Unit 1 Measurement: 2-D and 3-D
Grade 9 Applied
BIG PICTURE Students will: describe relationships between measured quantities; solve problems involving the areas and perimeters of composite 2-D shapes; develop volume formulas for pyramids cones and spheres; apply knowledge and understanding of 3-D formulas to solve simple problems in context.
Math Learning Goals Develop through investigation the formulas for volume of a pyramid, and a cone based on the volume of the corresponding prism or cylinder of the same radius and height. Develop through investigation the formula for volume of a sphere based on the volume of a cylinder/cone. Consolidate volumes of prisms, pyramids, cylinders, cones and spheres. Solve problems involving combinations of the figures using metric and imperial measure. Activate students’ prior knowledge of terminology related to identifying geometry shapes. Determine students’ readiness to identify geometric figures in composition, and use appropriate calculations for perimeter and area. Find area and perimeter of composite shapes. Solve composite area problems, e.g., logos, business signs, irregularly shaped gardens. Connect the geometric representation of the Pythagorean theorem to the algebraic representation a2 + b2 = c2. Relate their understanding of inverse operations to squares and square roots. Solve 2-D and 3-D problems using the Pythagorean theorem. Substitute into and evaluate algebraic expressions involving exponents.
Expectations MG2.01 MG2.02 MG2.03 MG2.04 MG2.05 NA2.01 NA2.02 NA2.03 NA2.04 NA2.08
1.1: Investigation – Comparing Volumes Purpose Compare volumes of shapes that have the same base and height.
Hypothesis I think that...
1.
× _________ =
2.
× _________ =
3.
× _________ =
Investigate How many times will the volume of the shape on the left fill the shape on the right? 1. Vcone × ___ = Vcylinder 2. Vsquare pyramid × _____ = Vsquare prism 3. Vtriangular pyramid × _____ = Vtriangular prism
Conclusion
1.2: Pair Share – Volume Partner A answers question A, partner B coaches. Partner B answers question B, partner A coaches.
A – Prisms Circle the shapes that are prisms.
Volume of a prism = area of __________ __________.
B – Pyramids Circle the shapes that are pyramids.
Volume of a pyramid = volume of a prism __________.
1.3: Pair Share – Volume Calculate the volume of the following figures. Show your work. A
B
Calculate the volume of the following figures. Show your work. A
B
8 mm
15 mm
12 cm
5 cm
1.4: Volume of Pyramids and Cones - Worksheet 1. a)
Find the volume of the following: b)
3.5 mm
4m
10 cm
9 cm
8 cm
2.
Find the volume given the following: a) a cone with a diameter of 4 cm and height of 15 cm.
3.
b) a pyramid with a base area of 72 cm2 and height 10 cm.
BAAAAH petting zoo allows visitors to feed animals grain from paper cones. a) How much grain can fit into a cone with a diameter of 7 cm and height of 8.5 cm?
b) Smoots the goat usually eats around 600 cm3 of grain for his afternoon snack. How many times must you fill the cone in order to give Smoots his snack?
4.
Fred’s Fries come in a cylindrical container for $4.00, while Chuck’s Chips come in a cone-shaped container for $2.00. Who offers the best deal and what assumptions must you make?
Fred’s
5.
Chuck’s
a) Calculate the capacity of the given coffee scoop. 4 cm
4.5 cm
b) Which do you think would provide a greater increase in capacity: doubling the radius, or doubling the height? Draw each of these new scoops and label dimensions.
c) Calculate the capacity of each new scoop in part b), and compare to your predictions in part b).
1.5: Frayer Model of a Sphere Definition
Formula
Sphere Examples
Non-Examples
Frayer Model of a Sphere (solution) Definition
Formula
Sphere – A 3D shape where every point on the surface is the same distance from the center. Notes V represents the Volume r represents the radius
Examples -
Baseball Basketball Planets (Earth) Dodgeball Orange Eyeball Bubble
Non-Example
Sphere
-
Circle Pizza Egg Drum Frisbee Stop sign Football
Developing the Formula for the Volume of a Sphere VolumeSphere.ppt.
(Presentation software file)
1
2
3
4
5
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7
1.6: Volume of a Sphere
Volume of a Sphere Formula:
4 V r 3
3
Examples: Find the volume of: a)
r = 2 cm
b) a sphere with a diameter of 15 cm.
1.7: Volume of a Sphere - Worksheet 1.
