Fraction Multiply TMD Tracker
Skills TMD & Data Collection Tool
Special Education Push-In & ICR Internalization Plan
Teacher:
Date/Pd:
Lesson Title: U3, L1–3: Multiply Fraction by Fraction Using Diagrams
Objective/Std: 5.NF.B.4a — Represent fraction multiplication as overlapping area models (\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)).
ICR Push-In Table Target:
Fluency with unit/non-unit partitioning & cognitive ownership
1. Yesterday's Exit Ticket Data & Gap Reflection
Overall ET Mastery: Low (<50%) Med (51–75%) High (>75%)
Based on this data, what is the primary academic gap that needs to be closed for today's lesson?
Students correctly partitioned vertical columns for the first fraction, but struggled to partition horizontally for "fraction of a fraction", often treating numerators as addition rather than intersecting region.
Strategic adjustments today (targeted scaffolds, easel visuals, small group pull & BIDs):
Dual-color shading strategy (yellow vertical columns + blue horizontal rows = green double-shaded product); use tactile laminated grid tiles for Tier 2 group.
2. Co-Teacher Collaboration (Daily Check-In & Alignment) Mandatory Daily Sync
Daily Skills Focus
Recognizing that "\(\frac{a}{b}\) of \(\frac{c}{d}\)" means taking a part of a part; connecting double-shaded area to total rectangular subdivisions.
Resources, Tools & Strategies
Pre-drawn square boundary boxes, 2-color highlighters, visual step checklist (Column first, Row second, Double-shade count), Reference Sheet.
Predicted Gaps & Scaffolds
Count confusion: counting only unshaded parts as denominator. Shared prompt: "How many total equal pieces make up the entire 1 whole pan?"
3. Student-by-Student Accommodations (8 Students) Push-In / ICR Individualized Supports
| # | Student Name | IEP / 504 Accommodations & Specific Scaffolds | In-Class Push-In Focus / Note |
|---|
| 1 | Student 1 | | |
| 2 | Student 2 | | |
| 3 | Student 3 | | |
| 4 | Student 4 | | |
| 5 | Student 5 | | |
| 6 | Student 6 | | |
| 7 | Student 7 | | |
| 8 | Student 8 | | |
✓
Visual Ref Sheet
Step checklist & grids
✓
Graphic Frame
Pre-drawn 1-unit squares
✓
2-Color Scaffolding
Vertical vs horizontal
✓
Reduced Workload
75% & 50% modified tiers
5. Heart of the Lesson (HOTL): Facilitation & CFUs Mode: GP (We Do) & Small Group Push-In
Keys to Success (Student-Facing Step Method)
- Partition & Shade 1st Fraction: Cut vertically into denominator parts; shade numerator columns.
- Partition & Shade 2nd Fraction: Cut horizontally into denominator parts; shade numerator rows across whole pan.
- Count & Name Product: Count double-shaded squares (numerator) over total squares in whole (denominator).
Guiding Questions ("We Do")
Q1: "Which fraction represents what was left in the pan initially?"
Q2: "Why do we count the tiny rectangles to find our new denominator?"
STAMP: "Multiplying fractions finds a part of a part. The overlap is the numerator; all pieces are the denominator."
Daily Routine: Co-Teacher Sync ✓ | Answer Key ✓ | Modified Packet ✓ Page 1 of 4 • Context, Co-Teacher Sync, Accommodations & HOTL
6. Moment of Productive Struggle #1 (MPS 1)
First Starred Independent Practice Question • Core Fluency Checkpoint
Problem / Item #: Task 2 / Problem 4 Full 6 Laps
Identify 2–3 Potential Errors & Break It Down (BID) Prompts to Respond
| Anticipated Error / Misconception | BID / Teacher Prompt (Scaffold Question) |
|---|
| 1. Adding denominators: Student adds denominators (\(2+3=5\)) instead of finding the product of subdivisions (\(2 \times 3 = 6\)). | "Look at the whole square: how many total equal parts do you see when both horizontal and vertical lines cross? Count them. Is it 5 or 6?" |
| 2. Shading only remaining: Student shades the second fraction only inside the first column, leaving rest of grid unpartitioned. | "To know the size of a fraction of the whole, extend your cut lines all the way across the entire square. What size pieces did we make?" |
| 3. Operation confusion: Student attempts to divide or gets stuck deciding which operation represents 'part of a part'. | "Check step 1 on your Reference Sheet: 'Taking \(\frac{1}{3}\) of \(\frac{1}{2}\) means multiplication!' Write \(\frac{1}{3} \times \frac{1}{2}\)." |
Plan a Mini-RTD for Most Likely Error (Easel Script)
"Pencils down in 3, 2, 1. Eyes on easel. Watch my marker: 1 vertical cut makes halves. Now I make 2 horizontal cuts all the way across. Count the total boxes with me: 1, 2, 3, 4, 5, 6! Why is our denominator 6?"
