Fraction Multiplication Exemplar Guide
Teacher Master Exemplar Grade 5 Math • Unit 3: Lesson 1
Multiply Unit Fractions Using Area Models
CCSS 5.NF.B.4.a
B&W High-Contrast Copy Edition
Key Vocabulary & Conceptual Anchor Definitions
Unit Fraction
A fraction with numerator 1 (\(\frac{1}{d}\)), showing 1 equal part of a partitioned whole.
Area Model
A 2D rectangle model where side lengths are factors and the overlap area is the product.
"Fraction of a Fraction"
\(\frac{1}{3} \times \frac{1}{2}\) means finding \(\frac{1}{3}\) OF an existing \(\frac{1}{2}\) portion of the whole.
Double-Shaded Region
The overlapping piece representing numerator (1) out of total equal parts in whole (denominator).
Anchor Context Task 1: "Of What?" (IM 3.1.1) Time: 20 min
Situation: At a family dinner, a pan of macaroni and cheese is \(\frac{1}{2}\) full. Lin eats \(\frac{1}{3}\) of the remaining macaroni and cheese in the pan.
Model Representation
½ pan left empty ½ ⅓ of ½ Lin: ⅙ Dashed lines extend to show 6 equal parts in whole
1. Model Student Diagram Explanation:
"First, divide the entire pan in half vertically to show the \(\frac{1}{2}\) pan left. Next, partition that half into 3 equal horizontal rows to show taking \(\frac{1}{3}\) of it. Shade 1 of those 3 rows."
2. Connecting \(\frac{1}{3} \times \frac{1}{2}\) to the Diagram & Context:
"The expression represents finding a 'fraction of a fraction'. The second factor (\(\frac{1}{2}\)) is what Lin started with. The first factor (\(\frac{1}{3}\)) tells what fraction of that half Lin ate: \(\frac{1}{3} \text{ of } \frac{1}{2}\)."
3. How Much Did Lin Eat? (Product Explanation):
Answer: Lin ate \(\frac{1}{6}\) of the whole pan.
"When partition lines are extended across the entire pan, the whole is split into 6 equal-sized pieces. Lin ate exactly 1 piece, which is \(\frac{1}{6}\)."
TEACHER MONITORING CALLOUT Addressing the Incomplete Partition Misconception (\(\frac{1}{4}\) Error)
Common Misconception: Students often count 1 shaded piece out of 4 visible pieces and incorrectly conclude Lin ate \(\frac{1}{4}\).
Pedagogical Move: Ask: "Are all pieces the same size? What must be true of fractional pieces?" Direct students to extend the partition lines through the empty half. This visually proves the whole pan has 6 equal-sized portions.
Lesson Stamp
When multiplying unit fractions, we use area diagrams to show a "fraction of a fraction". The rows represent one fraction, columns represent the other, and the double-shaded overlap represents the product!
Instructional Modeling
Guided Practice: Step-by-Step Area Models
Time: 15 min
I Do (Teacher Model) \(\frac{1}{5} \times \frac{1}{4} = \frac{1}{20}\)
¼ (4 columns) ⅕ (5 rows) Overlap = 1 / 20
Step-by-Step Exemplar Answer:
"First factor \(\frac{1}{5}\) means partition into 5 equal horizontal rows and shade 1 row. Second factor \(\frac{1}{4}\) means partition into 4 equal vertical columns and shade 1 column. The model now has \(5 \times 4 = 20\) equal pieces. Exactly 1 piece is double-shaded, so \(\frac{1}{5} \times \frac{1}{4} = \frac{1}{20}\)."
Teacher Script & Cold Call: "Cold call: How many total parts? 20. How many double-shaded? 1. So numerator is 1, denominator is 20!"
We Do (Guided Discourse) \(\frac{1}{2} \times \frac{1}{8} = \frac{1}{16}\)
⅛ (8 columns) ½ (2 rows) Overlap = 1 / 16
Step-by-Step Exemplar Answer:
"Create 2 horizontal rows and shade 1 row (\(\frac{1}{2}\)). Create 8 vertical columns and shade 1 column (\(\frac{1}{8}\)). The whole square has \(2 \times 8 = 16\) equal parts. Only 1 part is double-shaded. Therefore, \(\frac{1}{2} \times \frac{1}{8} = \frac{1}{16}\)."
Turn & Talk Prompt: "Where do you see the product in the model? 1 double-shaded piece out of 16 congruent fractional sections."
You Do (Independent Check) \(\frac{1}{3} \times \frac{1}{6} = \frac{1}{18}\)
⅙ (6 columns) ⅓ (3 rows) Overlap = 1 / 18
Step-by-Step Exemplar Answer:
"Divide the whole square into 3 equal rows and shade 1 row (\(\frac{1}{3}\)). Next, divide into 6 equal columns and shade 1 column (\(\frac{1}{6}\)). The grid has \(3 \times 6 = 18\) equal pieces. Exactly 1 piece has both colors overlapping, so the product is \(\frac{1}{18}\)."
