003. Mental Strategies and Laws for Addition and Subtraction

Learning Intentions

• To understand the commutative and associative laws for addition.
• use the mental strategies of partitioning, compensating, and doubling/halving to calculate a sum or difference of positive integers mentally.

Pre-requisite Summary

• Commutative and Associative Laws: Familiarity with basic addition facts and the concept that the order of numbers doesn't change the total in simple sums (e.g., $3+2$ is the same as $2+3$).
• Mental Strategies: A strong understanding of place value (identifying tens and ones) and the ability to recognize numbers close to multiples of ten (e.g., recognizing 19 is close to 20).

Worked Examples

Worked Example 1

a) Use the commutative law to rewrite $14+27$ to make it easier to solve.

b) Use the associative law to group numbers efficiently: $13+18+7$.

Worked Example 2

Use the partitioning strategy to calculate the following:

a) $42+35$

b) $87-34$

Worked Example 3

Use the compensating strategy to calculate the following:

a) $56+19$

b) $124-38$

Worked Example 4

Use doubling or halving strategies to calculate the following:

a) $35+36$

b) $160-80$

Problems

Problem 1

a) Use the commutative law to rewrite $12+39$ to make it easier to solve.

b) Use the associative law to group numbers efficiently: $15+26+5$.

Problem 2

Use the partitioning strategy to calculate the following:

a) $53+24$

b) $76-42$

Problem 3

Use the compensating strategy to calculate the following:

a) $47+29$

b) $145-28$

Problem 4

Use doubling or halving strategies to calculate the following:

a) $45+46$

b) $180-90$

Exercises

Understanding and Fluency

1. Apply the commutative law to swap the order:
   a) $17+82$
   b) $9+123$
2. Group the numbers using the associative law to solve:
   a) $22+14+8$
   b) $35+17+5$
3. Use partitioning to solve the following:
   a) $61+27$
   b) $44+55$
   c) $95-32$
4. Use the compensating strategy (round to the nearest ten):
   a) $36+39$
   b) $82-19$
   c) $155+48$
5. Use doubling or "near doubles" to solve:
   a) $25+26$
   b) $70+70$
   c) $14+13$

Reasoning

6. Explain why the commutative law works for addition ($a+b=b+a$) but does not work for subtraction ($a-b\ne b-a$). Provide an example.
7. A student calculates $45-19$ by doing $45-20=25$, then $25-1=24$. Identify the error in their compensating strategy and explain the correct adjustment.
8. Show how the associative law can be used to solve $16+25$ by breaking 16 into $(11+5)$.

Problem-solving

9. You buy two items at the canteen costing $\mathrm{\$}4.50$ and $\mathrm{\$}4.60$. Which mental strategy (partitioning, compensating, or doubling) is most efficient for finding the total? Show your working.
10. A cricket team scores 142 runs in their first innings and 198 runs in their second. Use the compensating strategy to find the total runs scored.
11. Sarah has 154 stamps. She gives 35 to her brother. Use partitioning to find how many stamps she has left.
12. A bus starts with 48 passengers. At the first stop, 12 get off and 15 get on. Use mental strategies to find the final number of passengers and explain which laws or strategies you used.

Potential Misunderstandings

• Subtraction Direction: Students may mistakenly believe the commutative law applies to subtraction (e.g., thinking $10-2$ is the same as $2-10$).
• Compensating Adjustment: When using compensation for addition (e.g., $+19$), students often forget whether to add or subtract the "adjustment" value at the end.
• Partitioning Subtraction: When partitioning in subtraction (e.g., $52-25$), students may subtract the smaller unit from the larger unit ($5-2$) rather than dealing with the negative result or regrouping mentally.

Next: 004. Written Algorithms for Addition and Subtraction