Transcript Geoboards - fcpssecmath

Got Geoboards?
They are not just for Geometry!!
September 2011
Math In-service
Geoboards -- Agenda
Types of Geoboards
Rubber bands
Mathematical Connections





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
Rational Numbers
Algebra/Graphing
Statistics/Probability
Geometry/Measurement
Geoboards Types
Geoboards Types
Geoboards Rubber bands
Geoboards
Rational Numbers
Geoboards and fractions

Using your Geoboard, divide it into
fourths as many ways as you possibly
can. Record your answers on the
given paper.
Geoboards and fractions
Geoboards and fractions
Geoboards and fractions
Geoboards and fractions
Addition
of Rational Numbers
To find the sum of 1/3 and 1/2, the student
models both in the same unit, then covers
the islands with other islands of a single
color that could be used to fill the unit. In
this model, a pink island represents 1/3,
and a brown island represents 1/2. Red
islands may be used to cover both of the
islands. The five red islands represent the
sum 5/6 in this unit. Another student may
cover both islands with yellow islands,
each of which covers one Geoboard
square, showing the sum as 10/12; but
5/6 and 10/12 are equivalent, so represent
the same rational number.
Geoboards and fractions
Subtraction?


How would you show:
1/3 – ¼?
1 – 1/6?
Geoboards and fractions
Multiplication of Rational
Numbers
Finding the product of 1/3 and
1/2 is modeled as “1/3 of 1/2.”
1/2 is modeled with one brown
island in this unit. Since three red
islands cover the brown island,
then one red island models onethird of the brown area. The one
red island represents 1/6 in this
unit; thus the product is 1/6.
Fractions and Geoboards



Problem: Mr. McGregor has a garden that is
a rectangle 3 units by 2 units. He wants to
plant flowers on half of the garden Show
how he would divide his garden on the
geoboard. Then draw it on the paper and
shade in the part that would be flowers with
one of the colors.
How much of the garden will be flowers?
How much is 1/2 of a whole?
Is the result more or less than the whole?
Decimals and Geoboards


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How would you show 0.34 on your
geoboard?
How would you show 0.3 on your
geoboard?
How would you show 1.34 on your
geoboard?
Decimals and Geoboards

How would you show:

0.34 + 0.25?

0.34 + 0.3?

0.45 + 0.55
Decimals and Geoboards

How would you show:

0.34 - 0.25?

0.7 - 0.35?

1 - 0.21?
Decimals and Geoboards

How would you show:

2 X 0.42?

3 X 0.3?

5 X 0.12?
Percents and Geoboards

How would you show:

25%?

40%?

105%
Percents and Geoboards

How would you show:




50% of 1?
50% of 2?
25% of 2?
20% of 4?
Rational Numbers and
Geoboards
Questions?
Geoboards
Algebra and Geoboards
Coordinates

Think of your Geoboard as the 1st
Quadrant
y
(0,0)
x
Coordinates

Graph the
following points on
your Geoboard:



y
Red dot (1,2)
Blue dot (5,3)
White dot (8,7)
(0,0)
x
Coordinates

We can also think of the Geoboard as
y
4 Quadrants
(0,0)
x
Coordinates

Graph the
following points on
your Geoboard:



BINGO
Red dot (-1,2)
Blue dot (3,-3)
White dot (-2,-3)
y
(0,0)
x
Algebra and Pre-Algebra

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
We can use the
Geoboard to help with
graphing equations and
determining slope:
Connect (-3,-3) and (3,3)
What is the slope of the
line that connects these
points?
y
(0,0)
x
Slope
What is the slope of
the line that passes
through:
(2, -3) and (-4, 3)?
y
(0,0)
x
Algebra
y


Graph the following points
on your Geoboard:
 Red dot (-1,1)
 Blue dot (3,-3)
Use a band to draw the
line that goes through
both points
(0,0)
What is the equation of the line that
goes through the two points? y=-x
x
Algebra

Graph the
following line on
your Geoboard:

y
y=2x-4
(0,0)
x
Algebra

Graph the following
line on your
Geoboard:
1
y   x 1
2
y
(0,0)
x
Algebra

Graph the following
line on your
Geoboard:
2
y  x3
3
y
(0,0)
x
Algebra
Connect (-3,-2) and (3,2)
 What is the slope of
the line that connects
these points?
Connect (-2,2) and (2,-4)
 What is the slope of
the line that connects
these points?
y
(0,0)
x
Algebra
Connect (-3,-2) and (3,2)
 What is the slope of the
line that connects these
points?
Connect (-4,-1) and (2,3)
 What is the slope of the
line that connects these
points?
y
(0,0)
x
Algebra