Show-Call Check: Verify students draw equal intervals and correctly count 18 total subdivisions before applying the algorithm.
Core Rule
When we partition a whole into equal parts in both directions, the double-shaded parts show the numerator and the total number of equal parts shows the denominator!
Independent Practice
Problems 1 through 5: Solutions & Analysis
Page 1 of 2
Context: \(\frac{1}{4}\) of \(\frac{1}{2}\) of a Pan of Macaroni & Cheese Problems 1, 2, 3
Diagram A (Vert)
Diagram B (Horiz)
1. Explain how each diagram represents \(\frac{1}{4}\) of \(\frac{1}{2}\):
"Both diagrams show the full pan cut in half horizontally (\(\frac{1}{2}\) remaining). In Diagram A, the half is divided vertically into 4 equal strips, with 1 strip shaded. In Diagram B, the half is divided into 4 equal rectangular blocks (2 vertical \(\times\) 2 horizontal). In both, exactly 1 out of 4 equal parts of the half is shaded."
2. Connection to \(\frac{1}{4} \times \frac{1}{2}\):
The expression means taking \(\frac{1}{4}\) OF \(\frac{1}{2}\). The base amount is \(\frac{1}{2}\), and multiplying by \(\frac{1}{4}\) partitions that half into 4 equal parts and keeps 1.
3. How much of the whole pan? Explain:
Answer: \(\frac{1}{8}\) of the whole pan.
"If you partition the other half into 4 equal parts as well, there are \(4 + 4 = 8\) equal parts in all. The shaded portion is 1 of those 8 equal parts."
Problem 4 & 5: Drawing & Evaluating \(\frac{1}{2} \times \frac{1}{5}\) Constructed Response
⅕ columns ½ rows Shaded: 1 of 10
4. Step-by-Step Drawing Instructions:
"Partition the unit square into 5 equal vertical columns and shade 1 column to represent \(\frac{1}{5}\). Next, partition the square horizontally into 2 equal rows and shade 1 row to represent \(\frac{1}{2}\). The double-shaded corner shows \(\frac{1}{2}\) of \(\frac{1}{5}\)."
5. Value of \(\frac{1}{2} \times \frac{1}{5}\) Based on Drawing:
Product: \(\frac{1}{10}\)
Reasoning: "Dividing into 5 columns and 2 rows creates \(5 \times 2 = 10\) congruent rectangles in the whole. Exactly 1 rectangle is double-shaded, which equals \(\frac{1}{10}\)."
LESSON MONITORING & ERROR PROTOCOL
Error 1: Unequal Partitioning
Students freehand irregular boxes. Move: Prompt student to check if parts are equal size: "Could everyone get an equal share?"
Error 2: Order Reversal
\(\frac{1}{4} \times \frac{1}{2}\) confused with \(\frac{1}{2} \times \frac{1}{4}\). Move: Ground in context: "We start with \(\frac{1}{2}\) pan, then take \(\frac{1}{4}\) of it."
Error 3: Over/Under Shading
Shading entire columns without intersecting. Move: Have students use two distinct pencil patterns (e.g. diagonal lines vs. dots).
Uncommon Schools 5th Grade Mathematics • Unit 3: Multiplying and Dividing Fractions • Lesson 1 Teacher Exemplar (B&W)
Independent Practice
Problems 6 through 8: Solutions & Analysis
Page 2 of 2
6. Contextual Word Problem: Sky's Fence Painting Multi-Step Modeling
Prompt: Sky and his brother are painting a fence together. Sky's responsibility is \(\frac{1}{2}\) of the fence. After one day he has completed \(\frac{1}{3}\) of his half. Draw a diagram, write a multiplication expression, and determine how much of the fence Sky has painted.
Sky's ½ Brother's ½ Sky painted: ⅙ of fence
a. Diagram Note: Rectangle split in half vertically. Left half split into 3 equal rows with 1 row shaded. Dashed lines extend across right half to show 6 equal segments.
b. Expression: \(\frac{1}{3} \times \frac{1}{2}\)
c. Answer: Sky has painted \(\frac{1}{6}\) of the total fence.
Reasoning: "Sky's job was \(\frac{1}{2}\). Taking \(\frac{1}{3}\) of \(\frac{1}{2}\) means the whole fence has \(2 \times 3 = 6\) equal parts. Sky finished 1 of those 6 parts."
7. Priya's Shaded Square Analysis Geometric Decomposition
1 unit 1 unit 5 cols \(\times\) 2 rows
a. Explain how \(\frac{1}{5} \times \frac{1}{2}\) represents the shaded area:
"The unit square has its width partitioned into 5 equal columns, making each column width \(\frac{1}{5}\). Its height is partitioned into 2 equal rows, making each row height \(\frac{1}{2}\). The shaded rectangle has dimensions \(\frac{1}{5} \times \frac{1}{2}\)."
b. Area of the shaded piece: \(\frac{1}{10}\) square unit.
