Maths with GeoboarDSfor Lower Secondary Maths
By Jon Molomby
An Overview for Teachers and Researchers
“ 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”
• 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)
Book : “Geoboard Geometry” ( 1971 )
by Caleb Gattegno
His main focus is Primary Maths
https://issuu.com/eswi/docs/1027_geoboard_geometry/32
Video : “Mathematics at your Fingertips” (1961 ) Gattegno teaching fractions to Grade Ones with
Cuisenaire rods ( on Youtube )
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 ?’ ”
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
Typical Lower Secondary Maths CurriculumGRADE 7 GRADE 8 GRADE 9
GEOMETRY – POLYGONS : names,perimeter, symmetry, angles, area, diagonals
NUMBER THEORYINTEGERSINDICES or EXPONENTS or POWERSANGLESGEOMETRY – USING A COMPASSDECIMALS and FRACTIONSESTIMATION and APPROXIMATIONORDERED PAIRS and GRAPHS POLYNOMIALSLINEAR EQUATIONS
RATIO and PERCENTAGEMEASUREMENTPIE CHARTSCONGRUENT TRIANGLESTRANSFORMATIONS REAL NUMBER & SQUARE ROOTPYTHAGORAS’ THEOREMVARIATIONLINEAR FUNCTIONGEOMETRY - PARALLEL LINES
VOLUMES & SURFACE AREAS of SOLIDS GRAPHING TWO LINEAR EQUATIONSSIMULTANEOUS EQUATIONSSIMILARITY SETSINEQUALITIESSTATISTICSPROBABILITYTRIGONOMETRYFACTORING POLYNOMIALSPARABOLASPOLYNOMIAL FRACTIONSMISC. GEOMETRY THEOREMS
KEY : RED + BOLD : geo-boards can be used for these topics
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.
PRIMARY - Polygons - study of quadrilaterals 1. Name of Shape -
Rectangle2. Perimeter3. Symmetry4. Angles 5. Area6. Diagonals
PRIMARY - Polygons - study of quadrilaterals
1. Name of Shape -Rectangle
2. Perimeter – 12 units 3. Symmetry 4. Angles5. Area6. Diagonals
PRIMARY - Polygons - study of quadrilaterals 1. Name of Shape -
Rectangle2. Perimeter – 12 units3. Symmetry – 2 lines4. Angles 5. Area6. Diagonals
PRIMARY - Polygons - study of quadrilaterals 1. Name of Shape -
Rectangle2. Perimeter – 12 units3. Symmetry – 2 lines4. Angles – 4 right angles5. Area6. Diagonals
PRIMARY - Polygons - study of quadrilaterals 1. Name of Shape -
Rectangle2. Perimeter – 12 units3. Symmetry – 2 lines4. Angles – 4 right angles5. Area – 8 sq. units6. Diagonals
PRIMARY - Polygons - study of quadrilaterals 1. Name of Shape -
Rectangle2. Perimeter – 12 units3. Symmetry – 2 lines4. Angles – 4 right angles5. Area – 8 sq. units6. Diagonals – equal in length, bisect each other, bisect the area
PRIMARY / SECONDARY Angles Make the following angles acute right obtuse
Possible Answerstraight reflex full rotation
SECONDARY Angles Interior angle sum of a hexagon :
Possible Answer
6 triangles – 360 deg. (6 – 2) x 180 deg.
SECONDARY Angles Interior angle sum of a octagon :
Possible Answer
8 triangles – 360 deg. (8 – 2) x 180 deg.
