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Foundations and Pre-Calculus Mathematics 10
Measurement FP 10.3 Demonstrate understanding of SI and imperial units of measurement including:
- Linear measurement - Surface area of spheres, and right cones, cylinders, prisms, and pyramids - Volume of spheres, right cones, cylinders, prisms, and pyramids - Relationships between and within measurement systems
*Adapted from Pearson’s Foundations and Pre-Calculus Mathematics 10
F10 – Unit 2 – Trigonometry Key Terms
Date: _______________
Key Terms Proportion – a statement that two ratios are equal. Imperial Units – measurement units such as the mile, yard, foot and inch commonly used in the United States and in some industries in Canada. Unit Analysis – a method of converting a measure in a given unit to a measure in a different unit by multiplying the measure by a conversion factor SI System of Measures – A system of units on powers of 10; the fundamental unit: of length is the metre (m), mass is the kilogram (kg), and time is the second (s). Referent – Used to estimate a measure. Surface Area – The total area of the surface of an object. Volume – The amount of space occupied by an object. Prism – An object with two bases. Right Prism – An object that has two congruent and parallel faces, and other faces that are rectangular. Right Rectangular Prism – A prism that has rectangular faces. Triangular Prism – A prism with triangular bases. Right Cylinder – An object with two parallel, congruent, circular bases. Pyramid – An object that has one face that is a polygon, and other faces that are triangles with a common vertex. Right Rectangular Pyramid – A pyramid that has a rectangular base. Slant Height – The distance from a point on the perimeter of the base of a cone to the apex of a cone to the apex of the cone. Apex – The vertical farthest from the base of an object.
F10 – Unit 2 – Trigonometry Name: _____________________ Unit Checklist Date:_______________________
Unit 1 - Checklist
Lesson 1 – Video Lesson 2 – Video Lesson 1 and 2 Worksheet Lesson 1/2 – Portfolio
Lesson 3 – Video Lesson 3 Homework: Page 11 #3, 7, 8, 9, 11, 12
Lesson 3 – Portfolio Questions
Lesson 4 – Video Lesson 4 Homework: Page 22 #4, 5, 6, 9, 10, 12
Lesson 4 – Portfolio Questions
Quiz Review: Page 25 #1, 3, 4, 7, 8
Quiz
Lesson 5 Video Lesson 5 Worksheet Lesson 5 – Portfolio Question
Lesson 6 – Video Lesson 6 Worksheet Lesson 6 – Portfolio Question
Lesson 7 – Video Lesson 7 Homework: Page 42 #4, 6, 8, 9, 11, 12
Lesson 7 – Portfolio Question
Lesson 8 – Video Lesson 8 Homework: Page 51 #3, 4, 5, 8, 9, 13
Lesson 8 – Portfolio Question
Lesson 9 – Video Lesson 9 Homework: Page 59 # 3bd, 5 (just volume), 8, 10
Lesson 9 – Portfolio Questions
Test Review: Page 64 #3, 4, 6, 7, 10, 13, 15, 17, 20, 22, 23, 25 (just volume)
Test
F10 – Unit 2 – Trigonometry Name: _____________________ Unit Checklist Date:_______________________
Lesson 1 – Proportional Reasoning Proportion – A fractional statement of equality between two ratios or rates Proportional Reasoning – The ability to understand and compare quantities that are related multiplicatively Proportional Reasoning 1) Set up two ratios equal with one unknown value 2) Solve for x (If you forget, you can remember to cross multiply and divide) Example 1:
3
4 16
x
2 16
5 x
5
5 25
x
Setting up a Proportion 1) Set up a ratio as a fraction 2) Set up a second ratio equal to the first fraction 3) If you are solving for an unknown, put x in the place of the unknown
** REMEMBER ** Keep your ratio the same on the top and bottom in both ratios Example 2: You are in charge of making punch for party. To make one batch of punch you need 2 cups of juice and 5 cups of sprite. How many cups of sprite do you need to make 28 cups of punch? Example 3: John has found that he can arrange the work cubicles of his employees’ best if the ratio between the length and width of a room is 3:2. If a room is 6 meters long, how wide should the room be?
F10 – Unit 1 – Measurement Date: _____________ Lesson 2 – SI Systems of Measurement
Lesson 2 – SI Systems of Measurement SI – Systeme International – The modern version of the metric system; uses the metric as the basic unit of length.
