NAME
DATE
Square Metre
1 m = 100 cm
10 cm
1 cm2 = 1 cm × 1 cm
1 m2 = 1 m × 1 m = 100 cm × 100 cm = 10 000 cm2
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1 m = 100 cm
NAME
DATE
Area and Perimeter — Review
one square centimetre
This shape covers 4 square centimetres.
Its area is 4 cm2.
1 cm = 1 cm2
This shape has 6 sides. The distance around it is
1 cm + 3 cm + 2 cm + 1 cm + 1 cm + 2 cm = 10 cm.
1 cm
Its perimeter is 10 cm.
1. Find the area of these figures in square centimetres.
a)
b)
Area = c)
cm2 Area = cm2 Area = cm2
2. Look at the rectangle in 1 c).
Write the length and width of the rectangle:
width = cm
Write a multiplication statement for the area: Write an addition statement for the perimeter: length = cm
cm2
cm
3.Using a ruler, divide each rectangle into squares 1 cm × 1 cm. (The sides of the first
two rectangles have already been marked in centimetres.) Write a multiplication
statement for the area of each rectangle in cm2. Write an addition statement for the
perimeter of each rectangle in cm.
a)
b)
c)
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4. On grid paper, draw 3 different shapes that have an area of 10 cm2 (the shapes don’t
have to be rectangles).
5. On grid paper, draw a rectangle with area 12 cm2 and perimeter 14 cm.
6. If you know the length and width of a rectangle, how can you find its area?
7. Find the area of each rectangle using the clues.
a) Width = 2 cm Perimeter = 10 cm Area = ? b) Width = 4 cm Perimeter = 18 cm Area = ?
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Dividing Decimals by Decimals
1. Estimate the quotients.
a) 9.7 ÷ 1.3≈ ÷ b) 8.24 ÷ 2.16 ≈ ÷ ≈ ≈ c) 19.6 ÷ 5.2 ≈ ÷ d) 420.8 ÷ 6.95≈ ÷ ≈ e) 16.34 ÷ 3.87 ≈ ÷ ≈ f) 99.5 ÷ 4.07≈ ÷ ≈ ≈ 2. Estimate the quotients.
a) 0.462 ÷ 0.208 b) 0.629 ÷ 0.346
≈ 0.4 ÷ 0.2 ≈
c) 0.084 ÷ 0.0426
≈ 0.6 ÷ ≈ ÷ 0.04
≈ ÷ ≈ ÷ ≈ ≈ ≈ 4
÷ 2
d) 0.0965 ÷ 0.0316
e) 1.1548 ÷ 0.1863
f) 12.3956 ÷ 0.2015
≈ ÷ 0.03 ≈ ÷ ≈ ÷ 0.02
≈ ÷ ≈ ÷ ≈ ÷ ≈ ≈ ≈ 3. Use estimation to place the decimal point correctly in each quotient.
a) 50.46 ÷ 2.14 ≈ 2 3 5 7 9 4 3 9 2 5
b) 0.684 ÷ 0.27 ≈ 2 5 3 3 3 3 3 3 3
c) 562.6 ÷ 0.047 ≈ 1 1 9 7 0 2 1 2 7
d) 16.9 ÷ 0.1287 ≈ 1 3 1 3 1 3 1 3 1 3 1
e) 35.04 ÷ 0.0381 ≈ 9 1 9 6 8 5 0 3 9 f) 1.75 ÷ 2.515 ≈ 6 9 5 8 2 5 0 4 9 7
4. The decimal point is in the wrong place. Estimate the quotient to correct the answer.
a) 5.43 ÷ 0.96 ≈ 56.5625
b) 252 ÷ 5.5 ≈ 4581.8181
c) 15.78 ÷ 3.75 ≈ 0.4208
5.Multiply both the dividend and divisor by 10, 100, or 1000 to change them to whole
numbers. Then divide using a calculator. Estimate the quotient to check your answer.
a) 18.72 ÷ 1.2
b) 2.921 ÷ 2.3
c) 37.26 ÷ 6.9
d) 1.264 ÷ 0.016
e) 3.192 ÷ 0.042
f) 12.194 ÷ 5.2
6.Calculate using a calculator. Estimate to check your answer.