Reasoning: "There are \(5 \times 2 = 10\) congruent rectangles that compose the whole unit square, and 1 is shaded."
8. Grid Expression & Fractional Area Determination Formal Expression Link
1 unit 1 unit 4 cols \(\times\) 4 rows
Multiplication Expression: \(\frac{1}{4} \times \frac{1}{4}\)
Shaded Amount: \(\frac{1}{16}\) of the whole square.
Full Reasoning: "The square is partitioned horizontally into 4 equal rows (\(\frac{1}{4}\)) and vertically into 4 equal columns (\(\frac{1}{4}\)). This forms a grid of \(4 \times 4 = 16\) congruent smaller squares. Since exactly 1 of the 16 squares is shaded, its area is \(\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\)."
Synthesis Bridge
Notice the generalization emerging across Problems 4–8: Denominator of product = (rows in whole) \(\times\) (columns in whole). This builds direct conceptual rationale for multiplying denominators before introducing the algorithm!
Uncommon Schools 5th Grade Mathematics • Unit 3: Multiplying and Dividing Fractions • Lesson 1 Teacher Exemplar (B&W)
Assessment & Practice
Exit Ticket, Homework & Mixed Review Key
Evaluation & Extension
Exit Ticket: "Macaroni and Cheese" (Mai's Portions) 5-10 min Mastery Check
Scenario: A pan of macaroni and cheese is \(\frac{1}{3}\) full. Mai eats \(\frac{1}{4}\) of the remaining macaroni and cheese in the pan.
⅓ pan empty ⅔ Mai ate: 1 / 12
a. Diagram Explanation: "The pan is partitioned into 3 columns (1 shaded for \(\frac{1}{3}\) full). That third is cut into 4 equal rows with 1 row shaded. Partition lines extended across the remaining \(\frac{2}{3}\) show 12 equal pieces in the whole."
b. Multiplication Expression: \(\frac{1}{4} \times \frac{1}{3}\)
c. How Much Did Mai Eat? Mai ate \(\frac{1}{12}\) of the whole pan. Reasoning: "There are 12 total equal parts in the entire pan, and Mai ate only 1 of them. Splitting \(\frac{1}{3}\) into 4 parts yields twelfths."
Homework: Problems 1 & 2 Solutions Independent Review
1. Figure Analysis (3 columns \(\times\) 4 rows):
a. Expression: \(\frac{1}{3} \times \frac{1}{4}\) (or \(\frac{1}{4} \times \frac{1}{3}\))
b. Shaded Amount: \(\frac{1}{12}\) of the whole square.
Reason: 3 vertical columns \(\times\) 4 horizontal rows = 12 equal parts; 1 is shaded.
2. Show \(\frac{1}{5}\), Shade \(\frac{1}{4}\) of \(\frac{1}{5}\):
Expression: \(\frac{1}{4} \times \frac{1}{5} = \frac{1}{20}\)
Shaded Amount: \(\frac{1}{20}\) of the whole square.
Method: Partition into 5 columns, shade 1; partition into 4 rows, shade 1. Overlap is 1 out of 20 equal rectangles.
Mixed Review: Problems 3 through 6 Solutions Spiral Standards
3. Hyenas in Zoos (Division):
"36 hyenas shared equally across 4 zoos."
Equation: \(36 \div 4 = 9\)
Answer: 9 hyenas in each zoo.
4. Flowers in Garden (Comparison):
"24 violets = 4 times as many violets as daisies."
Equation: \(4 \times d = 24 \implies 24 \div 4 = 6\)
Answer: 6 daisies in the garden.
5. Water Bottles (Fraction \(\times\) Whole):
"Each bottle holds \(\frac{7}{10}\) liter; 6 bottles total."
Equation: \(6 \times \frac{7}{10} = \frac{42}{10} = 4\frac{2}{10} = 4\frac{1}{5}\)
Answer: \(4\frac{1}{5}\) (or \(4.2\)) liters.
6. Area of Rectangle (Geometry):
"Dimensions: length = 12 cm, width = 8 cm."
Equation: \(\text{Area} = 12 \times 8 = 96\)
Answer: \(96\text{ cm}^2\).
Bridge to Lesson 2: In Lesson 2, students will transition from unit fractions (\(\frac{1}{b} \times \frac{1}{d}\)) to non-unit fractions (\(\frac{a}{b} \times \frac{c}{d}\)), tiling multiple shaded cells!
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Uncommon Schools 5th Grade Mathematics • Unit 3: Multiplying and Dividing Fractions • Lesson 1 Teacher Exemplar (B&W)