When do we use SI Systems of Measurement
Converting between SI units 1) Count how many “steps” between units 2) If you are getting larger, move decimal place as many “steps” you jump to the left 3) If you are getting smaller, move decimal place as many “steps” you jump to the right Example 1: a) 10 m to mm b) 150 km to m c) 175 m to km d) 24 cm to km e) 1.5 m to mm f) 0.5 m to
mm Milli
0.001
cm Centi 0.01
Dm Deci 0.1
m metre
1
dam Deca
10
hm Hecto
100
km Kilo 1000
Divide by 10
Multiply by 10
F10 – Unit 1 – Measurement Date: _____________ Lesson 1 – Proportional Reasoning Lesson 2 - SI Systems of Measurement
Lesson 1 and 2 Worksheet 1) Convert the following to cm: a)10 mm b) 27 m c) 1.5 km d) 39 hm
2) Convert the following to m: a) 1500 mm b) 145 km c) 32 cm d) 40 hm 3) Convert the following to km: a) 100 cm b) 4256 m c) 8374 mm d) 42 hm 4) Convert the following to mm: a) 12 km b) 50 m c) 12 cm d) 230 hm Use Proportional Reasoning to Solve the Following
5) 40
10 50
x 6)
12 18
16 x
F10 – Unit 1 – Measurement Date: _____________ Lesson 1 – Proportional Reasoning Lesson 2 - SI Systems of Measurement
7) 56
64 8
x 8)
18 36
27 x
9) 3
2056 4
x 10)
3 15
12 x
11) 3
5 460
x 12)
25 40
200x
13. A compound of two chemicals is mixed in the ratio of 3:10. If there are 45 L of the compound, how much of each chemical is in the mixture? 14. As a janitor you make a cleaning solution by mixing 30 g of concentrated powdered cleanser into 2 L of water. How much powder will you need for 5 L of water?
F 10 – Unit 1 - Measurement Date: _____________ Lesson3 - Imperial Measures of Length
Lesson 3 – Imperial Measurement
Imperial System – The system most commonly used in the United States; the standard unit of measurement for length is the foot. Also use the following units: Inch (in), Mile (mi), Yard (yd)
Referent: Used to estimate a measure
Convert Imperial to Imperial (Down)
1) Decide if you are getting bigger or smaller 2) Go down the staircase one step at a time multiplying 3) If you have a fraction, find that number as a smaller unit 3) Continue until you are at the new unit
Example 1: Convert the following
a) 5 miles to yards b) 5 miles to feet
Unit Referent Inch (in. or “) Thumb length (end to first knuckle) Foot (ft. or ‘) Foot or shoe length Yard (yd.) Arm span Mile (mi.) Distance walked in 20 min
Inch ″ in
Feet ′ ft
Yard yd
Mile mi
÷ 12
÷ 3
÷ 1760 x 12
x 3
x 1760 Common Conversions 12 inches = 1 Foot 36 inches = 1 yard 3 feet = 1 yard 5280 feet = 1 mile 1760 yards = 1 mile
F 10 – Unit 1 - Measurement Date: _____________ Lesson3 - Imperial Measures of Length
Imperial to Imperial (Up)
1) Use the staircase to find which number you are dividing by 2) Use long division to divide into the number 3) The remainder will be what is left in the smaller unit
Example 2:
a) 14 ft to yards and feet b) 62 in to feet and inches
c) 62 in to yards feet and inches d) a) 3520 yd to mi
Unit analysis: a method of converting a measure in a given unit to a measure in a different unit by multiplying the measure by a conversion factor.
Example 3: Tyrell has 4 yds. of cord to make friendship bracelets. Each bracelet needs 8” of cord. How many bracelets can Tyrell make? Use unit analysis to check the conversions.
F10 – Unit 1 - Measurement Date: _____________ Lesson 4 - Relating SI and Imperial Units
Lesson 4 - Relating SI and Imperial Units
Each measurement in the imperial system relates to a corresponding measurement in the SI system.
Proportion – a statement that two ratios are equal.
SI Units to Imperial Units Imperial Units to SI Units
1 mm = 4
100in. 1 in. = 2.54 cm
1 cm = 4
10in. 1 ft. = 30 cm
1 ft. = 0.3 m
1 m = 39 in. 1 yd. = 91.44 cm
1 m = 1
34
ft. 1 yd. = 0.9144 m
1 km = 6
10mi. 1 mi. = 1.6 km
Converting between SI and Imperial
1) Set up a ratio between the two units you are converting 2) Use proportional reasoning to solve for your new unit 3) Convert to the proper unit
Example 1: Convert to the nearest tenth of the unit:
a) 17 ft. to m b) 59 m to ft.
c) 5 mi. to km d) 104 cm to ft.
F10 – Unit 1 - Measurement Date: _____________ Lesson 4 - Relating SI and Imperial Units Example 2: The school librarian needs to reach a shelf that is 1.7 m above the floor. The librarian has a reach of 5’11”. Will the librarian be able to reach the shelf?
