In a recent fourth grade class students were having a difficult time placing fractions on the number line. They also were not recognizing the value of the reference marks placed on the line. I used the number line tool to introduce the students to several strategies they could use to help them placing the fractions on the number line correctly. Five fractions were presented, ¼, 1/8, 3/8, 2/3 and 2/8. There was only one reference mark (tic marks), ½.
I asked the students to arrange the fractions from the smallest to the largest and not to worry about the placement of the fractions on the number line just yet. They struggled a bit so I changed the fractions to their pie models. Sorting went more quickly with the students taking turns explaining where a particular model would be placed in the sequence. When the class showed me all thumbs up, I switched from the pie to the vertical bar model and asked the class how this model would help. Very quickly, hands went up. “Line up the fractions so the bottoms are even. You can tell the order by looking at the amount of shading, the more shading, the larger the fraction.” Another student, “Two of the fractions have the same amount of shading.” Strategy one, order the fractions first, revert to the models if necessary.
Looking at the fractions, now as numbers, they picked out ¼ as halfway between 0 and 1/2 and 2/8 as an equivalent to ¼. The explanation, the models show them as having the same amount of shading. Two fractions down, three to go. “If 2/8 is where ¼ is on the line, then 1/8 must be halfway between 0 and ¼. I remember that 1/8 had half the amount of shading that ¼ had.” It is the visual model again. “Look, 1/8, 2/8! Then 3/8 will be halfway between ¼ and ½. I remember that 4/8 is equivalent to 1/2, so 1/8, 2/8, 3/8, and 4/8.” “Forget all the fractions and the marks on the line, now use your fingers to divide the space between 0 and 1 into three equal parts. The last fraction, 2/3 will be at the second mark.”
Most students just needed a ‘kick start’ with strategy one. I did not give any of the other strategies I was thinking the students needed as they already had them. This was another exciting classroom with the students using math expressions and reasoning.