Further Materials

# Small Insets

Material Description:

Ten square green frames each containing red squares or triangles formed by dividing a  square vertically, horizontally and diagonally.  Each shape has a central knob to hold the inset by.

On the first shelf are the horizontally and vertically divided insets;

The first is a ‘whole’ square

The second a square vertically ‘halved’ into two rectangles

The third is further divide into four small squares ‘quarters’

The fourth has the ‘quarters’ vertically divided into eight rectangles

The fifth divides the ‘eighths’ into sixteen small squares

On the second shelf are the insets which also have diagonally divisions;

The first is a ‘whole’ square

The second of these is a square and divided into two right angled isosceles triangles

The third has the isosceles triangles with a  further diagonal division, to create four right angled isosceles triangles

The fourth has the ‘quartered’ triangles subdivided by a cross, to create eight triangles

The fifth has the ‘eight’ triangles subdivided by a rhombus, to create sixteen triangles

Each shape is either a square, a rectangle or a right angled isosceles triangle

The name of the fraction is not given to the child until much later, it is used here as a reference only

The insets are carried one frame at a time

Much later the child is given the language

Presentation:

• Introduce one inset and let the child look at all of them

• Introduce Similar
• When the child is familiar with all of the insets present by putting all of the square insets in front of the child, one at a time
• Remove the large square ‘Whole’ and place it on the mat in front of its frame.
• Ask the child to find an identical shape when she cannot show her that the third and fifth set contain similar plane figure, of different proportions, not by speaking but by removing the ‘quarters’ and ‘sixteenths’ and placing them on the mat in front of their frames, in a line with the ‘whole’ inset
• Repeat this with a rectangle of the second inset with the fourth inset; the ‘half’ with the ‘eighths’
• The child can do the same with the triangular insets; the ‘half’ triangle is the same shape as two ‘quarters’, four ‘eighths’ and six ‘sixteenths’

• Introduce Identical at a separate sitting
• Take two rectangles from the second frame ‘halves’ and place them back to back to show that they are identical
• Take two squares from frame three ‘quarters’ and show that they fit exactly onto the  back of the ‘half’ rectangle
• Let the child find other identical rectangles, squares and triangles with figures of the same plane

Encourage her to draw round and cut and paste the shapes when she is older, or make a booklet.

• Introduce Equivalent, much later than similar and identical
• Take the large square ‘whole’, from the frame and in it’s place put two ‘half’ rectangles from the second inset
• Show that they fit exactly, use the word ‘equivalent’
• Let the child use all the insets together for example, fit the second triangle inset ‘halves’ into the first ‘whole’ square frame
• Show that they are equivalent by placing the rectangle ‘half’ upside down and the triangle ‘half’ on top, complete showing that the ‘eighth’ triangle fits into the remaining gap of both the shapes.

When the child continues to use the materials like this she will discover equivalence.

Exercises:

• When the child later works with fractions she can put name labels on each plane figure
• When the child is able to use Geometric Instruments she can make similar, identical and equivalent shapes, using the set square

Language:

When the words ‘Similar’, ‘Identical’ and ‘Equivalent’ are known revise them in a Three Period Lesson

Criteria of Perfection (Control of Error):

• In the child’s ability, the frame and the insets

Direct Aim:

• To help the child understand smililar, identical and equivalence
• To further explore geometry

Age at Presentation:

Beginning at three to three and a half years, later used in fractions to find the common denominator

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