Script a Stamp (MPS 1 Takeaway)
Stamp 1: "When we cut a fraction into equal parts, both cuts must go across the whole unit. The total number of little rectangles is always the product of the denominators!"
Name Your AM Laps 1 – 6 (Specific Look-Fors for Each Lap)
Lap 1
Square setup & 1st vertical partition
Lap 2
Correct 1st shading (columns)
Lap 3
2nd horizontal partitions full width
Lap 4
Cross-shading / overlap identification
Lap 5
Total unit pieces counted (denominator)
Lap 6
Final expression & sentence explanation
Real-Time ICR Student Data Tracker (6 Laps • 8 Students) Record Live Checkpoints During Push-In
| Student Name | L1 | L2 | L3 | L4 | L5 | L6 | Error Code / Observation | BID Delivered |
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| 1. | ○ | ○ | ○ | ○ | ○ | ○ | | |
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| 8. | ○ | ○ | ○ | ○ | ○ | ○ | | |
Push-In Move: If 2+ students make the same error on Lap 2 or 3, pause small group and execute 90-second Mini-RTD. Page 2 of 4 • MPS 1 Error Analysis & 8-Student 6-Lap Tracking
7. Moment of Productive Struggle #2 (MPS 2)
Second Starred Independent Practice Question • Rigor & Transfer Checkpoint
Problem / Item #: Word Problem Transfer (Pizza/Fence) Full 6 Laps
Identify 2–3 Potential Errors & Break It Down (BID) Prompts to Respond
| Anticipated Error / Misconception | BID / Teacher Prompt (Scaffold Question) |
|---|
| 1. Non-unit numerator error: Multiplying \(\frac{2}{3} \times \frac{4}{5}\), student correctly identifies \(\frac{1}{15}\) unit piece but counts only 1 overlap rectangle instead of \(2 \times 4 = 8\). | "Put your finger on the green cross-shaded array. How many rows high is it? (2). How many columns wide? (4). What is \(2 \times 4\)?" |
| 2. Misidentifying the whole: Student writes the answer as a fraction of only the shaded part, rather than a fraction of the entire item. | "The question asks 'What fraction of the WHOLE pizza?' What does the entire square represent? How many total pieces make the whole?" |
| 3. Word problem translation stall: Student cannot extract fractions from text without visual cues. | "Highlight the 1st fraction with yellow: What part was available? Highlight the 2nd fraction with blue: What part was used? What operation connects 'part of a part'?" |
Plan a Mini-RTD for Most Likely Error (Easel Script)
"Stop and track the board. When we have non-unit numerators like \(\frac{2}{3} \times \frac{4}{5}\), we don't just have 1 tiny box shaded twice. We have an entire block of \(2 \times 4\) boxes! Count: 2, 4, 6, 8! Over 15 total boxes. Result = \(\frac{8}{15}\)."
Script a Stamp (MPS 2 Takeaway)
Stamp 2: "Numerator \(\times\) Numerator tells us how many pieces are double-shaded. Denominator \(\times\) Denominator tells us the total pieces in 1 whole. Multiply across!"
Name Your AM Laps 1 – 6 (Transfer & Non-Unit Numerators)
Lap 1
Context unpack & equation target
Lap 2
Vertical partitioning (5 columns)
Lap 3
Horizontal partition (3 rows)
Lap 4
Array overlap count (\(a \times c\))
Lap 5
Total grid calculation (\(b \times d\))
Lap 6
Simplification & unit label included
Real-Time ICR Student Data Tracker (6 Laps • 8 Students) Record Live Transfer Checkpoints
| Student Name | L1 | L2 | L3 | L4 | L5 | L6 | Error Code / Observation | BID Delivered |
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| 1. | ○ | ○ | ○ | ○ | ○ | ○ | | |
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Push-In Priority: Provide pre-drawn square diagram and yellow/blue markers to maintain cognitive focus on the math relationship. Page 3 of 4 • MPS 2 Error Analysis & 8-Student 6-Lap Tracking
8. Exit Ticket & Assessment Tracker
Backwards Planning Core • Mandatory In-Class Assessment Collection Tool
Mandatory Non-Negotiable
Exit Ticket Aggressive Monitoring Focus & Stamp Immediate Live Scans (3 Minutes)
ET Lap 1 (Initial Setup / Strategy)
Look for: Student partitions vertical columns for \(\frac{1}{3}\) brownies and horizontal rows for Mai's \(\frac{1}{4}\) cut across the whole box.
