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Fractions

4-8_3-8_7-8

four eighths plus three eighths equals seven eighths

5-5

five fifths

5-6

five sixths

5-8

five eighths

6-8

six eighths

6-9

six ninths

7-4_7div4_1_3-4

seven over four equals seven divided by four equals one and three quarters

7-4_mul12_21

seven over four mutiplied by twelve equals twenty-one

7-8

seven eighths

7-8q

seven eighths?

9-8

nine eighths

9div12_9-12

9 divided by 12 equals 9 twelfths which is three quarters

a-b_adivb

a over b equals a divided by b

a-b_adivb

a over b equals a divided by b

equivalent_fractions

one third equals two sixths, two thirds equals four sixths, three thirds equals six sixths

fraction strips

fraction strips 2

F_A_AP_AP1_S1_fig1

Board game, with cards showing fractions, decimal fractions, area models and number lines.

Fraction matching game.

F_A_AP_AP2_S1_fig1

A student’s response to five questions related to fractions teaching and learning.

Survey response

F_A_AP_AP2_S1_survey

A student’s response to five questions related to fractions teaching and learning.

Survey response.

F_A_AP_AP3_S1_fig1

Schedule completed by the interviewer, detailing a student’s answers and representations regarding fractions.

Interview schedule with notes

F_A_AP_AP4_S1_fig1

Addition of a pair of fractions, also shown with fraction bars, to equal a set target of 4/5.

Screen grab from L9811 Kick the goal: add proper fractions 1: assessment.
Source: © Education Services Australia Ltd, 2011 

F_A_AP_AP4_S1_fig2

Six questions, four responses marked as correct, two as incorrect. Teacher comments that student needs to work on adding fractions with different denominators.

Adapted from L9811 Kick the goal: add proper fractions 1: assessment.

F_BI_I1_eq1

one half

F_BI_I1_fig1.jpg

Triangle partitioned vertically into two equal parts. One half is white; the other half is shaded.

An area model showing halves.

F_BI_I1_figA1

the top half of a fraction is called the numerator; the bottom half of a fraction is called the denominator

F_BI_I2_eq1

one fifth

F_BI_I2_eq2

one tenth

F_BI_I3_E1_eq1

one quarter

F_BI_I3_eq1

fraction a over b equals a divded by b

F_BI_I3_eq2

fraction 2 over 3 equals 2 divided by 3

F_BI_I3_eq3

fraction 7 over 4 equals 7 divided by 4 equals one and three quarters

F_BI_I3_eq4

fraction 3 over 8 equals 3 divided by 8 equals 0.375

F_BI_I3_eq5

2 divided by 3 equals two thirds

F_BI_I4_E1_eq1

one half multiplied by three quarters

F_BI_I4_E1_fig1

A circle partitioned into four quarter-circles, one quarter-circle shaded. To the right of the circle are eight stars, two adjacent stars circled.

Area and discrete models of unit fractions

F_BI_I4_E1_fig1A

A circle partitioned in four quarter circles, with one quarter-circle shaded.

Area model of unit fractions.

F_BI_I4_E1_fig1B

Eight stars with two adjacent stars circled.

Discrete model of unit fractions.

F_BI_I4_E1_fig2

A circle partitioned into four quarter-circles, three quarter-circles shaded. To the right of the circle are eight stars, three pairs of two adjacent stars each circled.

Area and discrete models of non-unit fractions

F_BI_I4_E1_fig2A

A circle partitioned in three quarter-circles, with three quarter-circles shaded.

Area model of non-unit fractions.

F_BI_I4_E1_fig2B

Eight stars, three pairs of two adjacent stars are circled.

Discrete model of non-unit fractions.

F_BI_I4_eq1

three quarters

F_BI_I4_eq2

three quarters multiplied by twelve equals nine

F_BI_I4_eq3

seven over four mutiplied by twelve equals twenty-one

F_BI_I5_E1_eq1

four sixths

F_BI_I5_E1_eq2

six ninths

F_BI_I5_E1_eq3

four sixths equals twelve eighteenths

F_BI_I5_eq1

two thirds

F_BI_I5_eq2

three over two

F_BI_I5_fig1

Two squares of equal size grouped together on the left. Three triangles of equal size grouped together on the right.

Ratio of 2 to 3.

F_CM_M1_A1_fig1

Five paper shapes, different sizes, each folded a number of times to represent varying fractions. The fraction size is marked on each folded section.

Folded and labelled paper shapes.

