Problem
Shrinking self-similar shapes
Figure 1 is a fully shaded equilateral triangle. At each stage every shaded triangle splits into four and the middle one is emptied. So 3 of every 4 stay shaded each time. Find the shaded fraction of figure 4.
Reasoning · Grade 3-1 Meaning of Fractions
Your answer
Watch out Figure 4 tempts four multiplications, but figure 1 is already 1 — only three are needed.
How to solve
Strategy Look for a Pattern — Each step keeps 3 of every 4 shaded pieces, so the shaded amount is multiplied by 3/4 every time (Look for a Pattern). Working out the first few figures (Solve an Easier Related Problem) shows the multiplying rule, and treating one step as 'split into 4, keep 3' (Identify Subproblems) makes the multiplication clear.
1STEP 1
Figure 1: everything is shaded
Figure 1 is fully shaded: 1.
Figure 1 = 1
Starting from the simplest case, the whole shape, gives a clear base to grow the pattern from.
3.NF.A.1Solve An Easier Related Problem2STEP 2
Figure 2: keep 3 of 4
Emptying the middle makes figure 2 3/4.
1 × 3/4 = 3/4
Emptying 1 of 4 equal pieces leaves 3 of 4, which is just reading 'three out of four equal parts' as a fraction.
3.G.A.2Identify Subproblems3STEP 3
Figure 3: do it again to every shaded triangle
Again gives three quarters of that: 9/16.
3/4 × 3/4 = 9/16
The same rule on each piece means the whole shaded amount shrinks by the same 3/4 factor each step, a clean repeating pattern.
3.OA.D.9Look For A PatternApplying the same rule to every shaded triangle shrinks the shaded amount by the same three quarters at each step.
Why?
Every shaded piece loses the same share of itself, so the shaded total shrinks in the same ratio however many pieces there are.
🧱Proportional quantities scale togetherWhen two quantities keep a fixed ratio, doubling one doubles the other, so a ratio you know can stand in for one you do not.
Why?
Keeping 3 out of 4 equal pieces is exactly what the fraction three quarters names, so the step can be written as one fraction.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
4STEP 4
Figure 4: multiply by 3/4 one more time
A third 3/4 gives 27/64.
9/16 × 3/4 = 27/64
Counting shaded little triangles: figure 4 has 27 shaded tiny triangles out of 64 equal tiny triangles, which is exactly 27/64.
3.OA.C.7Look For A PatternAnswer
27/64
Each step removes some shaded area, so the fractions should shrink: 1, then 3/4, then 9/16 (about 0.56), then 27/64 (about 0.42). They steadily get smaller but stay above zero, which fits a shape that keeps emptying its middles. Counting also agrees: by figure 4 the triangle is cut into 4x4x4 = 64 tiny equal triangles and 3x3x3 = 27 of them are shaded.
Takeaway
Keep 3 of every 4 each step, so just multiply by 3/4 each time - 27 out of 64 tiny triangles, all Grade 3 thinking!
- Figure 1: everything is shaded
- Figure 2: keep 3 of 4
- Figure 3: do it again to every shaded triangle
- Figure 4: multiply by 3/4 one more time