Solved Exercises on Practicing Drawing Models in Grade 3

Master drawing models: bar models, tape diagrams, and visual representations for mathematical problem solving through these 10 detailed exercises.

Solution: Exercises 1 to 3
1 Bar Model Addition
Exercise 1
Sarah has 12 stickers. Her friend gives her 8 more stickers. Draw a bar model to show how many stickers Sarah has now.
Definition:

Bar model: A rectangular representation showing parts of a whole or relationships between quantities

Solution method:
  1. Draw a rectangle representing the starting amount
  2. Draw another rectangle for the added amount
  3. Combine them to show the total
Bar Model Representation
12
8
20
Step 1: Draw the first part

Draw a rectangle representing 12 stickers (Sarah's starting amount)

Step 2: Draw the second part

Draw another rectangle representing 8 stickers (the added amount)

Step 3: Show the total

Combine both rectangles to show the total of 20 stickers

12 + 8 = 20 stickers
Final answer:

Sarah has 20 stickers now

Applied rules:

• Visual representation: Rectangles show the relationship between quantities

• Part-whole relationship: Individual parts combine to form the whole

• Addition modeling: Two parts joined together represent addition

2 Bar Model Subtraction
Exercise 2
Tom had 25 marbles. He gave 9 marbles to his sister. Draw a bar model to show how many marbles Tom has left.
Definition:

Subtraction bar model: Shows the whole amount with a part removed

Bar Model Representation
25
16
9
Step 1: Draw the whole amount

Draw a rectangle representing 25 marbles (Tom's starting amount)

Step 2: Show the part being subtracted

Divide the rectangle to show 9 marbles being given away

Step 3: Show the remaining part

The remaining rectangle shows 16 marbles left

25 - 9 = 16 marbles
Final answer:

Tom has 16 marbles left

Applied rules:

• Whole-part relationship: Start with the whole and remove a part

• Subtraction modeling: Visually shows taking away from a total

• Proportional representation: Rectangle sizes reflect actual values

3 Tape Diagram
Exercise 3
There are 4 bags of apples. Each bag has 6 apples. Draw a tape diagram to show how many apples there are in total.
Definition:

Tape diagram: A rectangular strip divided into equal sections to represent equal groups

Tape Diagram Representation
6
6
6
6
24
Step 1: Identify equal groups

There are 4 bags with 6 apples each (equal groups)

Step 2: Draw equal segments

Create 4 equal sections in the tape diagram

Step 3: Label each segment

Write 6 in each segment to represent apples per bag

Step 4: Calculate the total

4 × 6 = 24 apples in total

4 × 6 = 24 apples
Final answer:

There are 24 apples in total

Applied rules:

• Equal groups: Each segment represents the same quantity

• Multiplication modeling: Shows repeated addition visually

• Segment labeling: Each part shows the same value

Solution: Exercises 4 to 5
4 Division Model
Exercise 4
Maria has 18 cookies. She wants to share them equally among 6 friends. Draw a model to show how many cookies each friend gets.
Definition:

Division model: Shows how a total amount is divided into equal groups

Division Model Representation
3
3
3
3
3
3
18
Step 1: Identify the total

Maria has 18 cookies to share

Step 2: Identify the number of groups

She wants to share among 6 friends

Step 3: Divide the total equally

18 ÷ 6 = 3 cookies per friend

Step 4: Draw the model

Create 6 equal segments, each with 3 cookies

18 ÷ 6 = 3 cookies per friend
Final answer:

Each friend gets 3 cookies

Applied rules:

• Equal sharing: Total is divided into equal parts

• Division modeling: Shows how many in each group

• Proportional segments: Each segment represents equal shares

5 Comparison Model
Exercise 5
Jake read 32 pages of a book. Emma read 25 pages of the same book. Draw a model to show how many more pages Jake read than Emma.
Definition:

Comparison model: Shows the difference between two quantities

Comparison Model Representation
32 (Jake)
25 (Emma)
7
Step 1: Draw Jake's amount

Draw a rectangle representing 32 pages

Step 2: Draw Emma's amount

Draw a shorter rectangle representing 25 pages

Step 3: Show the difference

Highlight the extra length to show 7 more pages

Step 4: Calculate the difference

32 - 25 = 7 pages difference

32 - 25 = 7 more pages
Final answer:

Jake read 7 more pages than Emma

Applied rules:

• Visual comparison: Longer bar shows greater quantity

• Difference highlighting: Extra length shows the difference

• Proportional representation: Bar lengths reflect actual values

Drawing Models Rules
Bar Model: Part + Part = Whole
Addition/Subtraction Relationships
Tape Diagram: Equal Groups × Items = Total
Multiplication/Division Relationships
📊
Bar Model
Shows part-whole relationships
📋
Tape Diagram
Shows equal groups
🔍
Comparison
Shows differences
1️⃣
Read the problem
2️⃣
Identify quantities
3️⃣
Choose model type
4️⃣
Draw and label
Key Concept: Proportional Representation

Rectangle sizes should reflect the actual values being represented.

