Fraction Mastery Practice Test
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Grade 4 Mathematics • Unit 3 Practice Test
Fraction Equivalence, Operations & Line Plots
Question 1 of 8
1
Select all the expressions equivalent to \( \frac{7}{4} \).
A \( \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} \)
B \( \frac{3}{4} + \frac{2}{4} + \frac{2}{4} \)
C \( 1 + \frac{3}{4} \)
D \( \frac{4}{4} + \frac{1}{4} + \frac{1}{4} \)
E \( \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7} \)
Explain or show your reasoning:
How did you know which expressions were equivalent to \( \frac{7}{4} \)? Explain your thinking.
Grade 4 Mathematics • Unit 3: Fractions Page 1 (Question 1)
Answer Key & 2-Way Solution Guide
Question 1 • Decomposing Fractions & Equivalence
Teacher Resource
Correct Selections: A, B, C
Options A, B, and C each equal \( \frac{7}{4} \) (or \( 1\frac{3}{4} \)). Option D equals \( \frac{6}{4} \). Option E equals \( \frac{4}{7} \), which confuses the numerator and denominator.
1
Method 1: Unit Fraction Addition
Recognize that \( \frac{7}{4} \) means exactly 7 groups of \( \frac{1}{4} \).
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Option A: Adding 7 unit fractions:
\( \frac{1}{4} \times 7 = \frac{7}{4} \)
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Option B: Grouping unit fractions with like denominators:
\( \frac{3+2+2}{4} = \frac{7}{4} \)
Key Insight: When denominators are identical, simply combine the counts of fourths.
2
Method 2: Whole Number Regrouping
Convert between improper fractions and mixed numbers using the whole \( 1 = \frac{4}{4} \).
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Decompose into Wholes:
\( \frac{7}{4} = \frac{4}{4} + \frac{3}{4} = 1 + \frac{3}{4} \)
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Evaluates Option C: Matches directly as \( 1\frac{3}{4} \).
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Evaluates Option D:
\( \frac{4}{4} + \frac{2}{4} = \frac{6}{4} \neq \frac{7}{4} \)
Key Insight: Recognizing \( \frac{4}{4} = 1 \) allows immediate validation of mixed number forms.
Teacher Guidance & Common Misconceptions
Students who pick E often see the digit 7 and create sevenths instead of keeping fourths as the fractional size. Emphasize that the denominator names the size of each piece, and the numerator counts how many pieces exist.
Grade 4 Mathematics • Unit 3: Fractions Page 2 (Answer Key 1)
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Grade 4 Mathematics • Unit 3 Practice Test
Fraction Comparison & Benchmark Values
Question 2 of 8
2
Select all expressions with a value greater than 1.
A \( \frac{2}{5} + \frac{2}{5} + \frac{2}{5} \)
B \( \frac{3}{6} + \frac{2}{6} + \frac{1}{6} \)
C \( 2\frac{2}{8} - 1\frac{5}{8} \)
D \( 4 \times \frac{2}{6} \)
E \( 8 \times \frac{1}{10} \)
Explain or show your reasoning:
Show your calculations or explain how you compared each expression to 1.
Grade 4 Mathematics • Unit 3: Fractions Page 3 (Question 2)
Answer Key & 2-Way Solution Guide
Question 2 • Comparing Expressions to Benchmark 1
Teacher Resource
Correct Selections: A, D
Expressions A (\( \frac{6}{5} \)) and D (\( \frac{8}{6} \)) are strictly greater than 1. Option B equals exactly 1 (\( \frac{6}{6} \)), Option C is less than 1 (\( \frac{5}{8} \)), and Option E is less than 1 (\( \frac{8}{10} \)).
1
Method 1: Direct Computation
Compute exact numerical values and check if numerator > denominator:
- A: \( \frac{2+2+2}{5} = \frac{6}{5} = 1\frac{1}{5} > 1 \) ✓
- B: \( \frac{3+2+1}{6} = \frac{6}{6} = 1 \) (Not > 1) ✗
- C: \( 2\frac{2}{8} - 1\frac{5}{8} = \frac{18}{8} - \frac{13}{8} = \frac{5}{8} < 1 \) ✗
- D: \( 4 \times \frac{2}{6} = \frac{8}{6} = 1\frac{2}{6} > 1 \) ✓
- E: \( 8 \times \frac{1}{10} = \frac{8}{10} < 1 \) ✗
Key Insight: A fraction \( \frac{a}{b} > 1 \) if and only if \( a > b \).
