Area and Perimeter

The new unit is over Covering and Surrounding (area and perimeter).  We used the formulas for area and perimeter or a rectangle to achieve this.

a = l x w                 p = 2 ( l + w )

We rearrange or use fact families to isolate (get a letter by itself) in order to solve for our missing letter.

As you can see, I have every variable by itself on one side of the equal sign.  We do this by doing the opposite of what it tells use to do on the other side of the equal sign.  Look at 

a = l x w

If we do the opposite of multiply (x)  which is divide, we can get 

a ÷ l = w     or    a ÷ w = l

Perimeter is a little more complicated because we have two operations: multiply and add.  Same rules apply though.  The 2 outside the parentheses means to multiply, so we just divide that 2 on the other side of the equal sign.

p ÷ 2 = l + w.

Now we just subtract one of the other letters.

p ÷ 2 - w = l     or   p ÷ 2 - l = w

In class we used the second one, so for this example that is what we will use.  It doesn't really matter though.  

I like for the students to organize their problems by variables on the left side of their problem using 

p =

l = 

w =

This keeps everything organized.  If we know a perimeter, we can substitute or replace the p with this number.  Let's use 36 cm.  Then we can guess at the length (l).  For the example, I started with 2 cm because 36 cm is even and so is 2 cm.  Look at the example.  Notice my organization on the left and my formula at the top.  Then follow the steps.

The only step I did here was to substitute what I know, or change p and l to 36 cm and 2 cm.  Next I will complete the math.

I did the math in two steps and kept in mind order of operations.  I completed 36 cm ÷ 2 and got 18 cm.  Then I subtracted 18 cm - 2 cm to get 16 cm.  This tells me that when my width is 2 cm, my length will be 16 cm for a rectangle with a perimeter of 36 cm.

In class, our job was to find ALL the possible lengths and widths for a given perimeter.  We will do this again.  This time, I will guess a length of 3 cm.

For a length of 3 cm and a perimeter of 36 cm, I will have a length of 15 cm.  Notice my organization on the left and each step done separately on the right.  Also notice that my equal signs are lined up.  THIS IS MEGA IMPORTANT TO KEEP YOURSELF ORGANIZED.  If we kept on doing this, we could complete a table.  Here is the finished table.

Dimensions	Length	Width	Perimeter	Area
1 x 17	1	17	36	17
2 x 16	2	16	36	32
3 x 15	3	15	36	45
4 x 14	4	14	36	56
5 x 13	5	13	36	65
6 x 12	6	12	36	72
7 x 11	7	11	36	77
8 x 10	8	10	36	80
9 x 9	9	9	36	81

Once the length and the width are the same or close to the same, you have found all possibilities.  You can then switch the lengths and the widths and have the same.  (so instead of 1 x 17, you can have 17 x 1 and have the same perimeter and area).