This 2-lesson unit builds a bridge from additive thinking to multiplicative reasoning by grounding multiplication and division in concrete models (equal groups, arrays, and area models) and establishing their inverse relationship. Students transition from counting all objects to recognizing units of units, developing algebraic intuition and procedural fluency.
2 lessons Γ 60 min Β· 16 tasks with answers Β· 3 differentiation profiles Β· 10 typical errors Β· 1 assessment
Multiplication and division form the single most critical cognitive leap in Grade 3 mathematics. According to the Common Core State Standards (CCSS), mastery of these operations underpins all subsequent work with fractions, ratios, proportional relationships, and algebraic functions in Grades 4β8.
CCSS.MATH.CONTENT.3.OA.A.1: Interpret products of whole numbers (e.g., interpret 5 Γ 7 as the total number of objects in 5 groups of 7 objects each).CCSS.MATH.CONTENT.3.OA.A.2: Interpret whole-number quotients of whole numbers (e.g., interpret 56 Γ· 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each).CCSS.MATH.CONTENT.3.OA.A.3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.CCSS.MATH.CONTENT.3.OA.B.5: Apply properties of operations as strategies to multiply and divide (Commutative and Distributive properties).CCSS.MATH.CONTENT.3.OA.B.6: Understand division as an unknown-factor problem.ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
β LEARNING OUTCOMES β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ€
β 1. Knowledge (Declarative) β
β β’ Define multiplication as combining equal groups (factors = groups β
β and group size; product = total). β
β β’ Define division as fair-sharing (partitive) or grouping (quotative). β
β β’ State the Commutative Property of Multiplication (a Γ b = β
β b Γ a). β
β β
β 2. Conceptual Understanding β
β β’ Explain how an array represents both multiplication and division β
β simultaneously. β
β β’ Articulate that division is the inverse operation of multiplication β
β (? Γ 4 = 24 24 Γ· 4 = ?). β
β β
β 3. Procedural Skills β
β β’ Model a word problem using an array, tape diagram, or equal groups. β
β β’ Write matching multiplication and division equations for a given β
β visual representation. β
β β
β 4. Cross-Cutting Competencies (Mathematical Practices) β
β β’ Make sense of problems and persevere in solving them (MP1). β
β β’ Construct viable arguments and critique reasoning of others (MP3). β
β β’ Model with mathematics using physical and visual tools (MP4). β
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Bloomβs Taxonomy Focus: Analysis & Evaluation (Level 4 & 5)
Students will not merely calculate answers; they will analyze quantitative structures, deconstruct numbers into factors, and evaluate the mathematical validity of different representations of the same product or quotient.
| Time | Phase | Teacher Actions | Student Actions | Materials |
|---|---|---|---|---|
| 0β10 min | Hook & Activate | Displays 3 plates with 4 apples each and 1 plate with 7 apples. Asks: "Which plate setup lets us use multiplication shorthand? Why?" | Analyze the images. Turn-and-talk: Identify that multiplication requires equal groups (3 Γ 4), whereas 7 is an unequal addend. | Projector/Slide with visual, Math Journals |
| 10β25 min | Concrete Exploration | Distributes 24 counters per pair. Prompts: "Arrange 12 counters into equal rows and columns. How many distinct rectangles can you build?" Records findings on board. | Build 1 Γ 12, 2 Γ 6, 3 Γ 4, 4 Γ 3, 6 Γ 2, 12 Γ 1. Physically rotate a 3 Γ 4 array 90^ to see it becomes 4 Γ 3. | 24 Two-color counters per pair, Grid whiteboards |
| 25β40 min | Representational to Abstract | Connects physical arrays to equations: 3 rows of 4 = 12 β 3 Γ 4 = 12. Introduces formal terms: factor, factor, product. Guides drawing of tape diagrams. | Draw arrays on grid paper. Label rows as groups and columns as size of group. Write corresponding commutative pairs: 3 Γ 4 = 12 and 4 Γ 3 = 12. | Grid paper, Colored pencils, Anchor chart |
| 40β50 min | Guided & Partner Practice | Circulates, asking probing questions: "What does the 4 represent in your drawing? Where is the total?" Facilitates 'Pass-the-Problem' activity. | In pairs: Partner A writes a word problem; Partner B builds the array and writes the matching equations. Switch roles. | Differentiated task cards |
| 50β60 min | Synthesis & Exit Ticket | Synthesizes key takeaway: Arrays show that changing factor order preserves total area/count. Distributes Exit Ticket. | Complete independent 2-question Exit Ticket. Perform "Fist-to-Five" self-assessment on confidence with arrays. | Exit Ticket Slip 1.1 |
| Time | Phase | Teacher Actions | Student Actions | Materials |
|---|---|---|---|---|
