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Reasoning

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Triangular array of billiard balls, five on each side and 15 in total.

Billiard balls.
Source: James Barker/Freedigitalphotos.net

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Teacher recording observations of student working.

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Row of eight small squares matched to row of five large squares. Student response to questions comparing the areas, explaining reasoning.

Row of squares task.

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A long box divided into eight square sections. Below it, a second box divided into five sqaure sections. Box long boxes measure 10 cm in length. 10

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Student drawing of the 5 dice faces, explanation that double 8 is 16 then counting on 5 more gives 21.

A learning story about addition reasoning.

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Each of the triangular numbers 6 and 10 is modelled with rows of squares. Then one shape is rotated 180 degrees so that the two triangular numbers fit together, creating a 4 by 4 square.

Proof by modelling.

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Student work; picture of human bones; text; length of paper streamer, predominantly white with a thin strip of black.

Mungo Man poster.
Source: © 2012 MathsIsFun.com

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Four triangles of different sizes, each labelled with two interior angles.

Similar or not?

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Lines on a rectangle showing that it is to be folded in half and half again, to make a smaller rectangle. Then a triangular notch is cut out of one of the folded edges.

One example of paper folding.

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Screen grab showing an isosocles triangle with a single line of symmetry.

Screen grab from L7803 Finding symmetry: one line: city.
© Education Services Australia, 2011

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Arrays for multiplication explained using an egg carton (a six by two array).

Screen grab from L108 The array: go figure.
© Education Services Australia, 2011

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Horizontal line with the numbers 80, 24 and 32 above the line and the numbers 15, 71, 11 and 13 below the line.

Starting point for activity.

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Three images: a single cube, a square made of 9 cubes, a square made of 49 cubes.

Models of square numbers.

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A block of 2 by 3 (6) cubes and a block of 7 by 2 (14) cubes.

Models of non-square numbers.

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Square, rectangle, parallelogram, trapezium all with same vertical height.

Four different quadrilaterals.

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A 5 x 5 section of a hundred square with 7 numbers covered by an H shape.

Numbers covered by H.

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Close up of the top of an analogue clock, showing time as five seconds past twelve.

Source: Grant Cochrane/Freedigitalphotos.net

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Full face of an analogue clock, showing time as two minutes to twelve.

Source: graur razvan ionut/Freedigitalphotos.net

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Three long rows of red apples. The first row has one black apple in the centre, the second row three black apples in the centre.

Rows of apples.

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19 red apples in a line with the 10th apple coloured black

Day 1

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19 red apples in a line with the 9th, 10th and 11th apples coloured black

Day 2

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Two large rectangular arrays of red apples. The first has one black apple in the centre of top row. The second has three black apples in the centre of the top row and one in the centre of the second row.

Trays of apples.

R_BI_I2_E1_A1_fig2a

8 rows of 19 red apples in lines with the 10th apple in the first row coloured black

Day 1

R_BI_I2_E1_A1_fig2b

8 rows of 19 red apples in lines with the 9th, 10th and 11th apples in the first row and the 10th apple in the second row coloured black

Day 2

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A person visualising a net of a box and the completed cardboard box.

Visualising.
Source: graur codrin/Freedigitalphotos.net

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