006. Multiplying by Powers of Ten and Algorithms

Learning Intentions

• multiply by a power of ten by adding zeros to the end of a number
• apply the multiplication algorithm to find the product of any two positive integers

Pre-requisite Summary

• Understanding place value and how digits shift when multiplied by $10$
• Recognising powers of ten: $10,100,1000,\dots$
• Recall of single-digit multiplication facts
• Ability to multiply a number by a single digit using the written algorithm
• Understanding column alignment and place value in written multiplication

Worked Examples

Worked Example 1

Multiply by powers of ten:

a) $34\times 10$
b) $582\times 100$
c) $7406\times 1000$

Worked Example 2

Multiply by powers of ten:

a) $209\times 10$
b) $45\times 1000$
c) $6708\times 100$

Worked Example 3

Use the multiplication algorithm:

a) $34\times 27$
b) $156\times 43$

Worked Example 4

Use the multiplication algorithm:

a) $2405\times 36$
b) $3178\times 24$

Problems

Problem 1

a) $47\times 10$
b) $693\times 100$
c) $8204\times 1000$

Problem 2

a) $305\times 10$
b) $72\times 1000$
c) $4609\times 100$

Problem 3

a) $26\times 35$
b) $142\times 28$

Problem 4

a) $3104\times 45$
b) $2786\times 32$

Exercises

Understanding and Fluency

1. Multiply by powers of ten:
   a) $23\times 10$
   b) $451\times 100$
   c) $6204\times 1000$

2. Multiply by powers of ten:
   a) $89\times 10$
   b) $3407\times 100$
   c) $52\times 1000$

3. Multiply using the algorithm:
   a) $24\times 16$
   b) $135\times 24$
   c) $402\times 32$

4. Multiply using the algorithm:
   a) $37\times 28$
   b) $264\times 35$
   c) $1208\times 14$

5. Multiply:
   a) $58\times 10$
   b) $58\times 100$
   c) $58\times 1000$

6. Multiply:
   a) $3206\times 21$
   b) $2478\times 13$
   c) $1905\times 42$

Reasoning

7. Explain why multiplying by $100$ adds two zeros to a number.

8. A student writes $46\times 100=460$. Explain the mistake.

9. Why must the second line in the multiplication algorithm shift one place left?

10. Compare $3205\times 40$ and $3205\times 4$. How are they related?

Problem-solving

11. A factory produces $245$ items per day. How many items in $36$ days?

12. A school buys $128$ boxes of pencils with $24$ pencils in each. How many pencils?

13. A warehouse stores $3405$ cartons with $32$ items each. How many items?

14. A printer produces $1250$ pages per hour. How many pages in $24$ hours?

15. A farm plants $276$ trees in each section. There are $35$ sections. How many trees?

Potential Misunderstandings

• Students may append zeros incorrectly (e.g., adding zeros to the wrong number)
• Students may believe multiplying by $10$ changes digits rather than shifting place value
• Students may forget to shift the second partial product in the algorithm
• Students may misalign digits when adding partial products
• Students may ignore multiplication by zero within a number (e.g., $304\times 26$)
• Students may incorrectly count the number of zeros when multiplying by powers of ten

Next: 007. Division with Remainders and Short Division