Explains how learning with geo-boards - in the style of Caleb Gattegno - can be extended into Lower Secondary Maths (Years 7, 8 and 9 )........ Following a standard curriculum, over 100 examples are given.......... specifically designed for teachers and researchers.................... For worksheets and more, go to www.mathswithgeoboards.com

1. Maths with GeoboarDS
for Lower Secondary Maths
By Jon Molomby
An Overview for Teachers
and Researchers

2. “ The way of teaching
is to inspire,
not to inform ”

3. “ It is often repeated
that ‘ I teach them, but
they don’t learn ! ’
Well, if you know that,
stop teaching : not
resign from your job,
stop teaching in the way
that doesn’t reach
…people”

4. • Egyptian born
• Son of a Spanish trader
• Largely self-taught
• Pioneered the use of geoboards and Cuisenaire rods
• Spent the latter part of his life going around the world
teaching , giving lectures and seminars.
Caleb Gattegno
(1911 – 1988)

5. Book :
“Geoboard Geometry” ( 1971 )
by Caleb Gattegno
His main focus is
Primary Maths
https://issuu.com/eswi/docs/1027_ge
oboard_geometry/32

6. Video : “Mathematics at your Fingertips” (1961 )
Gattegno teaching fractions to Grade Ones with
Cuisenaire rods ( on Youtube )

7. Gattegno, in the 1980s, criticizing teaching methods
“ There is no good system of learning, because we are
only concerned with one component : that is, the teacher,
and what the teacher does, and we give means to the
teachers, thinking that what the teacher does will make
the student do better, and we have not been able to
substantiate this hypothesis. What is required is to
ask the question : ‘How could I improve the learning ?’ ”

8. Gattegno’s “invention” was putting a small geo-board in
the hands of every student, and giving them exercises
Gattegno believed hands on Maths tools used like this
greatly accelerate students’ learning

9. LEARNING
with
GEOBOARDS
Every student
is given a
geo-board
(from a
class set)

10. LEARNING
with
GEOBOARDS
The teacher
sets the
students a
task

11. LEARNING
with
GEOBOARDS
It is very
easy for the
teacher to
correct
answers

12. Typical Lower Secondary Maths Curriculum
GRADE 7 GRADE 8 GRADE 9
GEOMETRY – POLYGONS : names,
perimeter, symmetry, angles,
area, diagonals
NUMBER THEORY
INTEGERS
INDICES or EXPONENTS or POWERS
ANGLES
GEOMETRY – USING A COMPASS
DECIMALS and FRACTIONS
ESTIMATION and APPROXIMATION
ORDERED PAIRS and GRAPHS
POLYNOMIALS
LINEAR EQUATIONS
RATIO and PERCENTAGE
MEASUREMENT
PIE CHARTS
CONGRUENT TRIANGLES
TRANSFORMATIONS
REAL NUMBER & SQUARE ROOT
PYTHAGORAS’ THEOREM
VARIATION
LINEAR FUNCTION
GEOMETRY - PARALLEL LINES
VOLUMES & SURFACE AREAS of SOLIDS
GRAPHING TWO LINEAR EQUATIONS
SIMULTANEOUS EQUATIONS
SIMILARITY
SETS
INEQUALITIES
STATISTICS
PROBABILITY
TRIGONOMETRY
FACTORING POLYNOMIALS
PARABOLAS
POLYNOMIAL FRACTIONS
MISC. GEOMETRY THEOREMS
KEY : RED + BOLD : geo-boards can be used for these topics

13. The rest of this presentation is
examples of geoboard exercises
• Over 140 slides follow, which are examples for each
relevant curriculum topic.
• It starts at the Primary level (Prathom) and goes
through to the end of Year 9 (M3)
• To get an overview, these slides can be viewed quite
quickly, pausing only to check on details.

