Grade 8 · Distinguish length and area multipliers · 740 of 750
Distinguish length and area multipliers · practice 88
Dilate a rectangle with sides 4.5 and 5.5 by length factor 3. Find the new side lengths and area. Follow the calculation, then test the quantities in the animated example.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Dilate a rectangle with sides 4.5 and 5.5 by length factor 3. Find the new side lengths and area.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
Apply the factor to both lengths before multiplying them.
Work through the reasoning
Step 1
Translate the givens
The original area is 4.5 × 5.5 = 24.75 square units.
Step 2
Calculate with the model
Scaled sides are 4.5 × 3 = 13.5 and 5.5 × 3 = 16.5.
Step 3
Check and interpret
New area = 13.5 × 16.5 = 222.75. Equivalently, 24.75 × (3)² = 222.75; the area multiplier is 9.
The answer
Sides = 13.5, 16.5; area = 222.75 square units.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
A dilation multiplies every length by the same positive factor. Ratios of corresponding sides stay equal, but area changes by the square of that factor because two independent dimensions both change. A scale below one shrinks the figure without changing its shape. Starting quantities: Original width = 4.5; Original height = 5.5; Length scale factor k = 3.
new lengths = k × old lengths; new area = k² × old area
03 · Reflect and transfer
Explain what changes and why.
Why is a length scale factor different from an area scale factor?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.