2nd Grade Fractions Introduction - Understanding Problem Solving with Solutions

Learning to solve fraction problems: problem analysis, planning, and step-by-step solutions through these 10 detailed exercises.

Solutions: Exercises 1 to 10
1 Problem Analysis
Exercise 1
Read this problem: "Sarah has 8 cookies. She gives 3 cookies to her friend. What fraction of her cookies does she give away?"
Difficulty: Beginner Time: ~4 minutes Skills: Problem Analysis
Definition:

Problem analysis: Breaking down a word problem to identify what is being asked and what information is given

Note: Understanding the problem is the first crucial step in problem solving.

Step-by-step problem analysis method:
  1. Read the problem carefully multiple times
  2. Identify the question being asked
  3. List the important information provided
  4. Determine what type of problem this is (finding a fraction)
Problem Breakdown:

Given Information: Sarah has 8 cookies, she gives away 3 cookies

What's Being Asked: What fraction of her cookies does she give away?

Key Words: "fraction", "give away"

Operation Needed: Division (part over whole)

1
Identify the total number of cookies: 8 cookies
2
Identify the number of cookies given away: 3 cookies
3
Set up the fraction: cookies given away over total cookies = 3/8
4
Verify: 3 out of 8 cookies is indeed 3/8 of the total
3/8 of the cookies were given away
Final answer:

Sarah gave away 3/8 of her cookies.

Applied rules:

• Problem breakdown: Identify what is given and what is asked

• Fraction definition: Part over whole

• Verification: Check that the answer makes sense in context

• Practice Tip: Underline key information in the problem to stay focused

Related Examples:
  • Out of 10 candies, eating 4 → fraction eaten = 4/10
  • From 12 toys, giving 5 → fraction given = 5/12
  • Of 6 cookies, eating 2 → fraction eaten = 2/6
Quick Tips:
  • Read the problem multiple times to understand it completely
  • Identify what the question is asking for
  • Look for key words like "fraction", "part", "out of"
Frequently Asked Questions:

Q: How do I know what the fraction should represent?
A: Look for what is being compared to the total amount.

Q: What if there are extra details?
A: Focus only on the information needed to answer the question.

2 Planning Solutions
Exercise 2
Plan how to solve: "A pizza is cut into 6 equal slices. John eats 2 slices. What fraction of the pizza is left?"
Difficulty: Beginner Time: ~4 minutes Skills: Planning Solutions
Definition:

Planning solutions: Developing a strategy before attempting to solve a problem

Note: Having a plan helps organize thoughts and prevents mistakes.

Step-by-step planning method:
  1. Identify the total amount (whole)
  2. Identify the amount used/consumed
  3. Determine what is being asked (remaining amount)
  4. Plan the calculation (total - used = remaining)
  5. Express the answer as a fraction
Solution Plan:

Step 1: Total pizza = 6 slices

Step 2: Slices eaten = 2 slices

Step 3: Slices remaining = 6 - 2 = 4 slices

Step 4: Fraction remaining = 4/6

1
Identify the total: 6 equal slices make up the whole pizza
2
Identify what was eaten: John ate 2 slices
3
Calculate remaining: 6 - 2 = 4 slices left
4
Express as fraction: 4 slices out of 6 total = 4/6
4/6 of the pizza is left
Final answer:

4/6 of the pizza is left.

Applied rules:

• Problem planning: Organize steps before calculating

• Subtraction: Remaining = Total - Used

• Fraction expression: Remaining over total

• Practice Tip: Write down your plan before starting calculations

Related Examples:
  • 8 pieces of candy, eat 3 → 8-3=5 left → 5/8 remains
  • 10 pages in book, read 4 → 10-4=6 left → 6/10 remains
  • 12 stickers, give away 5 → 12-5=7 left → 7/12 remains
Quick Tips:
  • Plan your approach before doing calculations
  • For "left" problems, subtract the used amount from the total
  • Always express the answer as a fraction of the original whole
Frequently Asked Questions:

Q: Do I always subtract to find what's left?
A: Yes, when something is taken away, use subtraction.

Q: Should I simplify the fraction?
A: Yes, 4/6 can be simplified to 2/3.