14. PRIMARY :
POLYGONS
Maths with Geoboards

15. PRIMARY
- Polygons
- Triangles

16. PRIMARY
- Polygons
- Quadrilaterals

17. PRIMARY
- Polygons
- Pentagons,
Hexagons,
Octagons

18. PRIMARY - Polygons - study of quadrilaterals
1. Name of Shape -
Rectangle
2. Perimeter
3. Symmetry
4. Angles
5. Area
6. Diagonals

19. PRIMARY - Polygons - study of quadrilaterals
1. Name of Shape -
Rectangle
2. Perimeter – 12 units
3. Symmetry
4. Angles
5. Area
6. Diagonals

20. PRIMARY - Polygons - study of quadrilaterals
1. Name of Shape -
Rectangle
2. Perimeter – 12 units
3. Symmetry – 2 lines
4. Angles
5. Area
6. Diagonals

21. PRIMARY - Polygons - study of quadrilaterals
1. Name of Shape -
Rectangle
2. Perimeter – 12 units
3. Symmetry – 2 lines
4. Angles – 4 right angles
5. Area
6. Diagonals

22. PRIMARY - Polygons - study of quadrilaterals
1. Name of Shape -
Rectangle
2. Perimeter – 12 units
3. Symmetry – 2 lines
4. Angles – 4 right angles
5. Area – 8 sq. units
6. Diagonals

23. PRIMARY - Polygons - study of quadrilaterals
1. Name of Shape -
Rectangle
2. Perimeter – 12 units
3. Symmetry – 2 lines
4. Angles – 4 right angles
5. Area – 8 sq. units
6. Diagonals – equal in
length, bisect each other,
bisect the area

24. PRIMARY /SECONDARY
ANGLES
Maths with Geoboards

25. PRIMARY / SECONDARY Angles Make the following angles
acute right obtuse
Possible
Answerstraight reflex full rotation

26. PRIMARY / SECONDARY Angles
Interior angle sum of a triangle & complementary angles
Possible
Answer

27. PRIMARY / SECONDARY Angles
Interior angle sum of a quadrilateral & supplementary angles

28. SECONDARY Angles Interior angle sum of a hexagon :
Possible
Answer
6 triangles – 360 deg. (6 – 2) x 180 deg.

29. SECONDARY Angles Interior angle sum of a octagon :
Possible
Answer
8 triangles – 360 deg. (8 – 2) x 180 deg.

30. SECONDARY Angles Make concave and convex polygons
Pentagons of area 12 Octagons of area 10
Possible
Answer
Possible
Answer

31. PRIMARY / SECONDARY
FRACTIONS and
DECimals
Maths with Geoboards

32. PRIMARY / SECONDARY
Fractions
Teacher : “ The geo-board
is made up of
36 small squares.
Let the whole geo-board
equal 1 ”

33. PRIMARY / SECONDARY
Fractions
Teacher :
“ This is 1 (in blue),
make ½
What is the rest of the
area ? ”
Answer : ½
Possible
Answer

34. PRIMARY / SECONDARY
Fractions
Teacher :
“ This is 1 (in blue),
make 1/4
What is the rest of the
area ? ”
Answer : 3/4
Possible
Answer

35. PRIMARY / SECONDARY
Fractions
Teacher :
“ This is 1 (in blue),
make 1/3
What is the rest of the
area ? ”
Answer : 2/3
Possible
Answer

36. PRIMARY / SECONDARY
Fractions
Teacher :
“ This is 1 (in blue),
make 5/12
What is the rest of the
area ? ”
Answer : 7/12
Possible
Answer

37. PRIMARY / SECONDARY
Fractions
Teacher :
“ This is 1 (in blue),
make 4/9
What is the rest of the
area ? ”
Answer : 5/9
Possible
Answer

