What is the whole, what fractions?

Number Sense is built from imagery for number. Fractions are numbers that express parts of wholes. The whole can be a single unit or a set of units. Diagramming can bring immediate understanding to fractional relationships.

This set of representations strongly supports incorporating understanding of the different meanings of fractions in to operations with fractions seamlessly because these representations are extremely flexible in their use as the complexity of the curriculum expectations build. The effectiveness of the consistent use of representations is supported by the findings of researchers Pirie and Kieran (1994), who found that students hold on to the representations they are initially exposed to as the grounding for their conceptual understanding. Since this is the case, it makes sense to select precise representations that have longevity and power. Student gaps and misconceptions are powerfully revealed through their drawn representations of fractions, and studies in this area provide evidence to suggest that the multitude of representations used, some of which are potentially distracting representations, do not help students build deep understanding (Kilpatrick, Swafford, & Findell, 2001). It is important to ensure that representations are chosen to fit the problem context, and the fact that fractions are singular quantities is emphasized (Son, 2011; Charalambous et al., 2010, Watanabe, 2007). Unlike the North American preoccupation with the ‘pizza model’ or other circular area models, East Asian countries use a combination of carefully selected models that have longevity in terms of their application in representing fractions and that reflect the notion of fraction as a quantity. In particular, partitioning circles equally is much more difficult with odd or large numbers whereas rectangular area models and number lines are more readily and accurately partitioned evenly for odd and large numbers of partitions (Watanabe, 2012). Gould, Outhred and Mitchelmore (2006) asked young children to represent one half, one third and one sixth using circle area diagrams. In this study, the researchers found that most students were accurate when shading in one half of the region of a circle, using either a horizontal or a vertical line to partition the circle into two equal parts. However, when children were asked to represent one third and one sixth, there were a wide range of incorrect responses where the partitioning of circles was uneven (non-equal parts) and the students relied on a count- wise ‘number-of-pieces’ approach (where the number of pieces in total and the number of pieces shaded was more important than size of pieces).

THE BASIC FACTS: Imagery underlies number and number sense. Images for fractions must include area models and linear models. “Seeing” and labelling fractional parts and wholes relates the symbols to the idea being represented. Fractions are divisions. You can keep splitting into equal groups until you hit zero. Adding and subtracting and finding equivalent fractions spontaneously emerges when you focus on visual imagery. Eventually fractions must become abstract, every model is limited so compare compare, compare.

Represent through imagery Link to Literacy: How do I learn math? Sensory Engagement Build, Fold, Cut Visual Engagement: Represent through imagery Link to Literacy: Represent with symbol system Intellectual Engagement: Be a critical thinker COMPARE & CRITIQUE

DO IT ..... OVER AND OVER AND OVER Repeat the cycle of engagement as many times as needed to gain fluency and accuracy, until you can do it without stopping to explain your thinking.

HABITUATE THE BEHAVIOUR HABITUATE THE THINKING Link the body to the mind. 28 day rule: You will still need support.

Can you describe this image using fractions?

If it is 3 and one half, what is the one? 3 1 2

Do you see 7 halves? 7 2

Could it be 2 and one third? 2 1 3

If the triangle is one third, 3 of them make a 1 1 3 + 1 3 + 1 3 =1

1 3 + 1 3 + 1 3 =1

The blue triangles make another 1 1 3 + 1 3 + 1 3 =1

The blue triangles make another 1 1 3 + 1 3 + 1 3 =1

The triangle that is left is one third of the next trapezoid. 1 3

The triangle that is left is one third of the next trapezoid. 1 3

I created 2 and one third wholes. 2 1 3

What if this was the whole?

How would you determine what fraction of the whole pie the cut piece is?

Follow the lines to divide it equally…

What fraction of this set of cupcakes is yellow?

I see 6 columns with 4 in each. The “whole” box is divided into 6equal parts or sections.

Each column represents 1 6

1 6 1 6 1 6 1 6 1 6 1 6 Can you see it?

2 6 of the cupcakes are yellow 1 6 1 6 1 6 1 6 1 6 1 6 There is a light yellow and a darker yellow. 2 6 of the cupcakes are yellow

1 6 1 6 1 6 1 6 1 6 1 6 Do you see how 2 6 of 24 is 8?

How much of the image is red?

Does this help?

I divided the whole shape into equal parts.

I see 3 12

This is 1

Same 1. What changed?

one divided into 3 equal parts 1 ÷ 3 = 1 3 1 ÷ 3 = 1 3 How do I know it is one third?

When I open it up I see three thirds.

1 3 3 3 = 1 1 3 1 3 three thirds is another way to express 1

Start with 2

Lay one on the other & fold together

1 3 Fold into 3 equal parts.

Open and separate. Still 2 sheets…..

1 3 1 3 1 3 1 3 1 3 1 3 But also see 6 thirds 6 3 = 2

1 3 1 3 1 3 1 3 How many thirds?

1 3 1 3 1 3 1 3 1 3 5 3

5 3 = 1 2 3 six thirds is also one and 2 thirds of the next one. 1 3 5 3 = 1 2 3

What would four thirds look like? 4 3 = ?

Start with 3 sheets

How many thirds can you make?

This is 1

Still 1. What changed?

one divided into 4 equal parts 1 ÷ 4 = 1 4 1 ÷ 4 = 1 4 How do I know it is one fourth?

When I open it up I see four fourths.

1 4 1 4 1 4 1 4 four fourths is another way to express 1. 4 4 = 1

Start with 2

Lay one on the other & fold together

1 4 Fold into 4 equal parts.

Open and separate, still 2 sheets…..

1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 But also see 8 fourths 8 4 = 2

1 4 1 4 1 4 1 4 1 4 1 4 How many fourths?

1 4 1 4 1 4 1 4 1 4 1 4 six fourths 6 4

6 4 = 1 2 4 six fourths is also one and 2 fourths 1 4 1 4 1 4 1 4 1 4 6 4 = 1 2 4 six fourths is also one and 2 fourths

1 4 1 4 1 4 1 4 1 4 1 4 Demonstrate & explain 7 4

How would you demonstrate 10 4 ?

This is 1

Still 1. What changed?

one divided into 5 equal parts 1 ÷ 5 = 1 5 1 ÷ 5 = 1 5 How do I know it is one fifth?

Open it up, see five fifths.

five fifths is another way to express 1. 1 5 five fifths is another way to express 1. 5 5 = 1

Start with 2

Lay one on the other & fold together

1 5 Fold into 5 equal parts.

Open and separate, still 2 sheets…..

1 5 1 5 But also see 10 fifths 10 5 = 2

Now how many fifths?

And now how many fifths?

How would you demonstrate 12 5 ? 14 5 ?