| 0β10 min | Hook & Connection | Presents an array of 15 dots (3 Γ 5). Covers one dimension with paper: "I have 15 dots in total. There are 3 rows. How many are in each row?" Writes: 3 Γ ? = 15. | Identify that the missing value is 5. Discuss how this relates to 15 Γ· 3 = 5. | Magnetic counters on whiteboard, card cover |
| 10β25 min | Dual Models of Division | Demonstrates two scenarios with 12 cookies: 1) Share equally among 4 friends (Partitive: find group size); 2) Put 4 cookies in each bag (Quotative: find number of groups). | Act out both scenarios using counters and sorting mats. Articulate: "In one we know the number of groups; in the other we know the size of the group." | Sorting mats (paper plates), Counters |
| 25β40 min | Fact Family Factoring | Introduces the "Fact Family Triangle" (e.g., top: 24, bottom corners: 4, 6). Models generating 2 multiplication and 2 division equations (4 Γ 6 = 24, 6 Γ 4 = 24, 24 Γ· 6 = 4, 24 Γ· 4 = 6). | Complete triad relationship boards. Highlight that the total (dividend) in division represents the product in multiplication. | Whiteboard Triangles, Dry-erase markers |
| 40β50 min | Contextual Application | Provides 3 real-world word problems (baking, classroom supplies, sports teams). Prompts students to identify whether the problem seeks the group count or the group size. | Solve problems on mini-whiteboards using tape diagrams. Justify answers using multiplication checks: "I know 18 Γ· 3 = 6 because 3 Γ 6 = 18." | Word Problem Activity Cards |
| 50β60 min | Closure & Formative Check | Facilitates student summary: "Why is division just 'multiplication backward'?" Collects Exit Ticket. | Share explanations aloud. Complete Exit Ticket 1.2 independently. | Exit Ticket Slip 1.2 |
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β Student Mistake β Prevention Strategy β
ββββββββββββββββββββββββββββββββββββββββΌβββββββββββββββββββββββββββββββββββββββββββββββββββββββββ€
β 1. Additive Confusion β Have students physically build 3 groups of 4 and β
β Writing 3 Γ 4 = 7. β 3 + 4 side-by-side using two-color counters to β
β β observe the difference in total quantities. β
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β 2. Row vs. Column Mislabeling β Teach the physical mnemonic: "Rows go across like the β
β Reading a 3x5 array as 5 rows β horizon; Columns stand tall like columns on a building"β
β of 3. β Always emphasize: Rows = Groups, Columns = Items. β
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β 3. Dividend Misplacement β Frame division conceptually: "You cannot divide what β
β Writing 4 Γ· 20 = 5 instead β you do not have; the total (dividend) must be what is β
β of 20 Γ· 4 = 5. β shared or grouped." β
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β 4. Commutativity Overgeneralization β Compare 12 Γ· 3 = 4 with 3 Γ· 12 physically β
β Believing 12 Γ· 3 = 3 Γ· 12 β using blocks to prove division is not commutative. β
β because 3 Γ 4 = 4 Γ 3.β β
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β 5. Array Grid Alignment Breakdown β Provide standard centimeter grid paper to support β
β Drawing staggered, unequal dots β spatial alignment until motor control is secure. β
β instead of an aligned matrix. β β
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β 6. Equal vs. Unequal Partitioning β Present non-examples: 12 split into 3, 4, 5. Ask: β
β Assuming any division makes β "Is this division?" to reinforce that groups MUST β
β equal groups automatically. β be congruent in size. β
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β 7. Blind Skip-Counting Off-by-One β Anchor skip-counting on a visual number line or track β
β Counting "3, 6, 9, 12, 15" but β fingers intentionally as units are counted. β
β recording 6 as the factor. β β
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β 8. Symbol Inversion (Γ vs +)β Highlight math symbols in contrasting colors; have β
β Misreading 5 Γ 1 as β students verbalize the operator: "times means groups β
β 5 + 1 = 6. β of, plus means combine." β
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β 9. Ignoring the Zero Property β Model 3 Γ 0 as "3 empty plates = 0 items" and β
β Claiming 4 Γ 0 = 4. β 0 Γ 4 as "0 plates with 4 items = 0 items." β
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β 10. Partitive/Quotative Mix-up β Require students to write the question being asked β
β Inability to tell if answer is β before calculating: "Am I looking for HOW MANY GROUPS β
β 'boxes' or 'items per box'. β or HOW MANY IN EACH GROUP?" β
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β LESSON ARCHITECTURE: 60-MINUTE FLOW β
β [0-10m] HOOK: The Classroom Bakery Problem (Curiosity & Disequilibrium) β
β [10-25m] DISCOVERY: Concrete Array Construction & The 90-Degree Turn β
β [25-40m] PRACTICE: Grid Modeling & Symbolic Bridges β
β [40-50m] TRANSFER: The Floor-Tile Challenge (Real-World Application) β
