005. Multiplication Laws and Algorithms

Learning Intentions

• To understand the commutative and associative laws for multiplication
• use mental strategies to find products
• apply the multiplication algorithm to find the product of a single digit number by a positive integer

Pre-requisite Summary

• Understanding multiplication as repeated addition
• Knowledge of basic multiplication facts ($1$–$10$ times tables)
• Understanding place value (ones, tens, hundreds, thousands)
• Ability to partition numbers (e.g., $34=30+4$)
• Understanding of distributive reasoning (breaking numbers apart)

Worked Examples

Worked Example 1

Use multiplication laws to simplify:

a) $4\times 7$ and $7\times 4$
b) $(3\times 5)\times 2$ and $3\times (5\times 2)$

Worked Example 2

Use mental strategies:

a) $6\times 18$
b) $4\times 25$

Worked Example 3

Use mental strategies:

a) $9\times 32$
b) $8\times 105$

Worked Example 4

Use the multiplication algorithm:

a) $7\times 346$
b) $4\times 2578$

Worked Example 5

Use the multiplication algorithm:

a) $9\times 4205$
b) $6\times 3784$

Problems

Problem 1

a) $5\times 8$ and $8\times 5$
b) $(2\times 6)\times 3$ and $2\times (6\times 3)$

Problem 2

a) $7\times 19$
b) $3\times 25$

Problem 3

a) $8\times 34$
b) $6\times 104$

Problem 4

a) $5\times 463$
b) $8\times 2147$

Problem 5

a) $7\times 3406$
b) $9\times 2685$

Exercises

Understanding and Fluency

1. Use the commutative law:
   a) $3\times 9$
   b) $9\times 3$
   c) Explain the relationship

2. Use the associative law:
   a) $(2\times 4)\times 5$
   b) $2\times (4\times 5)$
   c) Compare the results

3. Use mental strategies:
   a) $5\times 18$
   b) $4\times 27$
   c) $6\times 21$

4. Use mental strategies:
   a) $9\times 23$
   b) $8\times 14$
   c) $7\times 32$

5. Use the multiplication algorithm:
   a) $3\times 458$
   b) $6\times 2304$
   c) $8\times 1729$

6. Use the multiplication algorithm:
   a) $7\times 605$
   b) $4\times 3289$
   c) $9\times 2507$

Reasoning

7. A student says $6\times 25=25\times 6$. Explain why this is true.

8. Which is easier to calculate mentally? Explain:
   a) $4\times 25$
   b) $25\times 4$

9. Show two different ways to calculate $5\times 36$ using mental strategies.

10. Explain why breaking $8\times 103$ into $8\times 100+8\times 3$ works.

Problem-solving

11. A box holds $7$ pencils. How many pencils are in $346$ boxes?

12. A bus carries $8$ passengers per row. There are $25$ rows. How many passengers?

13. A teacher gives $6$ stickers to each of $1245$ students. How many stickers are given?

14. A factory makes $9$ toys per minute. How many toys in $305$ minutes?

15. A farmer plants $4$ trees in each row. There are $2378$ rows. How many trees?

Potential Misunderstandings

• Students may think multiplication is not commutative (believing $a\times b\ne b\times a$)
• Students may confuse associative and commutative laws
• Students may incorrectly regroup when multiplying (e.g., forgetting to carry)
• Students may multiply digits without considering place value
• Students may omit multiplication by $0$ in numbers such as $3405$
• Students may incorrectly partition numbers when using mental strategies

Next: 006. Multiplying by Powers of Ten and Algorithms