Download Analyzing Student Work I

Understanding Division for Fractions
Part A:
Use the Cuisinaire rods to answer each of the following questions. Make a ballpark
estimate of each quotient first.
Record data about the divisor, dividend and quotient in the chart provided.
Important Notes About Filling In The Chart
 Don’t reduce fractions or convert mixed numbers into improper fractions. The
patterns needed later will show up better if you leave fractions unreduced and in
mixed number form.
 Leave the last column (far right) blank for now.
Exploration Questions (Use attached chart for answers)
1. Let the orange rod represent the whole. What color rod would represent 1/5 of the
whole? How many groups of 1/5 are there in one whole?
2. Let the light green rod represent the whole. What color rod would represent 1/3
of the whole? How many groups of 1/3 are there in one whole?
3. Let the brown rod represent the whole. What color rod would represent 1/4 of the
whole? How many groups of ¼ are there in one whole?
4. Let the blue rod represent the whole. What color rod would represent 1/3 of the
whole? How many groups of 2/3 are there in one whole?
5. Let the brown rod represent the whole. How many groups of 3/4 are in one
whole?
6. Let the green rod represent the whole. How many groups of 5/6 are in one
whole?
7. Let the orange rod equal the whole. How many groups of 2/5 are in one whole?
How many groups of 3/5 are in one whole?
8. Let the black equal the whole. How many groups of 4/7 are in one whole? How
many groups of 4/7?
9. Let the purple rod equal the whole. What color rod would represent 1 ½ of the
whole? How many groups of 1 1 /2 are in one whole?
10. Let the dark green rod equal the whole. What color rod would represent 1 1/3 of
the whole? How many groups of 1 1 /3 are in one whole?
11. Let the dark green rod equal the whole. What color rod would represent 1 1/6 of
the whole? How many groups of 1 1/6 are in one whole?\
12. Let the yellow rod equal the whole. What color rod would represent 1 2/5 of the
whole? How many groups of 1 2/5 are in one whole?
NCTM 2002 Tracy Rusch
Part B:
Now you have data you can use to discover why the “invert and multiply” algorithm for
fraction division makes sense. It’s time to fill in the far right column (the last column) of
your chart. In the third column from the left you were to write the division statement
using mixed numbers. In the last column, write symbolic statement you wrote earlier
using ONLY improper fractions.
Look for a pattern. You may have discovered it as you were filling in the last column.
Did you find that you were able to rewrite the symbolic statement without going through
the steps to determine the improper fraction?
Describe the pattern that you discovered. It is best to articulate your description either
verbally to someone else or in writing.
There is something about “reciprocals” going on here. What assertions can we make
about the information the reciprocal provides us?

Look at each division question on your chart and ask yourself these two questions.
How many groups of the divisor are in one “unit”?
What is the relationship between the divisor and the number of groups of the
divisor in one unit?

Question: How many groups of one “unit” are there in the dividend?

Think about fraction multiplication: (number of Groups) * (Size of Group) =
(Total Quantity)
The dividend tells you the number of groups of one unit that you have:
The reciprocal of the divisor tells you how many groups of the divisor are in one unit.
Consider:
6  2/3
The dividend is 6. There are 6 groups of one unit in 6.
The divisor is 2/3. The reciprocal of 2/3 is 3/2. There are 3/2 (or 1 and ½) groups of 2/3
in one unit.
In this example we have six units, with 3/2 groups of the divisor in each unit:
NCTM 2002 Tracy Rusch
Color of the
Unit/Whole
Color of the
portion
Statement in Words
Statement in Symbols
Use Mixed Numbers
Orange
Red
How many groups of 1/5 are in 1?
1  1/5 = 5
NCTM 2002 Tracy Rusch
Leave blank until
Part B