38. PRIMARY / SECONDARY
Improper Fractions
Teacher :
“ This is 1 (in blue),
make 4/3
Possible
Answer
Possible
Answer

39. PRIMARY / SECONDARY
Improper Fractions
Teacher :
“ This is 1 (in blue),
make 5/2
Possible
Answer

40. PRIMARY / SECONDARY
Mixed Numbers
Teacher :
“ This is 11/3 (in blue)
make 22/3 ”
Possible
Answer

41. PRIMARY / SECONDARY
Mixed Numbers
Teacher :
“ This is 22/3 (in blue)
make 6
as a triangle”
Possible
Answer

42. PRIMARY / SECONDARY
Mixed Numbers
Teacher :
“ This is 3 (in blue),
make 21/2
as a trapezium ” Possible
Answer

43. SECONDARY :
ORDERED PAIRS and
GRAPHS
Maths with Geoboards

44. SECONDARY
Ordered Pairs
and Graphs
Teacher explains
the Cartesian plane,
(x,y) coordinates,
and the quadrants
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1

45. SECONDARY
Ordered Pairs
and Graphs
Teacher :
“Make the line
y = 1 ”
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1

46. SECONDARY
Ordered Pairs
and Graphs
Teacher :
“Make the line
x = 2 ”
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1

47. SECONDARY
Ordered Pairs
and Graphs
Teacher :
“Plot the points
(1,1), (3,1), (2,3)
and make a triangle”

48. SECONDARY
Ordered Pairs
and Graphs
Teacher :
“Plot the points
(-3,-3), (-2, 3), (2,2)
and make a triangle”

49. SECONDARY
Ordered Pairs
and Graphs
Teacher :
“Plot the points
(-3,-1), (-2,2), (2,-2)
and make a triangle”

50. SECONDARY
Ordered Pairs
and Graphs cont.
Teacher explains
the gradient
(rise over run )
and the y intercept
The form : y = mx + c
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
rise
run

51. SECONDARY
Ordered Pairs
and Graphs
Teacher :
“Graph the equation
y = x ”

52. SECONDARY
Ordered Pairs
and Graphs
Teacher :
“Graph the equation
y = 2x + 1 ”

53. SECONDARY
Ordered Pairs
and Graphs
Teacher :
“Graph the equation
y = 3x – 3 ”

54. SECONDARY
Ordered Pairs
and Graphs
Teacher :
“Graph the equation
y = - 6x – 3 ”

55. SECONDARY :
RATIO and
PERCENTAGE
Introduction only
Maths with Geoboards

56. SECONDARY
Ratio and Percentage
Teacher : “ Express the
white part as a ratio and a
percentage ”
The answer :
Ratio 1 : 3 , % 25%

57. SECONDARY
Ratio and Percentage
Teacher : Express the white
part as a ratio and a
percentage
The answer :
Ratio 1 : 4 , % 20%

58. SECONDARY
Ratio and Percentage
Teacher : “Express all
colours as ratios and %s ”
Answer : Red 3 : 7, 30%
Yellow 1 : 4, 20%
Blue 1 : 1, 50%

59. SECONDARY
Ratio and Percentage
Teacher :
“Express the blue area
as a ratio of
1 : 2 : 3
in different colours ”
Possible Answer

60. SECONDARY
Percentage
(increase and decrease)
Teacher :
“Starting with the figure
in blue, make 80 % ”
Possible
Answer

61. SECONDARY
Percentage
(increase and decrease)
Teacher : “Starting with
the figure in blue, make a
40% decrease ”
Possible
Answer

62. SECONDARY
Percentage
(increase and decrease)
Teacher : Starting with
the figure in blue, make
an increase of
20 % (= 120 %)
Answer

63. SECONDARY
Percentage
(increase and decrease)
Teacher : Starting with
the figure in blue,
make an increase of
25 % (= 125 %)
Possible Answer

64. SECONDARY
Percentage
(increase and decrease)
Teacher : Starting with
the figure in blue,
make an increase of
75 % (= 175 %)
Possible
Answer

65. SECONDARY
Percentage
(increase and decrease)
Teacher : Starting with
the figure in blue,
make a decrease of 25 %
also as a triangle
Possible
Answer

66. SECONDARY
Percentage
(increase and decrease)
Teacher : Starting with
the figure in blue,
make 80 % ,
also as a parallelogram
Possible
Answer
Possible
Answer

