Strategies for Solving Word Problems

Opening Hook

Have you ever read a maths problem and thought, “Where do I even start?” You’re not alone! Word problems can feel like puzzles written in code - but what if you had a secret decoder ring? Imagine being able to look at any word problem and immediately know exactly what to do, like a detective solving a mystery. The best mathematicians don’t just compute - they strategize! Welcome to the world of problem-solving strategies, where every word problem becomes a solvable puzzle.

Concept Explanation

Word problems are mathematical questions disguised as real-life stories. Instead of seeing “15 + 8,” you might read “Maria has 15 apples and buys 8 more. How many apples does she have now?” The challenge isn’t usually the maths itself - it’s translating the words into numbers and operations.

Why do word problems exist? They help you apply maths to real life. In the real world, nobody hands you “47 × 3” - instead, you need to figure out how much three $47 shirts will cost. Word problems train your brain to recognize when and how to use maths.

The CUBES Strategy - A powerful framework for solving any word problem:

C - Circle the numbers: Identify all numerical information U - Underline the question: What exactly are you being asked to find? B - Box key words: Highlight operation clues (more, less, total, each, etc.) E - Eliminate extra information: Cross out irrelevant details S - Solve and check: Perform calculations and verify your answer makes sense

Alternative: The READ Strategy

R - Read the problem carefully (at least twice!) E - Examine what you know and what you need to find A - Act: Make a plan and choose your operation D - Do the work and check your answer

Both strategies work! Choose the one that feels most natural to you, or combine elements from each.

Visual Explanations

The Word Problem Anatomy:

[SETTING/CONTEXT] → [GIVEN INFORMATION] → [QUESTION] → [YOUR SOLUTION]

Example:
"Sarah is baking cookies" → "She makes 24 chocolate and 18 vanilla"
→ "How many total cookies?" → "24 + 18 = 42 cookies"

Operation Detective Chart:

ADDITION Clues:          SUBTRACTION Clues:
• total, sum              • difference, left
• altogether, combined    • fewer, less than
• in all, plus            • how many more
• increased by            • remain, decrease

MULTIPLICATION Clues:    DIVISION Clues:
• times, product          • share equally
• each, per               • split, divide
• groups of               • each person gets
• array, area             • how many in each

Visual Problem-Solving Flow:

Read Problem → Identify Question → Find Numbers → Choose Operation
     ↓              ↓                  ↓               ↓
  (2 times)    (Underline it)    (Circle them)   (Key words help)
                                                       ↓
Check Answer ← Write Solution ← Do Calculation ← Set Up Equation
     ↓                ↓               ↓               ↓
(Make sense?) (Complete sentence) (Show work)   (Use symbols)

Teacher’s Insight

Word problems are where students transition from computational mathematics to applied problem-solving - a critical life skill. Research shows that students struggle not with the arithmetic, but with reading comprehension, vocabulary, and deciding which operation to use.

Key teaching strategies:

1. Explicit vocabulary instruction: Teach mathematical language alongside operations
2. Visual representations: Encourage drawings, diagrams, and models
3. Think-alouds: Model your problem-solving process verbally
4. Multiple exposures: Present the same mathematical concept in varied contexts

Students often rush to “find the numbers and do something with them” without understanding the problem’s context. Slow them down! Require them to restate the problem in their own words before solving. Have them predict whether their answer should be larger or smaller than the given numbers - this metacognitive step dramatically improves accuracy.

Watch for students who struggle with reading comprehension. These students may excel at computation but find word problems frustrating. Partner them with strong readers, or provide problems with visual supports.

Multiple Strategies

Strategy 1: Act It Out

Use physical objects to model the problem. If Jake has 5 apples and gives away 2, actually move 5 objects and remove 2.

Strategy 2: Draw a Picture or Diagram

Sketch what’s happening. Simple stick figures, boxes, or circles can clarify complex situations.

Strategy 3: Make a List or Table

Organize information systematically to spot patterns or relationships.

Strategy 4: Work Backwards

When you know the end result but need to find the beginning, reverse the operations.

Strategy 5: Guess and Check

Make an educated guess, test it, then refine based on the result. Especially useful for problems with multiple possible approaches.

