Sandwich Sharing Answer Key
Teacher Reference Guide
Share More Sandwiches: Answer Key & Breakdown
Grade 5 Math • Unit 2, Lesson 2 • Standards: 5.NF.B.3, 5.NBT.5, 4.OA.A.1, 4.NF.B.3
Complete Key
Page 1 of 4
Lesson Big Idea & Standard (5.NF.B.3)
Interpret a fraction as division of the numerator by the denominator (\(a \div b = \frac{a}{b}\)). When \(a\) items are shared equally among \(b\) groups, each group receives \(\frac{a}{b}\) of an item.
1
Do Now Problems (PDF Pages 1–2)
Problem 1 (4.OA.A.1) Multiplication Habits
324 × 320
324 × 0 = 0
324 × 20 = 6,480
+ 324 × 300 = 97,200
= 103,680
Habit note: Estimate first (\(300 \times 300 = 90,000\)). Verify trailing place-value zeros.
Problem 2 (4.NF.B.3) Division Terms
426 ÷ 6
Dividend: 426 (total quantity)
Divisor: 6 (equal groups)
Quotient: 71
Decomposition: \((420 \div 6) + (6 \div 6) = 70 + 1 = 71\).
Problem 3 (5.NBT.5) Answer: C
A theater has 36 rows with 125 seats in each row. How many seats are there?
36 × 125 = 36 × (1000 ÷ 8)
= 4.5 × 1000 = 4,500 seats
Or: (30 × 125 = 3750) + (6 × 125 = 750) = 4,500
Options: A. 4,250 | B. 4,375 | C. 4,500 [✓] | D. 4,625
Problem 5 (Spicy) Answer: C
A library has 2,184 books placed equally on 28 shelves. How many per shelf?
2,184 ÷ 28:
28 × 70 = 1,960 → 2,184 - 1,960 = 224
28 × 8 = 224 → 70 + 8 = 78 books
Options: A. 68 | B. 72 | C. 78 [✓] | D. 82
Adapted from Illustrative Math & Uncommon Schools U2, L2: Share More Sandwiches
"Excellence" Sprint Key & Task 1 Analysis
Page 2 of 4
Sprint Solutions (50 Multiplication Facts, PDF Page 3)
Format: Factor × Factor = Product
10×5 = 50
5×4 = 20
6×3 = 18
8×3 = 24
7×8 = 56
4×9 = 36
5×9 = 45
6×8 = 48
5×5 = 25
5×10 = 50
2×7 = 14
5×2 = 10
8×7 = 56
10×3 = 30
9×7 = 63
4×10 = 40
6×6 = 36
3×2 = 6
6×10 = 60
10×4 = 40
7×3 = 21
7×9 = 63
5×7 = 35
2×6 = 12
10×8 = 80
7×7 = 49
7×2 = 14
7×4 = 28
7×10 = 70
9×5 = 45
10×7 = 70
4×8 = 32
4×10 = 40
4×9 = 36
9×4 = 36
10×7 = 70
8×9 = 72
7×5 = 35
3×7 = 21
6×7 = 42
2×10 = 20
7×10 = 70
7×6 = 42
10×5 = 50
3×6 = 18
9×10 = 90
2×7 = 14
8×8 = 64
6×8 = 48
8×3 = 24
2
Task 1: One Sandwich (PDF Page 4)
| Sandwiches Being Shared | Number of People Sharing | Amount of Sandwich Each Gets | Division Expression |
|---|
| 1 | 2 | \(\frac{1}{2}\) | \(1 \div 2\) |
| 1 | 3 | \(\frac{1}{3}\) | \(1 \div 3\) |
| 1 | 4 | \(\frac{1}{4}\) | \(1 \div 4\) |
| 1 | 5 | \(\frac{1}{5}\) | \(1 \div 5\) |
Q1: Diagram Representation (Example: 1 Sandwich shared by 4 people)
1 Whole Sandwich partitioned into 4 fourths
\(\frac{1}{4}\)
\(\frac{1}{4}\)
\(\frac{1}{4}\)
\(\frac{1}{4}\)
Key explanation: The full rectangle represents 1 whole sandwich (dividend). Partitioning into 4 equal segments reflects sharing among 4 people (divisor). Each person gets 1 piece of size \(\frac{1}{4}\).
Q2: What patterns do you notice in the table?
- Fraction as Division: The dividend (1 sandwich) is always the numerator, and the divisor (number of people) is always the denominator: \(1 \div n = \frac{1}{n}\).
- Inverse Relationship: As more people share the sandwich, each person's portion decreases in size (\(\frac{1}{2} > \frac{1}{3} > \frac{1}{4} > \frac{1}{5}\)).
Adapted from Illustrative Math & Uncommon Schools U2, L2: Share More Sandwiches
Task 2: Card Sort & Independent Practice Part 1
Page 3 of 4
3 Task 2: Card Sort (Sandwich Match) — 2 Sandwiches Shared by 5 People
Portion: \(\frac{2}{5}\) Sandwich
Sandwich 1
Sandwich 2
Each sandwich is sliced into 5 equal fifths (\(\frac{1}{5}\) each). One person takes 1 slice from Sandwich 1 and 1 slice from Sandwich 2: \(\frac{1}{5} + \frac{1}{5} = \frac{2}{5}\) sandwich per person.
