Fraction February - Multiply Fraction by a Fraction

So I've already taught this unit, but never blogged about it.  And since it's Fraction February, what better time!  I taught it directly after teaching multiplication of decimals by decimals and multiply decimals with thousandths grid since they are so related.  Even though I knew this would be much more difficult than using the hundredths grid, I wanted to jump right in when the process of the math was very similar.

I decided to teach it from the basic progression... starting with more of an enactive (hands on) way of solving, to iconic (visual) before moving on to symbolic (algorithm).

Enactive Modeling of Multiplying Fractions by Fractions

I made these fraction squares on a Smart Notebook file before printing out onto transparencies.  I couldn't find any online that divided up a square in the SAME direction.  For example, fourths were quartered into 4 squares, but that would not work for my purposes here.

With these fraction squares, it also allowed me to get kids to find equivalent denominators without formally teaching it.  I would say things such as:

• Which fraction squares fit into the halves, thirds, fifths?  (They try to match with their fraction squares.)
• What pattern do you notice about which fraction pieces fit into sixths, tenths, etc.? (Knowing their factors really helped with this.)

With this model, students can easily see that many fractions can equal 1/2.  We did this with many other fraction pieces.  Their prior knowledge of basic times tables helped even my lowest kids feel confident.

Then using the same concept of multiplying decimals by decimals, we talked about taking a part of something smaller than one. Students used their fraction squares and wet erase markers to solve scenarios I gave them.

For example: 2/3 of your bedroom is for the play area (find thirds square & shade in 2 of them) and 1/4 of the play area is specifically for drawing (find fourths square, slide it in horizontally to cut the thirds into fourths, making twelfths  Then color 1/4 of the 2/3 area already shaded in a different color).  How much of the entire room is for the drawing area?  The answer is where the colors overlap.  We also practice reading all the other parts of the model and what they represent.

The red represents the 2/3 of the room for the play area.  the thirds will be divided into fourths so we can determine the portion of the room for the drawing area.

With thirds divided into fourths, we have a room with 12 sections.

1/4 of the PLAY AREA is for drawing.  So 2/12 of the entire room is for drawing.  No, I am not requiring students to simplify right now, but talk about it and how you can see it in the model.

Reading all the parts:
The whole square represents the entire room.
2/12 (blue & red) is the drawing area within the play area - also the answer.
The red area is the play area in the room. 2/3 = 8/12
The blank space is the rest of the room.  

I also asked questions such as:

• Given ____ model (I pre-colored in fraction squares), what is the equation?
• Is 4/6 x 3/4 = 1/2?  Explain using fraction squares.
• What pattern do you notice about the denominators in the problem and the answer? (This question leads to "ah ha" moments of factors & products.  But instead of saying 2 and 5 fit into 10, I try to say, "Yes, so that means halves and fifths can make tenths.")
• So knowing the pattern (above), can someone tell me what fractional pieces will make twelfths, twentieths, etc.?  

Most of this was done with partners.  It saved on cleaning time and I could pair up kids who needed help with those who I knew would struggle.  We talked about how to switch off, etc. so no one was sitting idle.


Iconic Modeling of Multiplying Fractions by Fractions

After spending a couple of days with the fraction squares, we took notes and started to transfer what we did onto paper.

I felt the major difference that made this more difficult than multiplying decimals was the fact that when we worked with tenth decimals, the hundredths grid was a perfect set up, even if they had to draw one themselves.  I had to really explain that fractions ARE decimals, but they are just wearing different clothes.  However, fractions may not always be nicely divided into tenths, so we just have to draw our parts based on the unit size we're working with.

Word problems I created were VERY similar to the decimal by decimal word problems.  Basically, I just changed the decimals to fractions.  And just to mix it up, I changed student names and parts of the scenarios.  I also opted to not do color this time around.  We did much of that for multiplying decimals by decimals that I thought they could handle shading alternate directions.

Once students grasped the word problems, I decided to switch it up.  Could they determine an equation and model if I only gave them the answer?  Some answers could have 2 equations, so I would ask things such as, "Can you find two ways to get 3/12?  It led to many other "teachable moments," especially if students made an improper fraction or a whole fraction such as 3/3.

I also had a few questions where I gave them only the visual model and they had to determine the equation and answer.

Being able to see fractions and relate them to 1/2 is so helpful that I threw this in as well a couple of times.  It stumped a few until I gave some hints.

Symbolic Solving of Multiplying Fractions by Fractions

So as we continued through this unit, I kept asking students if they saw the pattern between the expression and the answer.  Without looking at the model, how do we get from 2/3 x 3/5 to 6/15?  Students quickly saw that they could multiply the numerators and the denominators.  This was a great way for them to start checking their model and to see what was incorrect if answers did not match.

We did not spend much time on the symbolic method on its own since 5th grade Common Core math really wants kids to be able to model fractions.  However, as I conducted enters, I used online computer games to review with the symbolic method since they did know it.  One good one is the gregtangmath.com/ site because students could choose the easy (not simplified) or hard (simplified)  versions as they played.  We discussed how the symbolic method helps when the modeling is too tedious, such as fractions with large denominators.