Strategy 6: Look for Patterns

Identify repeating sequences or relationships that can lead to shortcuts.

Strategy 7: Simplify the Numbers

Replace difficult numbers with easier ones (like 100 instead of 87) to understand the problem structure, then solve with real numbers.

Strategy 8: Write an Equation

Translate words directly into mathematical symbols: “5 more than a number” becomes “n + 5”

Key Vocabulary

Word Problem: A mathematical question presented as a real-life scenario using words instead of just numbers and symbols

Key Information: The essential numbers and facts needed to solve the problem

Extra Information: Details included in the problem that aren’t necessary for solving (used to teach critical reading)

Operation: The mathematical action needed (addition, subtraction, multiplication, division)

Key Words: Specific vocabulary that hints at which operation to use (“altogether” suggests addition)

Question: What the problem is asking you to find - your target answer

Solution: The complete answer including both the numerical result and appropriate units/context

Reasonable Answer: A result that makes sense in the context of the problem (not just mathematically correct)

Multi-Step Problem: A word problem requiring more than one calculation or operation to solve

Variable: An unknown quantity represented by a letter or symbol

Worked Examples

Example 1: Basic Addition Word Problem

Problem: Emma collected 24 seashells on Monday and 17 seashells on Tuesday. How many seashells did she collect in total?

Solution: 41 seashells

Step-by-Step Using CUBES:

1. Circle numbers: 24, 17
2. Underline question: “How many seashells did she collect in total?”
3. Box key words: “in total” (suggests addition)
4. Eliminate: No extra information to remove
5. Solve: 24 + 17 = 41 seashells
6. Check: Does 41 make sense? Yes, it’s more than both 24 and 17 ✓

Example 2: Subtraction with Extra Information

Problem: Carlos has 48 trading cards. He is 10 years old. He gave 15 cards to his friend. How many cards does Carlos have left?

Solution: 33 cards

Step-by-Step Using CUBES:

1. Circle numbers: 48, 10, 15
2. Underline question: “How many cards does Carlos have left?”
3. Box key words: “have left” (suggests subtraction)
4. Eliminate: “He is 10 years old” - this doesn’t affect the card count!
5. Solve: 48 - 15 = 33 cards
6. Check: 33 is less than 48, which makes sense when giving away cards ✓

Example 3: Multiplication Problem

Problem: A bakery sells cupcakes in boxes of 6. If you buy 8 boxes, how many cupcakes do you have?

Solution: 48 cupcakes

Step-by-Step:

1. Identify: 6 cupcakes per box, 8 boxes total
2. Key word: “per” suggests multiplication
3. Visualize: 8 groups of 6
4. Calculate: 8 × 6 = 48 cupcakes
5. Check: We have more than 6 (one box), so 48 makes sense ✓

Example 4: Division Problem

Problem: There are 36 students going on a field trip. They need to travel in vans that hold 9 students each. How many vans are needed?

Solution: 4 vans

Step-by-Step:

1. Identify: 36 total students, 9 students per van
2. Key word: “each” and we’re splitting into groups (division)
3. Calculate: 36 ÷ 9 = 4 vans
4. Check: 4 vans × 9 students = 36 total ✓

Example 5: Multi-Step Problem

Problem: Ava bought 5 notebooks for $3 each and a backpack for$25. She paid with a $50 note. How much change did she receive?

Solution: $10

Step-by-Step:

1. Identify steps: First find total cost, then calculate change
2. Step 1 - Cost of notebooks: 5 × $3 =$15
3. Step 2 - Total cost: $15 +$25 = $40
4. Step 3 - Calculate change: $50 -$40 = $10
5. Check: $10 +$40 = $50, which is what she paid ✓

Example 6: Comparison Problem

Problem: Liam scored 87 points in a game. His sister Mia scored 94 points. How many more points did Mia score than Liam?

Solution: 7 points

Step-by-Step:

1. Identify: Comparing two amounts
2. Key phrase: “how many more” (subtraction for comparison)
3. Calculate: 94 - 87 = 7 points
4. Check: 87 + 7 = 94, so Mia did score 7 more ✓

Example 7: Problem Requiring Interpretation

Problem: A movie theater has 18 rows with 12 seats in each row. If 200 people attend a showing, will everyone have a seat?

