Rational Realm Worksheet
Math 7 • Unit 2 Rational Numbers Packet
Rational Realm: Fractions & Decimals
Name:
Date: Period:
Core Concept: Rational Numbers (\(\mathbb{Q}\))
Definition Any number that can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).
Terminating Decimal A decimal that ends because the division leaves a remainder of 0 (e.g., \(\frac{3}{8} = 0.375\)).
Repeating Decimal A decimal that continues infinitely with a repeating digit or block of digits (e.g., \(\frac{2}{3} = 0.\overline{6}\)).
Part 1: Do Now Warm-Up (8–10 Minutes) Score: / 10 pts
1. Classify each value as Rational (\(\mathbb{Q}\)) or Not Rational by circling your answer:
\(-\frac{7}{9}\) Rational | Not Rational
\(0.825\) Rational | Not Rational
\(\sqrt{5}\) Rational | Not Rational
\(-4\frac{1}{2}\) Rational | Not Rational
2. Use long division to convert \(\frac{5}{8}\) into a decimal. State whether it terminates or repeats.
Decimal Value:
Type: [ ] Terminating
[ ] Repeating
3. Plot and label the following rational numbers on the number line below: A: \(-1.5\) | B: \(\frac{3}{4}\) | C: \(-0.25\) | D: \(1\frac{1}{2}\)
-2
-1
0
1
2
Today's Target: Convert, compare, compute, and apply rational numbers in both fraction and decimal form. Self Check: [ ] Ready
Rational Realm • Unit Packet Page 1 • Do Now & Anchor Middle School Mathematics
Independent Practice • Level 1
Decimal & Fraction Conversions
Name: IP Page 1 of 5
Key Technique: To convert a fraction to a decimal, divide numerator by denominator (\(a \div b\)). Use bar notation (\(\overline{x}\)) over the repeating digits only!
Part A Convert each fraction or mixed number into a decimal. Show your division work.
1. \(\frac{7}{20}\) Terminating? [ Y / N ]
Answer:
2. \(-\frac{5}{6}\) Terminating? [ Y / N ]
Answer:
3. \(2\frac{3}{8}\) Terminating? [ Y / N ]
Answer:
4. \(-\frac{4}{11}\) Terminating? [ Y / N ]
Answer:
Part B Convert each decimal to a fraction or mixed number in simplest form.
5. \(0.45\)
Fraction:
6. \(-0.125\)
Fraction:
7. \(3.64\)
Fraction:
8. \(-2.075\)
Fraction:
Mathematical Reasoning & Error Analysis
Marcus converted \(\frac{1}{6}\) on his calculator and recorded his answer as \(0.16\). Is Marcus correct? Explain his error and write the mathematically precise representation of \(\frac{1}{6}\).
Rational Realm • Unit Packet Page 2 • Conversions Mastery Middle School Mathematics
Independent Practice • Level 2
Rational Numbers on the Number Line & Ordering
Name: IP Page 2 of 5
Part A Convert each value to a common form, then plot and accurately label each point.
Point A: \(-\frac{7}{4}\)
Point B: \(1.6\)
Point C: \(-0.75\)
Point D: \(2\frac{1}{4}\)
Point E: \(-\frac{1}{2}\)
Decimal equivalents: A = ______ | B = 1.6 | C = -0.75 | D = ______ | E = ______
-3
-2
-1
0
1
2
3
Part B Compare each pair using <, >, or =. Show your reasoning or common format below each.
1. \(-0.8\) \(-\frac{4}{5}\)
2. \(-\frac{5}{6}\) \(-0.85\)
3. \(|-1.75|\) \(1\frac{3}{4}\)
4. \(-2.3\) \(-2\frac{1}{3}\)
5. \(0.\overline{6}\) \(\frac{5}{8}\)
6. \(-\frac{9}{2}\) \(-4.25\)
Part C Arrange the rational numbers in the requested order.
7. Order from Least to Greatest: \(\{-0.9,\; \frac{2}{5},\; -1\frac{1}{4},\; 0.35,\; -\frac{1}{8}\}\)
Answer: ______ < ______ < ______ < ______ < ______
8. Order from Greatest to Least: \(\{2.8,\; -3.1,\; 2\frac{5}{6},\; -3\frac{1}{5},\; 0\}\)
Answer: ______ > ______ > ______ > ______ > ______
Rational Realm • Unit Packet Page 3 • Ordering & Number Lines Middle School Mathematics
Independent Practice • Level 3
Addition & Subtraction with Mixed Rational Forms
Name: IP Page 3 of 5
Strategic Choice: Convert terms to common form (both fractions OR both decimals). Remember: \(p - q = p + (-q)\). Watch your signs!
1. Evaluate: \(-\frac{3}{4} + 1.6\) Fraction or Dec
Final Answer:
2. Evaluate: \(-3.25 - 2\frac{1}{2}\) Fraction or Dec
Final Answer:
3. Evaluate: \(-\frac{5}{6} - (-1.5)\) Fraction form
Final Answer:
4. Evaluate: \(4\frac{1}{5} + (-6.75)\) Decimal form
Final Answer:
Part B Multi-Step Expressions & Missing Rational Values
5. Evaluate the 3-term expression: \(-1.8 + \left(-\frac{3}{5}\right) - (-2.45)\) Show sequential steps
Final Result:
6. Find the unknown value \(k\): \(k + \left(-\frac{5}{8}\right) = 1.375\) Inverse operations
\(k =\)
Rational Realm • Unit Packet Page 4 • Adding & Subtracting Rationals Middle School Mathematics
Independent Practice • Level 4
Multiplication & Division of Rational Numbers
Name: IP Page 4 of 5
Multiplication Rule: Same signs \(\to\) positive product. Different signs \(\to\) negative product. Simplify fractions before multiplying!
