Grade 7 · The Number System · MAT-07-NS-008
Solve Multi-Step Rational-Number Problems
Represent and solve solve multi-step rational-number problems problems using precise mathematical notation, models, and reasoning.
Learning objectives
What you should be able to do
- Represent and solve solve multi-step rational-number problems problems using precise mathematical notation, models, and reasoning.
- Explain why a method for solve multi-step rational-number problems 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
Solve Multi-Step Rational-Number Problems 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 solve multi step rational number problems 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.
Evaluate -10 + 3 × 2.
3 × 2 = 6; -10 + 6 = -4Use multiplication before addition.
Evaluate -9 + 4 × 3.
4 × 3 = 12; -9 + 12 = 3Use multiplication before addition.
Evaluate -8 + 5 × 4.
5 × 4 = 20; -8 + 20 = 12Use multiplication before addition.
Evaluate -7 + 6 × 5.
6 × 5 = 30; -7 + 30 = 23Use multiplication before addition.
Evaluate -6 + 7 × 6.
7 × 6 = 42; -6 + 42 = 36Use multiplication before addition.
Evaluate -5 + 8 × 2.
8 × 2 = 16; -5 + 16 = 11Use multiplication before addition.
Evaluate -4 + 9 × 3.
9 × 3 = 27; -4 + 27 = 23Use multiplication before addition.
Evaluate -3 + 3 × 4.
3 × 4 = 12; -3 + 12 = 9Use multiplication before addition.
Evaluate -2 + 4 × 5.
4 × 5 = 20; -2 + 20 = 18Use multiplication before addition.
Evaluate -1 + 5 × 6.
5 × 6 = 30; -1 + 30 = 29Use multiplication before addition.
Evaluate 0 + 6 × 2.
6 × 2 = 12; 0 + 12 = 12Use multiplication before addition.
Evaluate 1 + 7 × 3.
7 × 3 = 21; 1 + 21 = 22Use multiplication before addition.
Evaluate 2 + 8 × 4.
8 × 4 = 32; 2 + 32 = 34Use multiplication before addition.
Evaluate 3 + 9 × 5.
9 × 5 = 45; 3 + 45 = 48Use multiplication before addition.
Evaluate 4 + 3 × 6.
3 × 6 = 18; 4 + 18 = 22Use multiplication before addition.
Evaluate 5 + 4 × 2.
4 × 2 = 8; 5 + 8 = 13Use multiplication before addition.
Evaluate 6 + 5 × 3.
5 × 3 = 15; 6 + 15 = 21Use multiplication before addition.
Evaluate 7 + 6 × 4.
6 × 4 = 24; 7 + 24 = 31Use multiplication before addition.
Evaluate 8 + 7 × 5.
7 × 5 = 35; 8 + 35 = 43Use multiplication before addition.
Evaluate 9 + 8 × 6.
8 × 6 = 48; 9 + 48 = 57Use multiplication before addition.
Practice
Guided practice with hints
01Evaluate 10 + 9 × 2.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 9 × 2 = 18; 10 + 18 = 28
02Evaluate 11 + 3 × 3.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 3 × 3 = 9; 11 + 9 = 20
03Evaluate 12 + 4 × 4.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 4 × 4 = 16; 12 + 16 = 28
04Evaluate 13 + 5 × 5.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 5 × 5 = 25; 13 + 25 = 38
05Evaluate 14 + 6 × 6.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 6 × 6 = 36; 14 + 36 = 50
06Evaluate 15 + 7 × 2.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 7 × 2 = 14; 15 + 14 = 29
Practice
Independent practice
01Evaluate 16 + 8 × 3.
Answer: 8 × 3 = 24; 16 + 24 = 40
02Evaluate 17 + 9 × 4.
Answer: 9 × 4 = 36; 17 + 36 = 53
03Evaluate 18 + 3 × 5.
Answer: 3 × 5 = 15; 18 + 15 = 33
04Evaluate 19 + 4 × 6.
Answer: 4 × 6 = 24; 19 + 24 = 43
05Evaluate 20 + 5 × 2.
Answer: 5 × 2 = 10; 20 + 10 = 30
06Evaluate 21 + 6 × 3.
Answer: 6 × 3 = 18; 21 + 18 = 39
07Evaluate 22 + 7 × 4.
Answer: 7 × 4 = 28; 22 + 28 = 50
08Evaluate 23 + 8 × 5.
Answer: 8 × 5 = 40; 23 + 40 = 63
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
Evaluate 24 + 9 × 6.
9 × 6 = 54; 24 + 54 = 78
Evaluate 25 + 3 × 2.
3 × 2 = 6; 25 + 6 = 31
Evaluate 26 + 4 × 3.
4 × 3 = 12; 26 + 12 = 38
Evaluate 27 + 5 × 4.
5 × 4 = 20; 27 + 20 = 47
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Reasoning Sprint
Solve solve multi-step rational-number problems problems accurately and justify each result with a valid check.
Practice
Mastery check
01Evaluate 28 + 6 × 5.
Answer: 6 × 5 = 30; 28 + 30 = 58
02Evaluate 29 + 7 × 6.
Answer: 7 × 6 = 42; 29 + 42 = 71
03Evaluate 30 + 8 × 2.
Answer: 8 × 2 = 16; 30 + 16 = 46
04Evaluate 31 + 9 × 3.
Answer: 9 × 3 = 27; 31 + 27 = 58
05Evaluate 32 + 3 × 4.
Answer: 3 × 4 = 12; 32 + 12 = 44
06Evaluate 33 + 4 × 5.
Answer: 4 × 5 = 20; 33 + 20 = 53
07Evaluate 34 + 5 × 6.
Answer: 5 × 6 = 30; 34 + 30 = 64
08Evaluate 35 + 6 × 2.
Answer: 6 × 2 = 12; 35 + 12 = 47
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.