Examples of Mathematical Practices - Problem Solving Guide

Learn essential mathematical practices with visual examples, practice exercises, and interactive guides for 3rd grade math.

Mathematical Practices Examples
1 Make Sense of Problems
Practice 1: Make Sense of Problems
Understanding what the problem is asking before solving
Make Sense of Problems
Read carefully, understand what is being asked, and plan your approach before solving
Problem-Solving Process
1. Read the problem carefully
2. Identify what you know
3. Identify what you need to find
4. Plan how to solve it
5. Solve the problem
6. Check if your answer makes sense
1
Understand
What is the problem asking?
2
Plan
How will you solve it?
Step 1: Read the problem

Understand what information is given and what is needed

Step 2: Plan your approach

Decide what operations to use

Step 3: Solve systematically

Work through the solution step by step

Understand → Plan → Solve → Check
2 Persevere in Solving
Practice 2: Persevere in Solving
Keep trying different approaches when problems are difficult
Persevere in Solving
Don't give up! Try different strategies when one doesn't work
Persistence Strategies
1. Stay calm when problems seem difficult
2. Try a different approach if the first doesn't work
3. Use drawings or manipulatives to help
4. Break the problem into smaller parts
5. Ask for help when needed
1
Stay Calm
Take deep breaths
2
Try Again
Different approach
3
Break Down
Smaller pieces
4
Keep Trying
Don't give up!
Step 1: Don't panic

Difficult problems are normal and solvable

Step 2: Try different methods

If one strategy doesn't work, try another

Step 3: Break it down

Solve smaller parts of the problem

Persist → Try Different Approaches → Succeed
3 Construct Arguments
Practice 3: Construct Arguments
Explaining your thinking and justifying your answers
Construct Arguments
Explain how you solved the problem and why your answer is correct
Argument Construction
1. Show your work clearly
2. Explain each step
3. Justify your reasoning
4. Check your answer
5. Explain why it makes sense
1
Show Work
Write steps clearly
2
Explain
Why you did this
3
Justify
Why it's correct
4
Verify
Check your answer
Step 1: Show your work

Write down each step you take

Step 2: Explain your thinking

Tell why you chose each step

Step 3: Justify your answer

Explain why your answer is correct

Show → Explain → Justify → Verify
Mathematical Practice Categories
Practice + Persistence = Proficiency
Mathematical Growth
1
Problem Solving
Understand and solve
2
Reasoning
Think logically
3
Arguments
Justify thinking
4
Modeling
Apply math to real situations
5
Tools
Use appropriate tools
6
Precision
Be accurate and clear
7
Structure
See patterns and relationships
8
Patterns
Look for regularity
Tip 1: Always explain your thinking
Tip 2: Use drawings to help solve problems
Tip 3: Check if your answer makes sense
Tip 4: Don't give up on difficult problems
Mathematical Practices:

• Problem Solving: Make sense of problems and persevere in solving them

• Reasoning: Reason abstractly and quantitatively

• Arguments: Construct viable arguments and critique reasoning

• Modeling: Model with mathematics

Why It Matters:

Deeper Understanding: Helps truly understand math concepts

Real-World Application: Applies to everyday situations

Communication: Develops ability to explain mathematical thinking

Practice Applications & Techniques
Think → Act → Reflect
Mathematical Process
Problem-Solving Framework
1. Understand the problem (read carefully)
2. Plan your solution (choose strategy)
3. Execute the plan (solve systematically)
4. Check your work (verify reasonableness)
5. Explain your thinking (construct arguments)
Verification: Always check if answer makes sense
Organization: Keep work neat and organized
Questioning: Ask "Does this answer make sense?"
Best Practices:

• Read problems carefully before solving

• Plan your approach before starting

• Show your work clearly

• Explain your thinking

• Check your answers

• Don't give up on difficult problems

Complete Summary & Study Guide
Key Definitions:

Mathematical Practices: Eight core practices that describe how students should engage with mathematics

Problem Solving: Understanding and solving mathematical challenges systematically

Mathematical Reasoning: Using logical thinking to make sense of mathematical concepts

Constructing Arguments: Explaining and justifying mathematical thinking

Mathematical Modeling: Applying math to real-world situations

Mathematical Practice Process:
  1. Make Sense: Understand the problem completely
  2. Persevere: Keep working even when it's challenging
  3. Reason: Think abstractly and quantitatively
  4. Argue: Construct viable arguments and critique reasoning
  5. Model: Use mathematics to represent real situations
  6. Use Tools: Apply appropriate mathematical tools strategically
  7. Attend to Precision: Communicate precisely and accurately
  8. Look for Structure: See patterns and relationships
Problem Solving: The foundation of all mathematical work
Reasoning: Thinking logically about mathematical concepts
Communication: Explaining mathematical thinking clearly
Tip 1: Always read the entire problem before starting
Tip 2: Draw pictures or diagrams to help visualize
Tip 3: Explain your thinking to someone else
Tip 4: Check your work by using a different method
Simple Example

Problem: Sarah has 12 apples. She gives 5 to her friend. How many does she have left?

Practice: Make sense of problem, subtract, check answer

Solution: 12 - 5 = 7 apples

Check: 7 + 5 = 12 ✓

Advanced Example

Problem: A store has 24 boxes. Each box has 6 toys. How many toys in total?

Practice: Understand problem, multiply, check reasonableness

Solution: 24 × 6 = 144 toys

Check: 144 ÷ 6 = 24 ✓

Essential Rules for Success:

• Always understand the problem before solving

• Show your work and explain your thinking

• Check your answers for reasonableness

• Use multiple strategies when appropriate

• Practice perseverance when problems are challenging

• Look for patterns and relationships in math

Questions & Answers

Question: How do mathematical practices differ from mathematical content in 3rd grade?

Answer: Mathematical practices and content serve different purposes:

  • Mathematical Content: Specific topics like addition, subtraction, multiplication, geometry, etc.
  • Mathematical Practices: Habits of mind and approaches to solving problems
  • Content focuses on: What to learn (skills and concepts)
  • Practices focus on: How to think mathematically
  • Relationship: Practices support and enhance content learning

Practices help students become better mathematical thinkers across all content areas.

Question: Why do I need to explain my math thinking? Can't I just get the right answer?

Answer: Explaining your thinking is important for several reasons:

  • Understanding: It helps you understand the math better
  • Mistake catching: You can spot errors in your reasoning
  • Communication: Others can follow your thinking
  • Memory: Explaining helps you remember concepts
  • Deeper learning: It builds stronger mathematical knowledge

The answer is important, but understanding HOW you got it is even more valuable!

Question: My child struggles with mathematical practices. How can I support their development?

Answer: Support mathematical practices development through:

  • Model thinking: Verbalize your own problem-solving process
  • Ask questions: "How did you solve that?" "Why did you choose that method?"
  • Encourage explanation: Have your child explain their thinking
  • Provide opportunities: Give problems that require reasoning
  • Celebrate process: Praise effort and thinking, not just correct answers

Focus on the process of mathematical thinking, not just the final answer.