Grade 1 · Operations · MAT-01-OPS-005
Associative Grouping for Addition
Regroup three addends without changing their sum.
Learning objectives
What you should be able to do
- Regroup three addends without changing their sum.
- Use grouping to make easier combinations such as doubles or make-10 pairs.
- Explain that grouping changes which numbers are combined first, not the addend values or final sum.
Prerequisite check
Make sure the foundation is ready.
Checks equivalent-addition reasoning.
Checks strategies used when regrouping.
Learn
Build the idea from meaning, not memorization.
Three addends can be grouped in different ways
For 2+3+4, you can first combine 2+3 to get 5, then 5+4=9. Or first combine 3+4 to get 7, then 2+7=9. The total stays the same.
Grouping can make mental math easier
In 6+4+2, combining 6+4 first makes 10, then 10+2=12. Strategic grouping reduces work.
Grouping is different from changing order
Commutative property changes order; associative property changes which addends are grouped first. In practice they can work together, but the ideas are distinct.
Parentheses can show grouping
(2+3)+4 means combine 2 and 3 first. 2+(3+4) means combine 3 and 4 first. Grade 1 learners can reason with grouping even before formal algebra.
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.
- drag parentheses
- make ten first
- double first
- compare groupings
Worked examples
Twenty different ways to see the concept work.
(1+2)+3
61+2=3, then 3+3=6.
1+(2+3)
62+3=5, then 1+5=6.
(2+3)+4
95+4=9.
2+(3+4)
92+7=9.
(4+4)+1
9Use double 4+4=8, then one more.
4+(4+1)
94+5=9; same total.
(6+4)+2
12Make ten first.
6+(4+2)
124+2=6, then 6+6=12.
(5+5)+3
13Double five makes ten, then add three.
5+(5+3)
135+8=13.
Which grouping is efficient for 7+3+2?
(7+3)+2Make ten first.
Which grouping is efficient for 4+4+2?
(4+4)+2Use a known double.
Does regrouping change addend values?
NoOnly which pair is combined first.
Does regrouping change total?
NoAssociative property preserves sum.
Compare (2+5)+3 and 2+(5+3).
Both 10Different grouping, same addends.
Use grouping for 1+9+4.
(1+9)+4=14Make ten first.
Use grouping for 3+2+3.
3+(2+3)=8 or (3+3)+2 after commutationChoose an easy pair.
Explain parentheses.
They show which addition is done firstGrouping is visible.
Can subtraction be regrouped the same way?
Not generallyAssociative property is not a general subtraction rule.
State associative idea.
Changing grouping of addends does not change sumSame numbers, different first pair.
Practice
Guided practice with hints
01Choose grouping for 8+2+3.
Hint: Find a make-10 pair.
Answer: (8+2)+3
02Choose grouping for 5+5+2.
Hint: Find a double/make-ten.
Answer: (5+5)+2
03Compute 2+(3+5).
Hint: Do parentheses first.
Answer: 10
04Compute (2+3)+5.
Hint: Do parentheses first.
Answer: 10
05What changes in associative property?
Hint: Which addends are combined first.
Answer: Grouping.
06What stays same?
Hint: Addend values and total.
Answer: The sum.
Practice
Independent practice
01(1+4)+5
Answer: 10
021+(4+5)
Answer: 10
03(6+4)+3
Answer: 13
046+(4+3)
Answer: 13
05Best grouping for 7+3+1
Answer: (7+3)+1
06Best grouping for 4+4+2
Answer: (4+4)+2
07Does regrouping change sum?
Answer: No
08What do parentheses show?
Answer: Which part is grouped/combined first.
Common mistakes
Learn to catch the error, not just the answer.
Keep all addend values exactly the same.
Order is commutative; grouping is associative.
Test simple subtraction to see regrouping changes results.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Group game scores to make easy totals.
- Combine small purchase or object counts by making tens first.
Challenge problems
Find easiest grouping for 9+1+6.
(9+1)+6=16
Why are (3+4)+5 and 3+(4+5) equal?
They contain same addends and addition is associative.
Can commutative and associative properties be used together?
Yes, to reorder and regroup addends for easier computation.
Show subtraction counterexample to associativity.
For example (8−3)−2=3, but 8−(3−2)=7.
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Group Smart
Choose the most useful first pair among three addends, then solve and compare another grouping.
Practice
Mastery check
01(2+3)+4
Answer: 9
022+(3+4)
Answer: 9
03Best grouping for 8+2+5
Answer: (8+2)+5
04What changes?
Answer: Grouping
05What stays same?
Answer: Addends and sum
06Do parentheses show order of grouping?
Answer: Yes
07Does associative property apply generally to subtraction?
Answer: No
08State the property.
Answer: Changing grouping of addends does not change sum.
Terminology
Words to know
- associative property
- A property stating that changing how addends are grouped does not change the sum.
- grouping
- Choosing which quantities to combine first.
- parentheses
- Symbols used to show grouping.
- strategy
- A planned method chosen to solve a problem.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Common Core State Standards for Mathematics — Grade 1 Operations & Algebraic Thinking
Grade 1 addition/subtraction fluency, properties, and strategy progression.