β [50-60m] REFLECTION: Exit Ticket & Synthesizing Metacognition β
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
3 rows of 6 (3 x 6 = 18) 6 rows of 3 (6 x 3 = 18)
β β β β β β β β β
β β β β β β ββββΊ β β β
β β β β β β Rotate β β β
90Β° β β β
β β β
β β β
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β CARD: WHY THIS LESSON IS EXCELLENT, NOT AVERAGE β
β βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ£
β 1. Grounded in Concrete-Representational-Abstract (CRA): Moves from β
β real egg cartons and counters to grid models and algebraic equations. β
β 2. Addresses Core Mathematical Structures: Emphasizes the Commutative β
β Property as an intrinsic spatial property of 2D space (arrays), not an β
β arbitrary memorized rule. β
β 3. Integrated Formative Feedback: Features 4 distinct checkpoint moments β
β (turn-and-talk, array rotation check, mystery factor, exit ticket). β
β 4. High Cognitive Demand: Incorporates missing-factor problems that β
β naturally lay the groundwork for division in Lesson 2. β
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Context: The 3rd grade class at Lincoln Elementary School in Ohio is raising money for a field trip to the science center. They are selling cups of lemonade and boxes of baked cookies.
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β LEMONADE & COOKIE PRICING β
β β’ 1 Box of Cookies contains 4 cookies and costs 3 β
β β’ Lemonade is sold in packs of 6 cups for 12 β
β β’ Single cups of lemonade cost 2 each β
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| Task | Full Marks (2 pts) | Partial Marks (1 pt) | Zero Marks (0 pts) |
|---|---|---|---|
| Real-Life e (Decision & Justification) | Correctly calculates 5 Γ 6 = 30 cups, multiplies by $2 to get $60, compares with $50, and concludes it is worse for the class. | Computes 30 cups and $60 correctly, but fails to make a clear comparison or draws an incorrect final conclusion. | Incorrect calculation of total cups (e.g., adds 5+6=11) and invalid financial conclusion. |
| Task 14A (Distributive Decomposition) | Correctly identifies unknown factor as 3, evaluates 35 + 21, and sums to 56 with clear structural notation. | Identifies 3 but makes an addition error in 35 + 21, or gets 56 without showing the decomposed products. | Fails to decompose factor 8; fills in incorrect numbers unrelated to 5 + 3. |
| Task 15A (Partitive vs. Quotative) | Writes two distinct, clear word problems: one explicitly seeking the number of groups (quotative) and one seeking group size (partitive). | Writes two problems, but both represent the same division type (e.g., both are fair-sharing). | Problems do not match the numbers or represent addition/subtraction instead of division. |
Diagnostic Item: "What is 24 Γ· 4?"
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β DIMENSION β 1: BEGINNING β 2: DEVELOPING β 3: PROFICIENT β 4: EXEMPLARY β
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β Procedural β Frequent errors β Computes facts β Computes within β Rapid, accurate β
β Fluency β in facts within β within 100 with β 100 accurately β computation; β
β β 20; confuses β skip-counting β using known β uses properties β
β β Γ and +. aid; minor slips.β strategies. β flexibly. β
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β Conceptual β Views β Recognizes β Explains β Proves inverse β
β Meaning β multiplication β equal groups; β division as β nature of β
β β as random β struggles to β missing factor; β operations and β
β β memorization. β model division. β models arrays. β explains 0 & 1. β
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β Transfer & β Unable to draw β Draws arrays β Translates β Flexibly shifts β
β Representations β an array from a β with scaffoldingβ between story, β among arrays, β
β β word problem. β but cannot writeβ array, diagram, β tape diagrams, β
β β β matching family.β and equation. β and contexts. β
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Name: _______________________ Date: _____________ Class: _____________
Check the box that matches your thinking today:
SKILL / CONCEPT π’ I can do this π‘ I need a little π΄ I am stuck
& explain it! help/practice and need help
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1. I can draw an array to match a
multiplication sentence (like 3 x 5). [ ] [ ] [ ]
2. I can turn an array 90 degrees and write
its commutative twin (5 x 3 = 3 x 5). [ ] [ ] [ ]
3. I can solve a division problem by asking
"What times this number equals the total?" [ ] [ ] [ ]
4. I can write all 4 equations in a
Fact Family triangle. [ ] [ ] [ ]
My Goal For Tomorrow: ______________________________________________________________________________
Act as a Common Core Grade 3 Mathematics specialist. I need 4 differentiated word problems for CCSS.MATH.CONTENT.3.OA.A.3 (Multiplication and Division within 100).
