Grade 2 · Operations · MAT-02-OPS-010
Unknowns in All Positions
Solve addition and subtraction equations with the unknown in varied positions.
Learning objectives
What you should be able to do
- Solve addition and subtraction equations with the unknown in varied positions.
- Use inverse operations and part-whole reasoning.
- Explain why the equals sign represents a relationship, not a signal to calculate only on the left.
Prerequisite check
Make sure the foundation is ready.
Checks conceptual readiness for symbolic unknown placement.
Learn
Build the idea from meaning, not memorization.
The unknown can appear anywhere in a relationship
Equations can ask for a result, a missing part, or a starting amount. □+7=15, 8+□=15, and15−□=8 all describe related quantities.
Use part-whole structure
In addition, two parts combine to make a whole. If the whole and one part are known, subtraction can find the missing part.
Inverse operations uncover unknowns
To solve □+9=16, subtract9 from16 or ask9+what=16. To solve □−5=12, rebuild the start with12+5=17.
Equality means both sides have the same value
The equals sign means 'has the same value as.' It can appear in equations such as15=8+7 just as correctly as8+7=15.
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.
- missing addend
- missing subtrahend
- missing minuend
- equality both directions
Worked examples
Twenty different ways to see the concept work.
□+7=15
815−7=8.
8+□=15
715−8=7.
15−□=8
78+7=15.
□−7=8
158+7 rebuilds start.
9+6=□
15Result unknown.
□+12=30
1830−12.
14+□=25
1125−14.
40−□=17
2317+23=40.
□−18=22
4022+18.
□=35−9
26Right side evaluates to26.
27=□+8
1927−8.
50=63−□
1363−50.
□+0=24
24Zero changes nothing.
□−0=31
31Subtracting zero changes nothing.
20−□=20
0Remove nothing.
□+□ not allowed here; one unknown symbol represents same number?
It could, but requires a new equation type beyond this lessonClarify symbol meaning.
Check □+7=15 with8.
8+7=15Substitution verifies.
Check □−5=12 with17.
17−5=12Substitution verifies.
Why is15=8+7 true?
Both sides equal15Equality is symmetric.
Story: after adding6, total20. Start?
14+6=20Unknown start maps to□+6=20.
Practice
Guided practice with hints
01□+9=17
Hint: Subtract9 from17.
Answer: 8
0212+□=20
Hint: Find missing part.
Answer: 8
0325−□=11
Hint: What plus11 gives25?
Answer: 14
04□−8=13
Hint: Rebuild the start.
Answer: 21
0530=□+12
Hint: Use equality both ways.
Answer: 18
06Why can the unknown be on the left?
Hint: Equality is a relationship.
Answer: Both sides simply must match in value.
Practice
Independent practice
01□+6=14
Answer: 8
029+□=18
Answer: 9
0322−□=15
Answer: 7
04□−9=12
Answer: 21
0525=□+7
Answer: 18
06□=40−16
Answer: 24
0730−□=30
Answer: 0
08Explain '='.
Answer: Both sides have the same value.
Common mistakes
Learn to catch the error, not just the answer.
Identify whether the unknown is a part, whole, start, or removed amount.
Read it as 'has the same value as.'
Unknown position does not change equality.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Represent missing starting amounts, changes, or totals in real situations.
- Use unknown equations as early algebraic reasoning.
Challenge problems
Solve□−14=29.
43
Solve52−□=18.
34
Why do□+7=15 and15−7=□ describe the same relationship?
They use inverse operations on the same part-whole structure.
Write a story for□−6=20.
Any story with unknown start, remove6, remain20.
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 6
Game mode
Mystery Box Equations
Find the unknown regardless of where the box appears.
Practice
Mastery check
01□+8=21
Answer: 13
0215+□=27
Answer: 12
0332−□=18
Answer: 14
04□−7=19
Answer: 26
0540=□+16
Answer: 24
06□=50−23
Answer: 27
07What does '=' mean?
Answer: Both sides have equal value.
08How do you check an unknown?
Answer: Substitute it into the original equation.
Terminology
Words to know
- unknown
- A value that has not yet been determined.
- equation
- A statement that two expressions have equal value.
- inverse operation
- An operation that undoes another.
- substitute
- Replace an unknown symbol with a proposed value to check it.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Common Core State Standards for Mathematics — 2.OA.A.1
Unknowns in addition/subtraction problem positions.