F_CM_M1_A2_fig1

Vertical number line divided into ninths used to measure the relative height of birds and a human as a fraction.

Screen grab from L119 Cassowary ecology quiz.
Source: © Education Services Australia Ltd, 2011

F_CM_M1_A2_fig2

Pictures of cassowaries in a key representing varying fractions, in order to match the area of a holding pen.

Screen grab from L155 Cassowary fractions.
Source: © Education Services Australia Ltd, 2011

F_CM_M1_E1_fig1

Six shapes, divided into sections. The first, fourth and sixth shapes each have one third shaded black, and are circled. The second has one half shaded and is also circled.

Responses to area diagram question.

F_CM_M1_E1_fig2

A rectangle with a diagonal drawn. To its right, a circle halved with a curving line from top to bottom.

Shapes showing halves but not line symmetry.

F_CM_M2_A1_fig1

Fraction bars with corresponding fractions and a comparative fraction slider.

Screen grab from L2800 Fraction fiddle: tool.
Source: © Education Services Australia Ltd, 2011 

F_CM_M2_E1_fig1

A group of shapes consisting of three stars and five squares. Question asking for the fraction of the group that are stars. Student answer is 3/5, which is incorrect.

F_CM_M2_E1_fig2

A circle divided by vertical diameter. One half divided in half again. Question asking for fractions seen. Student answer is thirds, which is incorrect.

F_CM_M2_E1_fig3

Five fraction cards, from left to right 1/2, 1/3, 1/4, 1/6, 1/8. Question asking to order from smallest to largest. Student response incorrect.

F_CM_M2_E1_fig4

Question asking ‘Which fraction is larger 7/8 or 5/6? Explain’. Student answer incorrect, ‘They are the same because both have just one more to make a whole’.

F_CM_M2_fig1

A circle divided into three equal parts radiating from the centre.

F_CM_M2_fig2

A square divided into four equal parts by its diagonals. The top and bottom quarters are shaded.

F_CM_M3_A1_fig1

Two circles. On the left, a smaller circle divided into quarters, with one cut out and marked with 1/4. On the right, a larger circle divided into quarters, with one cut out and marked 1/4.

Quarters cut from different sized circles.

F_CM_M3_A1_fig2

Student poster with cut out shapes marked with fractions and explanations.

Sample poster.

F_CM_M3_E1_fig1

Hand drawn student work stating 1/3 is bigger than 1/6. 1/6 represented by a rectangle divided into six equal boxes, one coloured; 1/3 represented by a circle divided into three equal parts, with one coloured.

Different shaped wholes.

F_CM_M3_E1_fig2

Hand drawn student work responding to ‘Draw fractions 1/4, 1/3 and 1/5 in order’. The fractions are represented correctly by rectangles divided into equal boxes with one box shaded. The rectangles are different sizes.

Different length wholes.

F_CM_M3_E1_fig3

Hand drawn student work to determine whether 1/3 or 1/4 is larger. Two circles the same size are side by side; the left divided into four equal parts with one part shaded, the right divided into three equal parts with one part shaded. Correct written explanation.

Same sized wholes.

F_CM_M4_A2_fig1

Non-unit fractions to represent the breakfast of two birds, in order to determine the larger fraction.

Screen grab from L2803 Fraction fiddle: comparing non-unit fractions.
Source: © Education Services Australia Ltd, 2011 

F_GT_T1_E1_A1_fig1

One section of a patterned paper plate is covered by a section of a plain paper plate to visually represent a fraction.

Fraction wheel made from paper plates.

F_GT_T1_E1_A2_fig1

A slice of cake represented as one fifth matched to the correct fraction.

Screen grab from L2801 Fraction fiddle: matching cake fractions.
Source: © Education Services Australia Ltd, 2011 

F_GT_T1_E1_fig1

A circle divided into four equal parts.

Area diagram showing quarters.

F_GT_T1_E1_fig2

A circle missing one quarter.

Diagram to prompt visualisation of the whole.

F_GT_T1_E2_A1_fig1.png

Line from zero to one divided into fifths, labelled at the end of the interval. Another line from zero to one divided into fifths, labelled in the centre of the interval. Second line incorrect.

Correctly labelling number lines.

F_GT_T1_E2_A1_fig2.jpg

Fraction cards pegged to a rope to show intervals of quarters between zero and five quarters. Four quarters is also marked as 1.

Quarters pegged on a rope.

F_GT_T1_E2_A2_fig1

One bird with half of a worm, another bird with a quarter of a worm. The fractions compared on a number line to find which is smaller.