3 Step-by-Step Approach
Exercise 3
Solve this problem step by step: "There are 10 balloons. 3 are red, 4 are blue, and the rest are yellow. What fraction of the balloons are yellow?"
Difficulty: Intermediate Time: ~5 minutes Skills: Step-by-Step Approach
Definition:

Step-by-step approach: Solving problems by breaking them into smaller, manageable parts

Note: This systematic approach helps prevent errors and ensures all steps are completed.

Step-by-step solving method:
  1. Identify the total amount
  2. Identify the amounts of known categories
  3. Calculate the unknown category
  4. Express as a fraction
  5. Simplify if possible
1
Total balloons = 10
2
Red balloons = 3, Blue balloons = 4
3
Yellow balloons = Total - Red - Blue = 10 - 3 - 4 = 3
4
Fraction of yellow balloons = Yellow/Total = 3/10
3/10 of the balloons are yellow
Final answer:

3/10 of the balloons are yellow.

Applied rules:

• Systematic approach: Tackle one piece at a time

• Subtraction: Find unknown amount by subtracting known amounts from total

• Fraction creation: Part over whole

• Practice Tip: Organize information in a table or list to keep track

Related Examples:
  • 12 marbles: 5 red, 4 blue, rest green → 12-5-4=3 green → 3/12 = 1/4
  • 15 fruits: 6 apples, 5 oranges, rest bananas → 15-6-5=4 bananas → 4/15
  • 20 toys: 8 cars, 7 dolls, rest blocks → 20-8-7=5 blocks → 5/20 = 1/4
Quick Tips:
  • Organize information in categories
  • Use subtraction to find the missing part
  • Always check that all parts add up to the total
Frequently Asked Questions:

Q: What if the known parts don't add up to the total?
A: Then there's an error in the problem or your calculation.

Q: How do I check my work?
A: Verify that all parts add up to the total amount.

Solutions: Exercises 4 to 5
4 Visualization
Exercise 4
Draw a picture to solve: "A rectangle is divided into 8 equal parts. 3 parts are shaded. What fraction is shaded? What fraction is not shaded?"
Definition:

Visualization: Using drawings or diagrams to understand and solve problems

Note: Visual representations make abstract concepts more concrete and understandable.

Step-by-step visualization method:
  1. Draw the whole shape described in the problem
  2. Divide it into equal parts as specified
  3. Shade the specified number of parts
  4. Count the shaded parts for the first fraction
  5. Count the unshaded parts for the second fraction
1
Draw a rectangle and divide it into 8 equal parts
2
Shade 3 parts (green) and leave 5 parts unshaded (red)
3
Shaded fraction = 3 parts out of 8 total = 3/8
4
Unshaded fraction = 5 parts out of 8 total = 5/8
Shaded: 3/8, Unshaded: 5/8
Final answer:

3/8 of the rectangle is shaded and 5/8 is not shaded.

Applied rules:

• Visual representation: Diagrams help understand the problem

• Counting: Count specific parts for each fraction

• Verification: Shaded + unshaded = total (3/8 + 5/8 = 8/8 = 1)

• Practice Tip: Visual models make abstract concepts more concrete

Related Examples:
  • Circle divided into 6 parts with 2 shaded → 2/6 shaded, 4/6 unshaded
  • Rectangle with 10 parts, 7 shaded → 7/10 shaded, 3/10 unshaded
  • Square divided into 4 parts with 1 shaded → 1/4 shaded, 3/4 unshaded
Quick Tips:
  • Draw neat, equal-sized parts for accurate counting
  • Use different colors or patterns to distinguish parts
  • Always verify that all parts add up to the whole
Frequently Asked Questions:

Q: Do I always need to draw pictures?
A: Not always, but it helps with understanding, especially with fractions.

Q: What if parts aren't equal?
A: Then it's not a valid fraction model - parts must be equal.

5 Checking Work
Exercise 5
Solve: "A bag has 12 candies. Emma eats 1/4 of them. How many candies did Emma eat? Check your work."
Definition:

Checking work: Verifying that your solution is correct using alternative methods

Note: Verification helps catch errors and builds confidence in problem-solving abilities.

