Welcome to Scale: Enlarging and Reducing Shapes!
Have you ever used a smartphone or a tablet to zoom in on a photo to see a tiny detail, or zoomed out to see the whole picture? Or have you ever built a miniature model of a car or looked at a map of your town? If so, you have already worked with scale!
In this chapter of Shape and space, we will explore how shapes can grow bigger (enlargement) or shrink smaller (reduction) while keeping their exact same shape and proportions. Don't worry if this sounds a bit tricky at first — we will take it step-by-step with plenty of clear examples!
Note: Earlier in Shape and space, you explored whether shapes have the exact same shape (similarity). Here, we focus specifically on using scale and ratio to change their size!
---What Does "Scale" and "Ratio" Mean?
When we change the size of a shape in mathematics, we must do it fairly so the shape does not look squished or stretched out of proportion.
Key terms to remember:
• Scale: The rule or relationship that tells us how much larger or smaller a shape becomes compared to the original.
• Ratio: A way to compare two quantities. In scale drawings, a ratio compares the length of a side on our drawing to the length of the matching side on the original shape.
• Enlargement: Making a shape larger by multiplying every side length by the same amount.
• Reduction: Making a shape smaller by dividing every side length by the same amount (or multiplying by a fraction).
Did you know? When you enlarge or reduce a shape properly, all the angles inside the shape stay exactly the same! Only the lengths of the sides change.
---Stage 1: Making Shapes Bigger (Enlargement)
Imagine you have a small rectangle drawn on grid paper. It is \(2\text{ units}\) wide and \(3\text{ units}\) tall.
Now, let's say we want to make it twice as big. This means our scale ratio is \(2 : 1\) (read as "two to one"), meaning every \(1\text{ unit}\) on the original becomes \(2\text{ units}\) on the new enlarged shape.
Step-by-step guide to enlarge a shape:
Step 1: Measure or count the units of each side on the original shape.
Original width = \(2\text{ units}\)
Original height = \(3\text{ units}\)
Step 2: Multiply every single side length by the scale multiplier (here, we multiply by \(2\)).
New width = \(2 \times 2 = 4\text{ units}\)
New height = \(3 \times 2 = 6\text{ units}\)
Step 3: Draw your new shape. Notice how it looks identical in form, just bigger!
Memory trick to remember: Think of an enlargement like a balloon being blown up evenly. Every part expands together at the same rate!
Key Takeaway for Enlargement: To enlarge a shape, multiply every side length by the scale factor. Never add to the sides — always multiply!
---Stage 2: Making Shapes Smaller (Reduction)
What happens when an object is too big to fit on a piece of paper, like a full-sized room or a playground? We need to create a reduction!
Imagine we have a large square with sides that are each \(8\text{ cm}\) long. We want to reduce it using a scale ratio of \(1 : 4\) (read as "one to four"). This means that every \(4\text{ cm}\) on the original will be represented by \(1\text{ cm}\) on our drawing.
Step-by-step guide to reduce a shape:
Step 1: Find the length of the original sides.
Original side = \(8\text{ cm}\)
Step 2: Divide the original side lengths by \(4\) (which is the same as multiplying by \(\frac{1}{4}\)).
New side = \(8 \div 4 = 2\text{ cm}\)
Step 3: Draw the smaller square with sides of \(2\text{ cm}\). It keeps the exact same square shape, but fits neatly on your page!
Key Takeaway for Reduction: To reduce a shape, divide every side length by the scale factor (or multiply by a fraction smaller than \(1\)).
---Writing Scale as a Ratio
A ratio is written using a colon symbol \(:\) between two numbers.
• Ratio for Enlargement: The first number is larger than the second number.
Example: \(3 : 1\)
This means the new shape is \(3\text{ times}\) as long as the original shape.
• Ratio for Reduction: The first number is smaller than the second number.
Example: \(1 : 5\)
This means the new shape is \(\frac{1}{5}\) the size of the original shape (every \(5\text{ units}\) on the original is shrunk down to \(1\text{ unit}\)).
• Same Size: A ratio of \(1 : 1\) means the shape stays the exact same size!
---Common Mistakes to Avoid
Mistake 1: Adding instead of multiplying or dividing.
Incorrect: "I want to double a rectangle of \(2\text{ cm}\) by \(4\text{ cm}\), so I will add \(2\text{ cm}\) to each side to get \(4\text{ cm}\) by \(6\text{ cm}\)."
Why it's wrong: If you add, the proportions change and the shape gets distorted!
Correct: Multiply both sides by \(2\): \(2 \times 2 = 4\text{ cm}\) and \(4 \times 2 = 8\text{ cm}\).
Mistake 2: Changing only one side.
Incorrect: Enlarging the width by multiplying by \(3\), but forgetting to change the height.
Why it's wrong: Your shape will look stretched out like silly putty!
Correct: Every single side must be multiplied or divided by the exact same number.
Mistake 3: Changing the angles.
Remember: When you enlarge or reduce a shape, the corners and angles never change! A right angle (\(90^\circ\)) remains a right angle, no matter how big or small the shape becomes.
Real-World Inquiry: Where Do We See Scale?
Scale and ratio are powerful tools we use to understand the world around us:
• Maps and Atlases: A map uses a scale ratio like \(1 : 100\) so that an entire park or city can fit onto a paper map or a screen.
• Architectural Blueprints: Builders draw plans of houses using scale so they can design an entire building on a tabletop.
• Science and Biology: Scientists use microscopes to create enlarged scale drawings of tiny plant cells or bacteria so they can study them clearly.
---Quick Review: Check Your Understanding!
1. Quick Question: A triangle has sides of length \(3\text{ cm}\), \(4\text{ cm}\), and \(5\text{ cm}\). If you enlarge it by a scale ratio of \(2 : 1\), what are the lengths of the new sides?
Answer: Multiply each side by \(2\):
\(3 \times 2 = 6\text{ cm}\)
\(4 \times 2 = 8\text{ cm}\)
\(5 \times 2 = 10\text{ cm}\)
2. Quick Question: A rectangle has a length of \(12\text{ cm}\) and a width of \(6\text{ cm}\). If you reduce it using a scale ratio of \(1 : 3\), what are the new measurements?
Answer: Divide each side by \(3\):
Length = \(12 \div 3 = 4\text{ cm}\)
Width = \(6 \div 3 = 2\text{ cm}\)
Remember: Keep your proportions fair, multiply to enlarge, divide to reduce, and keep those angles unchanged!