67. SECONDARY
Percentage
(increase and decrease)
Teacher : Starting with
the figure in blue,
make 75 % ,
also as a trapezium
Possible
Answer
Possible
Answer

68. SECONDARY :
CONGRUENT
TRIANGLES
Introduction only
Maths with Geoboards

69. SECONDARY Congruent Triangles Teacher :
“Make two congruent triangles, facing different ways.
(i) right (ii) isosceles (iii) obtuse ”

70. SECONDARY :
TRANSFORMATIONS
( Translations, Reflections,
Rotations )
Maths with Geoboards

71. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Translations
Teacher :
“ Make an object :
(1, 1), (2 ,1), (3, 3)”

72. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Translations
Teacher :
“ Translate the object
back 4 units ”

73. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Translations
Teacher :
“ Translate the object
down 4 units ”

74. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Translations
Teacher :
T-4,-4 (x,y)

75. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Reflections
Teacher :
“ Make an object :
(2,2), (1, -1), (3,-3)”

76. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Reflections
Teacher :
“Reflect over
the y axis ”

77. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Reflections
Teacher :
“Reflect over
the x axis ”

78. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Reflections
Teacher :
“ Reflect over
x = 1 ”

79. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Reflections
Teacher :
“ Reflect over
x = 2 ”

80. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Reflections
Teacher :
“ Reflect over
y = x ”

81. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Rotations
Teacher :
“ Make an object :
(2,3), (0, 1), (3,-1)”

82. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1
SECONDARY
Transformations :
Rotations
Teacher : “ Rotate
900 clockwise
about the origin ”

83. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Rotations
Teacher : “ Rotate
900 , 1800 , 2700
about the origin
i.e. Order 4 ”

84. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Rotations
Teacher : “ Rotate
900 clockwise about
one of its vertices,
(0,1)”

85. -3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Rotations
Teacher : “ Rotate
900 clockwise about
a point inside the
object , (1,1) ”

86. SECONDARY :
PYTHAGORAS’
THEOREM
Maths with Geoboards

87. SECONDARY
Pythagoras
Theorem
Teacher :
“ What is the length
of the hypotenuse ? ”
Answer : 5 units

88. SECONDARY
Pythagoras
Theorem
Teacher :
“ What is the length
of the hypotenuse ?
Answer :
√32 = 4√2 units

89. Answer :
√40 = 2√10 units
SECONDARY
Pythagoras
Theorem
Teacher :
“ What is the length
of the hypotenuse ?

90. SECONDARY
Pythagoras Theorem :
Converse of the Theorem
Teacher :
“ Is this a right triangle ? ”
Answer : No
(√37)2 ≠ (√20)2 + (√25)2
.˙. It is not a right triangle

91. SECONDARY
Pythagoras Theorem :
Converse of the Theorem
Teacher :
“ Is this a right triangle ? ”
Answer : No
(√45)2 ≠ (√17)2 + (√26)2
.˙. It is not a right triangle

92. SECONDARY :
PYTHAGORAS’
THEOREM - proofs
Maths with Geoboards

93. SECONDARY
Pythagoras Theorem
Geometric Proof
C2
C2
C2
a2
b2

94. SECONDARY
Pythagoras
Theorem –
Proofs
Geometric
Proof:
( Euclid’s )

95. SECONDARY
Pythagoras
Theorem –
Proofs
Algebraic Proof
c2 =
4( ½ ab ) + (b – a)2

96. SECONDARY
Pythagoras
Theorem – Proofs
Algebraic Proof:
( Garfield’s )
[(a + b)/2 ] x (a + b)
= 2( ½ ab ) + ½ c2

97. SECONDARY :
PARALLEL LINES
Introduction only
Maths with Geoboards

98. SECONDARY
Parallel Lines
Teacher : “ The
lines are parallel,
cut by another
straight line
( a transversal )”
Possible
Answer

