Grade 7 · The Number System · MAT-07-NS-005
Add and Subtract Rational Numbers
Represent and solve add and subtract rational numbers problems using precise mathematical notation, models, and reasoning.
Learning objectives
What you should be able to do
- Represent and solve add and subtract rational numbers problems using precise mathematical notation, models, and reasoning.
- Explain why a method for add and subtract rational numbers works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
The new lesson depends on this prerequisite as active knowledge.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
Mathematical meaning
Add and Subtract Rational Numbers extends arithmetic to signed, fractional, decimal, or coordinate quantities. Operations must preserve both magnitude and direction.
Represent the relationship
The Math Box uses rational number line to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify with an inverse operation, equivalent representation, estimate, or second method so both the calculation and reasoning are auditable.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Rational Number Line
Move through positive and negative halves and compare distance from zero.
- Model one example of add and subtract rational numbers in the Math Box.
- Change one input, predict the effect, and test the prediction.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
Compute -3/4 + 1/4.
-3/4 + 1/4 = -1/2Equal denominators use equal-sized parts, so combine signed numerators.
Compute -2/5 + 3/5.
-2/5 + 3/5 = 1/5Equal denominators use equal-sized parts, so combine signed numerators.
Compute -1/6 + 5/6.
-1/6 + 5/6 = 2/3Equal denominators use equal-sized parts, so combine signed numerators.
Compute 0/7 + 1/7.
0/7 + 1/7 = 1/7Equal denominators use equal-sized parts, so combine signed numerators.
Compute 1/8 + 3/8.
1/8 + 3/8 = 1/2Equal denominators use equal-sized parts, so combine signed numerators.
Compute 2/9 + 5/9.
2/9 + 5/9 = 7/9Equal denominators use equal-sized parts, so combine signed numerators.
Compute 3/10 + 1/10.
3/10 + 1/10 = 2/5Equal denominators use equal-sized parts, so combine signed numerators.
Compute -3/11 + 3/11.
-3/11 + 3/11 = 0/1Equal denominators use equal-sized parts, so combine signed numerators.
Compute -2/4 + 5/4.
-2/4 + 5/4 = 3/4Equal denominators use equal-sized parts, so combine signed numerators.
Compute -1/5 + 1/5.
-1/5 + 1/5 = 0/1Equal denominators use equal-sized parts, so combine signed numerators.
Compute 0/6 + 3/6.
0/6 + 3/6 = 1/2Equal denominators use equal-sized parts, so combine signed numerators.
Compute 1/7 + 5/7.
1/7 + 5/7 = 6/7Equal denominators use equal-sized parts, so combine signed numerators.
Compute 2/8 + 1/8.
2/8 + 1/8 = 3/8Equal denominators use equal-sized parts, so combine signed numerators.
Compute 3/9 + 3/9.
3/9 + 3/9 = 2/3Equal denominators use equal-sized parts, so combine signed numerators.
Compute -3/10 + 5/10.
-3/10 + 5/10 = 1/5Equal denominators use equal-sized parts, so combine signed numerators.
Compute -2/11 + 1/11.
-2/11 + 1/11 = -1/11Equal denominators use equal-sized parts, so combine signed numerators.
Compute -1/4 + 3/4.
-1/4 + 3/4 = 1/2Equal denominators use equal-sized parts, so combine signed numerators.
Compute 0/5 + 5/5.
0/5 + 5/5 = 1/1Equal denominators use equal-sized parts, so combine signed numerators.
Compute 1/6 + 1/6.
1/6 + 1/6 = 1/3Equal denominators use equal-sized parts, so combine signed numerators.
Compute 2/7 + 3/7.
2/7 + 3/7 = 5/7Equal denominators use equal-sized parts, so combine signed numerators.
Practice
Guided practice with hints
01Compute 3/8 + 5/8.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 3/8 + 5/8 = 1/1
02Compute -3/9 + 1/9.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: -3/9 + 1/9 = -2/9
03Compute -2/10 + 3/10.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: -2/10 + 3/10 = 1/10
04Compute -1/11 + 5/11.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: -1/11 + 5/11 = 4/11
05Compute 0/4 + 1/4.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 0/4 + 1/4 = 1/4
06Compute 1/5 + 3/5.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 1/5 + 3/5 = 4/5
Practice
Independent practice
01Compute 2/6 + 5/6.
Answer: 2/6 + 5/6 = 7/6
02Compute 3/7 + 1/7.
Answer: 3/7 + 1/7 = 4/7
03Compute -3/8 + 3/8.
Answer: -3/8 + 3/8 = 0/1
04Compute -2/9 + 5/9.
Answer: -2/9 + 5/9 = 1/3
05Compute -1/10 + 1/10.
Answer: -1/10 + 1/10 = 0/1
06Compute 0/11 + 3/11.
Answer: 0/11 + 3/11 = 3/11
07Compute 1/4 + 5/4.
Answer: 1/4 + 5/4 = 3/2
08Compute 2/5 + 1/5.
Answer: 2/5 + 1/5 = 3/5
Common mistakes
Learn to catch the error, not just the answer.
Use a number line, context, or inverse operation.
For negatives, values closer to zero are greater.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Model temperature, elevation, debt, gains and losses, coordinates, and measurements beyond whole numbers.
- Compute accurately with fractions, decimals, and signed quantities while checking magnitude and sign.
Challenge problems
Compute 3/6 + 3/6.
3/6 + 3/6 = 1/1
Compute -3/7 + 5/7.
-3/7 + 5/7 = 2/7
Compute -2/8 + 1/8.
-2/8 + 1/8 = -1/8
Compute -1/9 + 3/9.
-1/9 + 3/9 = 2/9
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Reasoning Sprint
Solve add and subtract rational numbers problems accurately and justify each result with a valid check.
Practice
Mastery check
01Compute 0/10 + 5/10.
Answer: 0/10 + 5/10 = 1/2
02Compute 1/11 + 1/11.
Answer: 1/11 + 1/11 = 2/11
03Compute 2/4 + 3/4.
Answer: 2/4 + 3/4 = 5/4
04Compute 3/5 + 5/5.
Answer: 3/5 + 5/5 = 8/5
05Compute -3/6 + 1/6.
Answer: -3/6 + 1/6 = -1/3
06Compute -2/7 + 3/7.
Answer: -2/7 + 3/7 = 1/7
07Compute -1/8 + 5/8.
Answer: -1/8 + 5/8 = 1/2
08Compute 0/9 + 1/9.
Answer: 0/9 + 1/9 = 1/9
Terminology
Words to know
- rational number
- A number expressible as a ratio of two integers with nonzero denominator.
- absolute value
- Distance from zero on a number line.
- opposite
- A number the same distance from zero on the other side.
- coordinate
- A number locating a point relative to an axis.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Middle-school instructional guidance, mathematical practices, modeling, and coherent progression.Common Core State Standards for Mathematics — Grade 7
Grade-level expectations for ratios, number systems, algebra, geometry, statistics, and probability.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.