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Q&A #117


7th grade - learning percents

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From: Cindy Wilkins (for Teacher2Teacher Service)
Date: May 24, 1998 at 23:37:17
Subject: Re: 7th grade - learning percents

I use base ten blocks followed by a diagram to model percent problems with my 
students (7th and 8th grade). Since e-mail is not set up to handle diagrams, 
I'll do my best to describe the procedure (it's really much easier to do than 
to explain).

Part A: using base ten blocks (estimated time: 5-10 minutes)

  the 100 square stands for 100
  the 10 bars represent 10's
  the units represent 1's

I start by asking my students to restate 20% as a ratio. When I get the answer 
20/100, I ask them to model it with the blocks, EXACTLY AS WRITTEN. They place 
2 ten bars on the table with the 100 block directly below so that they have a 
rectangle 10 units wide and 12 units tall (the 100 plus the 2 tens). Do not 
stack the blocks on top of each other. Each block must remain in contact with 
the table. We do a few more so they are comfortable with the idea (it doesn't 
take long).

Give a simple problem, such as 20% of 30 is what? 

First, model 20% (already done from the introduction). Then, where is the most 
logical place to put the 3 ten bars, which represent 30? The vertical axis of
the 100 block is already being used by the 20%, so you must use the horizontal 
one. Place the ten bars along the left edge of the 100 square (which side 
really doesn't matter - just be consistent). You now have a rectangle that 
measures 12 units up and 13 units over, except for a space in the top left 
corner. The number of units it takes to fill in the space (for this problem, 
it's 6) is the answer to the problem.

Give 2 or 3 more problems like this one (e.g. 40% of 20; 30% of 10; 60% of 40). 
Note that each number is a multiple of 10.

Second step - give problems like what 20% of what number is 6? Model the 
problem. You already know how to model 20%. Align the 6 unit cubes with the 
2 ten bars. You now have a space that needs filling. Since 3 ten bars will fill 
the space, the answer is 30.

Third step - what percent of 30 is 6? Place the 100 square, align the 3 ten
bars to the left, and align the 6 unit cubes on top. To complete the rectangle 
you need 2 ten bars placed over the 100 square, so the answer is 20%.

Part B:  What about all the other numbers? (estimated time: 10-15 minutes plus 
practice problems)

Model the entire equation 20% of 30 = 6. You have a rectangle that can 
logically be divided into 4 parts: the 100 part, the % part, the 'of' part, and 
the 'is' part. Draw a square and divide the square into 4 equal squares.  Label 
the squares (I'll try to draw a picture).

   ------------
   / is /  %  /
   / of / 100 /
   ------------

Draw another square just like the first and label it with the parts of the 
problem. The 100 always goes in the bottom right corner.

   ------------
   / 6  / 20  /
   / 30 / 100 /
   ------------

Cover up one of the problem numbers (e.g. 6)  How can you use the three numbers 
that remain to get the answer 6?  In a classroom I get many answers and I 
require that the exact answer work when I cover up a different number. After a 
brief discussion, the students realize that all they need to do is multiply the 
numbers on the diagonal and divide by the one left over. So, if the 6 is 
missing, you say 30 x 20 / 100. If the 20 is missing, you say 6 x 100 / 30.  
If the 30 is missing you say 6 x 100 / 20.

Once the students see the 'box' method, as they call it, they always ask why I 
made them do the blocks first when this is so easy. I've found that the box 
method is only easy if the blocks are used first, because otherwise the 
students don't know where to place each number or why. I also like this method 
because you don't have to worry about remembering a bunch of rules such as 
moving decimals.

If this explanation isn't clear, please let me know and I'll try to clarify it.  
Last year, one of my teachers used this with her lowest functioning class.  
After one day of instruction and practice, she gave a test and had 100% of her 
students pass.  We were all very impressed!

Cindy

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