Sandwich Sharing Teacher Guide
5th Grade Math • Unit 2: Lesson 1
Sharing Sandwiches
Teacher Instructional Facilitation Guide • Adapted from Illustrative Math
STANDARD CCSS 5.NF.B.3
Lesson Objective
SWBAT interpret and represent equal sharing contexts relating division and fractions in ways that make intuitive sense to them.
Core Lesson Stamp
We can represent equal sharing situations with division expressions.
Instructional Narrative & Mathematical Horizon
In earlier grades, students mastered division as either grouping (15 items in groups of 3 = 5 groups) or sharing (15 items shared into 3 groups = 5 items each), always yielding whole numbers or whole numbers with remainders.
Unit 2 Lesson 1 opens the horizon to situations where portions do not evenly divide into whole numbers (e.g., \(15 \div 6\)). Before formalizing the equivalence \(a \div b = \frac{a}{b}\), students use intuitive sandwich sharing to build a conceptual bridge:
Quotient > 1
More sandwiches than people. Each person receives at least one full sandwich plus fractional parts (e.g., \(3\) sandwiches for \(2\) people).
Quotient < 1
Fewer sandwiches than people. Each person receives strictly less than a whole sandwich; wholes must be partitioned (e.g., \(2\) sandwiches for \(3\) people).
Recommended Lesson Pacing
| Component | Time | Core Focus & Mini-Stamp |
|---|
| Do Now | 5 min | Multiplicative comparison, fraction addition review, multi-digit operations. |
| Oral Drill & Excellence | 10 min | Vocabulary precision (dividend, divisor, product) and rapid multiplication facts. |
| Warm-Up (WODB) | 10 min | Mini-Stamp: Sharing means splitting a total into equal groups. |
| Task 1: Share Sandwiches | 20 min | Mini-Stamp: We can represent equal sharing situations with division expressions. |
| Independent Practice | 20 min | Modeling sharing > 1 vs < 1, mapping dividend (total) and divisor (shares). |
| Exit Ticket | 5-10 min | Assessment of fair sharing model: 3 sandwiches shared among 4 people (\(3 \div 4\)). |
Vocabulary Progression & Materials
Before Discourse: fraction share equal
During Discourse: divide fair size of piece # of pieces
Materials: Colored pencils (for visual shares).
Uncommon Schools • 5th Grade Math • U2 L1 Teacher Facilitation Guide • Page 1 of 3
Activity Facilitation Protocols
Step-by-step launch, monitoring pathways, and discussion synthesis
Instructional Execution
Warm-Up: Which One Doesn’t Belong? (10 Minutes)
IM Task 5.2.1 Routine
Purpose: Allow scholars to informally engage with sandwich sharing before interpreting division expressions.
1. Launch (1-2 min) Display images A–D. Give 1 min quiet think time. Direct scholars to record their choice and justification.
2. Partner Share (2-3 min) Scholars compare choices. Prompt: "Did your partner look at the number of pieces or whether they were equal?"
3. Synthesis (5 min) Record student responses. Emphasize that there are no wrong answers if mathematically justified.
Target Monitoring Look-Fors & Justifications:
• A doesn’t belong: Only one cut into many rolled pieces/wraps.
• B doesn’t belong: Only one that is whole / not cut at all.
• C doesn’t belong: Only one split cleanly into 2 equal halves.
• D doesn’t belong: Only one cut into unequal pieces (some large, some small).
🔑 Mini-Stamp: "Sharing means splitting a total into equal groups."
Task 1: Share Sandwiches (20 Minutes)
Notice & Wonder + Small Group
Phase 1: Numberless Launch (Notice & Wonder)
Display ONLY: "_____ sandwiches are shared equally by _____ people." (Do not open student packet yet).
What do you notice? What do you wonder?
- No numbers are given.
- How many sandwiches? How many people?
What numbers make sense? Why?
- Equal amounts (1 sandwich per person).
- More people than food (sharing at a picnic).
Phase 2: Independent (2 min) & Partner Work (6 min)
Scholars select two numbers from {2, 3, 5} once each. Monitor actively and select two distinct scholars to share in the whole-group discussion:
STRATEGY 1: Quotient > 1 (e.g., 3 sandwiches, 2 people)
Scholar draws 3 wholes, assigns 1 whole to each person, then splits 3rd sandwich in half.
Prompt: "Does each person get more than 1 full sandwich? How do you know?"
STRATEGY 2: Quotient < 1 (e.g., 2 sandwiches, 3 people)
Scholar draws 2 wholes, partitions each into 3 equal pieces, giving 1 piece of each sandwich to each person (\(\frac{2}{3}\)).
Prompt: "Why must each person get less than 1 whole sandwich?"
Phase 3: Whole-Group Discourse (8 min)
Teacher Script: "We have equal sharing situations. How can we represent sharing mathematically? In math, when we split a total into equal groups, what operation is that? (Division!) What division expressions can we write for these scenarios?"
Record expressions: \(2 \div 3\), \(3 \div 2\), \(2 \div 5\), \(5 \div 2\), \(3 \div 5\), \(5 \div 3\). Label the Dividend (sandwiches/total) and Divisor (people sharing).
🔑 Lesson Stamp: We can represent equal sharing situations with division expressions.