ET Lap 2 (Calculation / Evidence)
Look for: Student writes \(\frac{1}{4} \times \frac{1}{3} = \frac{1}{12}\) and identifies 1 square out of 12 total squares in the whole brownie pan.
Conceptual Stamp / Key Rule
"The whole is 1 complete pan. Multiplying the denominators gives the size of each part. Overlap gives how many parts were eaten."
ICR Student Assessment Data Collection Tool (8 Students) Fill during collection / grading • Live Mastered vs. Gap Tally
| Student Name | Score / % | Level | Primary Misconception / Gap | Tomorrow's Push-In Action |
|---|
| 1. | ____ / 2 | L / M / H | | |
| 2. | ____ / 2 | L / M / H | | |
| 3. | ____ / 2 | L / M / H | | |
| 4. | ____ / 2 | L / M / H | | |
| 5. | ____ / 2 | L / M / H | | |
| 6. | ____ / 2 | L / M / H | | |
| 7. | ____ / 2 | L / M / H | | |
| 8. | ____ / 2 | L / M / H | | |
Next-Day Push-In Priority & Strategy
Priority Standard / Gap to Target:
5.NF.B.4a — Moving from unit fractions (\(\frac{1}{b} \times \frac{1}{d}\)) to mixed non-unit fraction multiplication arrays (\(\frac{a}{b} \times \frac{c}{d}\)) and checking fluency through standard algorithm multiplication facts.
Planned Grouping / Push-In Delivery:
Tier 2 Station: 5-minute fluency warmup on basic multi-digit and single-digit factors; transparent fraction overlay manipulatives for students below 60% mastery on ET.
Data Cycle Complete: Use this Exit Ticket data to fuel Tomorrow's Page 1 Reflection. Page 4 of 4 • Exit Ticket & Assessment Tracker
Fraction Diagrams Reference Sheet
Fraction Diagrams Reference Guide
IEP Math Anchor Tool • Visual Fraction Multiplication Steps
5th Grade Math Standard 5.NF.B.4a
\(\times\)
The Big Math Idea: "Fraction of a Fraction"
When you multiply two fractions less than 1, you are finding a part of a part. The product is always smaller than either starting fraction! \(\frac{\text{Numerator } a}{\text{Denominator } b} \times \frac{\text{Numerator } c}{\text{Denominator } d} = \frac{a \times c \text{ (Double-Shaded Overlap)}}{b \times d \text{ (Total Equal Squares in 1 Whole)}}\)
4-Step Action Checklist (How to Draw Any Diagram)
STEP 1 Setup
Unpack Word Problem
Ask: "What fraction do I start with?" and "What fraction of it is being used?"
\(\frac{1}{3}\) of \(\frac{1}{2} \rightarrow \frac{1}{3} \times \frac{1}{2}\)
STEP 2 Vertical
Cut Columns (1st Frac)
Cut the square vertically into equal columns (denominator). Lightly shade numerator columns.
Cut down \(\downarrow\) (e.g., 2 halves)
STEP 3 Horizontal
Cut Rows (2nd Frac)
Cut horizontally all the way across into equal rows. Shade numerator rows in opposite direction.
Cut across \(\rightarrow\) (e.g., 3 thirds)
STEP 4 Answer
Count & Name
Numerator = Overlap squares.
Denominator = Total squares in whole.
Product = \(\frac{\text{Overlap}}{\text{Total}}\)
Example 1: Unit Fractions (\(\frac{1}{3} \times \frac{1}{2}\)) 1 Overlap Tile
★ \(\frac{1}{6}\)
row
col
col
1. Vertical cuts: 2 columns (\(\frac{1}{2}\) pan shaded).
2. Horizontal cuts: 3 rows (\(\frac{1}{3}\) cut across).
3. Overlap (★): Exactly 1 piece double-shaded.
4. Total pieces: \(3 \times 2 = 6\) equal squares.
\(\frac{1}{3} \times \frac{1}{2} = \frac{1 \times 1}{3 \times 2} = \frac{1}{6}\)
Example 2: Non-Unit (\(\frac{2}{3} \times \frac{4}{5}\)) Array of 8 Overlaps
1. Vertical cuts: 5 columns, 4 shaded yellow.
2. Horizontal cuts: 3 rows, 2 shaded blue.
3. Overlap array: \(2 \text{ rows} \times 4 \text{ cols} = 8\) green.
4. Total pieces: \(3 \times 5 = 15\) total rectangles.
Fraction Diagrams 75 Percent Packet
Modified Packet • 75% Workload
U3, L1–3: Multiply Fraction by Fraction Using Diagrams
Name:
Date: Team:
IEP Step Checklist: 1. Unpack fractions \(\rightarrow\) 2. Vertical cuts & shade \(\rightarrow\) 3. Horizontal cuts & shade \(\rightarrow\) 4. Overlap over Total.
