Solved Exercises on Model-Based Problem Solving in Grade 1

Master model-based problem solving: visual models, concrete representations, mathematical modeling, and problem-solving strategies through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Visual Model: Addition
Exercise 1
Sarah has 4 apples. Her mom gives her 3 more apples. How many apples does Sarah have now? Draw a picture to solve.
Definition:

Visual Model: Using pictures or drawings to represent a math problem

Modeling Method:
  1. Draw the starting amount
  2. Draw the added amount
  3. Count all the items
  4. Write the answer
🍎🍎🍎🍎 + 🍎🍎🍎 = 🍎🍎🍎🍎🍎🍎🍎
Step 1: Draw Sarah's original apples

Draw 4 apples: 🍎🍎🍎🍎

Step 2: Draw the apples her mom gave her

Draw 3 more apples: 🍎🍎🍎

Step 3: Count all the apples together

Count: 1, 2, 3, 4, 5, 6, 7

Step 4: Write the number sentence

4 + 3 = 7

Sarah has 7 apples
Final answer:

Sarah has 7 apples altogether.

Applied rules:

• Visual Representation: Draw pictures to show the problem

• One-to-One Correspondence: Each picture represents one object

• Addition Model: Combine groups to find the total

• Counting Strategy: Count all items to find the sum

Tip: Draw neat rows to make counting easier.
Tip: Use the same symbol for the same type of object.
2 Visual Model: Subtraction
Exercise 2
Tom has 8 cookies. He eats 3 cookies. How many cookies does Tom have left? Draw a picture to solve.
Definition:

Subtraction Model: Using visual representations to show taking away

🍪🍪🍪🍪🍪🍪🍪🍪 - 🍪🍪🍪 = 🍪🍪🍪🍪🍪
Step 1: Draw all the cookies Tom starts with

Draw 8 cookies: 🍪🍪🍪🍪🍪🍪🍪🍪

Step 2: Cross out the cookies he eats

Cross out 3 cookies: 🍪🍪🍪~~🍪🍪🍪🍪🍪~~

Step 3: Count the remaining cookies

Count: 1, 2, 3, 4, 5

Step 4: Write the number sentence

8 - 3 = 5

Tom has 5 cookies left
Final answer:

Tom has 5 cookies left.

Applied rules:

• Subtraction Model: Remove items to show taking away

• Visual Removal: Cross out or separate items being subtracted

• Remaining Count: Count what's left after removal

• Number Sentence: Connect visual model to mathematical notation

Tip: Cross out items to show they're being taken away.
Tip: Always count what remains, not what was taken away.
3 Comparison Model
Exercise 3
Lisa has 6 stickers. Maya has 9 stickers. Who has more stickers? How many more? Draw to compare.
Definition:

Comparison Model: Using visual arrangements to compare quantities

Lisa: 🌟🌟🌟🌟🌟🌟
Maya: 🌟🌟🌟🌟🌟🌟🌟🌟🌟
Maya has 3 more
Step 1: Draw Lisa's stickers in one row

Draw 6 stickers: 🌟🌟🌟🌟🌟🌟

Step 2: Draw Maya's stickers in another row

Draw 9 stickers: 🌟🌟🌟🌟🌟🌟🌟🌟🌟

Step 3: Compare the rows visually

Maya's row is longer, so she has more

Step 4: Count the difference

Maya has 3 more than Lisa (9 - 6 = 3)

Maya has 3 more stickers
Final answer:

Maya has more stickers. She has 3 more than Lisa.

Applied rules:

• Side-by-Side Comparison: Arrange items in rows to compare

• Visual Difference: The extra items show how much more

• Subtraction Connection: Difference found by subtracting smaller from larger

• Quantity Recognition: Longer row represents greater quantity

Tip: Draw items in neat rows to make comparison clear.
Tip: Match up items to see the difference easily.
Model-Based Problem Solving Methods
Model → Solve → Check → Explain
Problem Solving
1️⃣
Draw or build a model of the problem
2️⃣
Use the model to find the answer
3️⃣
Check if your answer makes sense
4️⃣
Explain how you used the model
Addition Model
Draw + Draw = Total
Combine Groups
Subtraction Model
Start - Take Away = Left
Remove Items
Comparison Model
Bigger - Smaller = Difference
Find Difference
Key definitions:

Model: A representation of a problem using pictures, objects, or diagrams

Visual Model: Using drawings to show what's happening in a problem

Concrete Model: Using real objects to represent the problem

Mathematical Model: Connecting visual representations to number sentences

Complete methodology:
  1. Understand: Read the problem carefully and identify what you need to find
  2. Model: Create a visual or concrete representation of the problem
  3. Solve: Use the model to find the answer
  4. Check: Make sure your answer makes sense
  5. Explain: Describe how you used the model to solve the problem
Tip 1: Always draw the starting amount first, then show what changes.
Tip 2: Use the same symbol for the same type of object.
Tip 3: Keep your drawings neat and organized.
Tip 4: Always write the number sentence that matches your model.
Common errors: Miscounting, not drawing all parts of the problem, forgetting to check answers.
Test preparation: Practice drawing models, counting accurately, connecting visuals to numbers.
Formulas to know by heart:

• Addition: Start + More = Total

• Subtraction: Start - Take Away = Left

• Comparison: Bigger - Smaller = Difference

• Modeling: Draw → Count → Solve → Explain

Solution: Exercises 4 to 5
4 Shape Model: Pattern Recognition
Exercise 4
The pattern is: Circle, Square, Triangle, Circle, Square, Triangle... What shape comes next after Triangle? Draw the pattern to show.
Definition:

Pattern Model: Using visual arrangements to identify and continue sequences

⭕ □ △ ⭕ □ △ ⭕ □ △
Step 1: Draw the repeating pattern

Circle, Square, Triangle: ⭕ □ △

Step 2: Continue the pattern

Continue: ⭕ □ △ ⭕ □ △ ⭕ □ △

Step 3: Identify what comes after Triangle

After each Triangle, the next shape is Circle

Step 4: Verify the pattern continues correctly

Circle → Square → Triangle → Circle → Square → Triangle → Circle

Next shape: Circle
Final answer:

Circle comes next after Triangle in the pattern.

Applied rules:

• Pattern Recognition: Identify the repeating sequence

• Visual Continuation: Extend the pattern using the same sequence

• Position Analysis: Understand what comes after specific elements

• Pattern Consistency: Maintain the same order throughout

Tip: Circle or highlight the repeating part to see the pattern clearly.
Tip: Say the pattern out loud to hear the rhythm.
5 Ordering Model: Sequencing
Exercise 5
Draw 4 towers with 2, 5, 1, and 3 blocks each. Put them in order from shortest to tallest.
Definition:

Ordering Model: Using visual representations to arrange items by size or quantity

1 block
🧱
2 blocks
🧱🧱
3 blocks
🧱🧱🧱
5 blocks
🧱🧱🧱🧱🧱
Step 1: Draw each tower with correct number of blocks

Tower A: 2 blocks, Tower B: 5 blocks, Tower C: 1 block, Tower D: 3 blocks

Step 2: Compare heights visually

Tower C (1) < Tower A (2) < Tower D (3) < Tower B (5)

Step 3: Arrange from shortest to tallest

1 block, 2 blocks, 3 blocks, 5 blocks

Step 4: Verify the order is correct

Each tower is taller than the one before it

Order: 1, 2, 3, 5 blocks
Final answer:

The towers in order from shortest to tallest: 1 block, 2 blocks, 3 blocks, 5 blocks.

Applied rules:

• Visual Comparison: Taller objects represent larger numbers

• Sequential Ordering: Arrange from smallest to largest

• Height Correlation: More blocks = taller tower = larger number

• Ascending Sequence: Values increase from left to right

Tip: Draw towers with equal-sized blocks to make comparison fair.
Tip: Start with the smallest number and work up to the largest.
Comprehensive Summary: Model-Based Problem Solving
Complete Learning Summary

Definitions of Key Concepts:

  • Model: A representation of a problem using pictures, objects, or diagrams to help understand and solve it.
  • Visual Model: Using drawings or pictures to show what's happening in a math problem.
  • Concrete Model: Using real objects or manipulatives to represent the problem.
  • Mathematical Model: Connecting visual representations to number sentences and equations.
  • Problem Solving: Finding solutions to mathematical challenges using systematic approaches.
  • Pattern Recognition: Identifying regular sequences or repeated elements in problems.

Core Rules and Principles:

  • One-to-One Correspondence: Each picture or object represents exactly one item in the problem.
  • Visual Representation: Drawings should accurately reflect the quantities and relationships in the problem.
  • Model Consistency: The model should maintain the same relationships as the original problem.
  • Addition Rule: Combine groups to find totals (Start + More = Total).
  • Subtraction Rule: Remove items to find what remains (Start - Take Away = Left).
  • Comparison Rule: Find differences by subtracting smaller from larger (Bigger - Smaller = Difference).

Step-by-Step Methods:

  • Basic Modeling Method: 1) Read the problem carefully, 2) Draw a picture showing what happens, 3) Count or calculate using your model, 4) Check your answer, 5) Explain how you solved it.
  • Addition Modeling: 1) Draw the starting amount, 2) Draw what's added, 3) Count all items together, 4) Write the number sentence.
  • Subtraction Modeling: 1) Draw all items initially present, 2) Cross out or separate items being removed, 3) Count remaining items, 4) Write the number sentence.
  • Comparison Modeling: 1) Draw both quantities side by side, 2) Compare visually, 3) Count the difference, 4) Express as a number sentence.

