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**Annotation of Student Work**

Annotating student work is an excellent process for understanding student thinking in terms of the students’ developmental progress, their misconceptions, and the next steps which would help to extend their learning to the next level. The following slides annotate the work of three grade 5 boys who solved the problem in the segment entitled, “Equivalent Fractions – Grade 5.” Problem: Place the tiles on the 48-square grid so that: 2/6 of the rectangle is red 1/4 of the rectangle is blue 3/8 of the rectangle is green The rest of the rectangle is yellow How many ways can you describe the fraction of the rectangle that is red? Blue? Green? Yellow?

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**Place tiles on the rectangle so that 2/6 of the rectangle is red**

Observations: The boys saw that the whole was divided into 6 equal groups, namely 6 rows, so they decided to fill in two of those equal divisions. Assets: The boys understand that the whole of an area model must be divided into equal-sized sections. They were visually able to see the six equal groups and could solve this part of the problem without counting individual squares.

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**1/4 of the rectangle is blue**

Observations: The boys calculated how many individual squares there were by multiplying 6 rows by 8 columns. They then figured out that 48 divided into four equal groups, or fourths, would be 12 squares. Assets:. Until now the boys did not have to count individual squares but changed strategies when they did not readily see the whole divided into four sections. They could flexibly change strategies according to the fraction and the whole.

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**3/8 of the rectangle is green**

Observations: The boys reverted back to their first strategy since they could see the whole divided into 8 columns, or 8 equal groups. They then decided to fill in three columns of 8 but had to move some blues and reds to a different location in order to do so. They later coloured the remaining squares yellow. Assets: The boys were flexible and chose the strategy which made the most sense to them, depending on the fractions. By moving the red and blue tiles to make room for the green tiles, they demonstrated that changing the shape that each colour covers does not change the value of the fraction since it still take up the same area.

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**Create equivalent fractions**

Blue /48 Green 18/48 Red /48 Yellow 2/48 Observations: For each of the colours, the boys started making equivalent fractions with a denominator of 48 since they realized that the 48 square make up the whole. They then counted the various coloured squares to determine the numerator. Assets: They keep the whole of 48 squares in mind when creating the fractions and realize that the denominator represents the whole in each case. They also understand that the numerator represents a part of that whole.

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**Create equivalent fractions**

Blue /48 = 6/24 = 3/12 = 1/4 Red /48 = 8/24 = 4/12 = 2/6 = 1/3 Observations: The boys use a halving strategy to find more equivalent fractions for blue and red. They correctly divide both the numerator and denominator each time by 2. Assets: Their strategy reveals that they have the beginning understanding of the multiplicative relationship of the numerator and denominator. Wondering: Do the students realize that the equivalent fractions represent the same area, and that each is just divided into a different number of same-sized pieces of the whole?

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**Create equivalent fractions**

Green /48 = 9/24 Observation: The boys once again use a halving strategy to find another equivalent fraction. They are puzzled about what to do with 9/24 since the numerator is odd and cannot be divided evenly. They contemplate using a fraction or decimal in the numerator but decide against it. Assets: The boys question the validity of a fraction or decimal in the numerator. Challenges: The boys have not realized that they could divide both numerator and denominator by any common factor, in this case 3, in order to find 3/8. Wondering: Are they just calculating fractions procedurally, or do they understand that the all of the equivalent fractions represent the same area?

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**Create equivalent fractions**

Yellow 2/48 = 1/24 = 0.5/12 Observations: The boys use the halving strategy until they reach an odd number in the numerator. They readily know that half of one is ½ or 0.5, which is much easier than calculating half of 9 in 9/24, so they try using a decimal in the numerator. They realize that it is correct mathematically but question whether fractions can be written this way in standard notation. Assets: They connect to their previous experiences with fractions and question whether numerators can be decimals or fractions. Wondering: Do they understand what one half or 0.5 of 12 would look like?

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Next Steps Give the boys more experiences with odd numerators to see if they can discover that dividing by any common factor will produce equivalent fractions. Let the boys create their own grid question, deciding upon the size of the grid to use and the fractions that each colour should represent of the whole. This can then be given to other students in the class to see if they have been able to correctly construct the question. The creation of the problem would help to reveal the depth of their understanding or any areas that still need more development.

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Not Whole Numbers I: Fractions Presented by Frank H. Osborne, Ph. D. © 2015 EMSE 3123 Math and Science in Education 1.

Not Whole Numbers I: Fractions Presented by Frank H. Osborne, Ph. D. © 2015 EMSE 3123 Math and Science in Education 1.

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