Fraction Operations Review

Item 8 • 5.NF.B.7.a Correct: A, B, E

Solution: Dividing \(\frac{1}{3}\) mile into 4 equal segments is represented by \(\frac{1}{3} \div 4\) (B), the partitioned number line from 0 to \(\frac{1}{3}\) (A), and multiplying by the reciprocal \(\frac{1}{3} \times \frac{1}{4}\) (E).

Misconception: Choosing C (number line up to 4) or D (multiplying by 4).

Unit 3 Assessment Review • Teacher Answer Key Page 1 of 2

Unit 3 Review • Solutions (Items 9 – 16)

Diagnostic Explanations & Instructional Strategies

Items 9–16 Key

Quick-Scan Key: Items 9 to 16

Q9A

Q1066 oz

Q11C

Q12A

Q13D

Q14½ lb

Q15D

Q16C

Item 9 • 5.MD.B.2 Correct: A

Solution: Frequency table from data: \(\frac{1}{4}\) (2), \(\frac{2}{4}\) (3), \(\frac{3}{4}\) (4), \(\frac{4}{4}=1\) (1). Total = 10 runners. Plot A matches precisely.

Distractors: B swaps counts of \(\frac{1}{4}\) and 1; C shows uniform counts; D misses one mark on \(\frac{3}{4}\).

Item 10 • 5.NF.B.6 Correct: 66 oz

Solution: \(4\frac{2}{5} \times 15 = \frac{22}{5} \times 15 = 22 \times 3 = 66\) ounces. Alternatively: \((4 \times 15) + (\frac{2}{5} \times 15) = 60 + 6 = 66\).

Accept: 66 or 66 ounces.

Item 11 • 5.NF.A.2 Correct: C

Solution: Collected: \(5\frac{4}{12} + 4\frac{9}{12} = 9\frac{13}{12} = 10\frac{1}{12}\) lbs. Needed: \(13\frac{6}{12} - 10\frac{1}{12} = 3\frac{5}{12}\) lbs.

Distractor D: Student found the total collected (\(10\frac{1}{12}\)) but omitted the subtraction step.

Item 12 • 5.NF.B.6 Correct: A

Solution: \(\frac{1}{2} \times \frac{4}{5} = \frac{4}{10} = \frac{2}{5}\). The area model has 10 total cells with 4 cells shaded for carrots (\(\frac{4}{10}=\frac{2}{5}\)).

Misconception: Choice B adds numerators and denominators (\(5/7\)); Choice C ignores the \(\frac{4}{5}\) boundary.

Item 13 • 5.NF.A.1 Correct: D

Solution: Rename 14 as \(13\frac{3}{3}\). \(13\frac{3}{3} - 6\frac{1}{3} = 7\frac{2}{3}\).

Misconception: Choices A & B calculate \(14 - 6 = 8\) without regrouping to create a fractional part.

Item 14 • 5.MD.B.2 Correct: ½ lb

Solution: Total weight = \(\frac{2}{8} + \frac{3}{8} + \frac{3}{8} + \frac{5}{8} + \frac{5}{8} + \frac{6}{8} = \frac{24}{8} = 3\) lbs. Divide by 6 bags: \(3 \div 6 = \frac{3}{6} = \frac{1}{2}\) lb.

Common error: Finding the sum (3 lbs) but forgetting to redistribute by dividing by 6.

Item 15 • 5.NF.B.7.b Correct: D

Solution: Number of days = \(14 \div \frac{1}{4} = 14 \times 4 = 56\) days.

Distractor A: Divided \(14 \div 4 = 3.5\) instead of dividing by unit fraction \(\frac{1}{4}\).

Item 16 • 5.NF.B.7.c Correct: C

Solution: \(\frac{1}{8} \div 3 = \frac{1}{8} \times \frac{1}{3} = \frac{1}{24}\). Puffins occupy 1 equal share of 3 within \(\frac{1}{8}\) of the total zoo.

Distractor A: Added denominators \(8 + 3 = 11\) (\(1/11\)).

Targeted Review & Remediation Plan

Mixed Number Subtraction Practice renaming whole numbers (e.g. \(14 = 13\frac{3}{3}\)) and regrouping with fraction tiles before using purely abstract algorithms.

Unit Fraction Division Reinforce that whole ÷ unit fraction yields a larger count (\(14 \div \frac{1}{4} = 56\)), whereas unit fraction ÷ whole yields a smaller slice (\(\frac{1}{8} \div 3 = \frac{1}{24}\)).