5.NF.B.4a / 5.NBT.B.5
Do Now: Fluency & Visual Review (4 Minutes) Warm-Up Check
1. Fluency Check: Solve using your standard or area model steps.
\(14 \times 12 =\) _____
(Show work in box)
2. Fraction Visual: What fraction of the square below is shaded?
Fraction:
Task 1: Of What? (Guided Practice) IM 3.11 Adapted
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.
1. Draw a diagram of 1 whole pan:
1 Whole Pan
2. The expression \(\frac{1}{3} \times \frac{1}{2}\) represents the situation. How does this connect to your diagram?
3. How much of the whole pan of macaroni and cheese did Lin eat?
Lin ate: of the whole pan.
Task 2: More Macaroni and Cheese Non-Unit Partitions
At another dinner, a pan of macaroni and cheese is \(\frac{2}{3}\) full. Brad eats \(\frac{1}{2}\) of the remaining macaroni and cheese in the pan.
Draw Brad's diagram:
1 Whole Pan
1. Write a multiplication expression for this situation:
Expression:
2. How much macaroni and cheese did Brad eat? Explain using total pieces.
Answer: of the whole pan.
Grade 5 Math • U3 L1–3 (75% Workload) Page 1 of 4 • Do Now & Tasks 1–2
Independent Practice • Part 1
Analyzing Area Diagrams & Word Problems
★ Core Independent Practice
Self-Check Steps: Did you count the double-shaded overlap for the numerator and all rectangles for the denominator? Keep Solving!
1. Painting the Fence (Real-World Context) MPS 1 Checkpoint
Sky and his brother are painting a fence together. Sky's responsibility is \(\frac{1}{2}\) of the fence. By noon, Sky completed \(\frac{2}{5}\) of his half.
a. Draw a diagram of the fence:
1 Whole Fence
b. Write a multiplication expression:
Expression: ______________________
c. Determine what fraction of the whole fence Sky has painted:
Fraction Diagrams 50 Percent Packet
Modified Packet • 50% Workload Reduction
U3, L1–3: Multiply Fraction by Fraction Using Diagrams
Name:
Date: Team:
IEP Step Checklist: 1. Unpack fractions \(\rightarrow\) 2. Vertical cut/shade \(\rightarrow\) 3. Horizontal cut/shade \(\rightarrow\) 4. Overlap over Total
Reference Sheet Ready
Do Now: Fluency & Visual Review (4 Minutes) Warm-Up Check
1. Fluency Check: Use your multi-digit steps to calculate.
\(16 \times 12 =\) _____
Show partial products in box:
2. Fraction Visual: What fraction of the square is shaded?
Fraction:
Task 1: Of What? (Guided Practice) Core Conceptual Model
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.
1. Draw a diagram of the pan:
1 Whole Pan
Vertical cuts first, horizontal cuts second
2. The expression \(\frac{1}{3} \times \frac{1}{2}\) represents this situation. How does this connect to your diagram?
3. How much macaroni and cheese did Lin eat from the whole pan?
Lin ate: of the whole pan.
💡 Pause & Check: Did you partition the columns all the way across? Does the overlap show \(\frac{1}{6}\)? Ready for Independent Work
Grade 5 Math • U3 L1–3 (50% Workload) Page 1 of 3 • Do Now & Task 1
Independent Practice • 50% Reduced Selection
Core Fluency & Transfer Practice
2 High-Leverage Problems
Remember Your Steps: You choose the cuts! Cut vertically for the 1st fraction, horizontally for the 2nd fraction. Count the overlap for your numerator. Cognitive Ownership
Problem 1: Painting the Fence MPS Checkpoint 1
Sky and his brother are painting a fence together. Sky's responsibility is \(\frac{1}{2}\) of the fence. One day he completed \(\frac{2}{5}\) of his half.
a. Draw a diagram of the fence:
1 Whole Fence
b. Write a multiplication expression:
Expression: ______________________
c. Determine how much of the whole fence Sky has painted:
Sky painted _______ of the whole fence.
Explain how your diagram proves your answer:
Problem 2: Area Model Analysis Square Unit = 1
Use the square figure below to answer the following questions.
Dark region = Overlap
