Shaded Area Solution Guide Shaded Region Area Guide
Unit Square Partitioning & Geometric Probability
Unit Square (\(1 \times 1 = 1\))
Problem: A unit square is partitioned into 3 horizontal rows and 4 vertical columns . If 2 rows and 3 columns are shaded, what is the area of the shaded region?
Grid Partitioning Basics
Total Cells = \(3 \times 4 = 12\) identical rectangles • Area of each cell = \(\frac{1}{3} \times \frac{1}{4} = \mathbf{\frac{1}{12}}\)
1 Grid Cell = \(\frac{1}{12}\text{ units}^2\)
Interpretation A: Overlap
Multiplication Model
Common in fraction multiplication : shading represents the overlapping region of \(\frac{2}{3}\) and \(\frac{3}{4}\).
✓ ✓ ✓ ✓ ✓ ✓
Overlapping cells: \(2 \times 3 = 6\text{ cells}\)
Fraction calculation: \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12}\)
Calculated Area: \(\mathbf{\frac{1}{2}}\) (or \(0.5\))
Interpretation B: Union
Full Set Union
If 2 full rows and 3 full columns are completely shaded, we calculate their combined total area.
WHITE
Inclusion-Exclusion: \(8 + 9 - 6 = 11\text{ cells}\)
Complement trick: \(1 - \frac{1}{12} = \frac{11}{12}\)
Calculated Area: \(\mathbf{\frac{11}{12}}\) (\(\approx 0.917\))
Summary & Quick Reference
Overlap Question?
If "shaded by both", the area is 1/2 .
Union Question?
If "shaded by either", the area is 11/12 .
Single Cell Value
Each grid rectangle represents exactly 1/12 .
Mixed Number Area Worksheet Grade 5–6 Mathematics • Area & Fractions
Mixed Number Area Models
Name:
Date:
Score:
4-Step Method:
1. Mixed Dimensions → 2. Improper Fractions → 3. Multiply Numerators & Denominators → 4. Convert Back to Mixed Number
Worked Example: Partitioned Grid with Mixed Numbers
Study Before Solving
Width = 1 1/2 Height = 1 1/3
Step 1 & 2: Convert to Improper \(1\frac{1}{3} = \frac{4}{3}\) • \(1\frac{1}{2} = \frac{3}{2}\)
Step 3: Multiply Fractions \(\text{Area} = \frac{4}{3} \times \frac{3}{2} = \frac{12}{6}\)
Step 4: Convert to Mixed / Whole \(\frac{12}{6} = \mathbf{2\text{ units}^2}\)
1
Partitioned Garden Bed (Thirds & Fourths)
4 Points
A garden bed has a height of \(1\frac{2}{3}\) yards (5 rows of thirds) and a width of \(1\frac{1}{4}\) yards (5 columns of fourths). Find the total area.
A. Improper Height
\(1\frac{2}{3} =\)
B. Improper Width
\(1\frac{1}{4} =\)
C. Multiply Fractions
\((\text{Improper Product})\)
D. Final Mixed Number
\((\text{Simplified } \text{yd}^2)\)
2
Tiled Floor Section (Halves & Thirds)
4 Points
A hallway rug has dimensions \(2\frac{1}{3}\) meters by \(1\frac{1}{2}\) meters . Calculate the area covered by the rug.
A. Improper Length
\(2\frac{1}{3} =\)
B. Improper Width
\(1\frac{1}{2} =\)
C. Multiply Fractions
\((\text{Improper Product})\)
D. Final Mixed Number
\((\text{Simplified } \text{m}^2)\)
Mixed Number Area Models • Page 1 of 2 Turn page for word problems & challenge →
Applied Area & Reverse Reasoning
Show complete steps: Mixed → Improper → Multiply → Simplified Mixed
Page 2 of 2
3
Stone Patio Construction
4 Points
A builder lays flagstone for an outdoor patio measuring \(2\frac{2}{3}\) meters in length and \(1\frac{3}{4}\) meters in width.
