Introduction
Dividing 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 dividing integers and rational numbers.
What Is Dividing Integers and Rational Numbers?
Dividing Integers and Rational Numbers means using place value, operations, and equations to reason accurately with numbers.
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 Dividing Integers and Rational Numbers
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
Visual Model 1
Question: Use the sign-rule chart below to answer: Is \((-15)\div3\) positive or negative?
| Dividend sign | Divisor sign | Quotient sign |
|---|---|---|
| \(+\) | \(+\) | \(+\) |
| \(+\) | \(-\) | \(-\) |
| \(-\) | \(+\) | \(-\) |
| \(-\) | \(-\) | \(+\) |
- A. Positive; \((-15)\div3=5\)
- B. Negative; \((-15)\div3=-5\)
- C. Positive; \((-15)\div3=-5\)
- D. Zero; quotient is undefined
Why it works: Using the chart: negative \(\div\) positive \(=\) negative. So \((-15)\div3=-5\).
Answer: Negative
Visual Model 2
Question: On a number line, if \(x\div2=-3\), what is \(x\)?
- A. \(-6\)
- B. \(-3\)
- C. \(3\)
- D. \(6\)
Why it works: Division as inverse: \(x\div2=-3\) means \(x=-3\times2=-6\). The dot on the number line confirms this.
Answer: \(-6\)
Worked Examples
Example 1
Question: Reciprocal partitioning: To divide by \(\frac{2}{5}\), multiply by \_\_\_\_\_\_.
- A. \(\frac{2}{5}\)
- B. \(\frac{5}{2}\)
- C. \(\frac{1}{5}\)
- D. \(2\)
- The reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\).
- Division by a fraction equals multiplication by its reciprocal.
Answer: \(\frac{5}{2}\)
Example 2
Question: In a sign-rule table, the quotient \((-)\div(+)\) should be marked as:
| Dividend | Divisor | Quotient |
|---|---|---|
| \((-)\) | \((+)\) | ? |
- A. Positive \((+)\)
- B. Negative \((-)\)
- C. Zero
- D. Undefined
- Sign rule: negative divided by positive always gives negative.
Answer: Negative \((-)\)
Example 3
Question: Number-line model: Where does \(\frac{5}{6}\div\frac{1}{6}\) map on the number line?
- A. \(0\)
- B. Between 0 and 1
- C. At \(5\)
- D. Beyond \(5\)
- \(\frac{5}{6}\div\frac{1}{6}=\frac{5}{6}\times\frac{6}{1}=5\).
Answer: At \(5\)
Real-World Word Problems
Problem 1
Question: A submarine descends \(240\) meters in \(8\) hours at a constant rate. What is the descent per hour?
- A. \(30\) meters per hour
- B. \(-30\) meters per hour (descent)
- C. \(240\) meters per hour
- D. \(8\) meters per hour
Why it works: A descent of \(240\) m is \(-240\). Rate: \((-240)\div8=-30\) m/hr. The negative sign indicates downward motion.
Answer: \(-30\) meters per hour
Problem 2
Question: A freezer temperature drops from \(0°\text{C}\) to \(-30°\text{C}\) over \(10\) hours. What is the average temperature drop per hour?
- A. \(30°\text{C}\) per hour
- B. \(-3°\text{C}\) per hour
- C. \(3°\text{C}\) per hour
- D. \(-30°\text{C}\) per hour
Why it works: Change: \(-30-0=-30°\text{C}\). Rate: \((-30)\div10=-3°\text{C}/\text{hour}\).
Answer: \(-3°\text{C}\) per hour
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 \(-36\div(-9)\)?
- A. \(-4\)
- B. \(-3\)
- C. \(3\)
- D. \(4\)
Question 2
What is \(48\div(-6)\)?
- A. \(-8\)
- B. \(-1\)
- C. \(1\)
- D. \(8\)
Question 3
Evaluate \(-72\div8\).
- A. \(-9\)
- B. \(-8\)
- C. \(8\)
- D. \(9\)
Question 4
What is \(\frac{3}{4}\div\frac{1}{2}\)?
- A. \(\frac{3}{8}\)
- B. \(\frac{2}{3}\)
- C. \(\frac{3}{2}\)
- D. \(2\)
Question 5
Compute \(\frac{5}{6}\div\frac{5}{9}\).
- A. \(\frac{25}{54}\)
- B. \(\frac{3}{2}\)
- C. \(\frac{25}{30}\)
- D. \(\frac{3}{1}\)
Question 6
Which division expression equals \(2\)?
- A. \(\frac{1}{3}\div\frac{1}{3}\)
- B. \(\frac{2}{5}\div\frac{4}{5}\)
- C. \(\frac{4}{7}\div\frac{2}{7}\)
- D. \(\frac{3}{8}\div\frac{3}{4}\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(4\)
Negative divided by negative is positive. Calculate the magnitude: \(36\div9=4\). Apply sign rule: \((-)\div(-)=(+)\). Thus \(-36\div(-9)=4\).
Question 2
Answer: \(-8\)
Positive divided by negative is negative: \(48\div(-6)=-8\).
Question 3
Answer: \(-9\)
Negative divided by positive is negative: \(-72\div8=-9\).
Question 4
Answer: \(\frac{3}{2}\)
Multiply by the reciprocal: \(\frac{3}{4}\div\frac{1}{2}=\frac{3}{4}\times\frac{2}{1}=\frac{6}{4}=\frac{3}{2}\).
Question 5
Answer: \(\frac{3}{2}\)
\(\frac{5}{6}\div\frac{5}{9}=\frac{5}{6}\times\frac{9}{5}=\frac{45}{30}=\frac{3}{2}\).
Question 6
Answer: \(2\)
\(\frac{4}{7}\div\frac{2}{7}=\frac{4}{7}\times\frac{7}{2}=\frac{4}{2}=2\).
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
Dividing 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
Understand the model before choosing the operation.