Model Applications
  • Addition: Combine parts to show whole
  • Subtraction: Show whole with part removed
  • Multiplication: Equal groups representation
  • Division: Sharing into equal parts
  • Comparison: Showing differences
➕
Addition: Join bars
➖
Subtraction: Remove part
✖️
Multiplication: Equal groups
➗
Division: Split equally
Complete Guide to Drawing Models
Read → Model → Solve → Check
Modeling Process
Key definitions:

Bar model: A rectangular representation showing parts of a whole or relationships between quantities

Tape diagram: A rectangular strip divided into equal sections to represent equal groups

Visual model: A pictorial representation that helps understand mathematical relationships

Complete methodology:
  1. Read carefully: Understand the problem situation
  2. Identify quantities: Find the numbers and relationships
  3. Choose model type: Select appropriate visual representation
  4. Draw accurately: Create proportional rectangles or segments
  5. Solve: Use the model to find the answer
  6. Check: Verify the model reflects the problem
Tip 1: Use consistent sizing to maintain proportionality.
Tip 2: Label each part of the model clearly.
Tip 3: Use different colors or shading to distinguish parts.
Tip 4: Practice with simple numbers before complex problems.
Common errors: Incorrect proportions, missing labels, wrong model type.
Success strategies: Practice regularly, use grid paper, check accuracy.
Key rules to remember:

• Addition: Join bars together to show combination

• Subtraction: Show whole with part removed

• Multiplication: Create equal groups in tape diagrams

• Division: Split total into equal parts

• Comparison: Show difference with extra length

Modeling Techniques
📊
Addition
Bars join side by side
📋
Subtraction
Part removed from whole
🔍
Multiplication
Equal groups in tape
📏
Division
Equal parts of whole

Questions & Answers

Question: How do I know which type of model to draw for a word problem?

Answer: Look for clues in the problem:

  • Bar models: For addition/subtraction problems showing part-whole relationships
  • Tape diagrams: For multiplication/division problems with equal groups
  • Comparison models: When comparing two quantities or finding differences

Also consider what the problem is asking: combining, taking away, equal groups, or comparing.

Question: Do the rectangles in bar models need to be exact sizes?

Answer: The rectangles should be proportional to reflect the actual values:

  • Larger numbers should have longer rectangles
  • Smaller numbers should have shorter rectangles
  • The relationship between sizes should match the relationship between numbers

For example, if one number is twice as large, its rectangle should be approximately twice as long.

Question: How can drawing models help me solve word problems?

Answer: Drawing models provides several benefits:

  • Visualization: Makes abstract concepts more concrete
  • Understanding: Helps see relationships between numbers
  • Strategy: Provides a clear path to solve the problem
  • Error checking: Allows you to verify your answer makes sense

Models act as a bridge between the words in the problem and the mathematical operations needed.

Complete Summary: Practicing Drawing Models

Definitions of Key Concepts

Bar Model: A rectangular representation showing parts of a whole or relationships between quantities.

Tape Diagram: A rectangular strip divided into equal sections to represent equal groups.

Visual Model: A pictorial representation that helps understand mathematical relationships.

Proportional Representation: Drawing rectangles or segments that reflect the actual values being represented.

Part-Whole Relationship: The connection between individual parts and the total amount.

Equal Groups: Sets that contain the same number of items, used in multiplication and division models.

Core Rules and Principles

Addition Modeling: Join bars together to show combination of quantities.

Subtraction Modeling: Show whole with part removed to demonstrate taking away.

Multiplication Modeling: Create equal groups in tape diagrams to show repeated addition.

Division Modeling: Split total into equal parts to show sharing equally.

Comparison Modeling: Show difference with extra length to demonstrate how much more or less.

Proportional Accuracy: Rectangle sizes should reflect actual numerical values.

Clear Labeling: Each part of the model should be clearly labeled with values.

Step-by-Step Method
  1. Read Carefully: Understand the problem situation and identify what is being asked.
  2. Identify Quantities: Find the numbers and relationships mentioned in the problem.
  3. Choose Model Type: Select appropriate visual representation (bar model, tape diagram, etc.).
  4. Draw Accurately: Create proportional rectangles or segments that reflect actual values.
  5. Label Clearly: Add numbers and labels to each part of the model.
  6. Solve: Use the model to find the answer to the problem.
  7. Check: Verify that the model accurately represents the problem and the answer makes sense.
Examples (Simple to Advanced)

Simple Addition: "Tom has 3 apples and gets 2 more" → Draw 3-unit bar, then 2-unit bar, combine to show 5-unit total.

Simple Subtraction: "Sarah has 8 cookies and eats 3" → Draw 8-unit bar, shade 3 units, show 5 units remaining.

Simple Multiplication: "4 bags with 3 candies each" → Draw 4 equal segments, each labeled 3, total 12.

Simple Division: "12 cookies shared among 4 friends" → Draw 12-unit bar, divide into 4 equal parts of 3 each.

Advanced Comparison: "Jake read 25 pages, Emma read 18 pages" → Draw 25-unit bar above 18-unit bar, show 7-unit difference.

Tips, Tricks, and Common Pitfalls

Tips: Use grid paper for accuracy; label all parts clearly; maintain proportional sizes; use different colors for different parts.

Tricks: Start with simple numbers to practice the model type; compare your model to the problem to ensure accuracy.

Common Pitfalls: Drawing rectangles that don't reflect actual values; forgetting to label parts; choosing the wrong model type.

Memory Aid: "BART" - Bar models for Addition, Subtraction, and Relationships; Tape diagrams for equal groups.

Checking Strategy: Count units in your model to verify they match the problem numbers.

Key Notes for Memorization

Bar Model Uses: Addition (joining), subtraction (removing), comparison (showing differences).

Tape Diagram Uses: Multiplication (equal groups), division (sharing equally).

Proportionality Rule: Larger numbers need longer rectangles; smaller numbers need shorter rectangles.

Labeling Rule: Always write numbers on each part of the model.

Model Selection: Addition/subtraction → Bar model; Multiplication/division → Tape diagram.

Accuracy Check: Count total units in your model to verify against the problem.

Visual Benefits: Models make abstract concepts concrete and provide clear problem-solving pathways.