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Fraction Classroom · 2.4.0 · Teacher answers & responses

Rebuild a missing whole from a known fraction

Use a labelled fragment to reconstruct its reference whole, then prove that the reconstructed whole gives the original fraction.

The first three equal boxes represent the known 9 cm fragment. The fourth completes the same whole; it is the same size as the others.Each quarter = 3 cm3 cm3 cm3 cm?Known fragment → add an equal part to rebuild the whole
The first three equal boxes represent the known 9 cm fragment. The fourth completes the same whole; it is the same size as the others.

Build the idea

model. A 9 cm fragment is three quarters of a complete strip. How long was the complete strip?

  1. Read the fraction as a relationship to the missing whole. The 9 cm fragment is three quarters, not the entire reference strip.
  2. Draw three equal boxes for the known fragment and label their combined length 9 cm. Share that length across the boxes: each box represents 3 cm.
  3. Add one more equal box to complete four quarters. Four copies of 3 cm produce a whole strip of 12 cm.
  4. Check the reconstructed whole: splitting 12 cm into four equal parts gives 3 cm per quarter, and three quarters recover the original 9 cm fragment.

Worked example: 12 cm. The 9 cm fragment contains three equal quarter-parts, so each quarter is 3 cm. Four such parts make the whole.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

p1. A 10 cm fragment is 2/3 of a strip. Find the complete strip length in cm.

Answer: 15

Teacher response: Two equal parts total 10 cm, so one is 5 cm and all three total 15 cm.

p2. A 4 cm fragment is 1/4 of a strip. Find the complete length in cm.

Answer: 16

Teacher response: The known fragment is one of four equal 4 cm parts.

p3. An 18 cm fragment is 3/6 of a strip. Find the complete length in cm.

Answer: 36

Teacher response: One sixth is 6 cm; six sixths make 36 cm.

e1. A 14 cm fragment is 7/8 of a strip. Reconstruct the whole. Enter its length in cm and draw all equal parts.

Answer: 16

Look for:

  • Distinguishes the known fragment from the unknown whole.
  • Uses equal parts and checks that the stated fraction recovers the fragment.

Teacher response: Each eighth is 2 cm, so eight equal parts make 16 cm.

Transfer the idea

transfer. Can the same 6 cm piece be one half of one strip and one third of another? Draw both reference wholes.

Answer: Yes. A 12 cm whole has a 6 cm half, while an 18 cm whole has a 6 cm third. The piece stays the same; its relationship to the reference whole changes.

Run the 15-minute workshop

teach-1. 3 minutes: Show only the three-quarter fragment. Ask learners to say what the 9 cm label measures.

teach-2. 4 minutes: Find one equal part and reconstruct the whole with a fourth matching box. Check the fragment against the new whole.

teach-3. 5 minutes: Complete the first three reconstruction cards with a labelled strip for each.

teach-4. 3 minutes: Use the independent card and check the result by rebuilding its known fragment.

Respond to the learner

misconception. Take three quarters of 9 because the question contains three quarters.

Teacher response: Point to the label on the fragment. Nine is already the length of the selected three parts. First find one part, then build the missing whole.

support. Provide the correct number of empty boxes and mark how many belong to the known fragment.

extend. Choose a whole length and write two different fragment clues that both reconstruct it; include a diagram to verify each clue.

Teaching notes