Show Your Work:
Final Area:
Improper Product:
Final Mixed Number:
4
Rooftop Solar Array
4 Points
A rooftop solar panel array measures \(3\frac{1}{2}\) feet wide by \(2\frac{2}{5}\) feet tall. Find the total square footage of the array.
Show Your Work:
Final Area:
Improper Product:
Final Mixed Number:
Challenge: Reverse Dimensions
Bonus
An artist's canvas has a known width of \(1\frac{1}{4}\) feet and a total area of \(3\frac{1}{8}\) square feet . Use improper fractions to find the unknown length: \(\text{Length} = \text{Area} \div \text{Width}\).
Show Your Work:
Canvas Length:
Division (\(\frac{25}{8} \div \frac{5}{4}\)):
Mixed Number Length:
Self-Check: Did you convert to improper first, multiply straight across, and simplify your final mixed number?
Yes Needs Review
Mixed Number Area Answer Key Teacher Answer Key
Official Solutions
Mixed Number Area Models • Conversions & Calculations
Total Points: 20 Points
Solutions At A Glance
All units must be included for full credit
#
Problem Context
Improper Fractions
Improper Product
Final Mixed Number
Pts
1
Garden Bed (\(1\frac{2}{3} \times 1\frac{1}{4}\))
\(\frac{5}{3} \times \frac{5}{4}\)
\(\frac{25}{12}\)
\(2\frac{1}{12}\text{ yd}^2\)
4
2
Hallway Rug (\(2\frac{1}{3} \times 1\frac{1}{2}\))
\(\frac{7}{3} \times \frac{3}{2}\)
\(\frac{21}{6} = \frac{7}{2}\)
\(3\frac{1}{2}\text{ m}^2\)
4
3
Stone Patio (\(2\frac{2}{3} \times 1\frac{3}{4}\))
\(\frac{8}{3} \times \frac{7}{4}\)
\(\frac{56}{12} = \frac{14}{3}\)
\(4\frac{2}{3}\text{ m}^2\)
4
4
Solar Array (\(3\frac{1}{2} \times 2\frac{2}{5}\))
\(\frac{7}{2} \times \frac{12}{5}\)
\(\frac{84}{10} = \frac{42}{5}\)
\(8\frac{2}{5}\text{ ft}^2\)
4
5
Canvas Challenge (\(3\frac{1}{8} \div 1\frac{1}{4}\))
\(\frac{25}{8} \div \frac{5}{4} = \frac{25}{8} \times \frac{4}{5}\)
\(\frac{100}{40} = \frac{5}{2}\)
\(2\frac{1}{2}\text{ ft}\)
4
Page 1 Detailed Solutions
Problem 1: Garden Bed
• Step A: \(1\frac{2}{3} = \frac{(1 \times 3) + 2}{3} = \mathbf{\frac{5}{3}}\) (1 pt)
Problem 2: Hallway Rug
• Step A: \(2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \mathbf{\frac{7}{3}}\) (1 pt)
Page 2 Detailed Solutions
Problem 3: Flagstone Patio
• Convert: \(2\frac{2}{3} = \frac{8}{3}\); \(1\frac{3}{4} = \frac{7}{4}\)
Problem 4: Solar Panel Array
• Convert: \(3\frac{1}{2} = \frac{7}{2}\); \(2\frac{2}{5} = \frac{12}{5}\)
Problem 5: Challenge Reverse Canvas
• Area \(= \frac{25}{8}\); Width \(= \frac{5}{4}\)
Common Student Misconceptions & Teacher Notes
Misconception: Multiplying wholes & fractions separately Students often try \((2 \times 1) + (\frac{2}{3} \times \frac{3}{4})\), omitting cross terms. Emphasize that converting to improper fractions ensures all 4 partial areas are multiplied correctly.