Division Rule: Multiply by the reciprocal (\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)). Always convert decimals to fractions if repeating!
Part A Evaluate each product. State answers as fractions in simplest form or exact decimals.
1. \(\left(-\frac{2}{3}\right) \times 0.75\)
Product:
2. \(-1.2 \times \left(-3\frac{1}{3}\right)\)
Product:
3. \(\frac{5}{8} \times (-2.4)\)
Product:
Part B Evaluate each quotient. Show reciprocal setup or long division.
4. \(-4.8 \div \frac{3}{5}\)
Quotient:
5. \(\left(-\frac{7}{9}\right) \div (-0.35)\)
Quotient:
6. \(3\frac{3}{4} \div (-1.25)\)
Quotient:
Part C Order of Operations Challenge
7. Evaluate: \(\left(-\frac{1}{2}\right)^2 + 1.8 \times \left(-\frac{5}{9}\right)\). Express your answer as a simplified fraction.
Final Solution:
Rational Realm • Unit Packet Page 5 • Multiplying & Dividing Rationals Middle School Mathematics
Independent Practice • Level 5
Real-World Problem Solving & Modeling
Name: IP Page 5 of 5
1. Bank Balance & Transactions Financial Math
Elena's checking account balance was \(-\$34.50\). She deposited a birthday check of \(\$85\frac{1}{4}\), paid \(\$42.80\) for groceries using her debit card, and then incurred a bank maintenance charge of \(\$7\frac{1}{2}\). What is her final balance? Is her account positive or overdrawn?
Final Balance:
Status: [ ] Positive
[ ] Overdrawn
2. Carpentry & Material Cutting Measurement
A carpenter starts with a pine board that is \(9.6\) feet long. She cuts off \(3\) identical shelf pieces, each measuring \(2\frac{3}{8}\) feet in length. Each saw cut creates a waste kerf of \(0.05\) feet. How much wood remains from the original board?
Total Cut + Waste:
Remaining Board:
3. Deep-Sea Ocean Exploration Earth Science
At sea level, ocean surface water is \(18.4^\circ\text{C}\). A research submersible descends, and the water temperature drops at an average rate of \(1\frac{3}{4}^\circ\text{C}\) for every \(100\) meters descended. If the vessel descends to a depth of \(650\) meters, calculate the final water temperature.
Total Temp Drop:
Final Temperature:
Rational Realm • Unit Packet Page 6 • Real-World Applications Middle School Mathematics
Exit Ticket
Daily Mastery Check: Rational Numbers
Target Time: 10–12 Mins
Student Name:
Date:
Period:
1 Level 1: Quick Conversion & Comparison (2 pts) Foundational
Convert \(-\frac{7}{8}\) to a decimal and compare it to \(-0.85\) using <, >, or =.
Decimal Value:
\(-\frac{7}{8}\) [ ] \(-0.85\)
2 Level 2: Mixed Rational Operations (3 pts) Target Fluency
Evaluate: \(-2.4 + \left(-\frac{4}{5}\right) \div \frac{2}{3}\). Show all intermediate steps.
Final Answer:
3 Level 3: Conceptual Justification (3 pts) Application
Is the product of two rational numbers always a rational number? Explain why or why not using mathematical definitions (\(\frac{a}{b}\cdot\frac{c}{d}\)).
Student Self-Assessment & Growth Tracker
How confident do you feel with rational numbers?
1 (Need Help) 2 (Getting There) 3 (Mastered)
One question or muddy concept I still have:
Rational Realm • Assessment Page 7 • Exit Ticket Formative Evaluation Sheet
Independent Extension • Take-Home Sheet
Homework: Rational Number Mastery
Name:
Due Date:
Section 1: The Speedy Six (Computation Fluency) Show brief work for each
1. \(-\frac{4}{9} + \frac{5}{6}\)
Ans:
2. \(-3.75 - (-1.8)\)
Ans:
3. \(-2\frac{1}{2} \times 0.4\)
Ans:
4. \(\frac{7}{12} \div \left(-\frac{14}{15}\right)\)
Ans:
5. \(-5.6 \div (-0.08)\)
Ans:
6. \(\left(-\frac{3}{4}\right)^2 - 0.25\)
Ans:
Section 2: Real-World Application — The Mountain Hiker Modeling
Jamal starts hiking at base camp with an elevation of \(-125.5\) meters (below sea level in Death Valley). During the morning, he ascends \(450\frac{3}{4}\) meters. After lunch, a storm forces him to descend \(182.25\) meters.
a. Write an expression that models his total journey:
b. Calculate Jamal's final elevation:
Section 3: Which One Doesn't Belong? Higher-Order Thinking
Examine the four numbers below. Choose one number that does not belong with the others. State your mathematical reasoning clearly.
A. \(0.375\)
B. \(\frac{3}{8}\)
C. \(\frac{6}{16}\)
D. \(0.\overline{3}\)
Parent/Guardian Signature:
Time Spent: ______ min Effort: ★ ★ ★ ★ ★
Rational Realm • Unit Homework Page 8 • Take-Home Practice Middle School Mathematics