Solution: Yes, everyone will have a seat

Step-by-Step:

1. Find total seats: 18 × 12 = 216 seats
2. Compare: 216 seats available, 200 people attending
3. Answer: Yes, there are enough seats (216 > 200)
4. Extra seats: 216 - 200 = 16 empty seats

Common Misconceptions

Misconception 1: “I should just use all the numbers in the problem”

• Truth: Some numbers are extra information designed to test your reading comprehension
• Example: “Jake is 12 years old and has 8 marbles. He buys 5 more. How many marbles now?” The age (12) is irrelevant!
• Why it matters: Using irrelevant numbers leads to wrong operations and answers

Misconception 2: “Key words always tell me the operation”

• Truth: Key words are helpful clues but not foolproof rules
• Example: “Jane had 10 cookies and ate some. She has 3 left. How many did she eat?” Contains “left” but requires thinking, not just subtraction of 3
• Why it matters: Relying solely on key words without understanding leads to errors

Misconception 3: “The biggest number always comes first”

• Truth: Order depends on the operation and context
• Example: “From a 50-page book, you’ve read 32 pages. How many left?” is 50 - 32, not 32 - 50
• Why it matters: Wrong order in subtraction and division gives incorrect answers

Misconception 4: “If I get a number, I’m done”

• Truth: You need to check if your answer makes sense and use appropriate units
• Example: Finding that someone has -5 apples or that a person is 0.5 people should trigger review
• Why it matters: Unreasonable answers indicate errors in understanding or calculation

Misconception 5: “Word problems are just tricks to make maths harder”

• Truth: Word problems develop critical thinking and show how maths applies to real life
• Why it matters: This mindset creates resistance to learning valuable problem-solving skills

Memory Aids

CUBES Rhyme: “Circle numbers that you see, Underline the question, that’s the key, Box the words that help you know, Eliminate what’s just for show, Solve and check before you go!”

Operation Keyword Songs (to familiar tunes): Sing to “Twinkle Twinkle Little Star”: “More and plus and altogether, Total, sum - they go together, When you see these words appear, Addition is the answer here!”

Hand Signal Strategy:

• Thumbs up: Add (bringing together)
• Thumbs down: Subtract (taking away)
• Fingers spread: Multiply (groups spreading out)
• Chopping motion: Divide (splitting apart)

The 4-R Memory Chain: Read twice → Recognize the question → Retrieve the numbers → Respond with calculation

Acronym - SOLVE: Search for key information Organize what you know Line up your operation Verify your calculation Explain your answer

Tiered Practice Problems

Tier 1: Foundation (Getting Started)

1. A pet store has 23 fish in one tank and 17 fish in another tank. How many fish are there altogether? Answer: 40 fish (23 + 17 = 40)

2. Olivia had 45 stickers. She gave 12 to her friend. How many stickers does Olivia have now? Answer: 33 stickers (45 - 12 = 33)

3. There are 7 days in a week. How many days are in 4 weeks? Answer: 28 days (7 × 4 = 28)

4. A baker made 32 cookies and wants to put them equally into 4 boxes. How many cookies in each box? Answer: 8 cookies (32 ÷ 4 = 8)

Tier 2: Intermediate (Building Skills)

5. Marcus bought 6 packs of pencils with 8 pencils in each pack. He gave 15 pencils to classmates. How many pencils does he have left? Answer: 33 pencils (6 × 8 = 48; 48 - 15 = 33)

6. A rectangle garden is 12 metres long and 8 metres wide. What is the perimeter of the garden? Answer: 40 metres (12 + 8 + 12 + 8 = 40, or 2(12 + 8) = 40)

7. Sophie read 45 pages of her book on Saturday. On Sunday, she read 38 pages. If the book has 150 pages total, how many pages does she still need to read? Answer: 67 pages (45 + 38 = 83; 150 - 83 = 67)

8. Concert tickets cost $25 each. How much would tickets cost for a family of 5? _Answer:$125 (5 × $25 =$125)_

Tier 3: Advanced (Challenge Problems)

9. A school has 8 classrooms. Each classroom has 24 students and 4 teachers. How many people are in the classrooms altogether? Answer: 224 people (24 + 4 = 28 people per room; 8 × 28 = 224)

