Solved Exercises on Using Drawings and Models in Grade 3

Master using drawings and models: bar models, area models, number lines, and visual representations for problem solving through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Bar model
Exercise 1
Sarah has 15 stickers. Tom has 8 more stickers than Sarah. How many stickers does Tom have? Draw a bar model to solve.
Definition:

Bar model: A visual representation that uses rectangular bars to show the relationship between different quantities in a problem. It helps to understand addition, subtraction, and comparison problems.

Bar model method:
  1. Identify the quantities in the problem
  2. Draw a rectangle for each quantity
  3. Label each rectangle with the known values
  4. Use the model to find the unknown value
  5. Solve the problem based on the visual relationship
Sarah's stickers
15
Tom's stickers
15 + 8 = 23
Answer
23 stickers
Step 1: Draw the bar for Sarah

Draw a rectangle labeled "Sarah: 15 stickers"

Step 2: Draw the bar for Tom

Draw a longer rectangle labeled "Tom: Sarah's amount + 8"

Step 3: Calculate Tom's amount

Tom = Sarah's amount + 8 = 15 + 8 = 23

Step 4: State the answer

Tom has 23 stickers

Tom has 23 stickers
Final answer:

Tom has 23 stickers

Applied rules:

• Visual representation: Bars show the relationship between quantities

• Comparison: Longer bar represents larger quantity

• Relationship: Tom's bar is 8 units longer than Sarah's

2 Area model
Exercise 2
A garden has 4 rows of flowers. Each row has 6 flowers. How many flowers are there in total? Draw an area model to solve.
Definition:

Area model: A visual representation that uses a rectangle divided into rows and columns to show multiplication. The total area represents the product of the dimensions.

Rows
4
Columns
6
Total
4 × 6 = 24
Step 1: Draw the grid

Draw a rectangle with 4 rows and 6 columns

Step 2: Count the squares

Count all the small squares in the grid

Step 3: Calculate the total

Total = Number of rows × Number of columns = 4 × 6 = 24

Step 4: State the answer

There are 24 flowers in total

There are 24 flowers
Final answer:

There are 24 flowers in total

Applied rules:

• Grid representation: Rows and columns show multiplication

• Area concept: Total squares = length × width

• Visual multiplication: Counting squares demonstrates multiplication

3 Number line model
Exercise 3
Emma started at 0 on a number line. She walked forward 12 steps, then backward 5 steps. Where did she stop? Draw a number line to solve.
Definition:

Number line model: A visual representation that uses a line with evenly spaced numbers to show addition and subtraction. Moving right represents addition, moving left represents subtraction.

Start
0
Move forward
0 + 12 = 12
Move backward
12 - 5 = 7
Step 1: Draw the number line

Draw a horizontal line with numbers from 0 to 15

Step 2: Mark the starting point

Mark the starting position at 0

Step 3: Move forward 12 steps

Draw an arrow from 0 to 12

Step 4: Move backward 5 steps

Draw an arrow from 12 to 7 (12 - 5 = 7)

Step 5: State the answer

Emma stopped at 7

Emma stopped at 7
Final answer:

Emma stopped at 7 on the number line

Applied rules:

• Direction matters: Right = addition, Left = subtraction

• Sequential movements: Each movement starts from the previous position

• Visual path: Arrows show the sequence of operations

Rules and methods, laws,...
Visual Model = Drawing + Labels + Relationships
Model Building
Bar Model
Quantities as Bars
Compare values
Area Model
Grid Representation
Show multiplication
Number Line
Linear Representation
Show operations
Key definitions:

Visual model: A drawing or diagram that represents a mathematical problem to help understand and solve it.

Bar model: A visual representation using rectangles to show relationships between quantities.

Area model: A visual representation using grids or rectangles to show multiplication.

Number line: A visual representation of numbers on a straight line to show operations.

Concrete representation: Using physical objects or drawings to represent abstract mathematical concepts.