SECONDARY Angles Make concave and convex polygonsPentagons of area 12 Octagons of area 10
Possible Answer
Possible Answer
PRIMARY / SECONDARY
Fractions
Teacher : “ The geo-board is made up of
36 small squares. Let the whole geo-board
equal 1 ”
PRIMARY / SECONDARY
Fractions
Teacher : “ This is 1 (in blue),
make ½What is the rest of the
area ? ”Answer : ½
Possible Answer
PRIMARY / SECONDARY
Fractions
Teacher : “ This is 1 (in blue),
make 1/4
What is the rest of the area ? ”
Answer : 3/4
Possible Answer
PRIMARY / SECONDARY
Fractions
Teacher : “ This is 1 (in blue),
make 1/3
What is the rest of the area ? ”
Answer : 2/3
Possible Answer
PRIMARY / SECONDARY
Fractions
Teacher : “ This is 1 (in blue),
make 5/12
What is the rest of the area ? ”
Answer : 7/12
Possible Answer
PRIMARY / SECONDARY
Fractions
Teacher : “ This is 1 (in blue),
make 4/9
What is the rest of the area ? ”
Answer : 5/9
Possible Answer
PRIMARY / SECONDARY
Improper Fractions
Teacher : “ This is 1 (in blue),
make 4/3
Possible AnswerPossible Answer
PRIMARY / SECONDARY
Mixed Numbers
Teacher : “ This is 22/3 (in blue)
make 6as a triangle”
Possible Answer
PRIMARY / SECONDARY
Mixed Numbers
Teacher : “ This is 3 (in blue),
make 21/2
as a trapezium ” Possible Answer
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
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
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
SECONDARY
Ordered Pairs
and Graphs
Teacher : “Plot the points
(1,1), (3,1), (2,3) and make a triangle”
SECONDARY
Ordered Pairs
and Graphs
Teacher : “Plot the points
(-3,-3), (-2, 3), (2,2) and make a triangle”
SECONDARY
Ordered Pairs
and Graphs
Teacher : “Plot the points
(-3,-1), (-2,2), (2,-2) and make a triangle”
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
riserun
SECONDARY
Ratio and Percentage
Teacher : “ Express the white part as a ratio and a
percentage ” The answer :
Ratio 1 : 3 , % 25%
SECONDARY
Ratio and Percentage
Teacher : Express the white part as a ratio and a
percentage
The answer : Ratio 1 : 4 , % 20%
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%
SECONDARY
Ratio and Percentage
Teacher : “Express the blue area
as a ratio of 1 : 2 : 3
in different colours ”Possible Answer
SECONDARY
Percentage (increase and decrease)
Teacher : “Starting with the figure in blue, make 80 % ”
Possible Answer
SECONDARY
Percentage (increase and decrease)
Teacher : “Starting with the figure in blue, make a
40% decrease ”Possible Answer
SECONDARY
Percentage (increase and decrease)
Teacher : Starting with the figure in blue, make
an increase of20 % (= 120 %)
Answer
SECONDARY
Percentage (increase and decrease)
Teacher : Starting with the figure in blue,
make an increase of 25 % (= 125 %)
Possible Answer
SECONDARY
Percentage (increase and decrease)
Teacher : Starting with the figure in blue,
make an increase of 75 % (= 175 %)
Possible Answer
SECONDARY
Percentage (increase and decrease)
Teacher : Starting with the figure in blue,
make a decrease of 25 % also as a triangle
Possible Answer
SECONDARY
Percentage (increase and decrease)
Teacher : Starting with the figure in blue, make 80 % ,
also as a parallelogram
Possible Answer
Possible Answer
SECONDARY
Percentage (increase and decrease)
Teacher : Starting with the figure in blue,
make 75 % , also as a trapezium
Possible Answer
Possible Answer
SECONDARY Congruent Triangles Teacher : “Make two congruent triangles, facing different ways. (i) right (ii) isosceles (iii) obtuse ”
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Translations
Teacher : “ Make an object :
(1, 1), (2 ,1), (3, 3)”
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Translations
Teacher : “ Translate the object
back 4 units ”
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Translations
Teacher : “ Translate the object
down 4 units ”
-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)
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Reflections
Teacher : “ Make an object :
(2,2), (1, -1), (3,-3)”
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Reflections
Teacher : “Reflect over the y axis ”
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Reflections
Teacher : “Reflect over the x axis ”
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Reflections
Teacher : “ Reflect over
x = 1 ”
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Reflections
Teacher : “ Reflect over
x = 2 ”
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Reflections
Teacher : “ Reflect over
y = x ”
-3 -2 -1
-1
-2
-3
3
2
1
O 1 2 3
Q.4
Q.2
Q.3
Q.1SECONDARY
Transformations :
Rotations
Teacher : “ Make an object :
(2,3), (0, 1), (3,-1)”
-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 the origin ”
-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 ”
-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)”
-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) ”
SECONDARY
Pythagoras
Theorem
Teacher :“ What is the length of the hypotenuse ?
Answer : √32 = 4√2 units
Answer : √40 = 2√10 units
SECONDARY
Pythagoras
Theorem
Teacher :“ What is the length of the hypotenuse ?
