Problem
Shade a fraction of a shape
Four equal triangles are labelled 3/4, 5/6, 5/8 and 7/9. Each is cut into as many equal-area parts as its denominator. The number shaded must equal the numerator. Divide and shade each triangle.
Reasoning · Grade 3-1 Meaning of Fractions
Your answer
Watch out The pieces look like they must be identical, but only the areas need to match — the sixths need not be the same shape.
How to solve
Strategy Draw a Diagram — Each fraction is a separate small drawing job (Draw a Diagram), so I handle them one at a time (Identify Subproblems). The trick for each is to find an easy way to chop an equilateral triangle into that many equal pieces, building on simple cuts I already know like 'midpoints make 4' (Solve an Easier Related Problem).
1STEP 1
3/4 : make 4 equal parts, shade 3
Joining side midpoints gives quarters — shade 3.
3/4: 4 equal small triangles, shade 3
Connecting the side midpoints is the standard way to split a triangle into 4 equal smaller triangles.
3.G.A.2Draw A Diagram2STEP 2
5/6 : make 6 equal parts, shade 5
Three medians give sixths — shade 5.
5/6: 3 medians → 6 equal parts, shade 5
A median splits a triangle into two equal halves; all three medians together make 6 equal pieces of the same size.
3.G.A.2Identify SubproblemsDrawing all three medians cuts the triangle into 6 pieces of exactly the same size, so shading 5 of them shows five sixths.
Why?
A fraction only means anything when the whole has been cut into pieces that are all the same size, and the medians deliver exactly that.
🧱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.
Why?
The six pieces fill the triangle with no gap and no overlap, so together they are the whole and each one is a genuine sixth of it.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
5/8 : make 8 equal parts, shade 5
Halving each quarter gives eighths — shade 5.
5/8: 4 equal → halve each → 8 equal parts, shade 5
Starting from the easy 4 equal parts and cutting each in half doubles the count to 8 while keeping all pieces equal.
3.G.A.2Solve An Easier Related Problem4STEP 4
7/9 : make 9 equal parts, shade 7
Thirds on each side with parallels give ninths — shade 7.
7/9: split each side in 3 → 9 equal parts, shade 7
Splitting each side into 3 and drawing the parallel grid makes 9 identical little triangles, the same way 'each side in 2' made 4.
3.G.A.2Draw A Diagram5STEP 5
Check each fraction is shown correctly
Parts match the denominators and shading the numerators — all correct.
3/4, 5/6, 5/8, 7/9
Reading 'shaded parts out of equal parts' back as a fraction confirms each picture matches its label.
3.NF.A.1Identify SubproblemsAnswer
cut into 4, 6, 8 and 9 equal parts, then shade 3, 5, 5 and 7
Each shaded amount is less than the whole (3 ＜ 4, 5 ＜ 6, 5 ＜ 8, 7 ＜ 9), so every triangle is partly shaded, which is expected. The denominators 4, 6, 8, 9 are all numbers an equilateral triangle can be cut into with equal pieces, and the partitions use only equal cuts, so every part really is the same size.
Takeaway
The bottom number tells how many equal pieces to cut and the top number tells how many to colour - pure Grade 3 fraction sense!
- 3/4 : make 4 equal parts, shade 3
- 5/6 : make 6 equal parts, shade 5
- 5/8 : make 8 equal parts, shade 5
- 7/9 : make 9 equal parts, shade 7
- Check each fraction is shown correctly