Visual modeling methodology:
  1. Understand the problem: Identify what is being asked
  2. Choose the model: Select the most appropriate visual representation
  3. Create the drawing: Make the visual model accurate and clear
  4. Label the parts: Include all relevant information
  5. Use the model: Manipulate the drawing to solve the problem
  6. Check the answer: Verify with the visual representation
Tip 1: Use bar models for comparison and addition/subtraction problems.
Tip 2: Use area models for multiplication and division problems.
Tip 3: Use number lines for problems involving movement or sequential operations.
Tip 4: Always label your drawings with numbers and words to make them clear.
Common errors: Not labeling drawings, choosing the wrong model type, making drawings too small to see details, not maintaining proportionality in bar models.
Real-world applications: Understanding maps, measuring spaces, calculating areas, comparing quantities, visualizing budgets and expenses.
Modeling principles:

• Bar models: Show relationships between different quantities

• Area models: Represent multiplication as rectangular arrays

• Number lines: Show operations as movements along the line

• Proportional representation: Bars should reflect relative sizes

• Clear labeling: Include all relevant information

Solution: Exercises 4 to 5
4 Part-whole model
Exercise 4
A pizza was cut into 8 slices. If 3 slices were eaten, how many slices are left? Draw a part-whole model to solve.
Definition:

Part-whole model: A visual representation that shows how a whole is divided into parts. It helps understand subtraction as finding a missing part when the whole and one part are known.

Whole pizza
8 slices
Slices eaten
3 slices
Slices left
8 - 3 = 5
Step 1: Draw the whole

Draw a circle representing the whole pizza with 8 slices

Step 2: Shade the eaten part

Shade 3 slices to show they were eaten

Step 3: Count the remaining part

Count the unshaded slices: 5 slices remain

Step 4: Calculate using the model

Whole - Eaten = Remaining: 8 - 3 = 5

Step 5: State the answer

There are 5 slices left

5 slices are left
Final answer:

There are 5 slices left

Applied rules:

• Part-whole relationship: Whole = Part + Part

• Visual subtraction: Removing parts shows what remains

• Concrete representation: Circle divided into sections

5 Mixed model problem
Exercise 5
A farmer has 24 apples. He packs them into bags of 6 apples each. Then he gives 2 bags to his neighbor. How many apples does he have left? Use any model to solve.
Definition:

Mixed operations model: A visual representation that combines multiple types of models to solve complex problems involving different operations.

Total apples
24
Apples per bag
6
Bags made
24 ÷ 6 = 4
Bags given away
2
Bags remaining
4 - 2 = 2
Apples remaining
2 × 6 = 12
Step 1: Draw groups of apples

Draw 24 circles grouped into bags of 6: [●●●●●●][●●●●●●][●●●●●●][●●●●●●]

Step 2: Count the bags

There are 4 bags of 6 apples each

Step 3: Remove bags given away

Cross out 2 bags: [●●●●●●][●●●●●●][●●●●●●][●●●●●●] (cross out last 2)

Step 4: Count remaining bags

2 bags remain

Step 5: Calculate remaining apples

2 bags × 6 apples per bag = 12 apples

Step 6: State the answer

The farmer has 12 apples left

Farmer has 12 apples left
Final answer:

The farmer has 12 apples remaining

Applied rules:

• Grouping model: Circles in groups show division

• Removal model: Crossing out shows subtraction

• Multiplication model: Groups × items per group

Complete Summary: Using Drawings and Models
Visual Model = Drawing + Labels + Relationships
Model Building
Key definitions:

Visual model: A drawing or diagram that represents a mathematical problem to help understand and solve it. Visual models make abstract concepts concrete.

Bar model: A visual representation using rectangles to show relationships between quantities, especially useful for comparison and part-whole problems.

Area model: A visual representation using grids or rectangles to show multiplication and division, demonstrating the concept of arrays.

Number line: A visual representation of numbers on a straight line to show operations, especially addition and subtraction.

Part-whole model: A visual representation that shows how a whole is divided into parts, useful for understanding addition and subtraction relationships.

Concrete representation: Using physical objects or drawings to represent abstract mathematical concepts, bridging the gap between concrete and abstract thinking.

