Grade 1 · Operations · MAT-01-OPS-003
Relationship Between Addition and Subtraction
Use part-part-whole models to connect addition and subtraction.
Learning objectives
What you should be able to do
- Use part-part-whole models to connect addition and subtraction.
- Generate related addition and subtraction equations from the same three numbers.
- Use an inverse operation to check a result.
Prerequisite check
Make sure the foundation is ready.
Checks basic facts needed for relationship work.
Checks part-whole structure.
Learn
Build the idea from meaning, not memorization.
Addition and subtraction describe the same whole from different directions
If 3 and 4 are parts of 7, addition joins them: 3+4=7. Subtraction starts with the whole and finds a part: 7−3=4 or 7−4=3.
Three numbers can form a fact family
The numbers 2,5,7 create 2+5=7, 5+2=7, 7−2=5, and 7−5=2. These equations express one part-whole relationship.
Inverse operations can check each other
After solving 8−3=5, add 5+3. Because it returns 8, the subtraction is verified. After solving 4+3=7, subtract 3 from 7 to return 4.
Understanding relationships is more powerful than memorizing isolated facts
A single known relationship can produce several facts and help solve missing-number equations.
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.
- build parts whole
- generate four facts
- check with inverse
- solve unknown
Worked examples
Twenty different ways to see the concept work.
Parts 2 and 3; whole 5.
2+3=5Addition joins parts.
Same relationship, reverse addends.
3+2=5Order changes but whole stays five.
Whole 5 minus part 2.
5−2=3Subtraction finds other part.
Whole 5 minus part 3.
5−3=2The other subtraction fact.
Fact family for 4,1,5.
4+1=5,1+4=5,5−4=1,5−1=4Same three numbers.
Fact family for 3,4,7.
3+4=7,4+3=7,7−3=4,7−4=3One part-whole structure.
Check 6+2=8.
8−2=6Subtraction reverses the addition.
Check 9−4=5.
5+4=9Addition reconstructs the whole.
If 2+6=8, then 8−6=?
2Remove one part to reveal the other.
If 7−3=4, then 4+3=?
7Recombine parts.
Complete 5+__=9.
4Because 9−5=4.
Complete 10−__=6.
4Because 6+4=10.
Parts 5 and 0, whole 5.
5+0=5 and 5−5=0Zero participates in the family.
Double 4+4=8. Related subtraction?
8−4=4Both parts are the same.
Why are there only two unique equations for 4,4,8?
Reversing equal addends changes nothing4+4 and 8−4 repeat themselves.
A whole 10 has parts 7 and 3.
7+3=10 and 10−7=3Addition/subtraction connect.
Use subtraction to solve missing addend 6+__=10.
10−6=4Inverse operation finds missing part.
Use addition to solve 9−__=5.
5+4=9, so missing part 4Reconstruction identifies removed part.
Why does 7−2=5 imply 2+5=7?
Removed part plus remainder equals original wholeThe relationship is reversible.
Explain inverse operations.
They undo or reverse each other's effectAddition can rebuild what subtraction separated.
Practice
Guided practice with hints
01Build fact family for 2,4,6.
Hint: Use two addition and two subtraction equations.
Answer: 2+4=6,4+2=6,6−2=4,6−4=2
02Check 8−3=5.
Hint: Use addition.
Answer: 5+3=8
03Check 4+5=9.
Hint: Use subtraction.
Answer: 9−5=4 or 9−4=5
04Complete 3+__=7.
Hint: Use 7−3.
Answer: 4
05Complete 9−__=2.
Hint: What part with 2 makes 9?
Answer: 7
06Why do fact families help?
Hint: One relationship gives several facts.
Answer: They reduce isolated memorization.
Practice
Independent practice
01Fact family for 1,6,7
Answer: 1+6=7,6+1=7,7−1=6,7−6=1
02Check 9−2=7
Answer: 7+2=9
03Check 3+4=7
Answer: 7−3=4
045+__=8
Answer: 3
058−__=5
Answer: 3
06If 4+2=6, then 6−4
Answer: 2
07If 10−3=7, then 7+3
Answer: 10
08What makes addition/subtraction inverse?
Answer: They can undo each other.
Common mistakes
Learn to catch the error, not just the answer.
A fact family uses the same three values.
Identify the whole first; subtraction starts from the whole.
Use the inverse operation to verify.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Use a known total and one category count to find the missing category count.
- Check change-in-quantity stories by recombining removed and remaining parts.
Challenge problems
Create fact family for 3,7,10.
3+7=10,7+3=10,10−3=7,10−7=3
Why does a fact family with equal parts have fewer unique equations?
Reversing equal addends or subtracting either equal part produces the same equation.
Use inverse reasoning to solve __+6=10.
10−6=4.
Use inverse reasoning to solve 10−__=8.
8+2=10, so 2.
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Fact Family House
Complete all valid addition and subtraction equations for a part-part-whole set.
Practice
Mastery check
01Give fact family for 2,3,5.
Answer: 2+3=5,3+2=5,5−2=3,5−3=2
02Check 7−4=3.
Answer: 3+4=7
03Check 6+3=9.
Answer: 9−3=6
04Solve 4+__=9.
Answer: 5
05Solve 8−__=3.
Answer: 5
06What is the whole in 2,5,7?
Answer: 7
07What does inverse mean here?
Answer: One operation can undo the other.
08Why are fact families useful?
Answer: They connect several facts through one part-whole relationship.
Terminology
Words to know
- inverse operations
- Operations that undo each other.
- fact family
- Related equations built from the same part-whole relationship.
- whole
- The complete quantity formed from parts.
- verify
- Check a result using reasoning or another operation.
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.