Solved Exercises on Problem-Solving Strategies for Grade 3

Master problem-solving strategies: guess and check, draw pictures, work backwards, and logical reasoning through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Guess and check strategy
Exercise 1
Sarah has 12 animals on her farm. Some are chickens and some are cows. She counts 32 legs total. How many chickens and how many cows does she have?
Definition:

Guess and check: Make an educated guess, check if it works, adjust and try again until you find the answer.

Guess and check method:
  1. Understand the problem: Chickens have 2 legs, cows have 4 legs
  2. Make a reasonable guess about the number of each animal
  3. Check if your guess fits both conditions (total animals and total legs)
  4. Adjust your guess and try again if needed
Total animals
12
Total legs
32
Solution
8 chickens, 4 cows
Step 1: Organize the information

Chickens: 2 legs each, Cows: 4 legs each

Total animals: 12, Total legs: 32

Step 2: Make a first guess

Let's guess 6 chickens and 6 cows

Check: 6×2 + 6×4 = 12 + 24 = 36 legs (too many!)

Step 3: Adjust our guess

We have too many legs, so let's try fewer cows

Try 7 chickens and 5 cows: 7×2 + 5×4 = 14 + 20 = 34 legs (still too many)

Step 4: Continue adjusting

Try 8 chickens and 4 cows: 8×2 + 4×4 = 16 + 16 = 32 legs ✓

Sarah has 8 chickens and 4 cows
Final answer:

8 chickens and 4 cows

Check: 8 + 4 = 12 animals ✓

8×2 + 4×4 = 16 + 16 = 32 legs ✓

Applied rules:

• Logical Adjustment: If too many legs, try fewer animals with more legs

• Systematic Approach: Keep track of guesses to avoid repetition

• Verification: Always check your answer against both conditions

2 Draw a picture strategy
Exercise 2
There are 4 friends sitting around a round table. If each person shakes hands with everyone else exactly once, how many handshakes happen in total?
Definition:

Draw a picture: Create a visual representation to help understand the problem and find the solution.

Friends
4
Handshakes
6
Visual
Lines connecting dots
Step 1: Draw 4 dots representing the friends

Label them A, B, C, D

Step 2: Draw lines between each pair

A-B, A-C, A-D, B-C, B-D, C-D

Step 3: Count the lines

There are 6 lines connecting the dots

Step 4: Verify with counting

Person A shakes hands with 3 others

Person B shakes hands with 2 more (already counted A-B)

Person C shakes hands with 1 more (already counted A-C, B-C)

Person D already shook hands with everyone

Total: 3 + 2 + 1 = 6 handshakes

There are 6 handshakes in total
Final answer:

6 handshakes

Applied rules:

• Visual Representation: Drawing helps organize complex information

• Systematic Counting: Avoid double-counting by keeping track

• Pattern Recognition: Visual patterns can reveal mathematical relationships

3 Work backwards strategy
Exercise 3
Tom finished his homework. He had 25 math problems to start with. He gave 5 to his friend, then his teacher gave him 8 more problems. At the end, he had 18 problems left. How many problems did he solve?
Definition:

Work backwards: Start from the end result and undo each step in reverse order to find the starting amount.

End result
18 problems
Reverse steps
-8, +5, -solved
Problems solved
10
Step 1: Start with the end result

Tom had 18 problems left at the end

Step 2: Undo the last action

His teacher gave him 8 more problems

Before receiving 8: 18 - 8 = 10 problems

Step 3: Undo the previous action

He gave 5 problems to his friend

Before giving away 5: 10 + 5 = 15 problems

Step 4: Find the solved problems

He started with 25 problems

After giving away 5: 25 - 5 = 20 problems to work with

He ended with 10 unsolved problems

So he solved: 20 - 10 = 10 problems

Tom solved 10 problems
Final answer:

Tom solved 10 problems

Check: Started with 25, gave 5 away (20 left), solved 10 (10 left), received 8 more (18 total) ✓

Applied rules:

• Reverse Operations: Addition becomes subtraction, subtraction becomes addition

• Track Changes: Keep careful record of each action

• Verification: Check your answer by going forward through the steps

Problem-Solving Strategies: Complete Guide
\(\text{Problem} \rightarrow \text{Strategy} \rightarrow \text{Solution}\)
Problem-Solving Process
Guess & Check
Make educated guesses
Try, verify, adjust
Draw Picture
Visual representation
Sketch to clarify
Work Backwards
Reverse the steps
Start from the end
Key definitions:

Problem-Solving: Finding a way to reach a goal when the path isn't immediately obvious

Strategy: A plan or method for approaching and solving problems

Algorithm: A step-by-step procedure for solving a problem

Heuristic: A rule of thumb or strategy that often works but isn't guaranteed

Logical Reasoning: Using facts and logic to reach conclusions

Four-Step Problem-Solving Process:
  1. Understand: Read the problem carefully, identify what is known and what needs to be found
  2. Plan: Choose an appropriate strategy based on the problem type
  3. Solve: Carry out the plan using the chosen strategy
  4. Check: Verify that the answer makes sense and satisfies all conditions
Tip 1: Always read the problem twice before starting.
Tip 2: Underline or circle important information.
Tip 3: If one strategy doesn't work, try a different one.
Tip 4: Always check your answer by substituting it back into the problem.
Tip 5: Look for patterns in the problem that might suggest a strategy.
Common errors: Misreading the problem, not checking answers, using the wrong strategy, calculation mistakes.
Success strategies: Careful reading, systematic approach, verification, flexibility in trying different methods.
Problem-Solving Strategies:

• Guess and Check: Make an educated guess, check if it works, adjust as needed

• Draw a Picture: Create visual representations to clarify relationships

• Work Backwards: Start from the end result and reverse operations

• Make a Table: Organize information systematically

• Find a Pattern: Look for repeated sequences or relationships

• Act It Out: Physically demonstrate or role-play the problem

Solution: Exercises 4 to 5
4 Make a table strategy
Exercise 4
A pizza place offers small, medium, and large pizzas. Small costs $8, medium costs $12, and large costs $16. If you have $35, what combinations of pizzas can you buy without going over budget?
Definition:

Make a table: Organize information in rows and columns to systematically explore possibilities.

Budget
$35
Pizza prices
S=$8, M=$12, L=$16
Valid combos
Multiple options
Step 1: Set up a table with possible combinations
Small Pizzas Medium Pizzas Large Pizzas Cost Valid?
4 0 0 $32 Yes
3 1 0 $36 No
2 1 0 $28 Yes
1 2 0 $32 Yes
0 2 0 $24 Yes
1 0 1 $24 Yes
0 1 1 $28 Yes
0 0 2 $32 Yes
Step 2: Systematically list all valid combinations

After checking the table, valid combinations under $35 are:

  • 4 small pizzas ($32)
  • 2 small + 1 medium ($28)
  • 1 small + 2 medium ($32)
  • 2 medium ($24)
  • 1 small + 1 large ($24)
  • 1 medium + 1 large ($28)
  • 2 large ($32)
Step 3: Verify each combination

Check that each option is ≤ $35

7 valid combinations: 4S, 2S+1M, 1S+2M, 2M, 1S+1L, 1M+1L, 2L
Final answer:

You can buy any of these combinations:

• 4 small pizzas ($32)

• 2 small + 1 medium ($28)

• 1 small + 2 medium ($32)

• 2 medium pizzas ($24)

• 1 small + 1 large ($24)

• 1 medium + 1 large ($28)

• 2 large pizzas ($32)

Applied rules:

• Systematic Organization: Tables help organize complex information

• Complete Exploration: Methodically check all possibilities

• Verification: Check each option against the constraint

5 Logical reasoning strategy
Exercise 5
Three friends - Alex, Ben, and Carol - are standing in a line. Alex is not first. Ben is not last. Carol is not in the middle. Who is in the middle?
Definition:

Logical reasoning: Using given clues and process of elimination to determine the answer.

Clues
Alex≠1st, Ben≠last, Carol≠middle
Positions
1st, 2nd, 3rd
Middle person
Ben
Step 1: List the constraints

• Alex is not first

• Ben is not last

• Carol is not in the middle

Step 2: Consider who can be in each position

First position: Can't be Alex (given), could be Ben or Carol

Middle position: Can't be Carol (given), could be Alex or Ben

Last position: Can't be Ben (given), could be Alex or Carol

Step 3: Use process of elimination

If Carol is first, then Alex or Ben is middle, and Ben or Alex is last

But Ben can't be last, so if Carol is first, Ben must be middle and Alex last

Check: Alex is not first ✓, Ben is not last ✗ (Ben would be middle)

This works!