Complete visual modeling methodology:
  1. Understand the problem: Read carefully and identify what is being asked
  2. Identify quantities: Find all the numbers and their relationships
  3. Choose the model: Select the most appropriate visual representation
  4. Create the drawing: Make the visual model accurate and clear
  5. Label the parts: Include all relevant information and numbers
  6. Manipulate the model: Use the drawing to solve the problem
  7. Check the answer: Verify with the visual representation
Tip 1: Use bar models for comparison problems and part-whole relationships.
Tip 2: Use area models for multiplication and division problems.
Tip 3: Use number lines for problems involving movement or sequential operations.
Tip 4: Always label your drawings with numbers and words to make them clear.
Tip 5: Make sure your visual model reflects the actual relationships in the problem.
Common errors: Not labeling drawings, choosing the wrong model type, making drawings too small to see details, not maintaining proportionality in bar models, forgetting to include all parts of the problem in the model.
Real-world applications: Understanding maps and distances, measuring spaces for furniture or gardens, calculating areas for painting or tiling, comparing quantities when shopping, visualizing budgets and expenses.
Key modeling principles:

• Bar models: Show relationships between different quantities with proportional rectangles

• Area models: Represent multiplication as rectangular arrays with rows and columns

• Number lines: Show operations as movements along the line (right = add, left = subtract)

• Part-whole models: Demonstrate how parts combine to form a whole

• Proportional representation: Bars and areas should reflect relative sizes

• Clear labeling: Include all relevant information and units

• Sequential operations: Show multiple steps in order when solving complex problems

Visual Guide to Mathematical Models

📊
Bar Model

Use for: Comparison problems

Shows: Relationships between quantities

Example: Sarah has 15, Tom has 8 more

Draw: Rectangles representing values

Benefit: Easy to see differences

Area Model

Use for: Multiplication problems

Shows: Rows × Columns

Example: 4 rows × 6 columns = 24

Draw: Grid of squares

Benefit: Visualizes multiplication

Number Line

Use for: Addition/subtraction

Shows: Movement along the line

Example: Start at 0, move +12, then -5

Draw: Horizontal line with numbers

Benefit: Shows sequence of operations

Part-Whole

Use for: Taking apart problems

Shows: Whole divided into parts

Example: 8 slices total, 3 eaten

Draw: Circle or rectangle divided

Benefit: Shows missing parts

Questions & Answers

Question: How do I know which type of drawing or model to use for a problem? There are so many different ones.

Answer: Here's how to choose the right model:

  • Bar models: Use when comparing quantities or finding parts of a whole ("more than," "less than," "total")
  • Area models: Use for multiplication problems (arrays, rows × columns)
  • Number lines: Use for problems with movement or steps ("forward/backward," "before/after")
  • Part-whole: Use when something is divided into pieces ("out of," "remaining")

Example: "Sarah has 15 stickers, Tom has 8 more" → Bar model (comparison)

Example: "4 rows of 6 flowers each" → Area model (multiplication)

Example: "Started at 0, moved forward 12, then back 5" → Number line (movement)

Try sketching different models - the one that makes the problem clearest is the right one!

Question: My drawings don't always look perfect, and sometimes the bars aren't exactly proportional. Does that matter?

Answer: The important thing is that your drawing shows the correct relationships, not that it looks perfect:

  • Focus on relationships: If one quantity is larger, make its bar longer
  • Labels matter more: Writing the correct numbers is more important than perfect proportions
  • Neatness helps: Clean drawings are easier to understand, but don't stress about perfection

Example: If Tom has 15 stickers and Sarah has 8 more, Tom's bar should be visibly longer than Sarah's, even if it's not exactly to scale.

The drawing is a tool to help you think - it doesn't need to be a work of art!

As long as you can see the relationships and use the model to solve the problem, it's effective.

Question: Sometimes I think I can solve the problem in my head, so why do I need to draw pictures?

Answer: Drawing models is helpful for several important reasons:

  • Check your thinking: A drawing helps you see if your answer makes sense
  • Build understanding: Visual models help you understand WHY the math works
  • Handle harder problems: As problems get more complex, models become essential
  • Communicate: Drawings help you explain your thinking to others

Example: For "Sarah has 15 stickers, Tom has 8 more," drawing shows you need to add (15 + 8), not subtract.

Even if you can solve it mentally, the model confirms your thinking and builds good habits for more challenging problems ahead!

Models are like training wheels - they help you learn, and you can rely on them when problems get harder.