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
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
SECONDARY
Pythagoras
Theorem – Proofs
Algebraic Proof: ( Garfield’s )
[(a + b)/2 ] x (a + b) = 2( ½ ab ) + ½ c2
SECONDARY
Parallel Lines
Teacher : “ The lines are parallel,
cut by another straight line
( a transversal )”
Possible Answer
SECONDARY
Parallel Lines Teacher : “ Using coins of the same size, mark equal angles “
Possible Answer
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 ? ”
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
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
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
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
SECONDARY
Volumes and
Surface Areas
Teacher : “Create your own 3D shape and
calculate the V. and S.A.”
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 …
SECONDARY
Graphing 2 Equations
Teacher : “ Name the two linear equations”
(Note : m1 = m2 )
Answer : (i) y = x + 2 (ii) y = x
SECONDARY
Graphing 2 Equations
Teacher : “ Name the two linear equations”
(Note : m1 = m2 )Answer :
(i) y = 2x + 1 (ii) y = 2x – 1
SECONDARY
Graphing 2 EquationsTeacher : “ Name the two linear equations” (note : m1 m2 = -1 )”
Answer : (i) y = x + 1 (ii) y = -x – 1
m1 m2 = (1)(-1) = -1
SECONDARY
Graphing 2 EquationsTeacher : “ 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
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 : 1area to area S.F.- 9 : 1
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
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
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
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
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
SECONDARY
Graphing Inequalities
Teacher : List the inequalities which surround and define the closed area
Answer : y > –x +2, y < -1, y > 2x + 1
SECONDARY
Graphing Inequalities
Teacher : List the inequalities which surround and define the closed area
Answer : y < –2, y > -x, y > -3x + 3
SECONDARY
Graphing Inequalities
Teacher : List the inequalities which surround and define the closed area
Answer : y < -6x - 3, y < -2, y > -2x + 1
SECONDARY
Trigonometry
Teacher : “Use SOHCAHTOA and your calculator to find the
measure of this angle.” Ans : arcsin (1/√37)
= 9.4620
SECONDARY
Trigonometry
Teacher : “Use SOHCAHTOA and your calculator to find the
measure of this angle.” Ans : arcsin (4/2√13)
= 33.690
SECONDARY
Trigonometry
Teacher : “This angle is 450
Use SOHCAHTOA to find the length of the hypotenuse.”
Ans : 8.485 units
SECONDARY
The Centre of a
TriangleThe Centroid :intersection of the
medians. Divides the triangle into 6 equal
areas. The medians are cut 1 : 2
SECONDARY
The Centre of a
TriangleThe Orthocentre :
intersection of the altitudes : perpendiculars dropped from each angle to their opposite side.
SECONDARY
The Centre of a
TriangleThe 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
Triangle Midpoint Theorem :
SECONDARY
Geometry Theorems
that can be tested
Line is parallel to c and ½ its length
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
A = i + b/2 - 1 = 15 + 18/2 - 1 = 23 sq. units
SECONDARY
Geometry Theorems
that can be tested Pick’s Theorem
Euclid’s OrchardFrom 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
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.
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
Geoboard Construction1. MDF board or plywood2. Lacquer + paint (optional)3. Matt black vinyl sticker4. Brass nails5. Odd number of nails, e.g. 7 x 76. Cheap to make
------------------------------See my video on Youtube, “ How to Make Geoboards ”
https://www.youtube.com/watch?v=9yhCxlk9fx4
My Links• My Youtube channel : “ Maths with Geoboards “
( for videos )• My website : www.mathswithgeoboards.com
( for worksheets )• My email : [email protected]
( for contact )
This presentation available as a video (in English) on YoutubeAs a slide show ( in English and Thai ) on SlideShare
Pythagoras’ TheoremPythagoras Theorem Vol. 1 Pythagoras Theorem Vol. 2
Proofs of Pythagoras Theorem - GeometricProofs of Pythagoras Theorem - Algebraic
Number TheoryPrime Numbers
Fibonacci Numbers Factors and Multiples
GeneralHow to Make Geo-boards
AnglesIntroductory Terms Easy Angle Problems
The World’s Two Hardest Easy Geometry Problems
IndicesIndices : Laws 1, 2, 3 and 4
Indices : Word Problems
ProbabilityThe Monty Hall Problem
Youtube Channel : “ Maths with Geoboards” videos so far
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