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Q&A #117 |
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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