2
Method 2: Benchmark Reasoning
Use visual reference points and mental estimation without full fraction conversions:
- For A: Three \( \frac{2}{5} \) pieces is \( 6 \) fifths. Five fifths makes 1 whole, so 6 fifths exceeds 1.
- For B: Notice \( 3+2+1 = 6 \). Six sixths makes exactly one whole, not greater than 1.
- For C: Starting at \( 2\frac{2}{8} \), subtracting \( 1\frac{5}{8} \) removes 1 whole plus \( \frac{5}{8} \). Since \( \frac{5}{8} > \frac{2}{8} \), the result dips below 1.
- For D: \( 3 \times \frac{2}{6} = 1 \) whole, so 4 times \( \frac{2}{6} \) must be greater than 1.
Key Insight: Benchmark reasoning builds deep number sense before computing.
Common Student Trap
Students frequently select B because it equals 1, missing the strict inequality requirement "greater than 1". Review the distinction between "equal to 1" and "greater than 1".
Grade 4 Mathematics • Unit 3: Fractions Page 4 (Answer Key 2)
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Grade 4 Mathematics • Unit 3 Practice Test
Fraction Multiplication & Factors
Question 3 of 8
3
Select all the expressions that are equivalent to \( \frac{6}{10} \).
A \( 6 \times \frac{1}{10} \)
B \( 2 \times \frac{3}{10} \)
C \( 3 \times \frac{3}{10} \)
D \( 3 \times 2 \times \frac{1}{10} \)
E \( 10 \times \frac{1}{6} \)
Explain or show your reasoning:
Explain how you know which expressions multiply to make \( \frac{6}{10} \).
Grade 4 Mathematics • Unit 3: Fractions Page 5 (Question 3)
Answer Key & 2-Way Solution Guide
Question 3 • Equivalent Product Representations of Fractions
Teacher Resource
Correct Selections: A, B, D
Options A, B, and D each multiply to equal \( \frac{6}{10} \). Option C equals \( \frac{9}{10} \). Option E equals \( \frac{10}{6} \).
1
Method 1: Repeated Addition of Fractions
Interpret each multiplication as repeated addition of equal groups:
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Option A: 6 groups of \( \frac{1}{10} \):
\( \frac{1}{10} + \dots + \frac{1}{10} = \frac{6}{10} \)
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Option B: 2 groups of \( \frac{3}{10} \):
\( \frac{3}{10} + \frac{3}{10} = \frac{6}{10} \)
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Option C: 3 groups of \( \frac{3}{10} \):
\( \frac{3}{10} + \frac{3}{10} + \frac{3}{10} = \frac{9}{10} \neq \frac{6}{10} \)
Key Insight: Multiplication by a whole number scales the number of fractional parts.
2
Method 2: Factoring & Associative Property
Decompose the numerator 6 into whole number factors:
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Since \( 6 = 3 \times 2 \), we can regroup factors:
\( \frac{6}{10} = 6 \times \frac{1}{10} = (3 \times 2) \times \frac{1}{10} \)
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Associative regrouping:
\( 3 \times \left(2 \times \frac{1}{10}\right) = 3 \times \frac{2}{10} \text{ or } 2 \times \left(3 \times \frac{1}{10}\right) = 2 \times \frac{3}{10} \)
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This proves why both B (\( 2 \times \frac{3}{10} \)) and D (\( 3 \times 2 \times \frac{1}{10} \)) are equivalent.
Key Insight: \( n \times \frac{a}{b} = (n \times a) \times \frac{1}{b} \).
Instructional Reflection
Students often struggle with D because it has three factors. Guide them to see that \( 3 \times 2 = 6 \), transforming the expression into \( 6 \times \frac{1}{10} \).
