BC Math 9 Proportional Reasoning

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Spatial proportional reasoning
Scale diagrams, similar triangles and polygons, linear unit conversions
Limited to metric units
Drawing a diagram to scale that represents an enlargement or reduction of a given 2D shape
Solving a scale diagram problem by applying the properties of similar triangles, including measurements
Integration of scale for First Peoples mural work, use of traditional design in current First Peoples fashion design, use of similar triangles to create longhouses / models

How many cm in 2 meters?
Solution
200
How many mm in a km?
Solution
1,000,000
How many inches in a mile given 1 mile equals 5280 feet?
Solution
$63,360$ in
If a 10 cm long toy car is at a scale of 1:20, how long is the car in real life?
Solution
200 cm = 2 m
A right triangle has side lengths of 5-12-13.
This triangle is enlarged by a factor of 2.
1. Sketch the larger triangle
  Solution
  $10-24-26$
2. What is the perimeter of the larger triangle?
  Solution
  $60$
3. How many times larger is the area of the larger triangle vs. the smaller triangle?
  Solution
  $4$ times
A picture of the virus is 5 cm long. If the virus is 100 nanometres long in real life, what is the scale of this picture? (ex. 200:1 or 1:2000, etc.)

Solution
500,000 : 1
The following toy car is 6 cm wide:

If the real racing car is 2 m wide, what is the scale factor?
Solution
$1:33.\bar{3}$
Your toy plane is 5 cm wide. It is at a scale of 2:170. How large is the plane in real life?
Solution
$4.25$ m
See triangle below:
1. Find $c$
  Solution
  $c=13$
2. Find $x$
  Solution
  $x=6$
Find $x$ in the diagram below:

Solution
$x=5$
Find $x$ and $y$ in the diagram below:

Solution
$x=5$ and $y=\frac{16}{3}$
The Burj Khalifa is about 830 m tall. The ruler measurement of a picture of this building is 6 cm. The ruler measurement of the picture of a monster is 2 cm.
1. How tall is the monster in real life?
  
  Solution
  $\frac{830}{3}$ m
2. What assumption are you making when estimating the height of the monster?
  Solution
  The monster is beside the building (not in front).
The perimeter of the small hexagon in the diagram below is 12 m.
1. Find the side length of the large hexagon
  Solution
  4 m
2. Express the perimeter of the smaller hexagon to the perimeter of the larger hexagon as a simplified ratio
  Solution
  $2:1$
Find $x$ in the diagram below:

Solution
$\frac{45}{7}$
See diagram below:
1. Under what conditions are the two triangles similar?
  Solution
  Line 4 and line $x$ are parallel
2. Find $x$
  Solution
  $\frac{48}{5}$
Draw a larger version of the picture below by dividing a square blank sheet of paper into 16 squares.
Challenge:
1. Suppose the Coast Salish artwork below is enlarged to fit on a square building face with a width of 10 m. Estimate the area of one fish only.
  
  Solution
  Less than 25 squared meters
2. 360 degrees is equal to $2\pi$ radians. The formula for the circumference of a circle is $C=2\pi r$ and the area of a circle is $A=\pi r^2$ . Show that the arc length of a sector of a circle is $\text{Arc }=\theta r$ .
3. Why is the area of the sector below $A_{\text{sector}}=\frac{\theta r^2}{2}?$
4. Challenge – See the Pit House below:
  1. How many times does the volume of the Pit House grow by doubling the dimensions?
    Solution
    $8$
  2. How many times does the area of the Pit House grow by doubling the dimensions?
    Solution
    $4$
  3. How large does the Pit House’s area scale up by increasing the dimensions by a factor of $n$ ?
    Solution
    $4n$
  4. How large does the Pit House’s volume scale up by increasing the dimensions by a factor of $n$ ?
    Solution
    $8n$
  5. Does this scaling ratio increase for all types of shapes?
    Solution
    Yes

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