Step-by-step checking method:
  1. Solve the problem using the primary method
  2. Use an alternative method to verify the answer
  3. Check that the answer makes sense in context
  4. Confirm that the answer satisfies the original conditions
1
Find 1/4 of 12: 12 ÷ 4 = 3 candies
2
Check: If Emma ate 3 candies, that's 3/12 of the total
3
Simplify 3/12: 3/12 = 1/4 ✓
4
Alternative check: 1/4 of 12 = 1/4 × 12 = 12/4 = 3 ✓
Emma ate 3 candies
Final answer:

Emma ate 3 candies.

Applied rules:

• Verification: Use multiple methods to confirm your answer

• Fraction of a number: Divide the number by the denominator

• Alternative check: Multiply fraction by whole number

• Practice Tip: Always verify your answer using a different approach

Related Examples:
  • 1/3 of 15 = 15÷3 = 5, check: 5/15 = 1/3 ✓
  • 1/5 of 20 = 20÷5 = 4, check: 4/20 = 1/5 ✓
  • 1/2 of 14 = 14÷2 = 7, check: 7/14 = 1/2 ✓
Quick Tips:
  • Always check that your answer makes sense
  • Use different methods to verify your solution
  • Simplify fractions to confirm they match the original problem
Frequently Asked Questions:

Q: What if my checks don't match?
A: Then you made an error and should review your work.

Q: How many ways should I check my work?
A: At least one alternative method to confirm your answer.

Solutions: Exercises 6 to 10
6 Multi-Step Problems
Exercise 6
"A baker made 24 cupcakes. On Monday, he sold 1/3 of them. On Tuesday, he sold 4 more. How many cupcakes does he have left? What fraction of the original number is this?"
Definition:

Multi-step problems: Problems requiring multiple operations or calculations to reach the solution

Note: Each step builds on the previous one, so accuracy at each stage is important.

Step-by-step solving method:
  1. Calculate the first operation
  2. Use the result in the next operation
  3. Continue through all required steps
  4. Express the final answer as requested
1
Calculate Monday's sales: 1/3 of 24 = 24 ÷ 3 = 8 cupcakes sold
2
Calculate remaining after Monday: 24 - 8 = 16 cupcakes
3
Calculate remaining after Tuesday: 16 - 4 = 12 cupcakes
4
Express as fraction of original: 12/24 = 1/2
12 cupcakes left, which is 1/2 of the original
Final answer:

The baker has 12 cupcakes left, which is 1/2 of the original number.

Applied rules:

• Sequential operations: Each step depends on the previous one

• Fraction of a number: Divide by the denominator

• Fraction simplification: Reduce to lowest terms

• Practice Tip: Keep track of changing quantities at each step

Related Examples:
  • 20 cookies: 1/4 on day 1 (5), 3 more on day 2 → 20-5-3=12 left → 12/20=3/5
  • 18 stickers: 1/3 on day 1 (6), 2 more on day 2 → 18-6-2=10 left → 10/18=5/9
  • 30 pencils: 1/5 on day 1 (6), 5 more on day 2 → 30-6-5=19 left → 19/30
Quick Tips:
  • Track quantities at each step
  • Perform operations in the order given
  • Simplify fractions when possible
Frequently Asked Questions:

Q: What if I make a mistake in an early step?
A: It will affect all subsequent steps, so check each step carefully.

Q: How do I know the order of operations?
A: Follow the sequence presented in the problem.

7 Real-World Context
Exercise 7
"At a birthday party, there are 2 pizzas cut into 8 slices each. If the children eat 10 slices, what fraction of the total pizza is left?"
Definition:

Real-world context: Problems set in practical, everyday situations that connect math to real life

Note: These problems help students understand the relevance of math in daily activities.

Step-by-step solving method:
  1. Calculate the total amount available
  2. Identify the amount used
  3. Calculate the remaining amount
  4. Express as a fraction of the total
1
Calculate total slices: 2 pizzas × 8 slices each = 16 slices
2
Identify slices eaten: 10 slices
3
Calculate remaining slices: 16 - 10 = 6 slices
4
Express as fraction: 6/16 = 3/8 of the pizza is left
3/8 of the total pizza is left
Final answer:

3/8 of the total pizza is left.