Graph the following
line on your
Geoboard:
1
y   x  1 and
2
1
y  x 1
2
y
(0,0)
What is the point of intersection? (2,0)
x
Geoboards and Algebra
Questions?
Suggestions?
Geoboards, Tangrams,
and Algebra
How can we relate the
Geoboard and
Tangrams with Algebra?
Geoboards and Tangrams
Use your Geoboard and bands to form a
special geometric shape following the steps
below.
1.
Band together: (0,0) (0,8) (8,8) and (8,0)
2.
Band together: (0,8) and (8,0)
3.
Band together: (0,4) and (4,0)
4.
Band together: (2,2) and (8,8)
5.
Band together: (2,2) and (2,6)
6.
Band together: (6,2) and (4,0)
Geoboards
Statistics and Geoboards
Bar Graphs
Create a bar graph
with the following
data on your
Geoboard:
A
Number of
students
27
B
24
C
24
D
30
E
30
Class
Scatter Plots
Create
a scatter plot
with the following data
on your Geoboard:
The table shows the
number of class
absences and final
exam scores for 9
students in Mr. Hayes’
class.
Absences
Final Exam
Score
0
1
3
90
100
80
3
4
5
70
80
60
6
7
10
60
50
40
Scatter Plots
y
x
Scatter Plots
Relationship?
y
Line of Regression
x
Back of Geoboard
Circle Graphs
Remember that
they need to be
in sections of ?
Back of Geoboard- Circle Graph
A group of 72 randomly selected 8th graders were
asked about their favorite ice cream flavor. The
results are shown below.
Flavor
Rocky Road
Chocolate
Chip
Chocolate
Oreo
Vanilla
Total
Frequency
12
18
24
12
6
72
Percent
Degrees
100%
360
Geoboards
Probability and Geoboards
Back of Geoboard
Creating spinners
The probability of landing
on an even number is ½
and the probability of
landing on an odd
number is ¼.
Back of Geoboard
Creating spinners
The probability of
landing on a multiple
of seven is 3/8 and
the probability of
landing on a multiple
of eight is 3/8.
Geoboards– Prob/Stat
Questions?
Suggestions?
Geoboards and
Geometry/Measurement
Parallel Lines and
transversals
Parallel Lines
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Graph y=1 and y=3
Graph transversal
line through (-2,3)
and (1,0)
Measure the angles
y
(0,0)
x
Parallel Lines
Connect (-3,-2) and (3,2)
Connect (-4,-1) and (2,3)
Create a transversal (-4,2)
and (3,-3)
 What angles are
congruent?
y
(0,0)
x
Transformations
Draw triangle WVY and
translate it (3,-1).
W(-1,0) V(-3,-3) Y(2,-3)
y
(0,0)
x
Transformations
Draw triangle RST and
reflect it over the y-axis.
R(-5,0) S(-2,-5) T(-1,-1)
y
(0,0)
x
Transformations
Draw triangle RST and
reflect it over the x-axis.
R(-5,0) S(-2,-5) T(-1,-1)
y
(0,0)
x
Transformations
Draw triangle RST and
rotate it 90° clockwise.
R(-5,0) S(-2,-5) T(-1,-1)
y
(0,0)
Can use graph paper too
x
Triangles
Find three locations
for a point P, above
segment AB, so that
triangle APB is a right
triangle.
A
B
Triangles
Find three locations
for a point P, above
segment AB, so that
triangle APB is an
isosceles triangle.
A
B
Triangles
Find three locations
for a point P, above
segment AB, so that
triangle APB is an
acute triangle.
A
B
Triangles
Find three locations
for a point P, above
segment AB, so that
triangle APB is an
obtuse angle.
A
B
Perimeter
Perimeter
Perimeter
Create another figure, that is NOT a
rectangle, with the same perimeter.
Perimeter
Create another figure, that IS a
rectangle, with the same perimeter.
Area

Establish that each “square” is 1 unit
Area

Establish that each “square” is 1 unit
Area and Perimeter

Create a rectangle whose
perimeter and area are the same
Area of triangles

Determine the
area of this
triangle as many
ways as you can-- discuss
Area of triangles
Determine the area
of this triangle as
many ways as you
can--- discuss
How efficient was
your approach?
Would you approach
it differently now?
Area of triangles
Determine the area
of this triangle.
Does your method
work for this triangle
too?
Area of quadrilaterals
Determine the area
of this polygon.
Does your method
from the triangle
work for this
polygon?
Area of quadrilaterals
Determine the area
of this polygon.
Does your method
from the triangle
work for this
polygon?
Area of quadrilaterals
Create these
trapezoids on your
Geoboard.
Prove the formula for
determining the area
of a trapezoid
(b1  b2 ) h
2
Area of quadrilaterals
Create a trapezoid
with an area of 8
square units
Geoboards and Tangrams
Use your Geoboard and bands to form a
special geometric shape following the steps
below.
1.
Band together: (0,0) (0,8) (8,8) and (8,0)
2.
Band together: (0,8) and (8,0)
3.
Band together: (0,4) and (4,0)
4.
Band together: (2,2) and (8,8)
5.
Band together: (2,2) and (2,6)
6.
Band together: (6,2) and (4,0)
Geoboards and Tangrams
What is the
area of each
piece?
Geoboards and Geometry
What other areas of
Geometry could we
use the Geoboard for
in our classrooms?