Grade 4 Mathematics • Unit 3: Fractions Page 6 (Answer Key 3)
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Grade 4 Mathematics • Unit 3 Practice Test
Word Problems • Whole Number by Fraction
Question 4 of 8
4
Diego needs 2 cups of almonds to make granola bars.
He has 3 bags of almonds. Each bag contains \( \frac{3}{4} \) cup of almonds.
Does Diego have enough almonds to make the granola bars?
Explain or show your reasoning:
Diego has enough almonds? (Circle one): YES NO
Grade 4 Mathematics • Unit 3: Fractions Page 7 (Question 4)
Answer Key & 2-Way Solution Guide
Question 4 • Real-World Context with Fraction Multiplication
Teacher Resource
Final Conclusion: YES, Diego has enough
Diego has \( 2\frac{1}{4} \) cups of almonds (or \( \frac{9}{4} \) cups), which is \( \frac{1}{4} \) cup more than the 2 cups needed.
1
Method 1: Repeated Addition & Visual Model
Draw or combine the 3 bags of \( \frac{3}{4} \) cup:
\( \frac{3}{4} + \frac{3}{4} + \frac{3}{4} = \frac{9}{4} \text{ cups} \)
Recompose fourths into whole cups:
\( \frac{4}{4} \) (1 whole) + \( \frac{4}{4} \) (1 whole) + \( \frac{1}{4} \) = \( 2\frac{1}{4} \) cups
Compare to the needed amount:
\( 2\frac{1}{4} \text{ cups} > 2 \text{ cups} \)
Key Insight: 9 fourths easily splits into two full groups of 4 fourths plus 1 leftover fourth.
2
Method 2: Whole Number Multiplication
Multiply the number of bags by the quantity per bag:
\( 3 \times \frac{3}{4} = \frac{3 \times 3}{4} = \frac{9}{4} \)
Convert target amount (2 cups) into fourths:
\( 2 \text{ cups} = \frac{8}{4} \text{ cups} \)
Direct comparison of numerators:
\( \frac{9}{4} > \frac{8}{4} \) (by \( \frac{1}{4} \) cup)
Key Insight: Converting both amounts to the same denominator (\( \frac{9}{4} \) vs \( \frac{8}{4} \)) makes comparing instantaneous.
Evaluation Rubric
Award full credit if the student reaches "Yes" with clear mathematical reasoning showing \( 3 \times \frac{3}{4} = \frac{9}{4} \) or \( 2\frac{1}{4} \) and explicitly compares it to 2 cups (or \( \frac{8}{4} \)).
Grade 4 Mathematics • Unit 3: Fractions Page 8 (Answer Key 4)
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Grade 4 Mathematics • Unit 3 Practice Test
Fractional Line Plots • Interpreting & Plotting Eighths
Question 5 of 8
5
The line plot shows the lengths of some wooden craft sticks in inches. The tick marks on the number line show fourths of an inch. Some data points fall between the tick marks.
Craft Stick Lengths
2 3 4 5 ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ ✕ Length in inches
a. The tick marks show fourths of an inch. What fraction of an inch does a point located halfway between two tick marks represent? What is the length of the shortest craft stick shown?
Halfway point represents:
Shortest stick length:
b. What is the difference in length between the longest craft stick and the shortest craft stick shown on the line plot? Show your reasoning.
c. How many craft sticks measure \( 3\frac{5}{8} \) inches or longer?
d. Lin measures two more craft sticks: one is \( 2\frac{5}{8} \) inches and one is \( 3\frac{7}{8} \) inches. Plot these two measurements with an ✕ on the line plot above.
Grade 4 Mathematics • Unit 3: Fractions Page 9 (Question 5)
Answer Key & 2-Way Solution Guide
Question 5 • Fractional Line Plots with Eighths & Fourths
Teacher Resource
Summary Answers:
a. Halfway: \( \frac{1}{8} \) in.