Round answers to two decimal places.
a) 2.174 ÷ 0.649
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b) 1 000 ÷ 0.068
c) 2 ÷ 54.873
d) 987.54 ÷ 3.13
Blackline Master — Measurement — Teacher’s Guide for Workbook 7.1
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NAME
DATE
Distance Between Parallel Lines
A. Measure the line segments with endpoints
on the two parallel lines with a ruler. Write the
lengths of the line segments on the picture.
B. Use a square corner to draw at least three perpendiculars from one parallel line to
the other, as shown.
Measure the distance between the two parallel lines along the perpendiculars.
What do you notice? C. Explain why all the perpendiculars you drew in part B are parallel.
D. A parallelogram is a 4-sided polygon with opposite sides parallel. You can draw
parallelograms by using anything with parallel sides, like a ruler. Place a ruler across
both of the parallel lines and draw a line segment along each side of the ruler. Use
this method to draw at least 3 parallelograms with different angles.
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E. Measure the line segments you drew between the two given parallel lines in part D.
What do you notice? F. To measure the distance between two parallel lines, draw a line segment
perpendicular to both lines and measure it. Does the distance between parallel lines
depend on where you measure it? Blackline Master — Measurement — Teacher’s Guide for Workbook 7.1
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DATE
Parallelograms
1
3
2
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4
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Area of Parallelograms
1.Albert cuts a parallelogram into two triangles and rearranges the pieces
to find the area.
a)The shaded triangle in the first diagram was moved. Shade
the same triangle where it was reattached in the second diagram. b) Explain why the vertical line segments in each diagram are equal.
c) Why are the areas of both parallelograms the same?
d)Does the formula for the area of a parallelogram (base × height) still
work when the height falls outside the base? Explain. 2.Regina wants to find the area of a parallelogram a different way. She cuts
her parallelogram in half and rearranges the pieces as shown.
a) Measure the side AB. AB = b) Does the area of the parallelogram change? A
B
c)Regina wants to use the horizontal side of the new
parallelogram as a base. What is the length of the
base of the new parallelogram? How is it related to AB? d) Draw and measure the height of the new parallelogram. Height = e)Which fraction of the height of the old parallelogram is the height of the new
parallelogram? f)Verify that the formula for the area of a parallelogram gives the same answer for
both the new parallelogram and the old parallelogram.
3. Find the area of EFGH in two ways.
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a) Use EH as the base. EH = Extend EH and draw a perpendicular to EH from vertex G.
What is the height of EFGH? Height = Area = b) Use EF as the base. EF = Draw the height to EF. Height = Area = Did you get the same answer both ways? H
F
G
Explain.
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Investigating Area of Parallelograms
Two rectangles with the same area and perimeter are congruent. Are
all parallelograms with the same area and perimeter congruent?
INVESTIGATION
A.Find all possible combinations of base and height for a parallelogram with area
24 cm2. The base and height are whole numbers of centimetres. Start with base 1 cm
and search systematically.
1
Base (cm)
2
3
4
6
Height (cm)
B.On 1 cm grid paper, draw all rectangles that have area 24 cm2 and sides that are
whole number of centimetres.
C. Find the perimeters of all rectangles in part B.
Length (cm)
1
2
3
4
6
Width (cm)
Perimeter (cm)
D. Are all rectangles also parallelograms? Explain. E.Copy these parallelograms onto 1- cm grid paper. Draw the next two parallelograms
in this pattern.
B
C
A
D B
A
B
C
D
C
A
Figure 1 Figure 2 D
Figure 3
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
P = P = P = P = P = A = A = A = A = A = Do these parallelograms have the same area? Do these parallelograms have the same perimeter? G.Look at your answers in parts C and F. Can you find two parallelograms that have
the same perimeter and area but are not congruent? Which ones? Use part E
to confirm your answer.
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F.Measure the sides to the nearest millimetre and find the perimeter (P) and the area (A)
of the parallelograms in the pattern.
NAME
DATE
Triangles
Right triangles
Acute triangles
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Obtuse triangles
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Subtracting Area
3 cm
Dianne wants to find the area of the shaded part. She notices that the
large rectangle can be divided into three parts, two of which can be joined
to form a small rectangle:
3 cm
1 cm
2 cm
2 cm +
=
The area of the large rectangle is 4 cm × 2 cm = 8 cm2.
