In this activity we are comparing each rod to each of the others to see the fractional relationship. It looks overwhelming, but it’s not. The first couple of rows or columns are the most challenging and after a pattern emerges, and it becomes much easier.  Having said that, this comparison did take us three to four days to complete. (At the time of this post she is “grade 4” and approaching 10 years of age. There is a video at the end of this post with the highlights from our work.

We used rods here for comparison so we can *see* and *touch* to learn.  We compared rods using an expression of fractions.  The row header was the numerator, and the column headers were our denominator.  

We also worked in both directions. We started out moving from left to right on the rows.  I would ask “how much is the white rod (1) of the red rod (2.). She would answer half. Then we moved onto the green rod.  “How much of the white rod of the green rod (1/3)?”  

We continued in this way until she realized that the numerator was staying the same and the denominator was increasing by one each time.  When she recognized this, she just filled in the rest of the row. After doing a couple rows, we stopped to notice and wonder.  We noticed the following

• The first row has a definite pattern – the numerator stays the same while the denominator increases by 1 each time.
• The second rows pattern is a little more challenging to see.  It is similar to the first, but our numerator stays constant as a 2 and the denominator grows by one each time. In some places where there are equivalent fractions, they are expressed by the lowest fraction.
• There were several fractions that were common in both rows.  

Here we took the time to make each of the common fractions in the two rows.  In the first row we used the white to red rod for 1/2. From the second row we used the red to purple rod (2/4) for 1/2.  She was able to see that the next in that series would be light green to dark green (3/6) for 1/2. We did not fill out that space yet though.

Working in another direction, we moved down the columns. The first column was easy to see that it simply counted down the page in whole numbers.  The second column counting by halves. At the 3^rd and 4^th columns I stopped us to examine the patterns. These two columns count by 1/3 and ¼ respectively.  She did not notice this initially, so we expressed these units with the rods.  For the column of 3, the light green rod served as our 1 and the white was 1/3.  It counts consecutively from 1/3 as the white, 2/3 as the red, and light green being one. This pattern repeats but this time with a whole green rod – 1 1/3, 1 2/3, 2.  Each third time we added another light green rod to our pattern. This helped her to see the pattern of the columns. The fourth column follows a similar pattern that you can see in the video below.

Near the end of the activity, we used the factor work that we had done earlier in the academic year to help us see the fractions more clearly.  When doing the 8^th column where the white rod is 1/8 and the brown rod acts as our whole, we were able to factor the eight with 8 white rods, 4 red (2) rods, and 2 purple (4) rods.  It is obvious with the factors present to see that red is ¼ and purple is ½.  We used these to quickly fill in the rest of the 8^thcolumn.  We factored 9 to quickly fill in the 9^th column and 10 to fill in the 10^th column.

This activity is useful in so many ways. For one it is a way to manipulate fractions without the drudgery of worksheets or repetitive calculations.  It’s most useful function perhaps would be the sense of fractions (number sense) that is gained from this activity.  Pulling other resources like factoring to find our fractions is a way of interweaving (bringing concepts previously learned into the current concept of studying and sometimes using them in a different way but always illustrating a relationship) and aiding in memory retention of both the concepts of factoring and fractions. Lastly it gave us a way of comparing equivalent fractions (which we will also correspond to ratios when we reach that point.)

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