Explorations in Perimeter: Using What You Know to Help Solve Problems

Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

I’m very excited to share that I am stepping into a new role on my campus next year.  I will be moving from the instructional coach role (working with teachers) to the academic support role (working with students).  I’m super psyched about the change!  I got a little glimpse of the future today as I subbed for the current academic support teacher and worked with a group of 5th graders on perimeter.

I started our session by asking students what they knew about perimeter. There were two reasons for this.  First, it’s important to activate prior knowledge, but it also served as a formative assessment for me.  Since I don’t routinely work with these kiddos, I wanted to quickly determine what they already knew about the topic.  There was also, actually, a third reason.  I love to hear students talk about math.  I want to give them opportunities to engage in mathematical discussions and develop academic vocabulary.

I got lots of response–it’s the distance all the way around the shape, you add all the sides to find it, there’s a formula for it, etc.

So I drew a rectangle on the whiteboard and labeled it as shown.

I told students that mathematicians use what they already know to help them solve problems, and then I asked them to tell me what they knew about this figure that might help them find the perimeter.  They knew it was a rectangle.  Great! So, what do you know about a rectangle?  Two sides are the same and the other two sides are the same.  We added that information to the drawing as I explained that mathematicians like to get all their thinking down in writing.

Notice how I continually stress the habits that mathematicians employ in their work.  With the model drawn and labeled, I asked the students to mentally calculate the perimeter.  I then asked each student, in turn, for their solution.  I didn’t just call on one student, I let each one respond.  After they told their solution (which most correctly calculated as 22 in.), I asked them to tell me how they calculated it. Some said they added 7 + 7 and then 4 + 4 and then added the sums together to get 22.  Others multiplied 7 x 2 and 4 x 2 and added the products together.  As they explained, I added that academic vocabulary.  Then I added, You know how I did it?  I added 4 + 7 to get 11.  I knew if these two sides had a sum of 11, then the other two sides did, too.  I like doubles, so I did 11 + 11 and also got 22.  They liked that.  Later, as we did additional problems and they told me how they solved it, I asked them how I would have solved it, and they were able to explain my doubling strategy. 🙂

Time for more of a challenge.  Next, I drew the following problem on the whiteboard.

Again, I asked them to share what they knew.  They said they knew the length was 8 inches.  They knew it was a rectangle, so they knew the other side was also 8 inches.  Through discussion, we determined that while they didn’t know what the right side measurement was, it had to be the same as the ? side.  They knew that 8 + 8 = 16.  Most determined that 16 + something had to equal 22–interesting that they thought of adding up instead of subtracting, right?  As we discussed getting our thinking down on paper, and the fact that adding all the sides had to equal 22, we wrote an equation that looked like this:
And we determined the ? had to equal 3.
What?  You need MORE of a challenge?  I can do that!  Hmmmm, how about this one?

Talk about wide eyes!  So I brought them back to what do you know?  Well, we know the measurements of the other sides, so we wrote them in.

I again asked them to mentally calculate the perimeter, and I called on each one in turn.  I got some WILD answers!  So I told them I was going to do something that would seem almost magical, and I covered up the fractions.

Then I asked, okay, what’s the perimeter now?  10. Then I uncovered one of the fractions.  How about now?  10 1/2.

I uncovered the other fraction.  And now?  [big smile!] 11!  Now, I can’t say that each of the groups got it that easily, which again informed my instruction.  With some groups, I had to do a mini-lesson on skip-counting by fractions. Which was a bonus!  In a lesson on perimeter, I got to help build their fraction number sense.
Here’s the point–I asked the kids why that last problem stumped them.  Why was it so much harder?  Of course they said it was because of the fractions.  Which set me up perfectly for again making my point–mathematicians use what they know to help figure out what they don’t know!  I reminded them they knew a LOT in this problem: they knew about the properties of rectangles, they knew that the perimeter is the distance all the way around, and they can easily find the perimeter with whole numbers–so we just ignored the fractions for a moment.
What a wonderful day!  Can you tell I’m going to like this new job?
Looking for meaningful area and perimeter practice?  Check out my Perimeter, Length, and Area Task Card set.


  1. Primary Teachspiration

    I can tell you will do a wonderful job in this position! I’m happy for you!

    • Donna Boucher

      Thanks so much! Remediation truly is my passion. I was truly so fortunate that this position opened up at my own campus!

    • B3achHouse

      Great that you’re so happy about your move! You have so many wonderful ideas to use with the lucky children! Congratulations!

  2. Rebecca H.

    I love the whole process you took the kids through. I wish we had you as an interventionist at our school! Congratulations on getting a position you will love!

    Ladybugs Lounge

    • Donna Boucher

      I find remediation fascinating, Rebecca. You never know exactly what direction it will take, because it’s guided on the spot by the students’ responses! It’s kind of like teaching improv. Ha ha.

  3. Resource Angel

    This explains the process so clearly! It is so important to build on prior knowledge!

    • Donna Boucher

      And it’s also so, SO important to communicate to students that they do know a lot! Kids in remediation in 5th grade are often defeated because it’s always about what they DON’T know. We need to remind them all the time that there is lots they DO know!

  4. Christy

    I’m so happy that you will be in this new role! Your kiddos will be so blessed. Can’t wait to read more about all the goings on in your classes!

    Teaching Tales Along the Yellow Brick Road

    • Donna Boucher

      Thanks, Christy! I’m sure I’ll have a lot to write about!

  5. Rochelle

    I LOVE this! We don’t have a math coach, or interventionist, at our school. Thank you for being so generous with your lessons and materials!

    • Donna Boucher

      It’s my pleasure, Rochelle! Sharing is what it’s all about!

  6. Tanya Smith

    Thanks for sharing! Its’ so important for children to make connections and recall what they already have learned. In my second grade classroom we often stop to practice thinking about what we know and then figure out what we need to find out. It is powerful to help children own their learning and make it meaningful.

  7. Kim Andrews

    I love the way you built on prior knowledge. I express to my 4th grade students that they know more than they think. I would love this job. Out of curiosity, how is your schedule set up? Do you have scheduled time to visit the classrooms or do you rotate through the classes at an intervention time? Thanks in advance!

    • Donna Boucher

      The kiddos actually come to me, Kim. We have “extended learning time” built into our schedule on my campus. This allows students who need pull-out services to be serviced without missing first-line instruction and the classroom teacher has time built into her schedule for differentiation.

      • Kim

        Thanks for the quick response. We also have ELT at the end of the day (30 mins). This has been helpful. I look forward to your future posts.


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