## Number Lines + Fractions

As a grade 4/5 teacher for the last 4 years, I generally find that my students enjoy working with fractions.  They like working with fraction manipulatives and approach visual representations for fractions with relative ease.  Naming and identifying fractions is rarely a problem.  But then we get to comparing and ordering fractions… and it all falls apart.

With my tutoring students (Grades 9-11), fractions are generally a disaster.  They are chugging along fine with whatever they are working on and then… a FRACTION!!!  Reducing fractions and finding common denominators is sometimes OK, but if a fraction is tossed into the middle of an algebraic expression, they don’t know what to do.  There seems to be no recognition that fractions are, in fact, numbers.

I have been doing a lot of thinking this year about how and why fractions seem to be a place of struggle for so many students as they advance through math.

Many of my students treat fractions as a completely new set of numbers, with no connection to whole numbers.  They are obviously not getting the big idea that all numbers are connected and have their own place on the number line.  Part of me wonders if this is because we (I?) over-use certain representations for fractions (pizzas and chocolate bars) and under-use others (number lines).  So, this year, I have really been trying to help students connect what they already know about numbers with the new information that they are learning about fractions.

One activity that I have used a few times this year – very successfully – is a fraction clothesline activity.  This week, I was invited to visit a Grade 4/5 class to introduce decimals by linking them to what the students already knew about fractions (they have been working on fractions for a couple of weeks).  I thought this was a perfect opportunity for me to combine two great things and I set the fraction clothesline up like a number talk (yes, I am obsessed with Number Talks).

I set up the clothesline at the front and we looked at the three benchmark cards I brought: 0, 1/2 and 1.  We hung the 0 and the 1 on the number line and then… NO ONE could tell me where 1/2 went.  Yikes.  This is the part of number talks that still makes me anxious… the WAITING… letting kids think… and HOPING… that SOMEONE… will come up with something to move the conversation forward.  My patience paid off… eventually someone suggested – “well, if it’s one-half, couldn’t we just put it in the middle?”  And, whew… yes.  Yes, we can.  We were rolling again.

So, once we had the benchmarks on the line, I handed out a fraction card to each student.  Because we were looking at connecting decimals and fractions, I gave each student a “tenth” – nothing tricky.  Just 1/10 – 9/10.  I asked the students to think for a moment and give me a thumbs up when they were pretty sure they knew where their fraction was supposed to go.  And I waited.  After a while, a few kids (probably about 1/2) had their thumbs up.  So I invited those students who had an idea to come and place their fractions, and others could just watch to see if it helped them figure out where theirs should go.  And… this is what I got:

Yikes.  Again.  But mistakes are SUCH a valuable opportunity to pinpoint misconceptions. The first student started by defending the location of 9/10.  He explained that because 9/10 was almost one whole, he put it close to the 1.  Whew.  Good start.

Then, the student who placed 1/10 wanted to explain.  She said she placed her fraction there because 10 is bigger than 2, so 1/10 must be bigger than 1/2.  Cool – good explanation, and a good misconception to tackle.  But, this is where I am still working on my “thinking-on-the-spot” skills, and trying to find the balance between “teaching” and letting the students help each other and wrestle with their own thinking.  I ended up drawing some pictures on the whiteboard behind the number line – I went back to the classic “pizza” shape.  I asked the students to help me draw 1/2 behind the 1/2 benchmark card and then 1/10 behind the 1/10 card.  Gasps all over the classroom.  I asked the student if she was still happy with where her card was and – NO, she was not!  She came and moved it to its correct location.  I asked if any other students wanted to move theirs and several more came up to make adjustments (accurately).  I asked the students to explain why they chose to move their cards and they were able to relate the spot on the number line with what the picture would look like.

I then encouraged the students who had not yet placed their cards to come up and find a reasonable spot for them.  This worked well – most of them were able to be successful in the location and were able to explain why.  We also talked about the spacing between fractions on the number line and how it is hard on a clothesline to be exact, but we know that fraction pieces all have to be the same size.

I was pretty happy with this number talk.  The students were pretty confident with ordering fractions with the same denominator, and I thought we got a good start at thinking about fractions with different denominators.  At the end, I gave a few students some “tricky” cards – 0/10, 11/10 and 12/10 and they were able to successfully place and explain these fractions as well.

This lesson really highlighted (again) for me the power of number talks and having opportunities for students to own and explain their thinking.  The real power of number talks is in giving students these types of opportunities on a DAILY basis.  This is what helps them to build up their number sense over time.

What other activities do you use to help students build a broad, connected understanding of fraction concepts?