095r. Mental and Written Strategies for Addition and Subtraction

Learning Intentions

• To understand the commutative and associative laws for addition
• use the mental strategies partitioning, compensating and doubling/halving to calculate a sum or difference of whole numbers mentally
• use the addition and subtraction algorithms to find the sum and difference of whole numbers

Pre-requisite Summary

• Understand that addition combines quantities and subtraction finds the difference between quantities
• Know the place value of digits in whole numbers
• Be able to split numbers into tens, hundreds and ones
• Recall basic addition and subtraction facts
• Understand that an algorithm is a step-by-step written method
• Be able to line up numbers correctly by place value

Worked Examples

Worked Example 1

Use the commutative law to rewrite:

a) $7+15$
b) $34+9$

Worked Example 2

Use the associative law to make the addition easier:

a) $(6+4)+13$
b) $25+(75+8)$

Worked Example 3

Use partitioning to calculate mentally:

a) $46+32$
b) $83-27$

Worked Example 4

Use compensating to calculate mentally:

a) $49+36$
b) $82-19$

Worked Example 5

Use doubling/halving to calculate mentally:

a) $25+26$
b) $48+49$

Worked Example 6

Use the addition algorithm:

a) $348+276$
b) $506+189$

Worked Example 7

Use the subtraction algorithm:

a) $754-328$
b) $900-457$

Problems

Problem 1

Use the commutative law to rewrite:

a) $12+8$
b) $41+27$

Problem 2

Use the associative law to make the addition easier:

a) $(9+1)+16$
b) $14+(6+24)$

Problem 3

Use partitioning to calculate mentally:

a) $57+21$
b) $94-35$

Problem 4

Use compensating to calculate mentally:

a) $68+29$
b) $71-18$

Problem 5

Use doubling/halving to calculate mentally:

a) $37+38$
b) $64+63$

Problem 6

Use the addition algorithm:

a) $427+358$
b) $609+274$

Problem 7

Use the subtraction algorithm:

a) $843-256$
b) $1000-468$

Exercises

Understanding and Fluency

1. Rewrite each addition using the commutative law:
   a) $5+14$
   b) $28+13$
   c) $101+9$

2. Use the associative law to add more easily:
   a) $(2+8)+17$
   b) $36+(14+4)$
   c) $(25+75)+19$

3. Use partitioning to calculate mentally:
   a) $43+25$
   b) $67+12$
   c) $88-24$

4. Use compensating to calculate mentally:
   a) $39+45$
   b) $57+19$
   c) $93-29$

5. Use doubling/halving to calculate mentally:
   a) $14+15$
   b) $56+57$
   c) $99+101$

6. Use the addition algorithm:
   a) $264+315$
   b) $708+186$
   c) $495+267$

7. Use the subtraction algorithm:
   a) $682-347$
   b) $800-256$
   c) $1002-478$

8. Choose a sensible mental strategy and calculate:
   a) $48+22$
   b) $61-19$
   c) $75+25$
   d) $87+88$

Reasoning

9. Explain why $18+27$ and $27+18$ have the same answer.

10. Mia says the associative law means $9+6=6+9$. Is she correct? Explain your answer.

11. Which mental strategy is most efficient for $49+18$: partitioning, compensating, or doubling/halving? Explain why.

12. A student solves $503-278$ and writes:

Describe the error.

Problem-solving

13. A shop sold $38$ notebooks in the morning and $27$ in the afternoon. How many notebooks were sold altogether?

14. A stadium had $950$ seats. If $486$ seats were filled, how many seats were empty?

15. A teacher buys $49$ pencils for one class and $51$ pencils for another class. How many pencils are bought altogether? Solve this using a mental strategy.

16. At a fun run, Sam ran $245$ m before lunch and $387$ m after lunch. How far did Sam run altogether?

17. A library had $1000$ stickers. It used $368$ stickers on new books and $247$ on displays. How many stickers were left?

18. Create two addition questions where:
    a) the commutative law is useful
    b) the associative law is useful

Potential Misunderstandings

• The commutative law applies to addition, but students may incorrectly assume it always applies to subtraction
• Students may confuse the commutative law with the associative law
• Students may think the associative law changes the order of numbers, when it actually changes the grouping of numbers
• In partitioning, students may split numbers incorrectly by place value
• In compensating, students may adjust one number but forget to compensate in the answer
• Students may overuse doubling/halving when it is not an efficient strategy
• In written algorithms, students may misalign digits and place values
• In subtraction with regrouping, students may subtract the smaller digit from the larger digit regardless of order
• Students may forget to regroup correctly across zeros
• Students may treat mental strategies and written algorithms as unrelated, rather than as different methods for the same operations