Introduction
Multiplying Integers and Rational Numbers is an important Grade 7 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with multiplying integers and rational numbers.
What Is Multiplying Integers and Rational Numbers?
Multiplying Integers and Rational Numbers means understanding equal groups, arrays, and repeated addition as multiplication.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Multiplying Integers and Rational Numbers
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Name the equal groups before choosing an operation.
- Use arrays, repeated addition, or related facts to explain the work.
- Connect multiplication and division as inverse operations.
- Check that the answer fits the story problem.
Visual Models
Visual Model 1
Question: The number line shows repeated additions. What does \(3\times (-2)\) equal?
- A. \(-8\)
- B. \(-6\)
- C. \(6\)
- D. \(8\)
Why it works: Three groups of \(-2\): \(3 \times (-2) = -6\). The arrows move left 2 units, 3 times, landing on \(-6\).
Answer: \(-6\)
Visual Model 2
Question: Which row correctly shows the sign rules for multiplication?
| Operation | Example | Result |
|---|---|---|
| A. \((+)\times(+)\) | \(5\times 3\) | \(8\) |
| B. \((-)\times(-)\) | \((-4)\times(-2)\) | \(-8\) |
| C. \((+)\times(-)\) | \(6\times(-2)\) | \(-12\) |
| D. \((-)\times(+)\) | \((-7)\times 3\) | \(21\) |
- A. Row A
- B. Row B
- C. Row C
- D. Row D
Why it works: Row C correctly demonstrates \((+)\times(-) = -12\): \(6\times(-2) = -12\) ✓. Row A error: \(5\times 3 = 15\) (not \(8\)). Row B error: \((-4)\times(-2) = 8\) (positive, not \(-8\)). Row D error: \((-7)\times 3 = -21\) (negative, not \(21\)).
Answer: Row C
Worked Examples
Example 1
Question: The image shows repeated groups of chips. How many chips in total? What multiplication does this represent?
- A. \(4\times 1 = 4\)
- B. \(4\times 2 = 8\)
- C. \(1\times 4 = 4\)
- D. \(2\times 4 = 8\)
- The diagram shows 4 groups of 1 chip each, totaling 4 chips.
- This represents \(4\times 1 = 4\).
Answer: \(4\times 1 = 4\)
Example 2
Question: What is \(-6\times\frac{2}{3}\)?
- A. \(-9\)
- B. \(-4\)
- C. \(4\)
- D. \(9\)
- \(-6\times\frac{2}{3}=\frac{-12}{3}=-4\).
- A negative times a positive is negative.
Answer: \(-4\)
Example 3
Question: What is \(5\times 4\)?
- A. \(9\)
- B. \(18\)
- C. \(20\)
- D. \(25\)
- Positive times positive equals positive. \(5 \times 4 = 20\).
Answer: \(20\)
Real-World Word Problems
Problem 1
Question: Temperature drops 2 degrees per hour for 6 hours. What is the total change?
- A. \(4\) degrees
- B. \(-4\) degrees
- C. \(-12\) degrees
- D. \(12\) degrees
Why it works: Drop of 2 degrees per hour for 6 hours: \((-2) \times 6 = -12\) degrees.
Answer: \(-12\) degrees
Problem 2
Question: Each burger costs $4.50. How much will 6 burgers cost?
- A. $24.00
- B. $25.50
- C. $27.00
- D. $30.00
Why it works: \(6 \times 4.50 = 27.00\). Each burger is $4.50, so 6 burgers cost $27.00.
Answer: $27.00
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
What is \(-7\times (-3)\)?
- A. \(-21\)
- B. \(-10\)
- C. \(10\)
- D. \(21\)
Question 2
What is \(3\times 0.5\)?
- A. \(0.5\)
- B. \(1.5\)
- C. \(3.5\)
- D. \(6\)
Question 3
What is \(-8\times 2\)?
- A. \(-16\)
- B. \(-6\)
- C. \(6\)
- D. \(16\)
Question 4
What is \(\frac{3}{4}\times 8\)?
- A. \(6\)
- B. \(8\)
- C. \(10\)
- D. \(12\)
Question 5
What is \(-\frac{1}{2}\times 10\)?
- A. \(-20\)
- B. \(-5\)
- C. \(5\)
- D. \(20\)
Question 6
What is \(0\times (-9)\)?
- A. \(-9\)
- B. \(0\)
- C. \(9\)
- D. \(-1\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(21\)
Negative times negative equals positive. \(-7 \times (-3) = 21\).
Question 2
Answer: \(1.5\)
\(3 \times 0.5 = 1.5\). Positive times positive is positive.
Question 3
Answer: \(-16\)
Negative times positive equals negative. \(-8 \times 2 = -16\).
Question 4
Answer: \(6\)
\(\frac{3}{4}\times 8 = \frac{3 \times 8}{4} = \frac{24}{4} = 6\).
Question 5
Answer: \(-5\)
\(-\frac{1}{2}\times 10 = \frac{-10}{2} = -5\). Negative times positive is negative.
Question 6
Answer: \(0\)
Any number multiplied by zero equals zero.
Connection to Standards
This lesson supports Grade 7 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Multiplying Integers and Rational Numbers becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Equal groups make multiplication make sense.