Applied rules:

• Total calculation: Multiply quantity by units per item

• Subtraction: Remaining = Total - Used

• Fraction simplification: 6/16 = 3/8

• Practice Tip: Connect math to familiar situations to enhance understanding

Related Examples:
  • 3 cakes with 6 slices each, 14 eaten → 18-14=4 left → 4/18=2/9
  • 4 sandwiches cut into 3 pieces each, 9 eaten → 12-9=3 left → 3/12=1/4
  • 2 loaves of bread with 10 slices each, 15 eaten → 20-15=5 left → 5/20=1/4
Quick Tips:
  • Calculate the total amount first
  • Identify what's being compared to what
  • Make sure units are consistent throughout
Frequently Asked Questions:

Q: What if the items are different sizes?
A: For fractions, we assume equal parts unless stated otherwise.

Q: Should I always simplify the fraction?
A: Yes, when possible, to express in simplest form.

8 Strategy Selection
Exercise 8
"Which strategy would be best for this problem: 'A garden has 15 flowers. 6 are red, some are yellow, and 4 are pink. What fraction are yellow?' Draw, calculate, or make a table?"
Definition:

Strategy selection: Choosing the most effective approach for solving a particular problem

Note: Different problems may benefit from different solution strategies.

Step-by-step strategy evaluation:
  1. Examine the problem requirements
  2. Consider which strategy would be most efficient
  3. Calculate using the selected strategy
  4. Verify that the strategy worked well
1
Analyze the problem: Need to find unknown quantity (yellow flowers)
2
Best strategy: Calculation - subtract known quantities from total
3
Calculate: Yellow = Total - Red - Pink = 15 - 6 - 4 = 5
4
Express as fraction: 5/15 = 1/3 are yellow
Calculation strategy; 1/3 are yellow
Final answer:

The calculation strategy is best. 1/3 of the flowers are yellow.

Applied rules:

• Strategy evaluation: Consider the problem's requirements

• Efficiency: Choose the most direct approach

• Arithmetic: Use subtraction to find unknown quantities

• Practice Tip: Match strategy to problem type for optimal efficiency

Related Examples:
  • For "compare fractions": Use visual models or common denominators
  • For "find part of whole": Use calculation (division/multiplication)
  • For "pattern recognition": Use organized lists or tables
Quick Tips:
  • For unknown quantities: Use calculation (addition/subtraction)
  • For comparisons: Use visual models or common denominators
  • For patterns: Use tables or organized lists
Frequently Asked Questions:

Q: What if multiple strategies could work?
A: Choose the most efficient one that leads to the answer quickly.

Q: How do I know which strategy to pick?
A: Look at what the problem is asking and choose accordingly.

9 Problem Modification
Exercise 9
"Original problem: 'A pizza has 8 slices. Tom eats 3.' Modify it to ask for the fraction remaining, then solve."
Definition:

Problem modification: Changing the question or parameters of a problem to explore different aspects

Note: This technique helps deepen understanding and practice different solution approaches.

Step-by-step modification method:
  1. Identify the original problem elements
  2. Modify the question or parameters
  3. Solve the modified problem
  4. Compare with the original approach
1
Original: Pizza with 8 slices, Tom eats 3
2
Modified question: What fraction of the pizza remains?
3
Calculate remaining: 8 - 3 = 5 slices
4
Express as fraction: 5/8 of the pizza remains
Modified problem: What fraction remains? Answer: 5/8
Final answer:

Modified problem: "What fraction of the pizza remains?" Answer: 5/8 of the pizza remains.

Applied rules:

• Problem variation: Changes perspective on the same scenario

• Subtraction: Remaining = Total - Consumed

• Fraction expression: Remaining over total

• Practice Tip: Modifying problems helps understand different solution approaches

Related Examples:
  • Original: 10 cookies, eat 4 → Modified: What fraction eaten? Answer: 4/10
  • Original: 12 marbles, lose 3 → Modified: What fraction remain? Answer: 9/12
  • Original: 15 books, read 5 → Modified: What fraction unread? Answer: 10/15
Quick Tips:
  • Change the question while keeping the scenario
  • Focus on what information is available
  • Practice multiple perspectives on the same problem
Frequently Asked Questions:

Q: Why modify problems?
A: It helps understand the same situation from different angles.

Q: Can I modify the numbers too?
A: Yes, changing numbers creates new practice opportunities.

10 Comprehensive Problems
Exercise 10
"A fruit basket contains 20 pieces of fruit. 1/4 are apples, 1/5 are oranges, and the rest are bananas. How many of each fruit are there? What fraction are bananas?"
Definition:

Comprehensive problems: Complex problems that require multiple problem-solving skills and concepts

Note: These problems integrate various mathematical concepts and require strategic thinking.