Theme: NASA Space Exploration and Rover Missions.
Requirements:
1. Problem 1 (Approaching Grade Level): Equal groups multiplication (factors 2, 5, or 10), requiring a tape diagram.
2. Problem 2 (On Grade Level): Quotative division (finding the number of groups) with total <= 40.
3. Problem 3 (Above Grade Level): Two-step problem combining multiplication and division (Distributive Property focus).
4. Problem 4 (ELL/Language Support): Simple, decodable syntax with visual cues (e.g., bracketed scaffolding) focusing on the Commutative Property.
For each problem, include:
- The full problem text
- The mathematical equation with an unknown variable
- The visual model description
- The complete step-by-step solution key
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β STUDENT DIFFERENTIATION MATRIX β
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β Profile β Concrete Adaptations & Instructional Strategy β
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β 1. The β β’ Name: Emily (High anxiety, avoids messy draft work). β
β Perfectionist β β’ Adaptation: Provide reusable dry-erase grid sleeves. β
β β Mistakes can be wiped away instantly. β
β β β’ Task Adjustment: Include "Error Analysis" tasks β
β β where Emily evaluates someone else's work, removing β
β β the personal risk of making mistakes. β
β β β’ Feedback: Praise strategy and resilience, not speed. β
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β 2. The Creative β β’ Name: Jackson (Doodles, resists rigid math steps). β
β Thinker β β’ Adaptation: "Math Story Author" role. Let Jackson β
β β invent comic strips where characters solve array- β
β β based puzzles (e.g., arranging alien spaceships). β
β β β’ Constraint: Must include a verified equation box. β
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β 3. The β β’ Name: Mateo (ELL, works well with hands, gets lost β
β Practitioner β in abstract algebraic word problems). β
β β β’ Adaptation: Provide concrete Unifix cubes and realia β
β β (egg cartons, coin rolls). β
β β β’ Scaffolding: Provide dual-language visual cards and β
β β have him demonstrate solutions with objects first. β
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β 4. The Deep β β’ Name: Ethan (Computes instantly, works alone, β
β Analyst β rushes through explanations). β
β β β’ Adaptation: Assign "Proof Tasks" instead of more β
β β worksheets: "Prove why an odd number times an odd β
β β number ALWAYS produces an odd product using arrays." β
β β β’ Role: Peer Coach using structured questions. β
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β 5. The Social β β’ Name: Chloe (Verbal, energetic, struggles to work β
β Learner β independently on written tasks). β
β β β’ Adaptation: Use the "Show-and-Tell Math" structure: β
β β Chloe explains her thinking verbally to a peer, who β
β β writes the equation, then they swap roles. β
β β β’ Reinforcement: Gamified partner review stations. β
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β VAK MULTI-SENSORY PATHWAYS β
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β ποΈ VISUAL PATHWAY β
β β’ Color-coded grid paper (Rows = Orange, Columns = Purple). β
β β’ Array dot cards flashed for 2 seconds to encourage subitizing equal β
β groups rather than single-unit counting. β
β β
β π AUDITORY PATHWAY β
β β’ Rhythmic skip-counting chants paired with clapping cadence. β
β β’ Choral Call-and-Response: Teacher says "4 rows of 5!"; Students reply β
β "20 in the hive!" β
β β
β β KINESTHETIC PATHWAY β
β β’ Floor Grid Math: Blue painter's tape on the floor making a 5x5 grid. β
β Students jump to coordinates to physically demonstrate arrays. β
β β’ Snap-cube arrays: Physically snapping together rows and rotating them. β
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khanacademy.org)desmos.com)phet.colorado.edu)geogebra.org)bbc.co.uk/bitesize)Copy and paste this prompt into any generative AI tool (e.g., ChatGPT, Gemini, Gamma, Canva AI, Microsoft Copilot) to generate a slide deck for this unit:
"Create an engaging 10-slide presentation for a Common Core Grade 3 Mathematics lesson on Multiplication, Arrays, and Division.
Slide Breakdown:
- Slide 1: Title: 'Unlocking Multiplication: The Power of Arrays!' (Fun, welcoming visual).
- Slide 2: The Bakery Problem (Hook: 3 rows of 6 cookies vs. 6 rows of 3 cookies; who has more?).