Step 4: Verify the solution

Carol is first, Ben is middle, Alex is last

• Alex is not first ✓ (Alex is last)

• Ben is not last ✓ (Ben is middle)

• Carol is not in the middle ✓ (Carol is first)

Ben is in the middle
Final answer:

Ben is in the middle

Arrangement: Carol (first), Ben (middle), Alex (last)

Applied rules:

• Process of Elimination: Remove impossible options systematically

• Constraint Satisfaction: Ensure all conditions are met

• Logical Deduction: Use implications of each clue

Advanced Problem-Solving Techniques
\(\text{Strategy} = f(\text{Problem Type}, \text{Given Information})\)
Strategy Selection Function
Advanced definitions:

Multi-step Problems: Problems requiring more than one operation or strategy

Open-ended Problems: Problems with multiple valid solutions

Constraint Problems: Problems with specific limitations or requirements

Optimization Problems: Finding the best solution among multiple options

Pattern-Based Problems: Problems that rely on identifying sequences or relationships

Advanced Problem-Solving Methodology:
  1. Problem Analysis: Identify the type of problem and key information
  2. Strategy Selection: Choose the most appropriate approach
  3. Multiple Approaches: Be ready to switch strategies if needed
  4. Systematic Execution: Follow the plan methodically
  5. Verification: Check the solution thoroughly
  6. Reflection: Consider alternative approaches and learn from the process
Tip 1: For complex problems, break them into smaller, manageable parts.
Tip 2: When stuck, try a simpler version of the same problem.
Tip 3: Look for connections between different problem-solving strategies.
Tip 4: Practice explaining your solution process to others.
Tip 5: Keep a problem-solving journal to track successful approaches.
Advanced errors: Jumping to conclusions without sufficient analysis, ignoring alternative approaches, failing to verify solutions.
Advanced strategies: Combining multiple strategies, using algebraic thinking, systematic enumeration, working with variables.
Strategy Selection Guidelines:

• Guess and Check: When you can easily verify answers (e.g., age problems, coin problems)

• Draw a Picture: For spatial problems, geometric problems, or relationship problems

• Work Backwards: When you know the end result and need to find the beginning

• Make a Table: For problems with multiple possibilities or systematic exploration

• Find a Pattern: For sequence problems or repetitive structures

• Logical Reasoning: For constraint satisfaction problems or puzzles

Questions & Answers

Question: How do I know which problem-solving strategy to use? There are so many different ones!

Answer: Great question! Here's how to choose the right strategy:

  • Guess and Check: Use when the numbers aren't too big and you can easily check your answer
  • Draw a Picture: Use when the problem talks about shapes, positions, or relationships
  • Work Backwards: Use when you know the final result and need to find the beginning
  • Make a Table: Use when you need to organize information or try multiple possibilities
  • Logical Reasoning: Use for puzzles or problems with clues about what's true/false

If you're not sure, start with the strategy that seems easiest, and switch if it's not working!

Question: My child gets frustrated when a problem-solving strategy doesn't work immediately. How can I help them persist?

Answer: Problem-solving frustration is normal! Here are supportive strategies:

  • Remind them that mathematicians often try multiple approaches
  • Encourage them to take a short break and come back to the problem
  • Ask guiding questions like "What do you know?" and "What are you trying to find?"
  • Praise their effort and persistence, not just correct answers
  • Model problem-solving by thinking aloud when solving problems together

Remember: struggling with problems builds resilience and deeper understanding!

Question: Can I use more than one strategy for the same problem? Sometimes I want to draw a picture AND make a table!

Answer: Absolutely! Many problems benefit from multiple strategies:

  • You might draw a picture to understand the problem, then make a table to organize information
  • You could use guess and check while also drawing a diagram to visualize progress
  • Working backwards might involve creating a table to track your reverse steps
  • Logical reasoning often benefits from drawing diagrams to visualize relationships

Using multiple strategies can help you check your work and deepen your understanding. The best mathematicians are flexible thinkers who use whatever tools help them solve the problem!