Simplification & Labeling Reminder Accept equivalent unsimplified mixed numbers (e.g. \(4\frac{8}{12}\) for Problem 3) for partial credit, but award full credit for simplified fractions and correct unit labels (\(\text{m}^2\), \(\text{yd}^2\), \(\text{ft}^2\)).
Poster Dimensions Worksheet Fractions & Measurement • Area of Rectangles
Poster Dimensions Worksheet
Name:
Date:
Score:
Quick Conversion Review
Mixed → Improper: Multiply whole by denominator, add numerator: \(W\frac{N}{D} = \frac{(W \times D) + N}{D}\)
Example: \(3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \mathbf{\frac{7}{2}}\)
Part 1: Fraction Conversion Warm-Up
6 Points
Convert to Improper: \(2\frac{3}{4} =\)
Convert to Improper: \(3\frac{1}{2} =\)
Convert to Improper: \(2\frac{1}{4} =\)
Convert to Mixed: \(\frac{17}{3} =\)
Convert to Mixed: \(\frac{25}{4} =\)
Convert to Mixed: \(\frac{63}{8} =\)
1
The Rectangular Poster Problem (Anchor)
Guided Focus
A rectangular movie poster measures \(3\frac{1}{2}\) feet wide by \(2\frac{1}{4}\) feet long . What is the total area of the poster in square feet?
Step 1: Improper Width
\(3\frac{1}{2}\text{ ft} =\)
Step 2: Improper Length
\(2\frac{1}{4}\text{ ft} =\)
Step 3: Multiply Fractions
\(\text{Numerator} \times \text{Denom}\)
Step 4: Final Mixed Area
\(\text{Square Feet } (\text{ft}^2)\)
2
Art Gallery Canvas
4 Points
An artist stretches canvas for an oil painting that is \(4\frac{1}{2}\) feet long and \(1\frac{2}{3}\) feet wide . Calculate the area of the canvas.
Show Your Work (Convert & Multiply):
Final Area:
Mixed Number (\(\text{ft}^2\)):
Poster Dimensions Worksheet • Page 1 of 2 Turn page for additional poster & banner problems →
Independent Practice & Multi-Step Challenge
Apply the 4-step method to find exact rectangular areas
Page 2 of 2
3
Vinyl Concert Banner
4 Points
A music band orders a vinyl stage banner measuring \(5\frac{1}{3}\) feet long by \(2\frac{1}{4}\) feet tall . What is the area of the banner?
Work Space (Improper Fractions):
Area (\(\text{ft}^2\)):
Answer:
4
Science Fair Trifold Board
4 Points
A student's science fair display board has dimensions of \(3\frac{1}{5}\) feet wide by \(2\frac{3}{4}\) feet high . What is its total display area?
Work Space (Improper Fractions):
Area (\(\text{ft}^2\)):
Answer:
5
Framed Photography Print
4 Points
A framed nature photo measures \(2\frac{2}{3}\) feet long by \(1\frac{7}{8}\) feet wide . Find the glass surface area needed to cover the print.
Work Space (Improper Fractions):
Area (\(\text{ft}^2\)):
Answer:
Challenge: Poster Cutout Border
Bonus
A graphic designer has a large poster board measuring \(4\frac{1}{2}\) feet by \(3\) feet . She cuts out a center photo window measuring \(2\frac{1}{4}\) feet by \(1\frac{1}{2}\) feet . What is the remaining border area of the board?
Show Total Area − Cutout Area:
Remaining Area:
Border (\(\text{ft}^2\)):
Final Verification: Did you label all final answers with square feet (\(\text{ft}^2\)) and simplify all fractions?
