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Division as an Unknown Factor

A 45-minute Grade 3 lesson that turns every division into a multiplication with a gap, so a pupil who knows their tables already knows their division facts.

Division as an Unknown Factor is a free 45-minute Grade 3 maths lesson plan for Common Core standard 3.OA.B.6 — see division as finding the missing factor. It is timed across 5 sections, gives the exact wording to use where the wording carries the mathematics, names 3 misconceptions with the teaching move that addresses each, and ends with an exit ticket and a printable worksheet with its answer key.

Grade:
Grade 3
Length:
45 minutes
Standard:
3.OA.B.6

Grade 3 · Operations & Algebraic Thinking

3.OA.B.6

Understand division as an unknown-factor problem.

Objective

Students will be able to solve a division within 100 by finding the unknown factor in the related multiplication.

Students are successful when they can

  • Rewrites 56 ÷ 8 as 8 × ___ = 56.
  • Uses a known multiplication fact to answer a division.
  • Writes all four facts of a multiplication and division family.

What you need

  • Fact family triangles
  • Squared paper for arrays
  • A multiplication grid for reference
  • The printable practice sheet linked at the end of this plan

The lesson

Warm-up — cover a corner

5 min
  1. Show a fact family triangle with 6, 8 and 48. Cover one corner and ask for the hidden number.
  2. Cover the 48 and it is a multiplication; cover a 6 or an 8 and it is a division. Same triangle.

Teach — the division you already know

14 min
  1. Write 56 ÷ 8. Ask how many pupils know it instantly. Then ask how many know 8 × 7.
  2. The second number is always larger, and that gap is the lesson.
  3. Rewrite it as 8 × ___ = 56. The question is now one they can answer.
  4. Draw the array: 8 rows, total 56, row length unknown. The division is asking for the missing side.
  5. Do three more, each time saying the multiplication out loud before the division.
  6. Finish with 63 ÷ 9 and let the class find it through 9 × 7 rather than by sharing anything out.
“You do not have a division fact to learn. You have a times table fact with one number hidden, and you already know it.”

Guided practice — flip and answer

12 min
  1. Pairs take a division card, write the matching multiplication with a blank, then answer it.
  2. The partner checks the multiplication matches the division before either solves it.
  3. Then write the full family of four facts.

Independent practice

11 min
  1. Students complete a think-multiplication page.
  2. Watch for pupils sharing counters out. They will be right and slow, and the strategy is the thing being taught.

Close

3 min
  1. Give 42 ÷ 6 and ask for the multiplication that answers it.
  2. 6 × 7 = 42, so the division is done. Stop there.

What goes wrong, and why

Each of these is a reasoning error rather than carelessness. The fix addresses the reasoning.

Writes 56 ÷ 8 as 8 × 56.

Why: The two visible numbers have been rearranged without deciding which is the total. Turning things round has worked for multiplication.

Fix: The fact family triangle, with the total always at the top. Fifty-six cannot be a factor because it is the whole.

Shares counters out for every division, including 81 ÷ 9.

Why: Sharing is the meaning of division that was taught first and it always works, so nothing forces a change of method.

Fix: Give 96 ÷ 8. Sharing is impractical and the multiplication route is quick. The numbers make the argument.

Knows 7 × 8 but cannot answer 56 ÷ 7.

Why: The fact has been stored as a one-directional string rather than as a relationship between three numbers.

Fix: Practise the triangle with each corner covered in turn. Three questions per fact, and the fact stops having a direction.

If they are not there yet

  • Work within the 2, 5 and 10 tables, where the multiplication is certain and only the reframing is new.
  • Keep a multiplication grid available. Looking up 8 × ? = 56 along a row is the same skill and makes the structure visible.

If they finish early

  • Answer a division with a remainder by finding the nearest multiple below.
  • Divide a two-digit number by a one-digit number that is not in the tables, using the distributive split.
  • Explain why every division fact has a multiplication fact but not every division has a whole number answer.

Exit ticket

Write the multiplication that helps you solve 72 ÷ 9, then answer it.

What to look for: 9 × 8 = 72 and the answer 8. A bare 8 with no multiplication shown may have been counted or shared — ask for the multiplication tomorrow before crediting the strategy.

Printable practice for 3.OA.B.6

Free to print and copy, answer key included. Each sheet can generate a fresh set of problems for a retake.

Questions about Division as an Unknown Factor

How long does the Division as an Unknown Factor lesson take?

45 minutes, split across 5 timed sections: warm-up 5 min, teach 14 min, guided practice 12 min, independent practice 11 min, close 3 min. The minutes are stated per section and add up to the stated length, so the plan can be cut or extended at a section boundary rather than abandoned halfway.

Which standard does Division as an Unknown Factor teach?

CCSS.MATH.CONTENT.3.OA.B.6 — 3.OA.B.6, Operations & Algebraic Thinking for Grade 3: “Understand division as an unknown-factor problem.” In plain terms, see division as finding the missing factor.

What do I need to teach this lesson?

Fact family triangles, squared paper for arrays, a multiplication grid for reference and the printable practice sheet linked at the end of this plan. Everything listed is either ordinary classroom equipment or a free printable from this site — there is nothing to buy and nothing to prepare beyond photocopying.

What mistakes should I expect in this lesson?

Three, each a reasoning error rather than carelessness: writes 56 ÷ 8 as 8 × 56; shares counters out for every division, including 81 ÷ 9 and knows 7 × 8 but cannot answer 56 ÷ 7. The plan gives the thinking behind each one and the teaching move that addresses the thinking rather than the symptom.

How do I know whether the lesson worked?

The exit ticket asks: “Write the multiplication that helps you solve 72 ÷ 9, then answer it.” 9 × 8 = 72 and the answer 8. A bare 8 with no multiplication shown may have been counted or shared — ask for the multiplication tomorrow before crediting the strategy. That tells you whether the class needs a reteach or a thirty-second conversation before you plan tomorrow.

Is there a worksheet to go with Division as an Unknown Factor?

Yes — 2 printable 3.OA.B.6 worksheets, linked at the end of the plan and free like the rest of the site. Each one comes with an answer key and 25 versions, so the practice after the lesson can be a different paper on every desk.

Is Division as an Unknown Factor free to use?

Yes. The plan, its printables and their answer keys are free to read, print and photocopy for your own class, with no account and no email capture. It was last reviewed on 5 September 2026. Please do not resell it or republish it as your own.

Last reviewed 2026-09-05. Aligned to 3.OA.B.6 of the Common Core State Standards for mathematics.