99. SECONDARY
Parallel Lines Teacher : “ Using coins of the
same size, mark equal angles “
Possible
Answer

100. SECONDARY Parallel Lines Teacher : “ Using
coins of different size, mark supplementary angles ”

101. SECONDARY :
VOLUMES and SURFACE
AREAS of SOLIDS
Maths with Geoboards

102. SECONDARY
Volumes and
Surface Areas
Answer :
V. = 64 cub. units
S.A. = 96 sq. units
Teacher : “What is the V.
and S.A. of this cube ? ”

103. SECONDARY
Volumes and
Surface Areas
Teacher :“ What is the V. and
S.A. of this rectangular prism
(the top is a square) ?”
Answer :
V. = 16 cub. units
S.A. = 48 sq. units

104. SECONDARY
Volumes and
Surface Areas
Teacher :“ What is the V. and
S.A. of this triangular prism
(the top is a square) ? ”
Answer :
V. = 30 cub. units
S.A. = 72 sq. units

105. SECONDARY
Volumes and
Surface Areas
Teacher : “If the original
perpendicular height of this
square pyramid was 6 units,
and its top half was removed,
what is the V. and S.A. ? ”
V. = 28 cub. units
S.A. = 20 + 12√10 sq. units

106. SECONDARY
Volumes and
Surface Areas
Teacher : “The ten sided base
of this regular pyramid has a
side of 2√2, an apothem of 2√2
and a side height of 5√2
V. and S.A.? ”
V. = 13 1/3 √42 cub. units
S.A. = 140 sq. units

107. SECONDARY
Volumes and
Surface Areas
Teacher : “Create your
own 3D shape and
calculate the
V. and S.A.”

108. SECONDARY :
PLATONIC SOLIDS
Maths with Geoboards

109. SECONDARY
Platonic Solids
Teacher explains the 5 Platonic Solids:
there is only one (the cube) that can be
truly made, but students can learn by
trying to make the others …

110. SECONDARY
Platonic Solids
Try to make its net
Teacher : Try to make a
Tetrahedron

111. SECONDARY
Platonic Solids
Make its net
Teacher : Make a
Cube

112. SECONDARY
Platonic Solids
Try to make its net
Teacher : Try to make a
Octahedron

113. SECONDARY
Platonic Solids
Teacher :
Try to make a
Dodecahedron

114. SECONDARY
Platonic Solids
Teacher :
Try to make a
Icosahedron

115. SECONDARY :
GRAPHING TWO
EQUATIONS
Maths with Geoboards

116. SECONDARY
Graphing 2 Equations
Teacher : “ Name the
two linear equations”
(Note : m1 = m2 )
Answer :
(i) y = x + 2 (ii) y = x

117. SECONDARY
Graphing 2 Equations
Teacher : “ Name the
two linear equations”
(Note : m1 = m2 )
Answer :
(i) y = 2x + 1
(ii) y = 2x – 1

118. SECONDARY
Graphing 2 Equations
Teacher : “ Name the
two linear equations”
(note : m1 m2 = -1 )”
Answer :
(i) y = x + 1 (ii) y = -x – 1
m1 m2 = (1)(-1) = -1

119. SECONDARY
Graphing 2 Equations
Teacher : “ Name the
two linear equations
(note : m1 m2 = -1 )”
Answer :
(i)y = 2x + 1 (ii)y = -½ x–1.5
m1 m2 = (2)(-½) = - 1

120. SECONDARY :
SIMILARITY
Introduction only
Maths with Geoboards

121. SECONDARY
Similarity (2D)
Teacher : “Find the
side to side scale factor
… (large : small ) and
the area to area S.F. ”
Answer :
side to side S.F. - 3 : 1
area to area S.F.- 9 : 1

122. SECONDARY
Similarity (2D)
Teacher : “Find the
side to side scale factor
… (large : small ) and
the area to area S.F. ”
Answer :
side to side S.F. - 5 : 2
area to area S.F.- 25 : 4