Examples from Simple to Advanced:

  • Simple: "2 + 3 = ?" → Draw 🍎🍎 + 🍎🍎🍎 = 🍎🍎🍎🍎🍎 → Count to get 5.
  • Intermediate: "8 - 3 = ?" → Draw 🍪🍪🍪🍪🍪🍪🍪🍪, cross out 3, count remaining to get 5.
  • Advanced: "Pattern: Circle, Square, Triangle repeated" → Draw the sequence to predict the 10th shape.

Tips, Tricks, and Common Pitfalls:

  • Tips: Always draw neatly, use consistent symbols, label your drawings, write number sentences that match your model.
  • Tricks: For addition, combine all drawings together; for subtraction, cross out items; for comparison, line up items side by side.
  • Common Pitfalls: Drawing incorrect quantities, miscounting, not representing the problem accurately, forgetting to check if the answer makes sense.

Key Notes for Memorization:

  • Memory Aid: "Draw → Count → Solve → Explain" - the four steps of model-based problem solving.
  • Operation Memory: Plus (+) means combine/add; Minus (-) means take away/remove.
  • Pattern Memory: Look for the smallest repeating unit in any sequence.
  • Comparison Memory: Bigger number minus smaller number equals the difference.

Explanation for Students:

Using models to solve problems is like having a helper that shows you exactly what's happening! When you draw pictures or use objects, you can see the problem clearly. This makes it much easier to understand and solve.

Think of it like this: instead of trying to imagine everything in your head, you're making a little movie of the problem with drawings. This way, you can count, add, subtract, or compare with real visual help!

Models help you slow down and think carefully about what the problem is asking, which makes you a better problem solver!

Key Concepts and Strategies for Model-Based Problem Solving
Model → Solve → Check → Explain
Problem Solving
Key definitions:

Model: A representation of a problem using pictures, objects, or diagrams

Visual Model: Using drawings to show what's happening in a problem

Concrete Model: Using real objects to represent the problem

Mathematical Model: Connecting visual representations to number sentences

Problem Solving: Finding solutions to mathematical challenges using systematic approaches

Complete methodology:
  1. Understand: Read the problem carefully and identify what you need to find
  2. Model: Create a visual or concrete representation of the problem
  3. Solve: Use the model to find the answer
  4. Check: Make sure your answer makes sense
  5. Explain: Describe how you used the model to solve the problem
Tip 1: Always draw the starting amount first, then show what changes.
Tip 2: Use the same symbol for the same type of object.
Tip 3: Keep your drawings neat and organized.
Tip 4: Always write the number sentence that matches your model.
Tip 5: Practice with different types of models to strengthen your problem-solving skills.
Common errors: Miscounting, not drawing all parts of the problem, forgetting to check answers.
Test preparation: Practice drawing models, counting accurately, connecting visuals to numbers.
Success strategies: Read slowly, draw carefully, count twice, connect to number sentences, explain your thinking.
Essential rules to remember:

• Modeling Rule: Draw → Count → Solve → Explain

• Addition Rule: Start + More = Total

• Subtraction Rule: Start - Take Away = Left

• Comparison Rule: Bigger - Smaller = Difference

• Accuracy Rule: Each drawing represents exactly one object

Questions & Answers

Question: Sometimes my drawings don't look perfect. Does that matter for solving the problem?

Answer: Don't worry about perfect drawings! What matters most is that your drawings help you solve the problem:

  • Recognition: Make sure others can tell what your drawing represents
  • Quantity: Draw the correct number of items
  • Claity: Keep your drawings neat enough to count easily
  • Purpose: Focus on helping yourself understand the problem

A wobbly circle is still a circle if you're using it to represent an apple. The math is what's important, not perfect artwork! Practice will make your drawings neater over time.

Question: My child struggles to connect their drawings to number sentences. How can I help?

Answer: Building connections between visual models and number sentences takes practice:

  • Parallel Language: Use the same words when describing drawings and writing equations
  • Write as You Draw: Narrate the equation while creating the visual model
  • Color Coding: Use different colors for different parts of the equation
  • Practice Translations: Go back and forth between drawings and numbers
  • Start Simple: Begin with basic addition/subtraction before moving to complex problems

Encourage your child to verbalize the connection: "I drew 4 apples and 3 more apples, so I write 4 + 3 = 7."

Question: When should I use a model? Can't I just solve the problem in my head?

Answer: Great question! Models are especially helpful when:

  • Problems are new or challenging
  • You're not sure what to do
  • You want to double-check your answer
  • The problem has multiple steps
  • You need to explain your thinking to someone else

Even if you can solve some problems in your head, models help you understand WHY the answer is correct. They're like training wheels that help you become a stronger problem solver! As you get more confident, you might need models less often, but they're always helpful tools to have.