Line Plot Equal Sharing Review the two-step routine: (1) calculate the total sum by multiplying each value by its frequency, (2) divide total sum by total count of items.

Unit 3 Assessment Review • Teacher Answer Key Page 2 of 2

Step 4: Combine into a mixed number:
Total = 8 inches

Key: Do not pick the tallest stack (most frequent); pick the greatest lengths!

Unit 3 Review Step-by-Step Guided Notes Page 1 of 4

Guided Notes

Unit 3 Step-by-Step Strategy & Notes

Page 2 (Q5 – Q8) • Area, Prime/Composite, Perimeter, Models

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Question 5 Strategy • Fractional Area 5.NF.B.6

Formula: Area = Length × Width
Measurements: 312 ft by 214 ft.

Step 1: Convert both mixed numbers to improper fractions:
312 = 2  and  214 = 4

Step 2: Multiply numerators: 7 × 9 = .
Multiply denominators: 2 × 4 = .

Step 3: Convert improper fraction back to a mixed number:
⁶³/₈ = 63 ÷ 8 = with remainder

Final Answer: 8 ft²

Never just multiply whole numbers (3×2=6 is wrong because parts are missing!).

Question 6 Strategy • Prime vs. Composite 5.OA.B.4

Definition: Prime = exactly 2 factors. Composite = more than 2 factors.

Step 1: List all factor pairs of 28:
1 × = 28
2 × = 28
4 × = 28

Step 2: Count the total number of factors of 28:
Factors: 1, 2, 4, 7, 14, 28 = total factors.

Step 3: Fill in the completion sentence:
Blank One: The number 28 is a number.
Blank Two: Because it has two factors.

Options to circle: Blank 1 = composite; Blank 2 = more than.

Question 7 Strategy • Perimeter of a Rectangle 5.NF.A.2

Formula: Perimeter = sum of ALL 4 sides

Step 1: Write down all 4 sides:
P = ³/₅ + ³/₅ + ¼ + ¼ = 5 + 4

Step 2: Note that ²/₄ simplifies to ½. Find the LCD of 5 and 2:
LCD =

Step 3: Convert to equivalent fractions:
⁶/₅ = 10  and  ½ = 10

Step 4: Add numerators: 12 + 5 = .
As mixed number: ¹⁷/₁₀ = 10 miles

Careful: Don't just add 2 sides! A rectangle has 4 sides in total.

Question 8 Strategy • Fraction Division Models 5.NF.B.7.a

Scenario: ⅓ mile is shared equally by 4 runners.

Model 1 (Division Expression):
Total distance ÷ number of runners = ⅓ ÷ (Choice B).

Model 2 (Multiplication by Reciprocal):
Dividing by 4 is equivalent to multiplying by ¼:
⅓ × 1 (Choice E).

Model 3 (Visual Number Line):
Look for the number line from 0 to 1 where the segment from 0 to ⅓ is divided into equal parts (Choice A).

The 3 correct models are: A (visual partition of ⅓), B (⅓ ÷ 4), and E (⅓ × ¼).

Unit 3 Review Step-by-Step Guided Notes Page 2 of 4

Guided Notes

Unit 3 Step-by-Step Strategy & Notes

Page 3 (Q9 – Q12) • Line Plotting, Scaling Fractions, Multi-Step Problems

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Question 9 Strategy • Frequency on Line Plots 5.MD.B.2

Data: ¼, ¼, ²/₄, ²/₄, ²/₄, ¾, ¾, ¾, ¾, ⁴/₄

Step 1: Count tally for each fraction value:
• How many ¼ values? (look for stack of this height)
• How many ²/₄ values?
• How many ¾ values?
• How many ⁴/₄ (1) values?

Step 2: Check total count:
2 + 3 + 4 + 1 = total runners.

Step 3: Match to the options. Plot shows exactly 2 Xs at ¼, 3 Xs at ²/₄, 4 Xs at ¾, and 1 X at 1.

Tip: Cross off numbers one-by-one as you count to prevent errors!

Question 10 Strategy • Mixed Number × Whole 5.NF.B.6

Problem: 425 ounces × 15 boxes = ?

