Secondary Maths with Geo-boards - an overview for teachers

Maths with GeoboarDSfor Lower Secondary Maths

By Jon Molomby

An Overview for Teachers and Researchers

“ The way of teaching is to inspire,

not to inform ”

“ 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

LEARNING

with

GEOBOARDS

Every student is given a geo-board (from a

class set)

LEARNING

with

GEOBOARDS

The teacher sets the

students a task

LEARNING

with

GEOBOARDS

It is very easy for the teacher to correct answers

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

Maths with Geoboards

PRIMARY

- Polygons

- Triangles

PRIMARY

- Polygons

- Quadrilaterals

PRIMARY

- Polygons

- Pentagons, Hexagons, Octagons

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


Rectangle2. Perimeter – 12 units3. Symmetry – 2 lines4. Angles 5. Area6. Diagonals


Rectangle2. Perimeter – 12 units3. Symmetry – 2 lines4. Angles – 4 right angles5. Area6. Diagonals


Rectangle2. Perimeter – 12 units3. Symmetry – 2 lines4. Angles – 4 right angles5. Area – 8 sq. units6. Diagonals


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


PRIMARY / SECONDARY Angles Make the following angles acute right obtuse

Possible Answerstraight reflex full rotation

PRIMARY / SECONDARY Angles Interior angle sum of a triangle & complementary angles

Possible Answer

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

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 / SECONDARYFRACTIONS and

DECimals


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


make 1/4

What is the rest of the area ? ”

Answer : 3/4

Possible Answer

PRIMARY / SECONDARY

Fractions


make 1/3


Answer : 2/3

Possible Answer

PRIMARY / SECONDARY

Fractions


make 5/12


Answer : 7/12

Possible Answer

PRIMARY / SECONDARY

Fractions


make 4/9


Answer : 5/9

Possible Answer

PRIMARY / SECONDARY

Improper Fractions


make 4/3

Possible AnswerPossible Answer

PRIMARY / SECONDARY

Improper Fractions


make 5/2

Possible Answer

PRIMARY / SECONDARY

Mixed Numbers

Teacher : “ This is 11/3 (in blue)

make 22/3 ”Possible Answer

PRIMARY / SECONDARY

Mixed Numbers

Teacher : “ This is 22/3 (in blue)

make 6as a triangle”

Possible Answer

PRIMARY / SECONDARY

Mixed Numbers


make 21/2

as a trapezium ” Possible Answer

SECONDARY :ORDERED PAIRS and

GRAPHS


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


(-3,-3), (-2, 3), (2,2) and make a triangle”

SECONDARY

Ordered Pairs

and Graphs


(-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

Ordered Pairs

and Graphs

Teacher : “Graph the equation

y = x ”

SECONDARY

Ordered Pairs

and Graphs


y = 2x + 1 ”

SECONDARY

Ordered Pairs

and Graphs


y = 3x – 3 ”

SECONDARY

Ordered Pairs

and Graphs


y = - 6x – 3 ”

SECONDARY : RATIO and

PERCENTAGEIntroduction only


SECONDARY

Ratio and Percentage

Teacher : “ Express the white part as a ratio and a

percentage ” The answer :

Ratio 1 : 3 , % 25%

SECONDARY


Teacher : Express the white part as a ratio and a

percentage

The answer : Ratio 1 : 4 , % 20%

SECONDARY


Teacher : “Express all colours as ratios and %s ”

Answer : Red 3 : 7, 30% Yellow 1 : 4, 20%

Blue 1 : 1, 50%

SECONDARY


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


Teacher : “Starting with the figure in blue, make a

40% decrease ”Possible Answer

SECONDARY


Teacher : Starting with the figure in blue, make

an increase of20 % (= 120 %)

Answer

SECONDARY


Teacher : Starting with the figure in blue,

make an increase of 25 % (= 125 %)

Possible Answer

SECONDARY



make an increase of 75 % (= 175 %)

Possible Answer

SECONDARY



make a decrease of 25 % also as a triangle

Possible Answer

SECONDARY


Teacher : Starting with the figure in blue, make 80 % ,

also as a parallelogram

Possible Answer

Possible Answer

SECONDARY



make 75 % , also as a trapezium

Possible Answer

Possible Answer

SECONDARY : CONGRUENT TRIANGLES

Introduction only


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

SECONDARY : TRANSFORMATIONS

( Translations, Reflections, Rotations )


-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


(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


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


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


(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


SECONDARY

Pythagoras

Theorem

Teacher :“ What is the length

of the hypotenuse ? ”

Answer : 5 units

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


SECONDARY

Pythagoras TheoremGeometric Proof

C2

C2

C2

a2

b2

SECONDARY

Pythagoras

Theorem –Proofs

Geometric Proof:

( Euclid’s )

SECONDARY

Pythagoras

Theorem –

Proofs

Algebraic Proof

c2 = 4( ½ ab ) + (b – a)2

SECONDARY

Pythagoras

Theorem – Proofs

Algebraic Proof: ( Garfield’s )

[(a + b)/2 ] x (a + b) = 2( ½ ab ) + ½ c2

SECONDARY :PARALLEL LINES

Introduction only


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

SECONDARY :VOLUMES and SURFACE

AREAS of SOLIDS


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


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

Platonic SolidsTry to make its net

Teacher : Try to make a Tetrahedron

SECONDARY

Platonic Solids

Make its net

Teacher : Make a Cube

SECONDARY

Platonic Solids

Try to make its net

Teacher : Try to make a Octahedron

SECONDARY

Platonic Solids

Teacher : Try to make a Dodecahedron

SECONDARY

Platonic Solids

Teacher : Try to make a Icosahedron

SECONDARY : GRAPHING TWO

EQUATIONS


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

Introduction only


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 (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


SECONDARY

Graphing Inequalities

Teacher : List the inequalities which surround and define the closed area

Answer : y > –x +2, y < -1, y > 2x + 1

SECONDARY



Answer : y < –2, y > -x, y > -3x + 3

SECONDARY



Answer : y < -6x - 3, y < -2, y > -2x + 1

SECONDARY : TRIGONOMETRY


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

Triangle


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 Circumcentre

intersection of the perpendicular bisectors.

SECONDARY

The Centre of a

TriangleThe Incentre :

intersection of the angle bisectors.

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

SECONDARY : MISCELLANEOUS

GEOMETRIC THEOREMS


Triangle Midpoint Theorem :

SECONDARY

Geometry Theorems

that can be tested

Line is parallel to c and ½ its length

SECONDARY

Geometry Theorems

that can be tested

Angle Bisector Theorem

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

CONCLUSIONS


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 CONSTRUCTION


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

https://www.youtube.com/watch?v=9yhCxlk9fx4

LINKS


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

http://www.mathswithgeoboards.com/

mailto:[email protected]

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

https://www.youtube.com/watch?v=ae0McT5WYa8


THE END

“ I don’t teach, I let them learn ” Caleb Gattegno

=========================

“ข้อมูลนีถู้กแปลเป็นภาษาไทยเพ่ือส่งเสริมการศึกษาไทย เพ่ือถวายเป็นพระราชกุศล แด่

พระบาทสมเดจ็พระปรมนิทรมหาภูมพิลอดุลยเดช รัชกาลที ่๙”

Secondary Maths with Geo-boards - an overview for teachers

Education

Transcript of Secondary Maths with Geo-boards - an overview for teachers