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One of the most difficult tasks as a math teacher is how to bring abstract math down into a more concrete, representational conceptual understanding. One of these daunting concepts can be found when we ask students to find "4 is 16% of what number". Instead of just doing the procedure of having set up the proportion 16/100 = 4/x and then cross-multiply, how do we get them to understand why 4 is 16% of 25. I have attached pictures to help explain this concept from a conceptual perspective.
First students must understand that when we say 4 is 16% of what number, that 16% is out of 100%. That is why I choose to use a blank 10-by-10 chart. This shows 100 pieces or the one in this case. So if we break down what we know: 4 is 16%, we can actually show this on the hundred chart. We can continue to find 16/100 and mark it as 4. When finished we can count how many fours we came up with. Another words, 4 x 6, but wait a minute, that's only 24. I thought the answer was 25! When there are four pieces left over we have to make the connection that those four pieces are ¼ of the 16 needed to make another four. So ¼ of four is the "1" that we add to 24.
Obviously I would start students with fractions that come out a little nicer. For example:
8 if 50% of what number?
Although we don't want students to draw this every time, they will need to for a while before they start to see, "Oh, I just need to double it." Or when working with 25% I just need to take the number multiplied by four. All these connections exist but it is best when students make the connections through guided practice and questioning of the teacher. This helps students to make the conceptual understanding so that when they move into the abstract they have a visual in their head of what the answer should be about. I've attached a blank 100 grid for your use. If you have other wonderful ideas of how to move students from procedure to concept, please don't hesitate to email me at firstname.lastname@example.org