Completed
Poster Dimensions Answer Key Teacher Answer Key
Official Solutions
Poster Dimensions Worksheet • Mixed Numbers to Improper & Area
Total Points: 26 Points
Part 1: Warm-Up Conversions (1 Pt Each = 6 Pts)
\(2\frac{3}{4} =\) \(\frac{11}{4}\)
\(3\frac{1}{2} =\) \(\frac{7}{2}\)
\(2\frac{1}{4} =\) \(\frac{9}{4}\)
\(\frac{17}{3} =\) \(5\frac{2}{3}\)
\(\frac{25}{4} =\) \(6\frac{1}{4}\)
\(\frac{63}{8} =\) \(7\frac{7}{8}\)
Problems 1–6 Solutions At A Glance
Requires correct units (\(\text{ft}^2\))
#
Problem
Improper Conversion
Improper Product
Final Area
Pts
1
Anchor Poster (\(3\frac{1}{2} \times 2\frac{1}{4}\))
\(\frac{7}{2} \times \frac{9}{4}\)
\(\frac{63}{8}\)
\(7\frac{7}{8}\text{ ft}^2\)
4
2
Art Canvas (\(4\frac{1}{2} \times 1\frac{2}{3}\))
\(\frac{9}{2} \times \frac{5}{3}\)
\(\frac{45}{6} = \frac{15}{2}\)
\(7\frac{1}{2}\text{ ft}^2\)
4
3
Concert Banner (\(5\frac{1}{3} \times 2\frac{1}{4}\))
\(\frac{16}{3} \times \frac{9}{4}\)
\(\frac{144}{12}\)
\(12\text{ ft}^2\)
4
4
Trifold Board (\(3\frac{1}{5} \times 2\frac{3}{4}\))
\(\frac{16}{5} \times \frac{11}{4}\)
\(\frac{176}{20} = \frac{44}{5}\)
\(8\frac{4}{5}\text{ ft}^2\)
4
5
Framed Print (\(2\frac{2}{3} \times 1\frac{7}{8}\))
\(\frac{8}{3} \times \frac{15}{8}\)
\(\frac{120}{24}\)
\(5\text{ ft}^2\)
4
6
Cutout Challenge
\(\frac{27}{2} - \frac{27}{8} = \frac{108}{8} - \frac{27}{8}\)
\(\frac{81}{8}\)
\(10\frac{1}{8}\text{ ft}^2\)
4
Problem 1 (Anchor Poster) Walkthrough 4 Points
• Width: \(3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \mathbf{\frac{7}{2}}\) (1 pt)Length: \(2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \mathbf{\frac{9}{4}}\) (1 pt)Multiply: \(\frac{7 \times 9}{2 \times 4} = \mathbf{\frac{63}{8}}\) (1 pt)Convert back: \(63 \div 8 = 7\text{ R }7 \rightarrow \mathbf{7\frac{7}{8}\text{ ft}^2}\) (1 pt)
Problem 6 (Cutout Challenge) Walkthrough 4 Points
• Total Board: \(4\frac{1}{2} \times 3 = \frac{9}{2} \times \frac{3}{1} = \frac{27}{2} = \mathbf{13\frac{1}{2}\text{ ft}^2}\)Window Cutout: \(2\frac{1}{4} \times 1\frac{1}{2} = \frac{9}{4} \times \frac{3}{2} = \frac{27}{8} = \mathbf{3\frac{3}{8}\text{ ft}^2}\)Subtract: \(13\frac{4}{8} - 3\frac{3}{8} = \mathbf{10\frac{1}{8}\text{ ft}^2}\) (or \(\frac{108 - 27}{8} = \mathbf{\frac{81}{8}}\))
Grading Rubric & Differentiation Tip
Cross-Simplification Strategy: Encourage students in Problems 3, 4, and 5 to simplify before multiplying (e.g., in Problem 3, cancel 4 into 16 to get 4, and cancel 3 into 9 to get 3, yielding \(4 \times 3 = 12\) directly).
Partial Credit Breakdown: Give 2 points for correct improper fraction conversions, 1 point for the improper fraction product, and 1 point for the correctly simplified mixed number with square feet (\(\text{ft}^2\)).