Uncommon Schools • 5th Grade Math • U2 L1 Teacher Facilitation Guide • Page 2 of 3
Independent Practice & Assessment Guide
Error analysis, oral drill questions, and exit ticket success criteria
Diagnostic Support
Targeted Independent Practice Interventions
Error #1: Drawing the Group or Size Instead of the Total Observed in IP #1a
Symptom: For 3 students sharing 18 sheets of paper, student draws 3 figures or sets of 3, instead of modeling the 18 total sheets.
Monitoring Question: "What do we have that we are sharing? Are we sharing people, or are we sharing paper? How many total items are being divided?"
Redirect Stamp: "Begin by drawing the total that we are sharing. Then make equal shares based on the number of people."
Error #2: Inverting Dividend and Divisor (Writing Larger Number First) Observed in IP #3, #5
Symptom: For 5 friends sharing 4 cookies, scholar writes \(5 \div 4\) (habitually putting the greater number first from whole number division).
Monitoring Question: "What is being cut up and shared? (The cookies). So what is our total? What tells us how many equal shares to make? (The friends)."
Redirect Stamp: "The dividend (first number) is what is being shared. The divisor (second number) is the number of shares: Total \(\div\) Shares."
Oral Drill Protocol & Spiral Review Keys (5 min)
| # | Focus | Question Prompt | Target Scholar Response |
|---|
| 1 | Spiral Review | What do we call the numbers being multiplied? | Factors |
| 2 | Spiral Review | What does the word "product" mean in mathematics? | The result/answer of multiplication |
| 3 | Spiral Review | In \(156 \div 12\), what is the divisor? | 12 (the number we divide by) |
| 4 | Spiral Review | In \(156 \div 12\), what is the dividend? | 156 (the number being divided) |
| 5 | Spiral Review | What operation checks if our quotient is correct? | Multiplication (\(\text{quotient} \times \text{divisor}\)) |
| 6-10 | Teacher RTD | Rapid target division/fraction review: e.g. \(18 \div 3\), \(24 \div 4\)... | Quick mental calculation & units |
Exit Ticket Analysis (3 Sandwiches Shared by 4 People)
Success Criteria (Mastery):
- Draws 3 wholes representing the 3 sandwiches.
- Partitions each sandwich into 4 equal quarters (or decomposes into halves + fourths).
- Allocates 1 piece from each sandwich to each person (totaling \(\frac{3}{4}\) sandwich per person).
- Clearly justifies equality: each person receives the same portion from each sandwich.
Next Day Instructional Action:
If >25% scholars invert the expression or fail to partition equal shares, launch Lesson 2 with a warm-up comparing a strip diagram of 3 shared by 4 (\(3 \div 4\)) with a strip diagram of 4 shared by 3 (\(4 \div 3\)).
Uncommon Schools • 5th Grade Math • U2 L1 Teacher Facilitation Guide • Page 3 of 3
Sandwich Sharing Answer Key
Teacher Answer Key & Exemplars
Sharing Sandwiches
Grade 5 Math • Unit 2: Lesson 1 • Complete Solutions & Models
STANDARD CCSS 5.NF.B.3
Do Now Answer Key (Packet pp. 5–6)
#1 (4.OA.A.1) Takara's Brother C
\(b = 4 \times 2\) (8 years old). Takara is 4, brother is 2 times as old.
#2 (4.NF.B.3) Remaining Pizza B
\(\frac{3}{8} + \frac{2}{8} = \frac{5}{8}\). Left pizza has \(\frac{3}{8}\) left; right has \(\frac{2}{8}\) left.
#3 (5.NBT.5) School Pencils C
\(24 \times 135 = 3{,}240\) pencils in all. Standard multi-digit multiplication.
#5 (Spicy) Apple Baskets C
\(1{,}575 \div 35 = 45\) apples per basket. \(35 \times 40 = 1{,}400\); \(35 \times 5 = 175\).
Excellence Fact Sprint Key (Packet p. 8 • 40 Problems)
Row-by-row solutions
R1: 16, 80, 15, 40, 35, 24, 4, 72, 54, 32
R2: 27, 24, 8, 12, 36, 16, 72, 12, 21, 18
R3: 42, 10, 30, 30, 6, 64, 90, 15, 28, 9
R4: 20, 18, 27, 54, 16, 90, 20, 42, 60, 32
R5: 30, 24, 81, 20, 8, 100, 40, 12, 18, 48
Warm-Up: WODB (p. 9)
- A: Only one cut into many pinwheels/wraps.
- B: Only one that is whole / not divided at all.
- C: Only one split cleanly in 2 equal halves.
- D: Only one cut into unequal pieces (unfair share).
Task 1: Share Sandwiches (p. 11)
Exemplar Choice A: 3 Sandwiches, 2 People (Quotient > 1) Expression: \(3 \div 2\) • Each person gets 1 whole sandwich + \(\frac{1}{2}\) sandwich = \(1\frac{1}{2}\) or \(\frac{3}{2}\).
Explanation: There are more sandwiches than people, so each person gets more than 1 whole.
Exemplar Choice B: 2 Sandwiches, 3 People (Quotient < 1) Expression: \(2 \div 3\) • Cut each sandwich into 3 thirds. Each gets 1 third from each = \(\frac{2}{3}\) sandwich.
Explanation: Fewer sandwiches than people, so each person gets less than 1 whole.
Uncommon Schools • 5th Grade Math • U2 L1 Answer Key & Exemplars • Page 1 of 3
Independent Practice Solutions (Part 1)
Step-by-step models, division expressions, and reasoning explanations
Packet pp. 12–14
Problem 1: Construction Paper & Glue (Packet p. 12)
