Day 4

Adding and Subtracting Fractions

Today was raining and dreary and so we returned to our circles.  We focused primarily on adding and subtracting fractions.  I began with our biggest fraction-the half- and we added those.

I began with 1/2 + 1/2 and then 1/2+1/2+1/2.  Then we moved onto to more complicated things like 3/2 + 1/2. Then we “reduced” by finding the mixed fraction.  We repeated this with thirds and then fourths.

After fourths we would pause to “notice.”  Though I saw that she understood that the fractions were added just as whole numbers, she didn’t express that.  What she did express is that she noticed that it takes the same number of fractions as the denominator to make one, i.e. it takes 8 of 1/8 to make one. I thought this was a significant observation.  I noted this in her math journal, and we moved on.

From here we moved onto subtraction.  We worked with the smaller fractions since there were more of them to work with.  I took a turn giving her a problem that she would work out and then find all the equivalents, and then she would give one to me.  After doing this a few times, we stopped to make more observations. I noted that when we added fractions, the numerator changed, but the denominator did not.  We have not added or subtracted fractions with different denominators. If it comes up in our play, I’ll go with it. Otherwise I’ll wait until next year.

Day 5

Multiplying fractions and whole numbers

In exploring the concept of multiplying fractions, instead of just showing her how this works with manipulative and numbers, I make a list of (boring, yet) systematic calculations.  We started with our largest fraction 1/2 and worked through our circle fractions of 1/3, and then 1/4 until a pattern of fractions in general began to emerge for her and also making sure to keep her attention. I will say, seeing all the calculations listed out was a little overwhelming to her, but I assured her once she found the pattern, it would go very quickly.

We begin with a column multiplying each whole number by our fraction.  In our example below, the fraction is thirds.  In doing this, she quick sees a pattern and is able to finish the rest of the problems with ease.

Then we move on to multiplying by 2 of our fractions, in this example 2/3 by the whole numbers 1 through 6. Again, a pattern begins to emerge.  She noticed that this time we are skip counting by 2’s with a third as the unit.

From here we move onto 3/3.  As you may have noted earlier, she had already noticed that it takes the number of fractions as the denominator to make a whole. In this example 3 of thirds to make one, so it came very quickly to her that we were just multiplying by 1.

An in our last column we multiplied with a fraction greater than 1, in this case 4/3. She noticed that here are we skip counting by 4’s or multiplying by 4 and the whole number in the units of thirds.

After we did this systematic calculations with halves, thirds, and fourths, we moved onto guessing and figuring out calculations with other fractions.  We did at least one calculation with each of our other fractions: sixths, eighths, twelfths, and sixteenths. After calculating, we would use our fractions to make sure our answer was correct.  Note in the picture of the dark blue sixths, that we did not have enough paper fractions and we move into drawing the figure.  This is a great tool to use for calculations through even high school and college. Many, like me, find visual representations very helpful.