which makes pacing tough), and tomorrow we’re going to conclude this lesson. Only one of my students asked for protractors and measured the angles; everyone else measured with their rulers. I found way more students measuring the slant, and in one of my classes, the students were NOT convinced by their classmates, so we had MANY MORE different rankings. About the only thing we could agree on was that F was the least steep and D was the most steep (and not even completely).

    We concluded on Friday with the notion that the individual measures of the parts weren’t useful, but that we could compare some by making ratios… They were considering which ratios might be useful this weekend, and trying out at least two to see if they could find one that supported their ranking.

    I’m trying to figure out how to guide the conversation tomorrow so that all students will come to see that we can use height/base or step height/step width to get a valid ranking that leads to seeing slope as change in y/change in x.

    My students are 8th graders, and they’re not used to discovering math on their own as much. I think this is some of their first year being pushed to think so much on their own (as opposed to being given formulas).

    Any suggestions about how to guide tomorrow’s conversation?

    • Fawn
      Posted October 20, 2014 at 12:50 am | Permalink

      Hi Kit. Thank you for sharing with me how this lesson is going! My favorite sentence from you is this: About the only thing we could agree on was that F was the least steep and D was the most steep (and not even completely). That’s GREAT! To me that means there’s potentially a lot that your kids can gain from this activity. I’d suggest having kids DISPROVE or BREAK someone’s conjecture by drawing a counterexample. If they cannot, then ask them for more examples of the conjecture, test it for all the staircases shown. I applaud you for pushing the kids to think and persevering with them. I know how tempting it is to lead them to the “answer,” but doing so only robs them of what they can discover on their own. Your questioning strategies are so key. Go ahead and do the calculations with them. Perhaps assign different groups to different sets of calculations, so no two groups are repeating each other’s work. But have them step back (even away from) the problem and look at gigantic hills and gentle slopes. Exaggerate these sketches if you don’t find real-life pictures. Ask them, “What do you notice? What do you wonder?” I’d love to hear back from you.

  2. Michael Waugh
    Posted April 26, 2018 at 12:29 am | Permalink

    I love your stuff. This is my third year using this activity to launch slope. This year, I added a little wrinkle inspired by Ron Lancaster’s work. I went around my school taking pictures of different ramps and stairways and put them in a slideshow. Before showing them the pictures, we went on a “field trip”, and I just walked up and down and around school. It was nice getting outside for bit! When we got back to class, we discussed which paths they chose and why. This led to some interesting discussions around what was easier, faster, shorter, following friends, etc. I explained that we make mental math calculations all of the time, and then handed out your worksheet. Can’t thank you enough for the inspiration!

  3. Farr Quinn
    Posted December 21, 2018 at 8:13 am | Permalink

    Hi, I am just curious, are most of your students operating “on grade level” when they get to your class? I would like to use this activity in my class to promote problem solving and reasoning about the meaning of slope and have a class where most of the students are not currently on grade level, and many who are far below grade level. Are there any scaffolds you would suggest to a teacher who is trying to cultivate problem solving in a classroom where students have generally not been successful with math in the past? Are there any scaffolds specific to this task that you would recommend. Thanks, I always enjoy learning and stealing from you!

    • Fawn
      Posted January 3, 2019 at 11:58 pm | Permalink

      Hello Farr. I really do have a bell-shaped curve class makeup in terms of being “on grade level.” The scaffolds are unique to each group of students. I gauge this by observing and listening to the conversations happening in the first 10-15 minutes of the task. More likely than not, at least one group has an idea that you can ask the group to share with the class. (I’m trying to stay out of it as long as I can, so I find a group that has some leads to share first before I intervene.) Thank you so much for dropping in and for your kind words, Farr.

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