Find the volume of the following spheres: a) b) r=8m
2.
d = 12.2 cm
A Whopper is a spherical shaped chocolate malt candy with a diameter of 1.75 cm. a) Determine the volume of one Whopper.
b) How much malt candy will you consume if you eat a package of 30 Whoppers?
3.
a) A sphere has a radius of 4 cm. Calculate its volume.
b) A second sphere has a radius double the radius of the first sphere. Predict how the volumes of the two spheres are related.
c) Calculate the volume of the second sphere. Compare this value to your prediction, and explain any differences.
4.
The diameter of the Earth is approximately 6400 km. Determine the volume of the Earth, in cubic kilometers.
5.
The perfect ice cream treat involves a completely filled cone with two perfectly spherical ice cream scoops loaded up on top of the cone. (see sketch) a) Find the volume of ice cream in the perfect ice cream treat.
6 cm 12 cm
b) If the ice cream is being scooped from a cylindrical container with a height of 14 cm and radius of 5.3 cm, find out how many perfect ice cream treats can be served from this container.
1.8: Applications of Volume Assignment Name:________________ 1. Calculate the volume of the perfume bottle.
15 cm 3 cm 8 cm
2. How much soup can this container hold?
7 cm
10 cm
3. Calculate the volume of the rectangular prism. 5 in. 4 in. 6 in.
4. Which popcorn container will hold more? Prove your answer by calculating the volume of each figure 16 cm 8 cm 15 cm
15 cm
5. How many times bigger is the second sphere?
2 cm
5 mm
Hint: Change both measurements to the same units!
6. How much more expensive should the large aquarium be than the small aquarium if the cost is based on the volume?
12 in. 16 in. 8 in.
24 in. 32 in. 16 in.
7. Determine the volume of the cabin. Show your work. 2m 3m
6m 6m
1.9: Design Details Name the geometric shapes contained in the following two objects. Use arrows to connect names of shapes to the diagram.
1.10: Frayer Model Definition
Examples
Facts/Characteristic s
Composite Figures
NonExamples
1.11: Scale Drawing Details 1. Consider the two composite figures. (a) Identify the geometric shapes in each. Write the names on the diagrams.
(b) Area For determining the area of the shaded regions, describe the features and calculations that are: i)
the same
ii) different
(c) Which of the two figures is larger? By how much? Justify your answer using pictures, symbols, and words.
(d) Perimeter Use a coloured pencil to outline the perimeters of the two figures. How do these two perimeters compare?
2. Use the diagram below to calculate the area and perimeter of the triangle.
a) What dimensions are needed to determine the perimeter?
b) What dimensions are needed to determine the area?
c) Calculate: Area
Perimeter
3. Provide an example in daily life of a figure that involves more than one geometric shape.
1.12: Area Challenge
On the diagram, draw line segments to subdivide the shape into simple shapes, and calculate the area.
On the placemat: In your space, list the steps required to determine the area of the figure.
As a group, discuss the steps, then provide one complete solution to the problem in the centre of the placemat.
1.13: Composite Figures – Worksheet #1
Name: 1. Find the area of a province.
Date:
2. What to charge?
3. Wall Painting...
4. The Big Bridge.
a. Find the area of the cross-section of this bridge.
b. Find the perimeter of the cross-section of this bridge.
1.14: Composite Figures – Worksheet #2 1.
For each composite figure shown, solve for the missing lengths and calculate the perimeter. b a) b) 34 m
8 cm 12 m a
a
b
a
20 cm 20 m
15 m
2.
a
For each composite figure below, calculate the total area by breaking the figure up into familiar shapes. a) b) 4m
12 m 6 cm
8m 4 cm
1.15: What Is This? What shapes are hiding in the drawing below?
1.16: Composite Figures
Composite Figures
(Presentation software file)
CompositeFigures.ppt.
Bill is painting his basement floor. a) Determine the total area.
2.3 m
The dimensions are:
2.5 m
If one can of paint will cover 3.5 m2, and each can of paint costs $15.25, what is his total cost including 15% GST and PST?
2.3 m
A1
2.5 m
Atotal = A1 + A2
A = 2.3 x 2.5 + 3.2 x 1.8 A = 5.75 + 5.76 3.2 m
A = 11.5
A2
3.2 m
m2
1.8 m
1.8 m
3
2
b) If one can of paint covers 3.5 m2, how many cans of paint will he need?