10. A farmer harvested 384 apples. He kept 48 for his family and divided the rest equally among 4 markets. How many apples did each market receive? Answer: 84 apples (384 - 48 = 336; 336 ÷ 4 = 84)

11. Train tickets cost $18 for adults and$12 for children. How much would it cost for 2 adults and 3 children? Answer: $72 (2 ×$18 = $36; 3 ×$12 = $36;$36 + $36 =$72)

12. A bookshelf has 5 shelves. Each shelf can hold 18 books. If 73 books are already on the shelves, how many more books can fit? Answer: 17 books (5 × 18 = 90 total capacity; 90 - 73 = 17)

Five Real-World Applications

1. Shopping and Budgeting

Every shopping trip is a series of word problems! “If jeans cost $45 and you have a 20$100. After buying shoes for $65, can you afford a$40 shirt?” Understanding these problems helps you make smart purchasing decisions, compare prices, calculate discounts, and stay within budget. Adults use these skills multiple times per week when grocery shopping, comparing deals, or planning large purchases.

2. Cooking and Recipe Scaling

Recipes are mathematical formulas in disguise. “This recipe serves 4 people but you need to feed 12 - how much of each ingredient?” involves multiplication and division. “The recipe calls for 2⅓ cups of flour, but you only have 1½ cups - how much more do you need?” is subtraction with fractions. Professional chefs and home cooks constantly solve these word problems to adjust serving sizes, convert measurements, and time multiple dishes to finish simultaneously.

3. Time Management and Scheduling

Planning your day involves constant word problems: “Soccer practice is 90 minutes, starting at 3:30 PM. What time will it end?” or “The movie is 2 hours 15 minutes long. If it starts at 7:00 PM, will it end before your 9:30 PM bedtime?” Students who master these problems manage homework time better, arrive on time for activities, and coordinate complex schedules as they grow older.

4. Sports Statistics and Games

Every sport is packed with word problems. “Your basketball team scored 78 points across 4 quarters. What was the average per quarter?” or “You ran 3 laps in 12 minutes. At this pace, how long would 7 laps take?” Athletes use these calculations to track performance, set goals, and develop strategy. Fantasy sports players solve complex multi-step problems involving player statistics to make team decisions.

5. Construction and Measurement Projects

Building anything requires word problem thinking: “You need 4 fence posts for every 6 metres of fence. For 30 metres, how many posts?” or “Paint covers 25 square metres per litre. For a 12m × 8m room, how much paint?” These practical applications show up in home improvement, art projects, garden planning, and any hands-on construction work. Contractors solve dozens of these daily!

Study Tips

1. Read SLOWLY and TWICE: Most errors come from misreading. Slow down and read every problem at least twice before calculating anything.

2. Restate in Your Own Words: Before solving, explain the problem to yourself or someone else. If you can’t explain it, you don’t understand it yet.

3. Predict Your Answer: Before calculating, estimate whether your answer should be bigger or smaller than the given numbers. This catches operation errors.

4. Draw It Out: Even simple sketches help visualize what’s happening. Quick stick figures, boxes, or circles take 10 seconds but prevent expensive mistakes.

5. Practice Different Problem Types: Don’t just repeat the same kind of problem. Seek variety in contexts (money, time, measurement, etc.) to build flexible thinking.

6. Learn from Mistakes: When you get one wrong, don’t just move on. Figure out exactly where your thinking went wrong.

7. Create Your Own Problems: Write word problems for friends or family to solve. Creating problems deepens your understanding of the structure.

8. Use Units in Your Answer: Never write just “25” - write “25 apples” or “25 metres.” Units help verify you solved the right question.

9. Build Vocabulary: Keep a list of key words and what operations they suggest. Review it regularly.

10. Time Yourself Gradually: Start with no time pressure, then gradually add reasonable time limits to build speed without sacrificing accuracy.

Answer Checking Methods

Method 1: Reverse Operation Check Work backwards using the opposite operation. If you added, subtract your answer to get back to the start. Example: 23 + 17 = 40 → Check: 40 - 17 = 23 ✓

Method 2: Reasonableness Test Ask: “Does this make sense in real life?” You can’t have 3.5 people or -10 apples.