Step-by-step comprehensive solving:
  1. Calculate each known fraction of the total
  2. Find the unknown quantity
  3. Express unknown as a fraction
  4. Verify that all parts add up to the whole
1
Calculate apples: 1/4 of 20 = 20 ÷ 4 = 5 apples
2
Calculate oranges: 1/5 of 20 = 20 ÷ 5 = 4 oranges
3
Calculate bananas: 20 - 5 - 4 = 11 bananas
4
Express banana fraction: 11/20 are bananas
5 apples, 4 oranges, 11 bananas; 11/20 are bananas
Final answer:

There are 5 apples, 4 oranges, and 11 bananas. 11/20 of the fruits are bananas.

Applied rules:

• Multiple fractions: Calculate each separately

• Unknown calculation: Subtract known amounts from total

• Fraction of whole: Part over total

• Practice Tip: Organize information to manage multiple components

Related Examples:
  • 15 toys: 1/3 cars (5), 1/5 dolls (3), rest blocks (7) → 7/15 blocks
  • 24 students: 1/4 boys (6), 1/3 girls (8), rest unspecified (10) → 10/24=5/12
  • 30 candies: 1/6 chocolates (5), 1/3 caramels (10), rest gummies (15) → 15/30=1/2
Quick Tips:
  • Calculate each known fraction separately
  • Use subtraction to find the unknown part
  • Organize information to avoid confusion
Frequently Asked Questions:

Q: What if fractions don't divide evenly?
A: The problem would need to specify how to handle remainders.

Q: How do I check my work with multiple parts?
A: Verify that all parts add up to the total amount.

Detailed Summary: Understanding Problem Solving with Fractions

Core Concepts

Problem Analysis: Breaking down word problems to identify key information and requirements.

Solution Planning: Developing a strategy before beginning calculations.

Step-by-Step Approach: Systematically working through problems in organized stages.

Visualization: Using diagrams and models to understand abstract concepts.

Verification: Checking work using alternative methods to ensure accuracy.

Core Rules & Laws

Rule 1: Always identify what the problem is asking before solving.

Rule 2: Use visual models when they help clarify the problem.

Rule 3: Check your work using a different method or approach.

Rule 4: Ensure all parts of a problem add up to the whole when applicable.

Rule 5: Express answers in the format requested by the problem.

Step-by-Step Methods

Method 1 - Problem Analysis: 1) Read carefully, 2) Identify question, 3) List given info, 4) Determine operation.

Method 2 - Solution Planning: 1) Organize information, 2) Plan approach, 3) Sequence steps, 4) Prepare calculations.

Method 3 - Step-by-Step Solving: 1) Execute planned steps, 2) Track quantities, 3) Verify each step, 4) Reach conclusion.

Method 4 - Verification: 1) Use alternative method, 2) Check reasonableness, 3) Confirm original conditions met.

Examples & Applications

Simple Examples: Finding fractions of sets, determining remaining amounts.

Advanced Examples: Multi-step problems, real-world applications, comprehensive scenarios.

Real World Applications: Sharing food, organizing groups, measuring portions.

Visual Models: Rectangles, circles, and sets to represent fractions.

Verification: Using different methods to confirm answers.

Tips & Memory Aids

Tip 1: Read the problem multiple times to understand it completely.

Tip 2: Underline or highlight key information.

Tip 3: Draw pictures or diagrams when helpful.

Tip 4: Always check that your answer makes sense in the context.

Memory Aid: "P-R-O-B-L-E-M" = Pause, Read, Organize, Begin, Link, Evaluate, Monitor.

Common Pitfalls

Pitfall 1: Not reading the problem carefully enough to understand what's being asked.

Pitfall 2: Performing calculations without a clear plan.

Pitfall 3: Forgetting to check if the answer makes sense.

Pitfall 4: Not verifying that all parts add up to the whole when needed.

Pitfall 5: Using inconsistent units throughout the problem.

Problem-Solving Strategies

Visual Strategy: Draw diagrams, use models, or create pictures to represent the problem.

Arithmetic Strategy: Perform calculations step-by-step with clear organization.

Logical Strategy: Use reasoning and elimination to narrow down possibilities.

