slower learners and upper elementary math

We use MUS and I’ve gotten 3 kids through that level so far. A couple ideas for that particular concept:

1. Make paper pizzas. Draw on paper a square for each 1 whole, draw other square cut into 3rds, 4ths, etc and have him color in how many he is starting with. So in your example problem he’ll have 8 whole squares, 1 squaree cut into 3rds (3 slices) with only 2 slices colored in. Then he’ll have 5 whole squares and one more square cut into 4ths (4 slices) with only 3 slices colored in. Remind him the rule is you can’t put things together that aren’t the same kind. So the 2/3s pizza and the 3/4 pizza are different size slices and can’t be put together. (REMEMBER – you can build 2/3 and 3/4 with the fraction overlays). So we need to change them into the same kind. To do that we cut/convert them to the same denominator (number of pieces in a whole pizza). In this case the easiest way is to change them to 12ths. What do we multiply 3 by in 2/3 to change it to 12? 4!  Whatever we do to the bottom number in a fraction we have to do to his partner on top. So the 2×4= 8, you have 8/12. What do we multiply the 4 in 3/4 by to change it to the same bottom number of 12? 3! Whatever we do to the bottom number in a fraction we have to do to his partner on top. So the 3×3=9, you have 9/12.

When you take 9 pieces of pizza and 8 pieces of pizza you have 17/12. That’s too many pieces to fit in the box (which only holds 12 slices). So we take 12 slices away from that 17 and make it 1 whole pizza (adding it to the 8 whole and 5 whole pizzas we already had =14 pizzas). We have 5 slices left over in a 12 slice box = 5/12.  So we have 14 and 5/12 pizzas.  (You can replace pizza with cake in this example if he tends to think of pizzas as round).

2. Another way to look at this: When we’re adding mixed numbers we add just the fractions first. To add fractions their bottom number must be the same kind (we can’t add pickles and fish, just like we can’t add 3rds and 4ths). But with fractions we can use an equivalent fraction to mean the same thing. Remember, 1/2 a cake is the same as 2/4 of a cake or 3/6 of a cake or 11/22 of a cake. We’re still eating half the cake, we just cut the pieces smaller and smaller each time. So we’re finding equivalent fractions for 2/3 and 3/4 that both have the same denominator. To do that we can skip count (since they are small numbers it’s fastest that way). What is the first number that 3 and 4 both have in their skip count? 2.4.6.8.10.12.14…. 3.6.9.12.15…. wait! It’s 12.  So we’re changing both to 12ths. Now we look at each fraction alone. 2/3 = ?/12. What did I multiply 3 by to change it to 12? 4. So I now multiply his partner/numerator by the same thing. 2×4=8. 8/12.  Looking at 3/4=?/12. What did I multiply 4 by to change it to 12? 3. So now I multiply his partner/numerator by the same thing. 3×3=9. 9/12.

We can add the same kind of thing, both are the same now (12ths), so we add how many 12ths we have (8+9=17) 17/12. We know when the top number is bigger than the bottom that we have too many pieces for the 12 to hold up on top. We are going to take away 12 pieces, which makes 1 whole cake/thing. It moves to a new place value space, the whole number. How many pieces are left? 5. What kind of pieces? 12ths. So we have 5/12.

Now we look at all the whole numbers in our problem. We have 8 and 5 from the original problem and 1 more from adding up our fractions. 8+5+1=14 whole.  14 and 5/12 is what we have.

3. Really, if he’s struggling with fractions (which is totally normal!) he can’t picture them in his head. Go back to practicing making equivalent fractions. Go back to building each problem’s fraction part with the fraction overlays for the new work – even if he hates using them. Because they make it so you can SEE that 4ths and 3rds are different but that if you cut them/multiply them by 3 and 4 respectively they are now the same kind of pieces (12ths) and can be combined.  And really, if possible, bake some pizzas or cakes and cut them into different fractions (equivalent ones). So one cake/square pizza gets cut in half. The next in 4ths. The next in 3rds. The next in 6ths. See how many pieces of one will make the same amount as another. 3rds and 6ths will work together. Halves and 4ths. or halves and 3 of the 6ths. You can do this with large pieces of square paper too, but let’s be honest, pizza or cake is going to be a lot more interesting to a teen.

Isn’t it amazing how different they each are?!