# Think Outside the Box

### 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.

Perimeter and area.  Two words that strike fear in the hearts of 4th and 5th grade teachers everywhere!  Common problems:

1. Kids find area when they are asked for perimeter and vice versa.
2. Kids see two measurements on a rectangle and only add those two measurements for perimeter.
3. Giant problem–kids can’t connect the two and see how knowing the area can help find the perimeter.
Consider this problem I presented my 3rd grade tutoring group this week: Find the perimeter of a square whose area is 25.

There was no picture, but they actually all knew they should represent (draw) the square (Yea!).  But what to do with that 25?  Of course, they all tried to write 25 on each side.  So I asked them if the problem said that each side was 25.  No, the area is 25.  Hmmm, what do we remember about area.  It’s the inside.  Right. So where would the number 25 go in our drawing? Inside the square (see Fig. 1). Exactly!  Do you remember how you’ve found the area in the past?  We counted squares.  Ah, how many squares would our rectangle need?  25!  Exactly.  Work with your partner and see if you can figure out what that would look like.  Great!  I see you have made 25 small squares covering the big square (see Fig. 2).  Now, how will we find the perimeter?
Note:  This was where it derailed!  They did not realize that the length of each side was 5 from looking at the drawing.  I questioned and probed and led them, but they did not make the connection that the lines they had drawn represented the units in the length of each side.  So, I asked them to pretend the lines they saw were the markings on a ruler, and then they got it (see Fig. 3)!  Fascinating to me, but then remediation always is.
For the rest of the lesson, I drew several more rectangles (see Fig. 4), and we explored the area and perimeter of each.  We noticed some cool things!
• Two rectangles can have the same perimeter and different areas.
• The perimeter and the area of a square can be the same.
• A square and a rectangle can have the same area.
• And, finally, there is a multiplicative relationship between the length and the width and the area (okay, so I helped them see that one).
What’s the take-away from this?  First, kids need concrete practice with both area and perimeter.  When they are using square tiles to cover a space, we should also have them find the perimeter.  But the bigger idea is that kids need to be able to explore and see connections and generalizations.
The slide in Fig. 4 is part of a SMART Notebook file I posted yesterday.  Click here to grab yours.
Happy Wednesday, all!
 Fig. 1
 Fig. 2
 Fig. 3
 Fig. 4

1. Hi These cues helped my students that were confusing the two -the word area looks like arena- how many fit inside the arena? ( rows?) and peRIMeter- has the word rim- the outside edge ( how far all the way around the rim/outside… back to the start?).

• Yes, I do hand movements, too. To show me perimeter, they trace around a square in the air with their finger. For area, they wave their hand like they’re coloring in the inside of the square.

2. Ooh, I like the RIM tip!

I also had my students just play a game where they moved around the room either in the area, or along the perimeter. I kept switching back and forth and giving them different directions just to get the vocabulary to stick in their heads (and to get them moving as a brain break)!

• Awesome! Anything to get the information into their brains in a different way, and movement is great.

3. Wow what a great idea! I could even modify it a little to fit my 3rd graders..
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