- Slide 3: What is an Array? (Definition: Rows go across, Columns stand tall; visual diagram).
- Slide 4: The 90-Degree Turn (Commutative Property: showing 3x4 turning into 4x3 with equal products).
- Slide 5: Let's Build! (Interactive challenge: How many arrays can we make with 12 tiles?).
- Slide 6: The Secret Link: Multiplication Meets Division (Showing an array and writing the 4-equation Fact Family).
- Slide 7: Fair-Sharing vs. Grouping (Visual cookie sharing problem).
- Slide 8: Real-World Math: The Lemonade Stand Fundraiser (Word problem with multi-step question).
- Slide 9: Common Mistakes to Avoid (Spotting the error in arrays and equations).
- Slide 10: Summary & Exit Ticket Challenge (Key takeaways and final thinking question).
Visual Style Guidelines: Clean, high-contrast, large fonts for elementary learners, minimal text per slide, warm color palette (blues, oranges, greens), and clean geometric grid layouts. If you need any clarification about the grade level or lesson focus, please ask before generating."
Multiplicative reasoning is not simply faster addition; it is a fundamental shift in how children view numbersβfrom counting individual items to grouping and scaling units of units. When students discover that multiplication and division are simply two ways of looking at the exact same array, memorizing facts transforms from a stressful chore into intuitive, visual sense-making.
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WORKSHEET β Multiplication & Division: Equal Groups and Arrays (Grade 3)
Name: ____________________________________ Date: _____________________________
Class: ___________________________________ Teacher: __________________________
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π± LEVEL 1: FOUNDATION (Equal Groups & Arrays)
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1. Look at the equal groups below:
[ β
β
β
β
] [ β
β
β
β
] [ β
β
β
β
]
a) How many groups are there? _______ groups
b) How many stars are in each group? _______ stars
c) Repeated addition sentence: ____ + ____ + ____ = _______
d) Multiplication sentence: ____ Γ ____ = _______
2. Draw an array that has 3 rows with 5 dots in each row.
Draw your array here:
Write the multiplication equation: ____ Γ ____ = _______
3. Write the missing numbers to show the Commutative Property:
a) 4 Γ 6 = 24 so 6 Γ ____ = 24
b) 2 Γ 8 = 16 so ____ Γ 2 = 16
πΏ LEVEL 2: INTERMEDIATE (Properties & Inverse Thinking)
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4. Fill in the Fact Family Triangle and write the 4 matching equations:
/ 28
/
/ 4 7
ββββββββββββββ
Multiplication: ____ Γ ____ = 28 ____ Γ ____ = 28
Division: 28 Γ· ____ = ____ 28 Γ· ____ = ____
5. Solve the unknown factor to find the division answer:
a) 24 Γ· 6 = ? Think: ? Γ 6 = 24 Answer: ? = _______
b) 35 Γ· 5 = ? Think: ? Γ 5 = 35 Answer: ? = _______
c) 18 Γ· 2 = ? Think: ? Γ 2 = 18 Answer: ? = _______
6. Read the story problem and solve:
Marcus has 30 stickers. He wants to share them equally among 5 of his friends.
a) Draw a model (tape diagram or equal circles) to show the problem:
b) Write the division equation: ________________________________
c) How many stickers does each friend get? _______ stickers
π³ LEVEL 3: CHALLENGE (Problem Solving & Analysis)
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7. Baker Hannah baked 24 muffins. She wants to arrange them in a rectangular array
on a single tray.
List THREE DIFFERENT ways she could arrange her muffins in equal rows and columns:
β’ Way 1: ____ rows of ____ muffins (Equation: ____ Γ ____ = 24)
β’ Way 2: ____ rows of ____ muffins (Equation: ____ Γ ____ = 24)
β’ Way 3: ____ rows of ____ muffins (Equation: ____ Γ ____ = 24)
8. Spot the Mistake!
Jordan looked at this problem: 20 Γ· 4 = ?
Jordan wrote: "20 Γ· 4 = 16 because 20 - 4 = 16."
Explain why Jordan is incorrect. What should the correct answer be?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
π EXTRA CHALLENGE: THE SUPER-SPLIT!
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Use the Distributive Property to solve 7 Γ 6 by splitting it into two easier facts:
7 Γ 6 = (5 Γ 6) + ( ___ Γ 6 )
= 30 + ____
= ____
Draw an array with a line splitting it to prove your work!
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π€ About this material: Content generated with AI assistance (Methodics AI) following pedagogical frameworks. Designed as a foundation for teacher adaptation β always review and customize for your students.