Estimation Strategy: Predict a reasonable answer before calculating.

Success Indicators

Indicator 1: Ability to identify the question being asked.

Indicator 2: Effective use of multiple problem-solving strategies.

Indicator 3: Consistent verification of answers.

Indicator 4: Ability to modify problems for additional practice.

Indicator 5: Success with comprehensive, multi-step problems.

Key Laws, Methods, Rules, and Definitions
\text{Fraction} = \frac{\text{Part}}{\text{Whole}}
Fraction Formula
Key definitions:

Problem Solving: The process of finding solutions to mathematical challenges

Problem Analysis: Breaking down a problem to understand its components

Strategy: A planned approach to solving a particular type of problem

Verification: Checking that a solution is correct using alternative methods

Complete methodology:
  1. Understand: Read and analyze the problem thoroughly
  2. Plan: Decide on the approach and sequence of steps
  3. Solve: Execute the planned solution methodically
  4. Check: Verify the solution using alternative methods
Tip 1: Always identify what the problem is asking before solving.
Tip 2: Use visual models to make abstract concepts more concrete.
Tip 3: Check your work using a different approach.
Tip 4: Organize information to prevent confusion in complex problems.
Common errors: Misreading problems, calculation mistakes, not checking answers.
Exam preparation: Practice with various problem types, focus on verification, review strategies.
Important concepts to understand:

• Problem decomposition: Breaking complex problems into simpler parts

• Strategy selection: Choosing the most effective approach for each problem

• Verification: Confirming solutions using alternative methods

• Real-world connection: Understanding how math applies to everyday situations

• Multiple perspectives: Viewing problems from different angles

Understanding Problem Solving with Fractions
\frac{\text{What you want}}{\text{Total amount}}
Problem-Solving Fraction

Problem-Solving Framework

Analysis: Understand what the problem is asking.
Planning: Develop a strategy to solve the problem.
Execution: Carry out the solution step-by-step.
Verification: Check that the solution is correct.

Key Takeaways

  • Always read the problem carefully to understand what's being asked
  • Develop a plan before starting calculations
  • Use visual models when they help clarify the problem
  • Check your work using alternative methods
  • Ensure your answer makes sense in the context of the problem

Questions & Answers

Question: I don't understand word problems. They seem too hard. How do I make them easier?

Answer: Here are strategies to make word problems easier:

  • Read the problem slowly and multiple times
  • Find the question worded as "What..." or "How many..."
  • Draw a picture to represent the problem
  • Identify the important numbers

Example: For "Sarah has 8 cookies and gives away 3", draw 8 circles, cross out 3, and count the remaining ones. This helps visualize the problem.

Tip: Start with simple problems and gradually work up to more complex ones.

Question: My child rushes through problems and makes mistakes. How can I help them slow down?

Answer: Here are techniques to encourage careful problem-solving:

  • Teach them to read the problem aloud slowly
  • Have them underline or highlight important information
  • Require them to draw a picture or diagram
  • Ask them to explain their thinking step by step

Create a structured routine: Read, Plan, Solve, Check. Practice this consistently until it becomes habit.

Tip: Use a checklist for each problem-solving step to ensure thoroughness.

Question: How can I help students who struggle with multi-step problems?

Answer: Here are effective approaches for multi-step problems:

  • Break the problem into smaller, single-step problems
  • Use graphic organizers to track information
  • Have students label each step clearly
  • Practice with problems that gradually increase in complexity

Teach students to pause after each step to check if their answer makes sense before continuing. Use color-coding to distinguish different parts of the problem.

Encourage students to verbalize each step to reinforce understanding!

Problem Solving Glossary

Problem Analysis
The process of examining a problem to understand what is being asked and what information is provided.
Solution Strategy
A planned approach to solving a particular type of problem.
Step-by-Step Approach
Solving problems by breaking them into smaller, sequential parts.
Verification
The process of checking that a solution is correct using alternative methods.
Multi-Step Problem
A problem requiring multiple operations or calculations to reach the solution.

Problem Solving Educational Team

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Our team of experienced elementary math teachers and curriculum developers creates research-based, student-friendly problem-solving resources. All content is aligned with Common Core State Standards and reviewed by mathematics education experts to ensure accuracy and pedagogical effectiveness for 2nd-grade learners developing problem-solving skills.