1. Sandwiches Shared Represented by 2 whole rectangles and the dividend 2 in \(2 \div 5\).
2. People Sharing Represented by the 5 equal partitions per bar and the divisor 5.
3. Final Portion Each person receives \(2 \times \frac{1}{5} =\) \(\frac{2}{5}\) sandwich.
4
Independent Practice (PDF Page 7)
Problem 1a: 4 Hikers, 3 Liters \(\frac{3}{4}\) Liter
Question: Four hikers equally share 3 liters of water. How much does each hiker drink?
Division: \(3 \div 4\)
Reasoning: Each liter cut into 4 fourths.
\(3 \times \frac{1}{4} = \mathbf{\frac{3}{4}\text{ liter}}\)
Problem 1b: 4 Hikers, 5 Liters \(\frac{5}{4}\) or \(1\frac{1}{4}\) L
Question: Four hikers equally share 5 liters of water. How much does each hiker drink?
Division: \(5 \div 4\)
Reasoning: Each gets 1 whole liter (\(4\div 4 = 1\)), plus \(\frac{1}{4}\) of the 5th liter.
\(5 \times \frac{1}{4} = \mathbf{\frac{5}{4}\text{ or } 1\frac{1}{4}\text{ liters}}\)
Problem 2: Story Problem for Quotient \(\frac{2}{5}\) Student Creation & Proof
Exemplar Story Problem:
"Five friends share 2 large personal pizzas equally after soccer practice. What fraction of a pizza does each friend get?"
Proof / Mathematical Reasoning:
Expression: \(2 \div 5\). Slice each of the 2 pizzas into 5 equal slices (\(\frac{1}{5}\) each). Each friend takes 1 slice from each pizza: \(\frac{1}{5} + \frac{1}{5} = \mathbf{\frac{2}{5}}\).
Adapted from Illustrative Math & Uncommon Schools U2, L2: Share More Sandwiches
Independent Practice (Cont.) & Exit Ticket
Page 4 of 4
Problem 3: 6 Pizzas, 7 Students \(\frac{6}{7}\) Pizza
a. Reasoning: Cut each pizza into 7 equal parts (\(\frac{1}{7}\) each). Each student gets 1 slice from each of the 6 pizzas: \(6 \times \frac{1}{7} = \mathbf{\frac{6}{7}}\).
b. Division Sentence: \(6 \div 7 = \frac{6}{7}\)
Problem 4: 5 Candy Bars, 8 Classmates \(\frac{5}{8}\) Bar
a. Reasoning: Each candy bar is cut into 8 equal pieces (\(\frac{1}{8}\) each). Each classmate gets one piece from each bar: \(5 \times \frac{1}{8} = \mathbf{\frac{5}{8}}\).
b. Division Sentence: \(5 \div 8 = \frac{5}{8}\)
Problems 5, 6, 7: Modeling & Solving \(7 \div 3\) Improper to Mixed Number
5. Model of \(7 \div 3\):
7 wholes ÷ 3 groups
Distribute 7 whole units into 3 equal groups: each group receives 2 whole units, and the remaining 1 unit is partitioned into 3 thirds (\(\frac{1}{3}\)).
6. Story Problem:
"Three artists share 7 pounds of clay equally for sculpting. How many pounds of clay does each artist receive?"
7. Solve \(7 \div 3\):
\(7 \div 3 = \frac{7}{3} = 2\frac{1}{3}\)
Each artist receives \(2\frac{1}{3}\) pounds (or \(\frac{7}{3}\) pounds) of clay.
Exit Ticket: "How much sandwich?" (PDF Page 10)
Mastery Check
Question 1: 4 sandwiches shared by 5 students
How much sandwich does each student get? Show/explain reasoning.
Answer: \(\frac{4}{5}\) of a sandwich
Reasoning: Divide each sandwich into 5 equal fifths (\(\frac{1}{5}\)). Each student takes 1 fifth from each of the 4 sandwiches: \(4 \times \frac{1}{5} = \frac{4}{5}\).
Question 2: Division Expression
Write a division expression to represent the situation.
Expression: \(4 \div 5\)
Note: Both the bare expression \(4 \div 5\) and equation \(4 \div 5 = \frac{4}{5}\) demonstrate conceptual fluency.
Teacher Watch-Out: Common Misconceptions
Inverting Divisor & Dividend: Students who are used to dividing larger numbers by smaller numbers may mistakenly write \(5 \div 4\) instead of \(4 \div 5\). Anchor students in the story: "What is being cut up? That is the dividend."
Remainder as a Whole Number: Students transitioning from 4th grade may write "\(7 \div 3 = 2\text{ R }1\)". Guide them to see that the remaining 1 item is divided among 3 groups to form \(\frac{1}{3}\), giving \(2\frac{1}{3}\).
Adapted from Illustrative Math & Uncommon Schools U2, L2: Share More Sandwiches
