Math Vale

🌱 Starter Village · Arithmetic

Multiplication

Multiply whole numbers, understanding multiplication as equal groups and as the area of a rectangle. Facts through 12 x 12 become automatic.

In short

  • Multiplication is repeated addition of equal groups, and a rectangle makes it visible.
  • Order never changes a product: a x b = b x a.
  • The distributive property lets you split a factor by place value and add the partial products.
  • A quick estimate using round numbers catches most multiplication errors.

Equal groups

Multiplication is repeated addition of equal groups. Six chests holding 4 crystals each is 4 + 4 + 4 + 4 + 4 + 4, which is written 6 x 4 = 24.

The word "equal" is doing the work. If the chests held different amounts, you would have to add them one by one; multiplication is the shortcut that equal groups make possible.

Like addition, multiplication does not care about order: 6 x 4 = 4 x 6. Six groups of four and four groups of six both come to 24, even though they are different pictures.

Arrays and area

Arrange the same 24 crystals in a rectangle of 6 rows with 4 in each row. Now the multiplication is visible: the number of rows times the number in each row gives the total.

This is the same idea as the area of a rectangle, which is why area is length x width. Seeing multiplication as a rectangle explains several later rules at a glance:

  • Turning the rectangle on its side shows why order does not matter.
  • Cutting the rectangle into two pieces shows why 6 x 14 = 6 x 10 + 6 x 4. That is the distributive property, and it is the engine of every written method.

Facts worth memorising

The facts up to 12 x 12 should eventually be instant. Until they are, rebuild them from ones you know rather than guessing:

  • x 4 is double, then double again.
  • x 5 is half of x 10.
  • x 9 is x 10 minus one group: 7 x 9 = 70 - 7 = 63.
  • The square numbers (1, 4, 9, 16, 25, 36, ...) are worth learning on their own — they turn up constantly in geometry and algebra.

Multiplying larger numbers with partial products

Split the bigger number by place value, multiply each part, then add the pieces back together.

For 34 x 7:

  • 30 x 7 = 210
  • 4 x 7 = 28
  • 210 + 28 = 238

For 23 x 45, split both numbers:

  • 23 x 40 = 920
  • 23 x 5 = 115
  • 920 + 115 = 1,035

The compact column method does exactly this; partial products just keep the pieces visible so a missing one is obvious.

Rates and scaling

Multiplication also scales a rate: an amount "per one" of something else.

If a quest pays 35 gold per quest, then 12 quests pay 35 x 12 = 420 gold. The rate stays constant and the total grows in proportion.

Check with a benchmark: 10 quests would pay 350, so 12 quests must be somewhat more than that. 420 fits; 3,500 does not.

Worked examples

Example 1

46 x 8 = ?

  1. Split 46 into 40 + 6.
  2. 40 x 8 = 320.
  3. 6 x 8 = 48.
  4. Add the partial products: 320 + 48 = 368.
  5. Estimate check: 50 x 8 = 400, so 368 is sensible.

Example 2

A vault holds 24 chests. Each chest holds 36 gold coins. How many coins are in the vault?

  1. Equal groups: 24 groups of 36, so the calculation is 36 x 24.
  2. Split 24 into 20 + 4.
  3. 36 x 20 = 720.
  4. 36 x 4 = 144.
  5. 720 + 144 = 864 coins. (Estimate: 36 x 25 = 900, so 864 is reasonable.)

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

4 x 3 = ?

Answer: 12

  1. 4 x 3 = 12

Problem 2

Difficulty 3 of 5

A storeroom stacks feathers in a rectangle: 13 rows with 4 in every row. How many feathers are there in total?

Answer: 52 feathers

  1. 13 rows, each with 4 feathers.
  2. Split 13 into 10 + 3.
  3. 4 x 10 = 40
  4. 4 x 3 = 12
  5. Add the partial products: 40 + 12 = 52
  6. There are 52 feathers.

Problem 3

Difficulty 4 of 5

156 x ? = 1,404

Answer: 9

  1. factor x factor = product, so missing factor = product / known factor.
  2. 1,404 / 156 = 9
  3. Check: 156 x 9 = 1,404

Common mistakes

  • Writing only one partial product in a two-digit multiplication, so 23 x 45 comes out as 115.
  • Adding the two factors instead of multiplying them.
  • Being one group out because a row of an array was miscounted.
  • Confusing the perimeter of a rectangle with its area.

What you should be able to do

  • Recall multiplication facts through 12 x 12.
  • Multiply a two-digit number by a one-digit number using place value.
  • Multiply two two-digit numbers with the partial-products method.
  • Model multiplication as equal groups and as an array or area.

Where this fits in the curriculum

Common Core

  • 3.OA.A.1

    Grade 3 — Interpret products of whole numbers as a number of equal groups.

  • 3.OA.C.7

    Grade 3 — Fluently multiply and divide within 100; know all products of two one-digit numbers from memory.

  • 4.NBT.B.5

    Grade 4 — Multiply a multi-digit number by a one-digit number, and two two-digit numbers, using place value.

Ontario

  • G4.B2.2

    Grade 4 — Recall and demonstrate multiplication facts from 1 × 1 to 10 × 10, and the related division facts.

  • G5.B2.2

    Grade 5 — Recall and demonstrate multiplication facts from 0 × 0 to 12 × 12, and the related division facts.

  • G5.B2.6

    Grade 5 — Represent and solve problems involving the multiplication of multi-digit whole numbers.

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