Before I started teaching my third graders to compare fractions, I wanted to assess what they already knew about fractions. True story. I asked my students to write in their journal and complete this sentence: A fraction is. . .
I got some very left-field responses and one good one. But then I got a gem.
↪ Teacher: What is a fraction?
↪ Student: When two bodies are rubbing up against each other.
↪ Teacher: WTH? Oh, you’re thinking of friction.
——>>> If you’re short on time, pin it for later!
Teaching Fractions is Like a Box of Chocolates
You never know what you’re going to get! From that, you can infer that I had a lot of teaching to do about fractions. I decided the class needed to have a hands-on approach to this. So I brought out the fraction strips.
Our math program came with a bag of fraction strips from a whole to twelfths for each student. I had the students take out the 1/2, 1/3, 1/4, 1/5, 1/6/, 1/8, 1/10 and 1/12. I also had on hand a recording sheet with three columns: less than 1/2, greater than 1/2 and equal to 1/2.
How Did We Start to Compare Fractions?
It’s important that the students have a benchmark number (fraction) firmly established. So we first identified the half. I asked them: would you rather have a half a bowl of ice cream or one-third of a bowl of ice cream? Many of the students voted for the one-third (which I knew they would!). I had them find the 1/3 fraction strip and put it under the one-half. I asked again, which was larger or more?
At first, I could see that many students were NOT lining up the fraction strips. I said, does it make a difference if the fraction strips are not lined up? Some said no.
What if I wanted to measure two students to see which was taller and I measured one sitting down and the other standing up? They understood that was not fair. So I explained that yes, it is important that when you compare fractions, the strips line up on one side.
Once we determined (by correctly lining up the strips) that 1/2 was larger than 1/3, I continued with another fraction. I asked them to compare fractions 1/2 and 1/4. Again, we talked about which was larger and smaller and why. The students continued to compare fractions 2 more times together. Then I asked them to start over but now record their findings on the paper by tracing the fraction strip in the correct column.
Compare Fractions and Record Your Answers
I circulated around while students were doing this to make sure they were lining up the fractions strips on the left side. There were about 4 students however, that were mightily confused about where to trace the fraction strip. I saw them tracing it in the greater than 1/2 column. At that point, I realized what they were doing.
So I stopped the class and then asked these particular students why they thought 1/5 was larger than 1/2. At first, they were hesitant because they assumed they were wrong (which they were), and I assured them it was ok. We just wanted to hear their thinking.
If you’ve guessed by now that the students were thinking that since the denominator was a larger number, it must be larger. You’re right! This is such a common misconception with children just learning about fractions. Of course, I corrected their misconception and we talked about how we know for sure that one fraction is larger or smaller than another.
I taught them to look at the size of the pieces directly and not to worry right now about the fraction numbers. Once all the fraction pieces had been recorded, we talked about why there weren’t any that were equal or greater than. They knew now that there are not pieces greater than 1/2 for unit fractions.
The Denominator is the Key to Compare Fractions
My objective with this guided exploration to compare fractions was to get my students to understand that denominators matter. The denominator is key to comparing fractions. Always look at the denominator first. For example, we tried some hypothetical questions:
? would you rather have 1/2 of a pizza or 1/8 of a pizza?
? would you rather have 1/3 of the money or 1/10 of the money?
? would you rather have 1/4 of an hour of free time or 1/6 of an hour of free time?
With those examples, students started understanding that the larger the denominator number, the smaller the unit fraction or piece!
Then I asked them to line up all the fraction strips together starting with the largest piece to the smallest piece.
I asked them what they noticed about the denominator numbers and the size of the pieces. We had a good mathematical conversation about how the larger the denominator number, the smaller the pieces. Objective met! Now it was time to cement the learning with some writing. I had these questions on a chart ready for them to answer in their math journals.
Wrapping it Up for the Day
Once the students had had enough time to write, we shared and concluded the lesson by turning our paper over and now using symbols to show inequalities. I wanted to end with this because I still had students who are confused about which is which (< or >). So we practiced recording some inequalities using just unit fractions.
This would then lead to the next day’s lesson about comparing fractions with the same denominator but different numerators, such as the example below.
Understanding that the denominator is key to comparing fractions will guide them on to comparing fractions with unlike denominators as well.
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