123. SECONDARY
Similarity (2D)
Teacher : “Find the
side to side scale factor
… (large : small ) and
the area to area S.F. ”
Answer :
side to side S.F. - 5 : 2
area to area S.F.- 25 : 4

124. SECONDARY
Similarity (3D)
Teacher “Find the following
ratios of the cubes:
(i) side to side S.F.
(ii) S.A. to S.A. S.F.
(iii) V. to V. S.F.
Answer :
(i) 2 : 1 (ii) 4 : 1 (iii) 8 : 1

125. SECONDARY
Similarity (3D)
Teacher “Find the ratios of
these rectangular prisms:
(i) side to side S.F.
(ii) S.A. to S.A. S.F.
(iii) V. to V. S.F.
(depth: lge. 2√5, sm.√5)
Answer :
(i) 2 : 1 (ii) 4 : 1 (iii) 8 : 1

126. SECONDARY
Similarity (3D)
Teacher “Find the ratios of
these triangular prisms:
(i) side to side S.F.
(ii) S.A. to S.A. S.F.
(iii) V. to V. S.F.
(depth: lge. 2√2, sm.√2)
Answer :
(i) 2 : 1 (ii) 4 : 1 (iii) 8 : 1

127. SecondARY :
GRAPHING
INEQUALITIES
Maths with Geoboards

128. SECONDARY
Graphing Inequalities
Teacher : List the
inequalities which surround
and define the closed area
Answer : y > –x +2,
y < -1,
y > 2x + 1

129. SECONDARY
Graphing Inequalities
Teacher : List the
inequalities which surround
and define the closed area
Answer : y < –2,
y > -x,
y > -3x + 3

130. SECONDARY
Graphing Inequalities
Teacher : List the
inequalities which surround
and define the closed area
Answer : y < -6x - 3,
y < -2,
y > -2x + 1

131. SECONDARY :
TRIGONOMETRY
Maths with Geoboards

132. SECONDARY
Trigonometry
Teacher : “Use
SOHCAHTOA and your
calculator to find the
measure of this angle.”
Ans : arcsin (1/√37)
= 9.4620

133. SECONDARY
Trigonometry
Teacher : “Use
SOHCAHTOA and your
calculator to find the
measure of this angle.”
Ans : arcsin (4/2√13)
= 33.690

134. SECONDARY
Trigonometry
Teacher :
“This angle is 450
Use SOHCAHTOA to find the
length of the hypotenuse.”
Ans : 8.485 units

135. SECONDARY :
THE CENTRE of a
Triangle
Maths with Geoboards

136. SECONDARY
The Centre of a
Triangle
The Centroid :
intersection of the
medians. Divides the
triangle into 6 equal
areas. The medians
are cut 1 : 2

137. SECONDARY
The Centre of a
Triangle
The Circumcentre
intersection of the
perpendicular bisectors.

138. SECONDARY
The Centre of a
Triangle
The Incentre :
intersection of the
angle bisectors.

139. SECONDARY
The Centre of a
Triangle
The Orthocentre :
intersection of the
altitudes : perpendiculars
dropped from each angle
to their opposite side.

140. SECONDARY
The Centre of a
Triangle
The Medial Triangle
Joining the midpoints
forms 4 congruent
triangles, all similar to the
original triangle with a side
to side scale factor of 2 : 1

141. SECONDARY :
MISCELLANEOUS
GEOMETRIC
THEOREMS
Maths with Geoboards

142. Triangle Midpoint
Theorem :
SECONDARY
Geometry Theorems
that can be tested
Line is parallel to c
and ½ its length

143. SECONDARY
Geometry Theorems
that can be tested
Angle Bisector
Theorem

144. a2 + b2 =
2(½ c)2 + 2d2
(√29)2 +(√41)2 =
2(3)2 + 2(√26)2
29 + 41 = 18 + 52
SECONDARY
Geometry Theorems
that can be tested
Apollonius’ Theorem