2.3 m
2.5 m
11.5 m2
c) If one can of paint costs $15.25, what is his total cost including 15% GST and PST?
2.3 m
2.5 m
11.5 m2
Paint cost: 4 x $15.25 = $61.00 3.2 m
He will need four cans of paint.
Taxes: = 0.15 x $61.00 = $ 9.15
3.2 m
Total cost: $70.15
1.8 m
1.8 m
4
5
a) Determine area of semi-circle.
Shelly wants to make curtains to cover her decorative window.
The radius for material is 1.10 metres.
In order to hem the curtains she will add 10 cm to each edge.
A = πr2 ÷ 2 A = π(1.10)2 ÷ 2 10 cm
If 1 m2 of curtain material cost $2.20, how much will it cost to make the curtains including 15% GST and PST?
2m
A ≈ 3.80 ÷ 2 A ≈ 1.90 m2
2m
2m 6
7
c) If 1 m2 of material costs $2.20, determine the total cost of the curtains including taxes?
b) Determine area of the square. The length of the sides of material is 2.20 metres.
Material:
A lw A (2.20)( 2.20) A 4.84 m 2 2 m 10 cm
2m
2m
Taxes:
$2.20 x 7 = $15.40
$15.40 x 0.15 = $ 2.31 2m
Total Cost: $17.71
2m
2m 2m
Total Area = Semi Circle + Square = 1.90 + 4.84 = 6.74 m2 or 7m2
8
9
1.17: Exploring Composite Shapes
Shape Divisions
Area Calculations
1.18: Use What You Know (use formula sheet) Knowledge and Skills
Reasoning and Proving
Calculate the area of the given circle.
Westview School has a track.
Show your work.
r = 2.5 cm
You want to run 2 km every day. Determine how many times you have to go around the track. Show your work. Hint: A r
2
Communicating
Connecting
Genna wants to tile her bathroom counter with mini tiles. She needs to determine the area of her counter space. Explain with words, diagrams, and symbols how she should determine the area.
This figure has a radius of r units.
r
Which of the following formulas could be used to determine the perimeter?
a)
2r
1 r r 4
b)
0.75r 2
c)
3 (2r ) r r 4
d)
2r
Give reasons for your answer.
1 4
1.19: Pythagorean Theorem 1
2
3
4
5
The Pythagorean Theorem
(Presentation software file)
PythagoreanTheorem.ppt.
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2
3
4
5
1.20: Coach and Be Coached A coaches B
x
5 mm
B coaches A
10 cm y
12 mm 24 cm
12 m
9m x
2 cm z
2 cm
3m 12 m 8m
m
p 8m
55 cm
60 cm
130 cm w 44 cm
A hydro pole casts a shadow that is 10 m long. A technician measures the wire that runs from the top of the pole to the end of the shadow and finds it to be 26 m. How tall is the pole?
h
Don is building a loft in his garage. The ladder he is using extends to 10 metres. The loft is 8 m from the floor. How far away from the wall should he anchor the ladder?
1.21: Pythagorean Worksheet 1. If the legs of an isosceles right triangle are 6 units long, find the length of the hypotenuse.
2. Eva wants to put an underground sprinkler system in her back yard. A drawing of the system is shown below. About how many feet of water pipe will Eva need?
3. Jackson is 54 miles east of Lazy R Resort. Ontario is 31 miles south of Jackson. A land developer proposes building a shortcut road to directly connect Ontario and Lazy R. Draw a picture and find the length of this new road.
4. A television screen measures approximately 15.5 inches high and 19.5 inches wide. A television is advertised by giving the approximate length of the diagonal of its screen. How should this television be advertised?
5. A 6-ft ladder is placed against a wall with its base 2 ft from the wall. How high above the ground is the top of the ladder?
1.22: The Pythagorean Theorem – Hidden Application The Geobellies Company wishes to make a new type of container for their product. The designer has created two containers: one the shape of a square-based pyramid, and the other a cone. Your job is to determine which container holds more.
Pyramid
Cone slant height 10.5 cm
base length 5.6 cm
slant height 10.5 cm
diameter is 5.6 cm
1) Use the Pythagorean Theorem to find the height of each shape
Pyramid
Cone
2) How much can each container hold? Show your work.
Pyramid
Cone
3) Identify the shape with the greater volume. 4) If the geobellies cost $0.005/cm3, how much will it cost to fill each container?