Shortest: \( 2\frac{3}{8} \) in.
b. Difference:
\( 2\frac{4}{8} \) or \( 2\frac{1}{2} \) in.
c. \( \ge 3\frac{5}{8} \) in.:
7 sticks
d. Plotted:
1 ✕ at \( 2\frac{5}{8} \), 1 ✕ at \( 3\frac{7}{8} \)
Part (b) Solutions: Difference between Longest (\( 4\frac{7}{8} \)) and Shortest (\( 2\frac{3}{8} \))
1
Method 1: Subtract Whole & Fractional Parts
Subtract whole numbers and numerators of eighths directly:
\( 4\frac{7}{8} - 2\frac{3}{8} = (4 - 2) + \left(\frac{7}{8} - \frac{3}{8}\right) \)
\( = 2 + \frac{7 - 3}{8} = 2\frac{4}{8} \text{ inches} \)
\( = 2\frac{1}{2} \text{ inches} \) (equivalent form)
Improper form: \( \frac{39}{8} - \frac{19}{8} = \frac{20}{8} = 2\frac{4}{8} \).
2
Method 2: Counting Up on Number Line
Count up in jumps along the eighths axis from \( 2\frac{3}{8} \) to \( 4\frac{7}{8} \):
- Jump +2 whole inches from \( 2\frac{3}{8} \) to \( 4\frac{3}{8} \)
- Jump \( +\frac{4}{8} \) inch from \( 4\frac{3}{8} \) to \( 4\frac{7}{8} \)
- Total distance: \( 2 + \frac{4}{8} = 2\frac{4}{8} = 2\frac{1}{2} \) inches
Direct visual jumps reinforce fraction number line structure.
Teacher Guidance: Understanding Eighths Between Tick Marks
Why halfway is an eighth: Each whole unit has 4 fourths-intervals (\(\frac{1}{4}\) each). Halving each fourth divides the whole into \(4 \times 2 = 8\) equal parts. Thus, each space between a tick mark and a halfway point equals \( \frac{1}{8} \) inch.
Part (c) Counting Breakdown:
Sticks at \( 3\frac{5}{8} \) (2), \( 4\frac{1}{4} \) (1), \( 4\frac{3}{8} \) (1), \( 4\frac{1}{2} \) (2), \( 4\frac{7}{8} \) (1) = 7 sticks. Note that \( 4\frac{1}{4} = 4\frac{2}{8} > 3\frac{5}{8} \).
Part (d) Plotting Accuracy:
Students should plot an ✕ midway between \( 2\frac{1}{2} \) and \( 2\frac{3}{4} \) (for \( 2\frac{5}{8} \)), and midway between \( 3\frac{3}{4} \) and \( 4 \) (for \( 3\frac{7}{8} \)).
Grade 4 Mathematics • Unit 3: Fractions Page 10 (Answer Key 5)
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Grade 4 Mathematics • Unit 3 Practice Test
Fraction Operations • Addition, Subtraction & Multiplication
Question 6 of 8
6
Find the value of each expression. Show your work or explain your thinking.
a. \( \frac{4}{6} + \frac{3}{6} + \frac{5}{6} \) Answer: ______
b. \( 1\frac{2}{4} - \frac{3}{4} \) Answer: ______
c. \( 3\frac{3}{5} + 2\frac{4}{5} \) Answer: ______
d. \( 7 - \frac{6}{10} \) Answer: ______
e. \( 4 \times \frac{5}{8} \) Answer: ______
Grade 4 Mathematics • Unit 3: Fractions Page 11 (Question 6)
Answer Key & 2-Way Solution Guide
Question 6 • Core Computational Fluency
Teacher Resource
Answer Summary:
a. \( \frac{12}{6} \) or \( 2 \)
b. \( \frac{3}{4} \)
c. \( 6\frac{2}{5} \) or \( \frac{32}{5} \)
d. \( 6\frac{4}{10} \) or \( 6\frac{2}{5} \)
e. \( \frac{20}{8} \) or \( 2\frac{4}{8} \)
Item (b): \( 1\frac{2}{4} - \frac{3}{4} \) Answer: \( \frac{3}{4} \)
Way 1 (Convert to improper):
\( 1\frac{2}{4} = \frac{4}{4} + \frac{2}{4} = \frac{6}{4} \)
\( \frac{6}{4} - \frac{3}{4} = \frac{3}{4} \)
Way 2 (Decompose subtrahend):
Split \( \frac{3}{4} \) into \( \frac{2}{4} + \frac{1}{4} \)
\( 1\frac{2}{4} - \frac{2}{4} = 1 \); then \( 1 - \frac{1}{4} = \frac{3}{4} \)
Item (c): \( 3\frac{3}{5} + 2\frac{4}{5} \) Answer: \( 6\frac{2}{5} \)
Way 1 (Add wholes then fractions):
\( 3 + 2 = 5 \); \( \frac{3}{5} + \frac{4}{5} = \frac{7}{5} = 1\frac{2}{5} \)
Total: \( 5 + 1\frac{2}{5} = 6\frac{2}{5} \)
Way 2 (Compensation to make a whole):
Move \( \frac{2}{5} \) from \( 3\frac{3}{5} \) to \( 2\frac{4}{5} \) to make 3.
