# Focus on Math

## Helping children become mathematicians!

### Reasoning About Fractions Using Benchmarks January 6, 2012

Filed under: General Math — Focus on Math @ 2:28 pm
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I first came across the idea of using benchmarks with fractions in John Van de Walle’s book Elementary and Middle School Mathematics: Teaching Developmentally (Pearson). I loved Van de Walle’s idea that students could quickly compare any proper fraction to the benchmarks of 0, 1/2, and 1 and decide which value the fraction would be closest to, thus giving a quick estimation of the fraction. By actually placing the fraction on a number line showing 0 to 1, students have the opportunity to develop the understanding that every fraction has a particular place on the number line (though it might share the spot with other fractions). Students generally have had real life experiences with fractions such as 1/2, 1/4, and 3/4 and thus I have found that all students can be successful thinking about fractions in this way.

Given the fraction 15/16, it is clear that having 15 of the 16 parts means that it is nearly the full amount, and thus is close to 1. Similarly for 19/24. A student might say here that there are 9 pieces more than half, and only 5 pieces away from being the whole amount, thus 19/24 is close to 1. (A word of caution here — make sure that students give sound reasoning. In the example of 19/24 to allow a student to say “it is only missing 5 pieces so it is close to 1” is not sound, for 3/8 is also missing 5 pieces but is close to 1/2. It is the relative amount missing that is important. Similarly 123/1000 has many fractional parts, but because the number of parts there is small relative to the number of parts in the whole, this fraction is closer to 0.)

Non traditional reasoning can be used when benchmarking fractions such as 4/9. A student might think that since 4 1/2 is exactly one half of 9, 4/9 must be close to 1/2. Elementary and middle school students are not usually exposed to fractions within fractions, but this reasoning is mathematically sound and is useful.

Once students are comfortable benchmarking fractions, they can move on to comparing fractions. For most of us, our experiences with fractions were such that if asked to compare 3/10 to 11/15, we would have changed the denominators to the common multiple of 30 and then compared the fractions. However, using benchmarks it is east to see that 3/10 is less than 1/2 and 11/15 is greater than 1/2, thus making 11/15 greater than 3/10. There is much to be gained by taking the time to explore fractions using benchmarks.

I have created pocket charts to go along with this activity, a large one to be used at the front of the class as well as smaller individual ones students can use. Having done this with a number of classes, including recently with my grade 4 and grade 4/5 classes at Charlie Lake Elementary, I have found that students enjoy using the little pocket charts. One teacher, after I had made and used pocket charts with her class, emailed me later that day to tell me the students were begging to take the pocket charts home so they could show them to their parents! Seriously, when was the last time your class begged to go home and do fractions with their parents?

I explored this topic more fully in an article published in the Summer 2011 issue of Vector (the journal of the BC Association of Mathematics Teachers). That issue was a special elementary edition of Vector and was sent out to all public elementary schools in BC. If you are in the province, hopefully you have access to that article!

I hope you try using fraction pocket charts with your students. Let me know how it goes!
Mathematically yours,
Carollee