Find multiples and common multiples of one-digit numbers

What you will learn

• Find multiples and common multiples of one-digit numbers

Models used: number sentences, stories, tables.

Before you start

• Find common factors of two numbers

Count equal jumps to find multiples

Which numbers can we reach by counting in threes? Which can also be reached by counting in fours?

1. Start at the marked number. A jump to the right adds; a jump to the left takes away. Read the size of each jump.

   A multiple of three is the total of some whole groups of three. One, two, three and four groups give 3, 6, 9 and 12.

   3, 6, 9, 12, …

   Successive multiples add one equal group each time. A common multiple appears in both lists.

2. Start at the marked number. A jump to the right adds; a jump to the left takes away. Read the size of each jump.

   Counting in fours gives a different list: 4, 8, 12, 16, 20, 24.

   4, 8, 12, 16, 20, 24

3. Look for values shared by both lists. Read each value with the heading above its column.

   A common multiple belongs to both lists. Twelve and twenty-four appear in both.

   12 = 4 × 3 = 3 × 4

   Look for values shared by both lists

   Multiples of 3	Multiples of 4
   3, 6, 9, 12	4, 8, 12
   15, 18, 21, 24	16, 20, 24

Worked example: A new example, together

What is the smallest positive common multiple of 3 and 5?

1. A product links factors and multiples. Read each value with the heading above its column.

   Follow the equal group size in each list and look for shared totals.

   Multiples are found by counting equal groups; common multiples occur in both lists.

   A product links factors and multiples

   Number	Some factor pairs
   15	1 × 15; 3 × 5

2. A product links factors and multiples. Read each value with the heading above its column.

   Write successive multiples of each given number.

   A product links factors and multiples

   Number	Some factor pairs
   15	1 × 15; 3 × 5

   What is the smallest positive common multiple of 3 and 5?

   15

3. A product links factors and multiples. Read each value with the heading above its column.

   Write successive multiples of each given number.

   A product links factors and multiples

   Number	Some factor pairs
   15	1 × 15; 3 × 5

4. A product links factors and multiples. Read each value with the heading above its column.

   For common multiples, keep numbers in both lists.

   A product links factors and multiples

   Number	Some factor pairs
   15	1 × 15; 3 × 5

5. A product links factors and multiples. Read each value with the heading above its column.

   15 ÷ 3 and 15 ÷ 5 are whole numbers.

   A product links factors and multiples

   Number	Some factor pairs
   15	1 × 15; 3 × 5

6. A product links factors and multiples. Read each value with the heading above its column.

   Check the result against the question we started with.

   15 ÷ 3 and 15 ÷ 5 are whole numbers.

   We found 15. Multiples are found by counting equal groups; common multiples occur in both lists.

   A product links factors and multiples

   Number	Some factor pairs
   15	1 × 15; 3 × 5

Common mistakes to discuss

• Confusing multiples with factors. Multiples are built by multiplying, so they grow: 4, 8, 12. Factors fit exactly into the number and cannot exceed it.

Check your understanding

• Find multiples and common multiples of one-digit numbers
• distinguish multiples and common multiples from arbitrary larger numbers
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.