Fraction Approximation - Visualizing fractions on a number line in the classroom

I would like to tell you a little (maybe a long) story about how we do this in our classroom.

I started out the lesson by 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 students decided that I needed to call that "distance." Then, we went on to talk about how we could choose where to mark our fractions on a number line. For example, the students told me that we either needed to "find the middle" or divide the line in half to find where half should be placed. 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 what I am telling you now is also written on the whiteboard in the last picture.  I have learned that if I write down the words said out loud, it helps keep them more focused on the lesson. For example, as soon as I said that the line was 39 inches long, one student said, "UGH? Why not 40!!" another said, "that's not a nice number." While neither student said "divisibility" - or really used any type of mathematical vocabulary - it was apparent that they had an understanding that 39 was not going to give them very "nice" answers when they started breaking the number line apart.

We started working on our division and found that half of 39 was 19.5 inches. So we used the measuring tape to measure 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 was not great, most said 'no' because it was too long or short. However, the students were surprised when I put the left end of the measuring tape on the 1/3 mark, and the right end landed on the 1.

Then a student asked if that would always work (and I started to get giddy!). 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. So we tried to figure out what to do with 3rds. 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 the 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 kids were super excited about this. Then one said, "Mrs. S, 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 halves, thirds, fourths, fifths, sixths, eighths, tenths, and twelfths without including decimals when we divided. At this point, students started thinking of numbers. 

One student 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 one said, "it's always either 'good' or '1 off'". So 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, not divisible by  6 (at this point, the '1 off' theory didn't work)... So while the approach didn't work, the conjecture was terrific!

Then we chatted about how guessing wasn't really working for us, and a student suggested we try multiples..... YES!!

So we each took a number 2-12 and found all the multiples up to 200 (I told them to stop there 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 smartboard in an Excel file. You can see below the color-coding we used.

12 is the LCM for the numbers  2, 3, 4, 6, and 12.

Likewise, 24 is the LCM  for the numbers  2, 3, 4, 6, 8, and 12.

We kept moving through multiples of 12 until we hit 120, which is the LCM for ALL the numbers.

The next day students 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 repeatedly counting each tick mark.

Then, as a class, we decided what color our number line should be - they chose purple. Next, we placed 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. Next, students took turns being the 'official gluer.' While one student decided where to place their given fraction, the official gluer ensured 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. Next, we had to decide what to do with 2/4 because it was equal to ½.  Students agreed that they wanted to see both numbers, so they" stacked" it on top of the ½ so you could visualize the equivalence.  At the end of day 3, our number line was taking shape!

 

Day four came along quickly, and the numbers grew and grew! The number line quickly became very colorful, and kids started to look for patterns as we completed the line.

Each student was given post-it notes to find patterns.  Some of their pattern discoveries were:

“Every denominator in the ½ column increases by 2”

“This 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 ½’s column”

“they all have zero in the numerator!”

”both numbers are the same!”

We ended up hanging our number line in the hallway, and kids were super proud of their work; after all this, it took 4 days.  I was super proud of the lightbulbs that took place.

Have you tried a giant number line in your classroom?  What would you do differently?

I have made this activity into a product on TPT you can find it here!

Thanks for reading, and I would love to hear from you! 

Yours truly,

Jameson