145. A = i + b/2 - 1
= 15 + 18/2 - 1
= 23 sq. units
SECONDARY
Geometry Theorems
that can be tested
Pick’s Theorem

146. Euclid’s Orchard
From the origin, how many
trees can you see? Group
together the trees seen plus all
those behind them (unseen).
Let each point be x over (x +
y). What do you notice ?
SECONDARY
Geometry Theorems
that can be tested
1 2 3 4 5 6Ans : Equivalent fractions

147. CONCLUSIONS
Maths with Geoboards

148. Q. : Why do geoboards work as a learning tool?
An answer from the student’s point of view
1. “ User friendly “ Fun to use. It is easy to attempt an answer,
easy to correct a mistake.
2. Geoboards help concentration : attention is kept easily on the task
3. Concepts are clear, and not difficult to understand
4. Integer based, adding to the simplicity
5. Can be used to test newly learned geometric theorems and shapes
6. Hands-on : this helps learning, and also helps recall,
or “retention”, as Gattegno called it.

149. 1. Students enjoy using geoboards : their interest in Mathematics is greater
2. Cure “fear” of Maths, and help others who find Maths “hard” or “boring”
3. Students learn quickly, advancing ahead of the curriculum
4. Repeated exercises increase students’ skill, understanding and confidence
5. “Hands-on” learning helps both understanding and retention of knowledge
6. Time efficient : geo-boards need only be used for 10 - 15 minutes of a class,
and answers in class can quickly be corrected by the teacher (vs. a worksheet)
7. Once students have done an exercise, and understand it, they can then be told
to make up their own examples, to answer themselves, or ask others (in groups).
8. Cheap to make, easy to maintain, a class set is portable (with a car)
Q. : Why do geoboards work as a learning tool ?
An answer from the teacher’s point of view

150. GEOBOARD
CONSTRUCTION
Maths with Geoboards

151. Geoboard Construction
1. MDF board or plywood
2. Lacquer + paint (optional)
3. Matt black vinyl sticker
4. Brass nails
5. Odd number of nails, e.g. 7 x 7
6. Cheap to make
------------------------------
See my video on Youtube,
“ How to Make Geoboards ”
https://www.youtube.com/watch?v=9yhCxlk9fx4

152. LINKS
Maths with Geoboards

153. My Links
• My Youtube channel : “ Maths with Geoboards “
( for videos )
• My website : www.mathswithgeoboards.com
( for worksheets )
• My email : mathswithgeoboards@gmail.com
( for contact )
This presentation available as a video (in English) on Youtube
As a slide show ( in English and Thai ) on SlideShare

154. Pythagoras’ Theorem
Pythagoras Theorem Vol. 1
Pythagoras Theorem Vol. 2
Proofs of Pythagoras Theorem - Geometric
Proofs of Pythagoras Theorem - Algebraic
Number Theory
Prime Numbers
Fibonacci Numbers
Factors and Multiples
General
How to Make Geo-boards
Angles
Introductory Terms
Easy Angle Problems
The World’s Two Hardest Easy Geometry Problems
Indices
Indices : Laws 1, 2, 3 and 4
Indices : Word Problems
Probability
The Monty Hall Problem
Youtube Channel : “ Maths with Geoboards” videos so far

155. Video : “ Mathematics at your Fingertips ” (1961) on Youtube.com
https://www.youtube.com/watch?v=ae0McT5WYa8
Caleb Gattegno on the internet
-----------------------------
Book : “ Geoboard Geometry “ ( 1971 )
https://issuu.com/eswi/docs/1027_geoboard_geometry/32

156. THE END
“ I don’t teach, I let them learn ”
Caleb Gattegno
=========================

157. “ข้อมูลนี้ถูกแปลเป็นภาษาไทยเพื่อส่งเสริมการศึกษาไทย
เพื่อถวายเป็นพระราชกุศล แด่
พระบาทสมเด็จพระปรมินทรมหาภูมิพลอดุลยเดช
รัชกาลที่ ๙”