Grade 7 · The Number System · MAT-07-NS-006
Multiply and Divide Rational Numbers
Represent and solve multiply and divide rational numbers problems using precise mathematical notation, models, and reasoning.
Learning objectives
What you should be able to do
- Represent and solve multiply and divide rational numbers problems using precise mathematical notation, models, and reasoning.
- Explain why a method for multiply and divide 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
Multiply and Divide 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 multiply and divide 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.
Multiply -1/2 × 3/6.
-1/2 × 3/6 = -1/4Multiply numerators and denominators, track the sign, then simplify.
Multiply 2/3 × 4/7.
2/3 × 4/7 = 8/21Multiply numerators and denominators, track the sign, then simplify.
Multiply -3/4 × 5/8.
-3/4 × 5/8 = -15/32Multiply numerators and denominators, track the sign, then simplify.
Multiply 4/5 × 1/9.
4/5 × 1/9 = 4/45Multiply numerators and denominators, track the sign, then simplify.
Multiply -5/6 × 2/3.
-5/6 × 2/3 = -5/9Multiply numerators and denominators, track the sign, then simplify.
Multiply 1/7 × 3/4.
1/7 × 3/4 = 3/28Multiply numerators and denominators, track the sign, then simplify.
Multiply -2/8 × 4/5.
-2/8 × 4/5 = -1/5Multiply numerators and denominators, track the sign, then simplify.
Multiply 3/2 × 5/6.
3/2 × 5/6 = 5/4Multiply numerators and denominators, track the sign, then simplify.
Multiply -4/3 × 1/7.
-4/3 × 1/7 = -4/21Multiply numerators and denominators, track the sign, then simplify.
Multiply 5/4 × 2/8.
5/4 × 2/8 = 5/16Multiply numerators and denominators, track the sign, then simplify.
Multiply -1/5 × 3/9.
-1/5 × 3/9 = -1/15Multiply numerators and denominators, track the sign, then simplify.
Multiply 2/6 × 4/3.
2/6 × 4/3 = 4/9Multiply numerators and denominators, track the sign, then simplify.
Multiply -3/7 × 5/4.
-3/7 × 5/4 = -15/28Multiply numerators and denominators, track the sign, then simplify.
Multiply 4/8 × 1/5.
4/8 × 1/5 = 1/10Multiply numerators and denominators, track the sign, then simplify.
Multiply -5/2 × 2/6.
-5/2 × 2/6 = -5/6Multiply numerators and denominators, track the sign, then simplify.
Multiply 1/3 × 3/7.
1/3 × 3/7 = 1/7Multiply numerators and denominators, track the sign, then simplify.
Multiply -2/4 × 4/8.
-2/4 × 4/8 = -1/4Multiply numerators and denominators, track the sign, then simplify.
Multiply 3/5 × 5/9.
3/5 × 5/9 = 1/3Multiply numerators and denominators, track the sign, then simplify.
Multiply -4/6 × 1/3.
-4/6 × 1/3 = -2/9Multiply numerators and denominators, track the sign, then simplify.
Multiply 5/7 × 2/4.
5/7 × 2/4 = 5/14Multiply numerators and denominators, track the sign, then simplify.
Practice
Guided practice with hints
01Multiply -1/8 × 3/5.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: -1/8 × 3/5 = -3/40
02Multiply 2/2 × 4/6.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 2/2 × 4/6 = 2/3
03Multiply -3/3 × 5/7.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: -3/3 × 5/7 = -5/7
04Multiply 4/4 × 1/8.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 4/4 × 1/8 = 1/8
05Multiply -5/5 × 2/9.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: -5/5 × 2/9 = -2/9
06Multiply 1/6 × 3/3.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 1/6 × 3/3 = 1/6
Practice
Independent practice
01Multiply -2/7 × 4/4.
Answer: -2/7 × 4/4 = -2/7
02Multiply 3/8 × 5/5.
Answer: 3/8 × 5/5 = 3/8
03Multiply -4/2 × 1/6.
Answer: -4/2 × 1/6 = -1/3
04Multiply 5/3 × 2/7.
Answer: 5/3 × 2/7 = 10/21
05Multiply -1/4 × 3/8.
Answer: -1/4 × 3/8 = -3/32
06Multiply 2/5 × 4/9.
Answer: 2/5 × 4/9 = 8/45
07Multiply -3/6 × 5/3.
Answer: -3/6 × 5/3 = -5/6
08Multiply 4/7 × 1/4.
Answer: 4/7 × 1/4 = 1/7
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
Multiply -5/8 × 2/5.
-5/8 × 2/5 = -1/4
Multiply 1/2 × 3/6.
1/2 × 3/6 = 1/4
Multiply -2/3 × 4/7.
-2/3 × 4/7 = -8/21
Multiply 3/4 × 5/8.
3/4 × 5/8 = 15/32
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Reasoning Sprint
Solve multiply and divide rational numbers problems accurately and justify each result with a valid check.
Practice
Mastery check
01Multiply -4/5 × 1/9.
Answer: -4/5 × 1/9 = -4/45
02Multiply 5/6 × 2/3.
Answer: 5/6 × 2/3 = 5/9
03Multiply -1/7 × 3/4.
Answer: -1/7 × 3/4 = -3/28
04Multiply 2/8 × 4/5.
Answer: 2/8 × 4/5 = 1/5
05Multiply -3/2 × 5/6.
Answer: -3/2 × 5/6 = -5/4
06Multiply 4/3 × 1/7.
Answer: 4/3 × 1/7 = 4/21
07Multiply -5/4 × 2/8.
Answer: -5/4 × 2/8 = -5/16
08Multiply 1/5 × 3/9.
Answer: 1/5 × 3/9 = 1/15
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.