Over the past few weeks my 6th grade math resource kiddos and I have been working hard to have a better conceptual understanding of fractions , we’ve been using concept sorts, fraction towers,
fraction wars. Spin and compare, anything we can create or get our hands on! But this week…. this week we did the culminating fraction activity…. the fraction number line! Last year I did this activity with my 7th and 8th graders and they really struggled. This year, I took a bit different approach to the activity, and because it was the culmination of a bunch of fraction work it went much smoother than before! I was so incredibly impressed with some of the math talk that occurred over this week!
I started out the lesson drawing a random line on the board (I really didn’t measure it until we started the conversation). As we chatted we talked about how much “space” the line took up and the kiddos decided that I needed to call that “distance” (so far MUCH better than last year!). We went on to talk about how we could decide where to mark our fractions on a number line. The students told me that to find where half should be placed, we either needed to “find the middle” or divide the line in half. At this point, I pulled out a tape measure and measured the line 39 inches (inside I was ecstatic that it was only going to be divisible by 3!)
You may notice that a lot of what I am typing is written on the board below. I’ve learned with my resource kiddos, that if I write down the words they say out loud, it helps keep them more focused. As soon as I said that the line was 39 inches long, one kiddo said “UGH!? Why not 40!!”. Another kiddo said “that’s not a ‘nice’ number”. While neither of the kiddos said anything about divisibility, or really used any type of mathematical vocabulary, it was clear that they had an understanding that 39 was not going to give them very ‘nice’ answers when they started working with breaking the number line apart.
We started working on our division and found that half of 39 was 19.5 inches. We used the measuring tape to measure out 19.5 inches and mark it on the number line, then the magic occurred. I asked if using that 19.5 inch length of measuring tape would ‘fit’ in the other half of the number line.
As their spatial reasoning is not great (still working on that!) most said ‘no’ because it was either too long or too short. They were super surprised when I put the left end of the measuring tape on the 1/2 mark and the right end landed on the 1 at the end.
Then a kiddo asked if that would always work (and I started to get giddy!). So we tried to figure out what to do with 3rds. We decided we needed to split the number line into three identical areas, so we divided by 3. 39 divided by 3 is 13 so we measured out 13 inches. We placed the measuring tape on the 0 and measured 13 inches and marked 1/3. Then we moved the tape to the right and measured 13 more inches. After some debate (and pulling out the fraction towers) we decided that second mark should be called 2/3 and we marked it on the line. Then we moved the measuring tape one more time and it landed on 1 again! The kiddos were super excited about this. Then one kiddo said, “Mrs. Riggs, of course that happened! 3/3 = 1” and then showed me that if she took 3 orange 1/3 pieces and stacked them she would get 1 whole piece.
YES… they were getting it! So we worked through 4ths and 5ths… and then someone said could we please not use decimals….
So I asked them how we could create a number line that would work for
without including decimals when we divided. At this point, kiddos started thinking of numbers. (see pic below)
One kiddo said “try 24!” but someone quickly responded that “5 didn’t divide into that number”, then someone suggested 35 but someone else said that 2 and 3 didn’t divide evenly into that number….
Then a kiddo said “it’s always either ‘good’ or ‘1 off'”. I asked them to elaborate and they showed me that 40 was divisible by 2, not by 3 (1 off as 3*13=39), divisible by 4, divisible by 4, not divisible by 6 (at this point the ‘1 off’ theory didn’t work)….
While the theory didn’t work, the conjecture was wonderful!
Then we chatted about how guessing wasn’t really working for us and a kiddo suggested we try multiples….. YES!!
So we each took a number 2,3,4,5,6,8,10,12 and found all the multiples up to 200 (I told them to stop their for time sake, but really that would have been fun to decide how far we needed to go as well)
And then we listed the multiples on the smart board in an Excel file.
You can see below the color coding we used.
12 is a LCM for the numbers 2,3,4,6, and 12.
24 is a LCM for the numbers 2,3,4,6,8,12.
We kept moving through multiples of 12 until we hit 120 which is a LCM for ALL the numbers.
The next day kiddos took a very long strip of adding tape and used a ruler to mark 120 inches along the length. They also labeled the tick marks from 0 to 120. This helped us quickly find our placement on the number line without having to count each tick mark out repeatedly.
Then, as a class, we decided what color our number line should be – they chose purple.
The kiddos made sure to place the 0, 1/2, and 1 markers on the number line to help visualize the benchmarks. You can also see that we needed to move into the hallway to give ourselves some more room.
Kiddos took turns being the ‘official gluer’. While one student decided where to place their given fraction, the official gluer made sure it was on the correct number of the 120 inches, then glued down the fraction card.
We placed all the halves, then all the thirds, then we got to the fourths. We had to decide what to do with 2/4 because it was equal to 1/2. The kiddos decided that they wanted to see both numbers so they “stacked” the 2/4 on top of the 1/2 so you could visualize the equivalence.
At the end of day 3 our number line was taking shape!! (see above!)
Day four came along quickly and the numbers grew and grew! the number line quickly became very colorful and kiddos started looking for patterns as the line was complete.
Each kiddo was given post it notes to ‘find patterns’ in the number line. You can see below where the colorful post it notes are all stuck to the paper.
Some of their pattern discoveries were:
“Every denominator [in the 1/2’s column] increases by 2”
“This [the 1/4’s column] pattern is going up by 4’s you can see it in the denominators”
“The numerator counts up as 1,2,3,4,5,6 [in the 1/2’s column]”
“[on the far left] they all have zero in the numerator!!”
“[on the far right] they all have the same number in the numerator and denominator”
None of the kiddos noticed the symmetry in the number line, or the patterns in the color coding…. but I think that will come next week when we hang the number line up on the wall and look at it from slightly farther away.
Super excited for how this lesson turned out! AND I’m even more excited to be apart of Meg Anderson’s new monthly link-up: LOVED THAT LESSON! I definitely LOVED this lesson! Make sure to click on the link below to go find other lessons people loved!
Interested in making a number line in your classroom? It’s super easy! Just grab some supplies, create some fractional pieces and go to town! However, if you want to save some time and have everything ready to print and cut, you can check out my fraction number line and sort here