Plan two cuts and account for the offcut

model. A 2-metre strip must provide pieces of 3/4 metre and 2/3 metre. Is there enough material, and how much remains?

1. Keep one measurement unit throughout: every length is measured in metres. Estimate first: each required piece is below one metre, so their sum is below the 2-metre stock.
2. Choose twelfths so both piece lengths can be represented with equal units. Three quarters is 9/12 and two thirds is 8/12.
3. Add the required lengths to get 17/12 metres. Subtract that total from 2 metres, or 24/12, to obtain an offcut of 7/12 metre.
4. Check the entire record: 9/12 + 8/12 + 7/12 = 24/12. Both required pieces and the offcut together must recover the original stock.

Worked example: Yes. The required pieces total 17/12 metres, leaving 7/12 metre from the original 24/12 metres.

p1. From 2 metres, cut pieces of 1/2 metre and 3/4 metre. Enter the remaining length in metres.

Answer: 3/4

Teacher response: The cuts use 1 1/4 metres, leaving 3/4 metre.

p2. From 1 1/2 metres, cut 2/3 metre and 1/4 metre. Enter the remaining length in metres.

Answer: 7/12

Teacher response: The stock is 18/12 and the cuts use 8/12 + 3/12, leaving 7/12.

p3. You have 1 metre but need separate pieces of 3/4 metre and 1/2 metre. How many additional metres are needed?

Answer: 1/4

Teacher response: The required 1 1/4 metres exceed the stock by 1/4 metre; there is a shortage rather than an offcut.

e1. From a 3-metre strip, plan cuts of 1 1/4 metres and 5/6 metre. Enter the offcut length and check that all pieces recover 3 metres.

Answer: 11/12

Look for:

• Uses one length unit and equivalent fractions before adding.
• Accounts for every piece and interprets a shortage correctly.

Teacher response: The cuts use 15/12 + 10/12 = 25/12 metres, leaving 11/12 of the original 36/12.

transfer. If a real cutting process removes a small length at every cut, can this exact model be used unchanged to order material? Explain what information is missing.

Answer: No. The model explicitly assumes no cutting loss. A practical plan needs the loss per cut and the number of cuts, then includes that extra material before deciding whether the stock is sufficient.

teach-1. 3 minutes: Predict whether the two required pieces fit before calculating an exact offcut.

teach-2. 4 minutes: Draw a strip in twelfths, mark both cuts and account for the remaining section.

teach-3. 5 minutes: Complete the first three cut records, distinguishing a leftover length from a shortfall.

teach-4. 3 minutes: Use the independent plan and add all outputs back to the original stock.

misconception. Add the numerators and denominators to find 5/7 metre of required material.

Teacher response: Quarters and thirds are different-sized units. Rename them as equal twelfths before counting; a total smaller than either large piece would also contradict the estimate.

support. Use a pre-partitioned strip with a common denominator, label the stock first and shade each required length once.

extend. For the worked two-piece plan, find the shortest exact stock length with no offcut: 17/12 metres, or 1 5/12 metres.