Week 3: Multiplication and Division of Fractions

This week was a busy week! We learned methods for multiplication and division, and a model for multiplication  The area model is very useful when multiplying fractions. In fact, it might be my favorite way to work through fractions now! It's quite easy. Area model for fractions is nearly identical to the area model with whole numbers.

Area model for whole numbers will generally be with an open grid:

The numbers inside the box, when added up, give the answer to the problem 67 X 60.

The area model for fractions is similar, where the top side of the box is one fraction and the left side is another fraction.

This area model represents 1/2 X 3/5. The top fraction, 1/2 in red, was the first cut to be made in the box. Then only 1/2 was shaded because the problem asks for one 1/2 pice . Then on the left side, the box was then cut into 1/5 pieces. The second number asks for three 1/5 pieces, and that is shaded in blue. The mixed-shaded colour, purple, is the answer, 3/10. 

The actual computation of 1/2 X 3/5 equals 3/10, and that is done without a visual by multiplying numerators together and denominators, respectively. 

Mixed numbers can also be represented in the area model. An open area model is best used for mixed numbers. 

The mixed numbers were broken up, much like whole numbers are on a mixed number. When multiplying whole numbers by fractions, simply putting a 1 underneath the whole number turns it into a fraction, so that multiplying across the top and bottom are easy. Then, just like in the whole numbers area model, you add up all of the numbers inside the box.

10 + 4/8 + 5/3 + 2/24 = 12 3/12 (simplified is 12 1/4) 

Next, we have division of fractions. The way our elementary teachers explained it to us a little kids (4th grade?) was to flip the second fraction and then multiply across. While I don't know the reason behind flipping the second fraction and multiplying across, it is easy to do. Although, an even easier method of division is to utilize common denominators. For example: 

When flipping the second fraction, 167 turn into 1 due to "cross cancelling", and then multiplying across leaves us with 9/3, which is just 3. Knowing number relations is very important for children to learn, as it makes problems such as this much easier. 

Week 2: Fraction Sense

This week we learned about fraction sense. There are four different methods to look at fraction sense:

Spacial Relationships: having a physical picture of the number, including where it lies on a number line.

This is the most common fraction sense and should be the first method that we teach children. Spatial relationships is using our big ideas of partitioning and unit iteration in picture form to look at fractions.

Take, for example, the spacial relationship of 5/4. In the picture below, the picture of 5/4 is show in the coloured boxes. Here the student can count each 1/4 piece to make 5/4.

Simiarly, spacial relationships can be shown on a number line. The number line sort of incorporates benchmarks as well, but it's mainly to show where it lies on the line. The picture below shows this.

One/Two (units) More and Less: essentially it's knowing what is one more or less of the given unit. 

O/T, M&L was difficult to understand because it allows us to think about fractions in terms that our teachers initially forbade us to. If we look at the number line above, we can see that 1 (4/4) is one less than 5/4, and 6/4 is one more than 5/4. 

Benchmarks of 0, 1/2, 1...: being able to recognize where the given  unit is in relation to the benchmarks of 1, 1/2, 1... etc. 

Benchmarks are particularly important when trying to distinguish what fractions are larger, which is illustrated in the picture below. 

It is easy to tell why the top two fractions are greater than. 5/8 and 17/32 are both larger than 3/7 and 11/24, respectively. The bottom two are more difficult to recognize, but this is where benchmarks come in. To understand these problems, we have to think about which piece is large. A 1/10 piece is smaller than a 3/8 piece. Both fractions are less than 1/2, but which one is closer? The unit of 1/10 is closer to 1/2 than a unit of 1/8. 

Part-Part-Whole: the ability to break up the given unit into friendlier numbers. 

Basically just decomposing. Using a number line or a "tree" diagram is the most efficient way to show this method. 

New Unit!

In the last unit of Math 251, we started fractions, decimals, and percentage. Personally, I think this will be the most helpful unit because partial numbers are always more difficult to think about than whole numbers, or even integers.

The big ideas in this unit are partitioning and unit iteration. We are very familiar with partitioning. For fractions, partitioning is again just splitting the whole unit into equivalent parts. (illustrated below) Partitioning is best used when you start with the whole unit.

Unit iteration is just as useful and common as partitioning, although it's not as obvious. Unit iteration is consistently repeating a unit to build a whole. (illustrated below) This method is best used when you only have part of the whole.

Lastly, the most efficient way to teach fraction names is to say it as: 3/4 = three 1/4 pieces. It's easy to confuse a child by saying fractions like "three out of four" because that is not always the case with all fractions!