Minds on.

Proportional Reasoning and Scale

In Unit 5, you saw many applications which demonstrated how reasoning proportionally allowed you to solve problems. Proportional reasoning helps governments and health care agencies serve the public. It allows doctors, pharmacies and nurses to determine correct dosages. It allows families to calculate expenses. In what other situations is proportional reasoning applied to real life?

Proportional Reasoning and Large Numbers



In these circumstances, proportional reasoning is used because an actual count would not be possible. A sample is taken, then proportional reasoning is used to estimate the population. To get a sense of how this happens, let's try counting elephants!

This is the discussion icon. Assignment: Counting Elephants

The image, taken from the air, shows a large heard of elephants.

Your task is to estimate the number of elephants in the image by using proportional reasoning.

Here is one method of estimating the population by using ratios:

  • Choose two rectangles randomly and carefully count the number of elephants in each rectangle square.
  • Find the total number of elephants in the two rectangles squares.
  • Determine how many rectangles squares there are in the grid.
  • Set up a ratio table (or use another method) and determine an estimate of how many elephants there are in the photograph.

This is the discussion icon. Consider: Thinking About Counting Elephants

Consider the following five questions.

  • If the total herd that was photographed that day required four photographs like the one you just saw, how would you estimate how many elephants were in the herd?
  • Estimating the number of elephants in a herd this way is an example of sampling. A sample is a group selected from the population. A sample may be chosen to represent an entire population. However, a sample may also be chosen to only include certain specific members of that population. Your sample was the two rectangles you selected. What might you do to get a better estimate of the number of elephants in the herd?
  • Why is selecting a sample and making an estimate a good way to count elephants in a herd?
  • What might be a problem with doing it this way?
  • What other scenarios can you think of where a count is needed but an actual count is not possible?

Proportional Reasoning and Scale

Proportional reasoning is used to interpret information from scale drawings. A scale drawing is often used to represent a 3-D object as a 2-D drawing (on paper, using computer software, etc) so that the object can be designed before it is constructed. Scale drawings are used when dimensions or distances are too large, or too small, to be represented on paper in their actual size.


Action.

Math and Construction

Surveying a space

Watch the following video that explains how a builder lays out right angles in his work.

In the video, Joe mentions that his method uses the Pythagorean Theorem. You first saw the Pythagorean relationship of right-angled triangles years ago. It says that the area of the square built on the longest side of a right-angled triangle is equal to the sum of the areas of the two squares built on the other two sides. In the image below, the area of the largest (green) square is equal to the areas of the two other squares (blue and yellow) added together.

Remember, the area of a square is the length of the side of the square times itself.

Making Mathematical Tools to Construct Right Angles

Ancient Egyptians did remarkable construction, all without the tools we have today. There is evidence that Ancient Egyptians used knotted ropes to create right angles.

This is the Think About It icon. Think About...

This is an example of how the knotted rope would have looked. If you had to describe this rope to someone, what would you say?

This is the question/answer icon. Question

Think about what you saw in the video. What connection can you make between the way Joe laid out a right triangle and the Egyptian rope knotted into 12 equal sections?

Hint...

The rope is knotted to make 12 sections that are equal in length.

Using Proportional Reasoning to Scale Up

Knowing the 3-4-5 rule for creating a right angle is nice, but what if you are making a patio that does not have a 3 meter length in it? How can I use 3-4-5 now?

The above patio starts at the house and goes 24 feet into the yard. It is 18 feet wide. How did the contractor use string to create the right angles at the corners?

To use the 3-4-5 rule to get the lengths you need, use proportional reasoning. We will use a ratio table.

1. Make a reasonably accurate diagram of the patio and label the known lengths:

2. Draw a right-angled triangle on the diagram:

3. Identify the three sides of the right-angled triangle, from shortest to longest:

4. In the 3-4-5 rule, the longest side is 5 so it will correspond to the longest length.
The middle number in the rule is 4 so it will correspond to the middle length, 24 feet.
The shortest side is 18 feet, so it will correspond to 3.

5. Use a ratio table to determine any unknowns:

6. Check your reasoning using the Pythagorean relationship of right-angled triangles:

The area of the square on the longest side:
30 x 30 = 900 sq feet

The area of the two squares:
18 x 18 = 324 sq feet
24 x 24 = 576 sq feet
The sum of the areas of the two squares:
324 + 576 = 900 sq feet

Looks like the reasoning is correct!

You should always check before you start working on something like the patio because, if you make a mistake, it will cost you -- in both time and money!

The string is cut to be 18 + 24 + 30 feet long (72 feet), and it is marked at 18 feet and at 24 feet. That's where the stakes need to be put.

This is the discussion icon. Assignment: Right Angles and Proportional Reasoning

  1. Use proportional reasoning to find the dimensions of four more right-angled triangles of different sizes.
  2. The dimensions of my 4 right-angled triangles:
    __________ by ______________ by _______________
    __________ by ______________ by _______________
    __________ by ______________ by _______________
    __________ by ______________ by _______________

How are Proportional Reasoning and Scale Connected?

If you want a new patio, a builder can help but first, they would need an accurate scale drawing of the design.

Scale Drawings

Scale drawings are used to design a large building project. Drawings may be made on paper or using computer drawing software.

Scale as a Ratio

The scale on a scale drawing is often indicated on the drawing, as can be seen on the following scale drawing.

It is represented as a ratio. The first number in the ratio is usually 1 and the second number represents the scale factor. (e.g., In this scale drawing, "Scale 1:50" means that every 1 unit represents 50 of the other unit. For example, if you measure the length of a table and it is 2 cm on the page, then the actual table will be 50 times longer, or 100 cm.)

One of the most important aspects of making a scale drawing is proportion:

The proportions of the objects in the drawing must be correct.

If a patio fireplace is to be 3 feet wide and 9 feet high, the drawing should show the proportion:

Width:Height = 3:9

The sketch must use a width : height ratio that is equivalent to 3:9.

This is the question/answer icon. Questions


In the following scale diagram, only one of the rectangles could not be used to represent the fireplace. Which one? Why?

Answer

The only rectangle that could not represent the fireplace is the green one (D). The lengths of these sides does not correspond to a 3:9 ratio.

Take a look at the following interactive. Click on each of the shapes to see how the ratio of each shape corresponds to the fireplace's 3:9 ratio... except shape D.

Long Description

This is the question/answer icon. Question

The diameter of the circular water feature is 1 m. It is placed to be in the centre of a square section of patio that has a width of 2 m.
Which of the following scale drawings could be used for this design?

Answer

The correct answer is b.

The ratio diameter : length has to be equivalent to 1 : 2.

The ratio in the first drawing Is 4 : 10. That is not equivalent to 1 : 2.

The ratio in the second drawing is 4 : 8, and that is equivalent to 1 : 2, so the correct answer is b.

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