The area of the small rectangle is 3 cm × 2 cm = 6 cm2.
So the area of the shaded part is 8 cm2 − 6 cm2 = 2 cm2.
1. Complete the chart. Show all the steps in your answers.
Area of large
rectangle
Parallelogram
7
Area of small
rectangle
2
3
9 × 3 = 27 cm2
7 × 3 = 7 cm2
Area of parallelogram
By subtraction
By formula of area
9×3−7×3
= (9 − 7) × 3 cm2
= 2 × 3 cm2
= 6 cm2
base = 2 cm
height = 3 cm
area = 2 cm × 3 cm
= 6 cm2
3
4
8
5
5
1
1
6
10
9
2 5
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2. Find the area:
4
5
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3
5
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Blackline Master — Measurement — Teacher’s Guide for Workbook 7.1
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Area of Parallelogram and Triangle
Parallelogram
A parallelogram is a quadrilateral with two pairs of parallel sides.
Any pair of parallel sides can be chosen to be the bases. The distance between these
two parallel sides is the height.
bases
height
The height is measured along a line perpendicular to the bases. This line can be drawn
anywhere. In these pictures, the thick black line is one of the bases and the dashed line
is the height.
Area of a parallelogram = base × height
Triangle
A triangle is a polygon with three sides.
Any side of a triangle can be the base. Draw a perpendicular from
the vertex opposite the base to the base. The distance from the
vertex to the base along that perpendicular is the height. height
base
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Sometimes the height is outside the triangle. In these pictures, the thick
black line is the base and the dashed line is the height.
Any triangle is half of a parallelogram with the same base and height.
Area of a triangle = base × height ÷ 2
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Area of Trapezoid
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
trapezoids
not trapezoids
The parallel sides are called bases. The distance between the bases is called the
height. The height is measured along a line perpendicular to the bases.
base 1
height
base 2
Sometimes the height is outside the trapezoid. In these pictures, the thick black lines are
the bases and the dashed line is the height.
Area of trapezoid = height × (base 1 + base 2) ÷ 2
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You can make a parallelogram from two copies of the same trapezoid. The base of
the parallelogram is the sum of the bases of the trapezoid, and the height of the
parallelogram is the height of the trapezoid.
NAME
DATE
Triangles to Circles
1. How many triangles are there for every 3 circles? Guess, check and revise to
find the ratio
:
a)
b)
triangles to circles = : 3 c)
d) triangles to circles = : 3
triangles to circles = : 3
triangles to circles = : 3
e)
triangles to circles = : 3
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2.In which parts above are there more triangles than circles? How can you tell that
from the ratio?
There are more triangles than circles in parts , , and because
3.The ratio of circles to triangles is 94 : 93. Are there more circles or triangles? How do
you know?
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Circles
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Facts About Circles
A circle with centre P: all the points on the circle are the same distance from P.
A radius: any line segment joining a point on the circle to its centre.
The plural of radius is radii. radii
centre
The radius (r): the distance between any point on a circle to P; the length
of a radius.
A diameter: a line segment between two points on a circle passing through the
centre of the circle.
diameter
The diameter (d): the distance between any two opposite points on a circle,
measured through the centre of the circle; the length of a diameter. A diameter
is twice as long as a radius, so d = 2r
An angle that has its vertex at the centre of a circle is called a central angle.
The sum of the central angles is 360°.
Example: ∠a + ∠b + ∠c = 360°
c
a
b
The distance around a circle is called the circumference (C).
The ratio C : d is the same for all circles. When this ratio is converted to
a unit rate and we look at it as a number, we see that it is special—it has an
infinite number of digits after its decimal point and there is no pattern in these
digits, so you cannot predict which digit is next. A Greek letter π (pronounced
“pie”) is used to identify this number. C
To the nearest hundredth, π = 3.14. To 5 decimal places, π = 3.14159.
Another good approximation is
22
22
. To three decimal places, the fraction
= 3.143,
7
7
and it is often used to approximate π.
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C : d = π : 1, so C = π × d = πd = 2πr.
The area of a circle of radius r is π × r2 = πr2.
Here is an acronym that helps to remember the value of π rounded to 6 decimal places.
The number of letters in each word in order gives the digit.
How I wish I could calculate pie!
3.
1 4 1 5
9
3
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