Staircases 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 or docx. 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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  • May 5, 2012 7:53 PM Mike wrote:
    I just wanted to thank you not just for your great lesson ideas and resources, but for your lesson synopses -- particularly the questions they asked and your responses. It helps to prepare for where my students might get stuck, as well as what I can (hope to) not have to suggest.

    So... Thanks!
    Reply to this
    1. May 6, 2012 8:43 AM fawnnguyen wrote:
      I'm so glad this is helpful. It'll be a good reminder for me too when I do the lessons again next year. Thank you so much, Mike, for leaving your kind words.
      Reply to this
  • May 6, 2012 4:28 PM Matt wrote:
    Thank you for sharing the digital documents and lesson idea. I'll utilize this resource in the next few days with my upper elementary students. What a great way to incorporate problem-solving skills in math class.
    Reply to this
    1. May 7, 2012 7:23 AM fawnnguyen wrote:
      You're very welcome, Matt. Wonderful that you could use this lesson! Thanks for stopping by.
      Reply to this
  • May 22, 2012 2:01 AM Kate Nowak wrote:
    Hi Fawn - this is great! I really like how you recorded data like how many kids changed their rankings, what observations they made - how do you do that?
    Reply to this
    1. May 22, 2012 10:03 AM fawnnguyen wrote:
      Thank you so much, Kate!
      Reply to this
  • December 3, 2012 1:17 PM Jeff wrote:
    Your use of questioning to direct student thinking is amazing. Thanks for including how you responded to students' incorrect thinking. I have to get better at that.
    Reply to this
    1. December 3, 2012 1:56 PM fawnnguyen wrote:
      Thank you, Jeff! I loved how this lesson turned out. The kids were the ones who drove the lesson where it went. Nothing was really planned out by me (shhhhh), I was just going with the flow. Appreciate your dropping in!
      Reply to this
  • May 9, 2013 3:26 PM Shane Loader wrote:
    Hi Fawn

    I have just recently completed this task with my class and it worked perfectly. What I really love about the task is it taps into students intuition about the idea of slope, they all know what is steep and not steep, it just gets them to think about the criteria that they are somewhat unconsciously using to rank them. The checking of their understanding with other students also allows them to refine their understanding of the concept and maybe be presented with a perspective that they may never have thought of. It was amazing to see how well the students understood the concept of slope without any explicit teaching in that area. These are the type of activities I love to regularly try to do with my classes, this is definitely one I will hang on to for a few years to come
    Reply to this
    1. May 9, 2013 7:35 PM fawnnguyen wrote:
      Hi Shane. You won't believe this. I was asked just last week to pick my one favorite lesson from my blog, and Stairs was the one I chose!! I love love this lesson. Just like you said, all kids understand that it's really tiring to walk or bike up a steep hill, but what makes it steep is something they haven't questioned or explored, and it's fun to be a part of this discovery. 

      I'm so glad I thought about them sharing their rankings with other classmates and re-ranking if they needed to. The conversations were gold!

      I so appreciate you dropping in to let me know, Shane. Thank you!

      Reply to this
  • September 10, 2013 2:28 AM Kathryn wrote:
    Thanks so much for sharing this. I had been talking with colleagues about introducing slope to our 8th graders by comparing staircases but we are in a single-story school! This is a great way to get the kids developing their concept of steepness in preparation for understanding slope. I'll blog about it when I've done the lesson. Thanks again!
    Reply to this
    1. September 12, 2013 8:03 PM fawnnguyen wrote:
      You're very welcome, Kathryn. I really loved this lesson. I think the kids got a lot out of it. Extra nice when someone else could use the lesson also. Look forward to your post on it! Thank you, Kathryn.
      Reply to this
  • November 21, 2013 4:16 PM Matt wrote:
    I decided to use this to pre-teach slope and the kids were begging to know how to define "steepness" and use rulers because they couldn't agree on which were steeper! Thanks for the awesome activity!
    Reply to this
    1. November 22, 2013 10:39 AM fawnnguyen wrote:
      Thanks for letting me know, Matt! YAY!!
      Reply to this
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