Example 3: A truck driver knows that his load is 15 ft. wide. Regulations along his route state that any load over 4.3 m wide must have wide-load markers and an escort with flashing lights. Does the truck driver need wide-load markers?
F10 – Unit 1 - Measurement Lesson 5 – Surface Area of Pyramids and Cones
Name: _______________
Lesson 5 – Surface Area of Pyramids
Net – The 2-dimensional figure of a 3-dimensional object. Square Pyramid – A pyramid that has a square as it’s base (four equal sides) Finding Surface Area of Pyramids 1. Draw the net of the pyramid 2. Label all lengths on the diagram 3. Label all the shapes that are similar 4. Calculate the area of each shape 5. Add them all together Example 1: Calculate the surface area of the following pyramid.
24m
18m 24m
h
b
A = ½ bh
w
l
A = l x w
F10 – Unit 1 - Measurement Lesson 5 – Surface Area of Pyramids and Cones
Name: _______________
Example 2:Calculate the surface area of the following pyramid. Example 3:Calculate the surface area of the following pyramid.
24m
24m
9m
10m
5m
4m
F10 – Unit 1 - Measurement Lesson 5 – Surface Area of Pyramids
Name: _______________
Lesson 5 Worksheet
1. Find the surface area of the following shape:
2. Calculate the surface area of a square pyramid with a base of 2 m and a slant height of 3 m. 3. Calculate the surface area of the following shape.
5 cm
10 c
m
5 cm
65 cm
30 cm
38 c
m
F10 – Unit 1 - Measurement Lesson 6 – Surface Area of Cones
Name: _______________
Lesson 6 – Surface Area of Cones
Calculating the Surface Area of a Cone 1. Determine the radius of the cone 2. Determine the slant height 3. Use the formula to solve for the surface area Slant Height: h2 + r2 = s2 Surface Area Formula: SA = 𝜋𝑟𝑠 + 𝜋𝑟 Example 1: Determine the surface area of the following cone. Example 2: A cone has a height of 5 cm and a radius of 3 cm. What is the surface area of the cone?
3m
4m
S
F10 – Unit 1 - Measurement Lesson 6 – Surface Area of Cones
Name: _______________
Lesson 6 Worksheet
1) Find the surface area of the following shape:
2) Find the surface area of the following shape:
3) Find the surface area of a cone that has a slant height of 82 cm and a radius of 28 cm.
4) Find the surface area of a cone with a diameter of 13.6 cm and a slant height of 9.8 cm.
20 cm
12 cm
1.45 cm
4 cm
F10 – Unit 1 - Measurement Lesson 7 – Volume Cones and Pyramids
Name: _______________
Lesson 7 – Volume of Cones and Pyramids
Find Volume of a Pyramid: 1. Determine the dimensions of the base of the pyramid (length and width) 2. Determine the height of the pyramid 3. Use the formula to solve for the volume
𝑉 = 𝑙𝑤ℎ
3
Example 1: What is the volume of the following pyramid? What is the capacity?
Find Volume of a Cone 1. Determine the radius of the cone 2. Determine the height of the cone 3. Use the formula to solve for the volume
𝑉 = 𝜋𝑟 ℎ
3
Example 2: What is the volume of the following cone? What is the capacity?
3 cm
4 cm
F10 – Unit 1 - Measurement Lesson 8 - Spheres
Name: _______________
Lesson 8 - Spheres
Sphere – A three-dimensional shape whose surface consists of points that are all the same distance from the centre of the shape. Surface Area of a Sphere:
A = 4𝜋r² Volume of a Sphere: V = 4𝜋r3
3 Example 1: What is the surface area and volume of the following sphere? Example 2: The diameter of a baseball is 7.5cm. What is the surface area of a baseball? What is the volume?
Example 3: A globe has surface area of 2735 cm2. Find the radius of the globe to the nearest tenth of a centimetre.
4 m
F10 – Unit 1 - Measurement Lesson 9 – Composite Objects
Date: _______________
Lesson 9 – Composite Objects
s
s
A = s2
h
b
A = ½ bh
w
l
A = l x w h
b
A = bh
r
A = πr2
a
b
h A = (a+b)h
2
C = 2πr
r
A = 4πr2 V = 4πr3
3
r
h s
A = πrs +πr2
V = πr2h
3
r
h m
V = πr2h
V = lwh
l
w
h 3 V = lwh
l w
h
F10 – Unit 1 - Measurement Lesson 9 – Composite Objects
Date: _______________
Composite Object - the result of combining two or more objects to make a new object Finding Volume of Composite Objects
1) Split the shape into different objects 2) Find the volume of each individual object 3) Add together the volumes of each shape
Example 1: Determine the volume of this composite object to the nearest tenth of a cubic centimeter.
Example 2: Determine the surface area of this composite object.