\( 3\frac{1}{5} + 3 = 6\frac{1}{5} + \frac{1}{5} = 6\frac{2}{5} \)
Item (d): \( 7 - \frac{6}{10} \) and Item (e): \( 4 \times \frac{5}{8} \) d: \( 6\frac{4}{10} \), e: \( \frac{20}{8} \)
For (d): Decompose 7 into \( 6 + \frac{10}{10} \). Then subtract: \( 6 + \left(\frac{10}{10}-\frac{6}{10}\right) = 6\frac{4}{10} \).
For (e): \( 4 \times \frac{5}{8} = \frac{4 \times 5}{8} = \frac{20}{8} \). Grouping in eighths: \( \frac{16}{8} + \frac{4}{8} = 2\frac{4}{8} = 2\frac{1}{2} \).
Grade 4 Mathematics • Unit 3: Fractions Page 12 (Answer Key 6)
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Grade 4 Mathematics • Unit 3 Practice Test
Tenths and Hundredths • Equivalence & Addition
Question 7 of 8
7
Find the value of each expression. Explain or show how you converted denominators to add.
a. \( \frac{9}{100} + \frac{6}{100} + \frac{3}{10} \) Value: _________
Show work / convert to hundredths:
b. \( \frac{4}{10} + \frac{25}{100} + \frac{1}{10} \) Value: _________
Show work / convert to hundredths:
c. \( \frac{16}{100} + \frac{5}{10} + \frac{24}{100} \) Value: _________
Show work / convert to hundredths:
Grade 4 Mathematics • Unit 3: Fractions Page 13 (Question 7)
Answer Key & 2-Way Solution Guide
Question 7 • Adding Fractions with Denominators 10 & 100
Teacher Resource
Answer Summary:
a. \( \frac{45}{100} \)
b. \( \frac{75}{100} \) or \( \frac{3}{4} \)
c. \( \frac{90}{100} \) or \( \frac{9}{10} \)
Part (a): \( \frac{9}{100} + \frac{6}{100} + \frac{3}{10} \) Answer: \( \frac{45}{100} \)
Way 1 (Algebraic Equivalence):
Convert \( \frac{3}{10} \) by multiplying numerator and denominator by 10:
\( \frac{3 \times 10}{10 \times 10} = \frac{30}{100} \)
Add numerators: \( \frac{9 + 6 + 30}{100} = \frac{45}{100} \)
Way 2 (Combine like terms first):
Combine hundredths: \( \frac{9}{100} + \frac{6}{100} = \frac{15}{100} \)
Know \( \frac{3}{10} \) is 3 tenths = 30 hundredths.
\( 15 \text{ hundredths} + 30 \text{ hundredths} = 45 \text{ hundredths} \).
Part (b): \( \frac{4}{10} + \frac{25}{100} + \frac{1}{10} \) Answer: \( \frac{75}{100} \)
Way 1 (Combine tenths first):
\( \frac{4}{10} + \frac{1}{10} = \frac{5}{10} \)
Convert to hundredths: \( \frac{5}{10} = \frac{50}{100} \)
\( \frac{50}{100} + \frac{25}{100} = \frac{75}{100} \)
Way 2 (Full simultaneous conversion):
\( \frac{40}{100} + \frac{25}{100} + \frac{10}{100} = \frac{40 + 25 + 10}{100} = \frac{75}{100} \)
(Students may also recognize this as \( \frac{3}{4} \)).
