Grade 2 · Operations · MAT-02-OPS-013
Use Properties to Explain Addition Strategies
Use commutative and associative reasoning to explain addition strategies.
Learning objectives
What you should be able to do
- Use commutative and associative reasoning to explain addition strategies.
- Decompose and recombine addends without changing the sum.
- Distinguish valid properties from changes that alter total value.
Prerequisite check
Make sure the foundation is ready.
Checks strategy knowledge that properties will justify.
Learn
Build the idea from meaning, not memorization.
Properties explain why strategies are valid
A strategy is more than a trick. Mathematical properties show why numbers can be reordered or regrouped in addition without changing the total.
Commutative property changes order
7+5 and5+7 have the same sum. Reversing addends does not change how many objects are present altogether.
Associative property changes grouping
For3+7+4, grouping3+7 first makes10, then10+4=14. Grouping7+4 first also reaches14. The grouping can change while addend order remains.
Decomposition works when total quantity is preserved
8+7 can become10+5 by moving2 from7 to8. The individual addends change, but their combined total remains15.
Interactive math lab
Change it. See what stays true.
Equation Balance
Adjust both sides. Equality means both expressions have the same value.
Balanced: both sides have equal value.
- swap addends
- regroup three addends
- compensate to ten
- spot invalid equivalence
Worked examples
Twenty different ways to see the concept work.
4+9 vs9+4.
both13Commutative property.
3+7+4 group3+7.
10+4=14Associative grouping.
5+5+2 group5+5.
10+2=12Use friendly number.
8+7 become10+5.
15Move2 while preserving total.
9+6 become10+5.
15Decompose6 as1+5.
2+8+6 group2+8.
16Make10 first.
6+4+7 group6+4.
17Associative.
12+5 vs5+12.
both17Commutative.
20+30 vs30+20.
both50Property applies to larger addends.
25+14 =25+10+4.
39Decompose14.
38+27 =38+2+25.
65Make40.
47+16 =47+3+13.
63Make50.
Is8+7=9+7 valid by commutativity?
NoOnly one addend changed; total increases1.
Is8+7=7+8 valid?
YesOnly order changed.
Is(2+3)+4=2+(3+4)?
YesAssociative property.
Does subtraction commute?
No8−3≠3−8 in whole-number context.
Why does moving2 from7 to8 preserve sum?
One addend gains2 while other loses2Net total unchanged.
Choose easier:6+9 or9+6.
Either;9+6 may cue make10Commutativity permits swap.
Explain25+37 as30+32.
Move5 from37 to25Total62 preserved.
Why properties matter?
They justify flexible equivalent strategiesReasoning replaces memorized tricks.
Practice
Guided practice with hints
01Rewrite7+9 using commutativity.
Hint: Swap addends.
Answer: 9+7
02Group2+8+5 efficiently.
Hint: Make10 first.
Answer: (2+8)+5=15
03Rewrite8+6 as10+4.
Hint: Move2 from6 to8.
Answer: 14
04Is5+7=7+5 valid?
Hint: Name property.
Answer: Yes, commutative
05Is(4+6)+3=4+(6+3)?
Hint: Name property.
Answer: Yes, associative
06Why not use commutative for subtraction?
Hint: Order changes result.
Answer: Subtraction is not commutative.
Practice
Independent practice
016+8 reversed
Answer: 8+6
02Group3+7+2 efficiently
Answer: (3+7)+2=12
039+5 as10+?
Answer: 10+4
04Is4+9=9+4?
Answer: Yes
05Is8−3=3−8?
Answer: No
0625+18 as30+?
Answer: 30+13=43
07Name order property
Answer: Commutative
08Name grouping property
Answer: Associative
Common mistakes
Learn to catch the error, not just the answer.
If one addend increases, decrease another by the same amount to preserve sum.
Commutativity applies to addition here, not subtraction.
Commutative changes order; associative changes grouping.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Rearrange mental addition to make tens.
- Break larger addends into friendlier place-value chunks.
Challenge problems
Explain38+27 as40+25.
Move2 from27 to38; sum stays65.
Find a friendly regrouping for19+26.
20+25=45.
Why is associative useful with three addends?
It lets you group convenient pairs first.
Create an invalid 'strategy' and explain why it changes total.
Any example that changes quantity without compensation.
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 6
Game mode
Strategy Switch
Transform addition expressions into equivalent easier forms and name the property used.
Practice
Mastery check
01Reverse5+9
Answer: 9+5
02Group4+6+7 efficiently
Answer: (4+6)+7=17
03Rewrite8+7 as10+5
Answer: Move2 from7 to8.
04Name order property
Answer: Commutative
05Name grouping property
Answer: Associative
06Does subtraction commute?
Answer: No
07Why does25+18=30+13?
Answer: 5 moved from18 to25, preserving total.
08Why use properties?
Answer: They justify flexible equivalent strategies.
Terminology
Words to know
- commutative property
- Changing addend order does not change the sum.
- associative property
- Changing grouping of addends does not change the sum.
- decompose
- Break a number into useful parts.
- equivalent
- Having the same mathematical value.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Common Core State Standards for Mathematics — Grade 2 Operations and Algebraic Thinking / Mathematical Practice
Properties and strategy explanation supporting fluency.