Saturday, February 27, 2010

Teaching Fractional Arithmetic Visually

I was recently asked to teach fractions to a group of kids in my daughter's second grade class. I think most kids are quick to learn the concept of a fraction, and as a consequence, they get bored with the endless drills of coloring exercises. Many stop paying attention, which leads to problems later on. Trying to figure out how to do things differently, I recalled an afternoon two years ago when I taught my daughter how to add and subtract fractions. I figured she that if she understood the concepts of a whole and a part of a whole, she would be able to tackle basic fractional arithmetic. I used visual representations, and she quickly learned how to add and subtract of homogeneous and heterogeneous fractions. Given my prior success, I decided to try this in my daughter's classroom.

Step 1:
Explain that fractions are the part of a whole that you are talking about. Give a ton of examples. This is a critical step before moving on. Kids must understand what a fraction is and how to read and write them using the standard notation.

Step 2: Review how to represent fractions as pictures and vice versa. For example,

Use a few more examples. Teach the kid how to draw squares divided into thirds, fifths, tenths, and a few other common denominators.

Step 3: Explain what it means to add fractions. Say that it means adding parts of a whole. Explain that if we add parts of a whole, it is easier to add pieces of the same size. Give the example of adding 1/2 and 1/4. 1/2 and 1/4 have different sizes. So, how do you tell how much of the whole you have when you put together 1/2 and 1/4? The best thing to do here is to use four blocks of the same size. Manipulate two of the four blocks as 1/2 of a whole. Give the kid enough time to realize that 1/4 is 1/2 of 1/2.

Step 4: Once the child understands that cutting a square into pieces of the same size is the key to adding fractions (i.e. making the fractions homogeneous), proceed with a few exercises such as the following. Draw the two fractions we used before.

Ask your child to figure out how to divide both fractions into pieces of the same size. Tell him that he is not allowed to erase lines already drawn. Tell him that he is allowed to draw new lines, but that the goal is both fractions to divided into pieces of the same size. Clearly, the answer here is to subdivide the 1/2 horizontally as follows:

Now (this is critical), make sure the child understands that how much is shaded in the square on the left does not change simply because we drew another line. It should be clear by now that 1/2 and 2/4 are the same fraction. Make sure this is clearly understood before proceeding.

Step 5: It is now time to learn how to convert heterogeneous to homogeneous fractions. I would suggest easy cases first, followed by slightly more complicated ones. Let's start with the following two fractions. Always draw one fraction using vertical lines and the other using horizontal lines.

The answer should be

By drawing vertical lines in one fraction and horizontal in the other, it becomes clear how to draw new lines to divide both pictures into pieces of the same size.

Let's try one more example.

The answer now is

Make sure your child understands that 3/5 is equal to 6/10. Likewise, make sure it is understood that 1/2 = 5/10.

Step 6: It is now time to introduce the pictorial representation of improper fractions. Ask your kid to draw the following fractions: 3/2, 5/4, 4/2, 5/2, etc. The answers follow:

Do as many examples as necessary until the child is proficient at drawing improper fractions.

Step 7: It is now time to bring it all together to add and subtract fractions visually. Tell the child to draw each fraction in a problem. Ask the child to complete the problem visually. Finally, ask the child to convert the drawing representing the answer to a written fraction.

As you probably realize, you can teach reduction to lowest terms visually as well.

I hope this blog entry helps you introduce fractions faster and earlier than is typically done in schools. It took only 40 minutes to teach a group of second graders in my daughter's school how to add and subtract homogeneous and heterogeneous fractions. All they knew before I taught them was what a fraction is and how to write them down in standard notation.

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