Screen grab from L2802 Fraction fiddle: comparing unit fractions.
Source: © Education Services Australia Ltd, 2011

F_GT_T2_E1_A1_eq1.svg

three quarters of 20 equals 15 or one fifth of 20 equals 4

F_GT_T2_E1_A1_fig1.jpg

Two dice, one die displaying 1/4 and the other 5. One set of 5 cubes in one colour, three sets of 5 cubes in another colour to make a total of 20.

Example: One quarter of my collection is 5 so altogether I have __

F_GT_T2_E1_A1_fig2

Diagrammatic explanation of the problem posed with calculations and written responses.

Student work sample

F_GT_T2_E1_A2_fig1

The equal division of 17 biscuits to 3 puppies, each receiving 5 2/3.

Screen grab from L2808 Divide it up: puppies.
Source: © Education Services Australia Ltd, 2011

F_GT_T2_E1_eq1

three times four equals twelve, twelve divided by three equals four, one third of twelve equals four

F_GT_T2_E1_eq2

three times four equals twelve, twelve divided by three equals four, one third of twelve equals four

F_GT_T2_E1_fig1.png

Three groups of four counters, each group a different colour.

Using 12 counters to model division.

F_GT_T2_E1_fig2.png

A total of twelve counters arranged in a 3 by 4 array next to a 3 by 4 blank grid.

An array of 12 and a grid of 12.

F_GT_T2_E1_fig3.png

Three groups of four counters, each group a different colour. The thirteenth counter is divided into three equal parts.

Using 13 counters to model division.

F_GT_T2_E2_A1_fig1

Three circular 'pancakes' made of modelling dough, each divided into four equal parts.

'Pancakes' divided into quarters.

F_GT_T2_E2_A1_fig2

Two circles divided into halves and one circle into quarters surrounded by four stick figures. A colour code shows that each will receive one half and one quarter.

A student’s sharing diagram.

F_GT_T2_E2_A1_M_fig1

Nine circles each divided into quarters with each quarter numbered. Written explanation of sharing.

A student’s sharing diagram.

F_GT_T2_E2_A1_M_fig2

Three groups of four stick figures, each group with three pancakes which are subdivided to represent the equal division.

Another sharing diagram.

F_GT_T2_E2_fig1

Each loaf is cut into three equal pieces and the six pieces arranged as three groups of two.

Two loaves shared evenly between three people.

F_GT_T3_E1_A1_fig1

Four horizontal strips of paper of equal length. First strip marked as one whole, the next folded in two and marked as halves, the next into quarters and the last into eighths.

A fraction wall.

F_GT_T3_E1_A1_fig2

Cards marked with equivalent fractions.

Pairs of equivalent fractions.

F_GT_T3_E1_A2_fig1

Game board showing one completely coloured fraction wall and one partially coloured fraction wall, and a completed record of the rolls and the colourings made.

A completed game.

F_GT_T3_E1_fig1

Number line marked with sixths below the line and thirds above the line to show equivalence.

Visualising equivalent fractions.

F_GT_T3_E2_A1_fig1

Selection of grid transparencies with one pair overlaid and equivalent fractions recorded.

Use of overlay templates.

F_GT_T3_E2_A2_fig1

Sample of a 4 by 3 grid with four parts shaded. The equivalent fractions 4/12 and 1/3 are recorded.

Screen grab from L134 Dynamic fractions.
Source: © Education Services Australia Ltd, 2011 

F_GT_T3_E2_fig1

Rectangle divided into thirds horizontally. Top layer divided into halves vertically, second into thirds, last into sixths.

A fraction wall.

F_GT_T3_E2_fig2

Two equally sized squares, first divided into halves vertically, second into thirds horizontally, then combined to show sixths.

Combining grids.

F_GT_T3_fig1

Three circles, one divided into halves, one into quarters, one into eighths.

Subdividing with related denominators.

F_GT_T4_E1_A1_fig1

Number line between 0 and 1,divided into eighths, showing ‘jumps’ from 0 to 4/8 then a further 3/8 to 7/8.

Student work sample showing addition of fractions with related denominators.

F_GT_T4_E1_A2_fig1

The addition of 1/6 and 5/6 to reach the target of 1, shown with fraction bars and a number line.

A screen grab from L2804 Fraction fiddle: hit the apple.
Source: © Education Services Australia Ltd, 2011 

F_GT_T4_E2_A1_fig1

Examples of addition and subtraction of fractions with related denominators using grids and number lines

Student work sample.

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