Part (c): \( \frac{16}{100} + \frac{5}{10} + \frac{24}{100} \) Answer: \( \frac{90}{100} \)
Way 1 (Friendly number combinations):
\( \frac{16}{100} + \frac{24}{100} = \frac{40}{100} \) (Notice \( 16+24=40 \))
Add \( \frac{5}{10} = \frac{50}{100} \):
\( \frac{40}{100} + \frac{50}{100} = \frac{90}{100} = \frac{9}{10} \)
Way 2 (Standard left-to-right):
\( \frac{16}{100} + \frac{50}{100} = \frac{66}{100} \)
\( \frac{66}{100} + \frac{24}{100} = \frac{90}{100} \)
Both answers \( \frac{90}{100} \) and \( \frac{9}{10} \) are fully correct.
Grade 4 Mathematics • Unit 3: Fractions Page 14 (Answer Key 7)
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Grade 4 Mathematics • Unit 3 Practice Test
Multi-Step Story Problem • Addition, Subtraction & Comparison
Question 8 of 8
8
Liam’s favorite podcast episode is \( 4\frac{1}{6} \) minutes long. Maya’s favorite podcast episode is \( 1\frac{5}{6} \) minutes long.
a. How many minutes does it take to listen to both Liam’s and Maya’s favorite episodes together?
b. How many minutes longer is Liam’s favorite episode than Maya’s favorite episode? Explain or show your reasoning.
c. Maya says she can listen to her episode two times in the same amount of time it takes Liam to listen to his episode once. Do you agree with Maya? Explain or show your reasoning.
Grade 4 Mathematics • Unit 3: Fractions Page 15 (Question 8)
Answer Key & 2-Way Solution Guide
Question 8 • Multi-Step Problem Solving with Mixed Numbers
Teacher Resource
Answer Summary:
a. Total time: 6 minutes
b. Difference: \( 2\frac{2}{6} \) or \( 2\frac{1}{3} \) min
c. Agree?: YES (takes \( 3\frac{4}{6} \) min < \( 4\frac{1}{6} \))
Part (a): \( 4\frac{1}{6} + 1\frac{5}{6} = 6 \) minutes
Way 1 (Wholes & Fractions):
\( (4 + 1) + \left(\frac{1}{6} + \frac{5}{6}\right) = 5 + \frac{6}{6} = 5 + 1 = 6 \) minutes.
Way 2 (Improper Fractions):
\( \frac{25}{6} + \frac{11}{6} = \frac{36}{6} = 6 \) minutes.
Part (b): \( 4\frac{1}{6} - 1\frac{5}{6} = 2\frac{2}{6} \) minutes
Way 1 (Regroup 1 whole):
\( 4\frac{1}{6} = 3 + \frac{7}{6} \)
\( 3\frac{7}{6} - 1\frac{5}{6} = (3-1) + \left(\frac{7}{6}-\frac{5}{6}\right) = 2\frac{2}{6} \) min.
Way 2 (Add on / Counting up):
From \( 1\frac{5}{6} \) add \( \frac{1}{6} \rightarrow 2 \); add 2 \( \rightarrow 4 \); add \( \frac{1}{6} \rightarrow 4\frac{1}{6} \).
Total: \( \frac{1}{6} + 2 + \frac{1}{6} = 2\frac{2}{6} \) minutes.
Part (c): Agree with Maya? (YES)
Way 1 (Repeated Addition / Double):
Maya's time doubled: \( 1\frac{5}{6} + 1\frac{5}{6} = 2 + \frac{10}{6} = 3\frac{4}{6} \) min.
Compare: \( 3\frac{4}{6} < 4\frac{1}{6} \) min. So she easily finishes twice!
Way 2 (Multiplication by 2):
\( 2 \times \frac{11}{6} = \frac{22}{6} \) minutes.
Liam's episode is \( 4\frac{1}{6} = \frac{25}{6} \) minutes.
Since \( \frac{22}{6} < \frac{25}{6} \), she has time left over (\( \frac{3}{6} \) min).
Grade 4 Mathematics • Unit 3: Fractions Page 16 (Answer Key 8)