Method A (Improper Fraction):
425 = (4×5)+25 = 5
Multiply: ²²/₅ × 15 = ²² × (15 ÷ 5) = 22 × =

Method B (Distributive Property):
(4 × 15) + (²/₅ × 15)
= + = ounces

Both methods give the exact same answer: 66 ounces!

Question 11 Strategy • Multi-Step Goal Problem 5.NF.A.2

Blake: 5⅓ lbs | Brother: 4¾ lbs | Goal: 13½ lbs

Step 1: Find LCD for denominators 3, 4, 2: LCD = .

Step 2: Add what they already collected:
5412 + 4912 = 91312 = 112 lbs collected

Step 3: Rename goal with LCD: 13½ = 1312.

Step 4: Subtract: 13⁶/₁₂ − 10¹/₁₂:
Whole numbers: 13 − 10 =  |  Numerators: 6 − 1 =
Still needed: 12 lbs (Choice C)

Don't stop after adding! The question asks for how much is STILL needed.

Question 12 Strategy • Fraction Area Model 5.NF.B.6

Scenario: ⁴/₅ for veggies. ½ of veggies for carrots.

Step 1: Count total grid squares in model:
5 columns × 2 rows = total equal parts.

Step 2: Count the dark shaded squares (carrots):
There are dark squares.

Step 3: Multiply fractions:
⁴/₅ × ½ = 4 × 15 × 2 = 10

Step 4: Simplify ⁴/₁₀ by dividing top & bottom by 2:
⁴/₁₀ = 5 (Choice A)

"Of" in mathematics usually means multiplication! (½ of ⁴/₅ = ½ × ⁴/₅).

Unit 3 Review Step-by-Step Guided Notes Page 3 of 4

Guided Notes

Unit 3 Step-by-Step Strategy & Notes

Page 4 (Q13 – Q16) • Whole Subtraction, Equal Sharing, Unit Fractions

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Question 13 Strategy • Subtracting from a Whole 5.NF.A.1

Problem: 14 − 6⅓ = ?

Step 1: 14 has no fraction part. You must rename 1 whole as a fraction with denominator 3:
14 = 13 + 1 = 13 + 3 = 133

Step 2: Set up the subtraction vertically:
1333 − 613

Step 3: Subtract whole numbers: 13 − 6 = .
Subtract numerators: 3 − 1 = .

Final Answer: 3 (Choice D)

Common Trap: Doing 14 − 6 = 8 and keeping ⅓ to get 8⅓ is incorrect!

Question 14 Strategy • Fair Share / Redistribution 5.MD.B.2

Combine all candy and divide equally into 6 bags.

Step 1: Find the total pounds by adding all 6 bags:
• 1 bag at ¼ (= ²/₈) = ²/₈
• 2 bags at ⅜ = 8  |  2 bags at ⅝ = 8
• 1 bag at ¾ (= ⁶/₈) = ⁶/₈

Step 2: Add all eighths:
²/₈ + ⁶/₈ + ¹⁰/₈ + ⁶/₈ = 8 = lbs total.

Step 3: Divide total weight by 6 bags:
3 ÷ 6 = ³/₆ = 1 pound in each bag

Fair share problems always have 2 steps: (1) Find sum, (2) Divide by count.

Question 15 Strategy • Whole ÷ Unit Fraction 5.NF.B.7.b

Tamara has 14 cups of food. Rabbit eats ¼ cup daily.

Key Question: "How many ¼ cups are inside 14 cups?"

Step 1: Think about 1 single cup:
How many ¼-cups are in 1 cup?

Step 2: Multiply by 14 cups:
14 × 4 = days.

Equation Rule:
14 ÷ ¼ = 14 × = days (Choice D)

When dividing by a unit fraction, your answer will be LARGER than the starting whole!

Question 16 Strategy • Unit Fraction ÷ Whole 5.NF.B.7.c

⅛ of the zoo is shared equally by 3 kinds of birds.

Step 1: Write the division expression:
⅛ ÷

Step 2: Rewrite dividing by 3 as multiplying by ⅓:
⅛ × ⅓ = 1 × 18 × 3 = 1

Step 3: Visual check with the diagram:
If each ⅛ strip is split into 3 parts, the whole zoo has 8 × 3 = equal parts.

Final Answer: 1 of the zoo (Choice C)

When dividing a fraction by a whole number, each piece gets SMALLER (¹/₂₄ < ⅛).

Unit 3 Review Step-by-Step Guided Notes Page 4 of 4