Method 3: Estimation Verification Round numbers to estimate before calculating precisely. Your exact answer should be close to your estimate. Example: 47 + 38 → Estimate: 50 + 40 = 90 → Actual: 85 (close enough ✓)

Method 4: Reread the Question Did you actually answer what was asked? Sometimes you solve correctly but answer a different question!

Method 5: Unit Check Does your answer have the right units? If the problem asks for dollars, your answer should be dollars, not number of items.

Method 6: Alternative Strategy Solve the same problem a different way (drawing vs. equation) and see if you get the same answer.

Extension Ideas

For Advanced Learners:

1. Create Multi-Step Challenge Problems: Write word problems requiring 3-4 operations. Trade with classmates.

2. Identify Missing Information: Present problems that don’t have enough information to solve, requiring students to identify what’s missing.

3. Multiple Solution Paths: Find problems that can be solved in more than one way and compare the efficiency of different approaches.

4. Real Data Projects: Use actual data from sports statistics, weather reports, or school surveys to create and solve authentic problems.

5. Problem-Solving Investigations: Explore open-ended scenarios like “Design a party for 30 people with a $200 budget” requiring multiple calculations and decisions.

6. Error Analysis: Present problems solved incorrectly and identify where the thinking went wrong.

7. Word Problem Without Numbers: Create problems using variables: “Jane has x apples, gives away y apples, how many left?” → “x - y”

8. Cross-Curricular Connections: Write word problems connecting to science (speed, distance, time), social studies (population, area), or art (symmetry, scaling).

Parent & Teacher Notes

For Parents:

Word problems bridge school maths and real life. Your child is developing critical thinking skills that apply far beyond the classroom. You can support their learning by:

At Home Practice:

• Point out word problems in daily life: “We need 2 eggs per pancake and we’re making pancakes for 4 people…”
• Let them help with real calculations: grocery budgeting, trip planning, recipe adjustments
• Read problems together, discussing what’s being asked before jumping to calculations

Common Struggles:

• Reading comprehension: If your child struggles with reading, word problems are doubly challenging. Read problems aloud together.
• Rushing: Many errors come from not reading carefully. Encourage slowing down.
• Operation selection: Practice identifying key words and what they mean.

Encouragement Tips:

• Celebrate the thinking process, not just correct answers
• Frame mistakes as learning opportunities
• Share how you use similar thinking in your work or daily life

When to Seek Extra Help: If your child consistently struggles after multiple attempts, consider asking their teacher for additional resources or tutoring. Sometimes a different explanation or approach makes everything click.

For Teachers:

Prerequisite Skills Check: Before introducing word problem strategies, ensure students have:

• Basic reading comprehension at grade level
• Fluency with the four operations
• Understanding of mathematical vocabulary
• Ability to identify relevant vs. irrelevant information in text

Differentiation Strategies:

For Struggling Students:

• Provide sentence frames: “I know _. I need to find _. I will _ because _.”
• Use visual supports and manipulatives extensively
• Start with very simple one-step problems before adding complexity
• Pair with stronger readers for comprehension support
• Highlight or pre-teach key vocabulary

For Advanced Students:

• Provide multi-step problems with multiple solution paths
• Encourage problem creation and peer teaching
• Introduce problems requiring additional research or data collection
• Challenge them to find the most efficient solution method

Assessment Strategies:

• Include problems requiring different operations to ensure true understanding
• Mix problems with and without extra information
• Ask students to explain their thinking, not just show calculations
• Provide partial credit for correct process even with calculation errors
• Use portfolios showing growth over time

Instructional Best Practices:

• Model think-alouds regularly, verbalizing your problem-solving process
• Use anchor charts displaying strategies and key word lists
• Practice collaborative problem-solving in small groups
• Connect to real contexts relevant to your students’ lives and interests
• Teach multiple strategies and let students choose their preferred approach

Common Teaching Pitfalls to Avoid:

• Over-reliance on key word lists (they’re helpful but not foolproof)
• Moving too quickly from concrete to abstract representation
• Focusing solely on getting right answers rather than understanding process
• Not providing enough varied practice across different contexts