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## 120 Math Word Problems To Challenge Students Grades 1 to 8

• Teaching Tools
• Subtraction
• Multiplication
• Mixed operations
• Ordering and number sense
• Comparing and sequencing
• Physical measurement
• Ratios and percentages
• Probability and data relationships

You sit at your desk, ready to put a math quiz, test or activity together. The questions flow onto the document until you hit a section for word problems.

A jolt of creativity would help. But it doesn’t come.

Whether you’re a 3rd grade teacher or an 8th grade teacher preparing students for high school, translating math concepts into real world examples can certainly be a challenge.

This resource is your jolt of creativity. It provides examples and templates of math word problems for 1st to 8th grade classes.

There are 120 examples in total.

The list of examples is supplemented by tips to create engaging and challenging math word problems.

## 120 Math word problems, categorized by skill

1. Adding to 10: Ariel was playing basketball. 1 of her shots went in the hoop. 2 of her shots did not go in the hoop. How many shots were there in total?

2. Adding to 20: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store to get 3 more pieces of gum. How many pieces of gum does Adrianna have now?

3. Adding to 100: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store and got 70 pieces of strawberry gum and 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

4. Adding Slightly over 100: The restaurant has 175 normal chairs and 20 chairs for babies. How many chairs does the restaurant have in total?

6. Adding to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In June, the hobby store sold 15,498 more trading cards than normal. In total, how many trading cards did the hobby store sell in June?

7. Adding 3 Numbers: Billy had 2 books at home. He went to the library to take out 2 more books. He then bought 1 book. How many books does Billy have now?

8. Adding 3 Numbers to and over 100: Ashley bought a big bag of candy. The bag had 102 blue candies, 100 red candies and 94 green candies. How many candies were there in total?

## Subtraction word problems

9. Subtracting to 10: There were 3 pizzas in total at the pizza shop. A customer bought 1 pizza. How many pizzas are left?

10. Subtracting to 20: Your friend said she had 11 stickers. When you helped her clean her desk, she only had a total of 10 stickers. How many stickers are missing?

11. Subtracting to 100: Adrianna has 100 pieces of gum to share with her friends. When she went to the park, she shared 10 pieces of strawberry gum. When she left the park, Adrianna shared another 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

## Practice math word problems with Prodigy Math

Join millions of teachers using Prodigy to make learning fun and differentiate instruction as they answer in-game questions, including math word problems from 1st to 8th grade!

12. Subtracting Slightly over 100: Your team scored a total of 123 points. 67 points were scored in the first half. How many were scored in the second half?

13. Subtracting to 1,000: Nathan has a big ant farm. He decided to sell some of his ants. He started with 965 ants. He sold 213. How many ants does he have now?

14. Subtracting to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In July, the hobby store sold a total of 20,777 trading cards. How many more trading cards did the hobby store sell in July compared with a normal month?

15. Subtracting 3 Numbers: Charlene had a pack of 35 pencil crayons. She gave 6 to her friend Theresa. She gave 3 to her friend Mandy. How many pencil crayons does Charlene have left?

16. Subtracting 3 Numbers to and over 100: Ashley bought a big bag of candy to share with her friends. In total, there were 296 candies. She gave 105 candies to Marissa. She also gave 86 candies to Kayla. How many candies were left?

## Multiplication word problems

17. Multiplying 1-Digit Integers: Adrianna needs to cut a pan of brownies into pieces. She cuts 6 even columns and 3 even rows into the pan. How many brownies does she have?

18. Multiplying 2-Digit Integers: A movie theatre has 25 rows of seats with 20 seats in each row. How many seats are there in total?

19. Multiplying Integers Ending with 0: A clothing company has 4 different kinds of sweatshirts. Each year, the company makes 60,000 of each kind of sweatshirt. How many sweatshirts does the company make each year?

20. Multiplying 3 Integers: A bricklayer stacks bricks in 2 rows, with 10 bricks in each row. On top of each row, there is a stack of 6 bricks. How many bricks are there in total?

21. Multiplying 4 Integers: Cayley earns $5 an hour by delivering newspapers. She delivers newspapers 3 days each week, for 4 hours at a time. After delivering newspapers for 8 weeks, how much money will Cayley earn? ## Division word problems Best for: 3rd grade, 4th grade, 5th grade 22. Dividing 1-Digit Integers: If you have 4 pieces of candy split evenly into 2 bags, how many pieces of candy are in each bag? 23. Dividing 2-Digit Integers: If you have 80 tickets for the fair and each ride costs 5 tickets, how many rides can you go on? 24. Dividing Numbers Ending with 0: The school has$20,000 to buy new computer equipment. If each piece of equipment costs $50, how many pieces can the school buy in total? 25. Dividing 3 Integers: Melissa buys 2 packs of tennis balls for$12 in total. All together, there are 6 tennis balls. How much does 1 pack of tennis balls cost? How much does 1 tennis ball cost?

26. Interpreting Remainders: An Italian restaurant receives a shipment of 86 veal cutlets. If it takes 3 cutlets to make a dish, how many cutlets will the restaurant have left over after making as many dishes as possible?

## Mixed operations word problems

27. Mixing Addition and Subtraction: There are 235 books in a library. On Monday, 123 books are taken out. On Tuesday, 56 books are brought back. How many books are there now?

28. Mixing Multiplication and Division: There is a group of 10 people who are ordering pizza. If each person gets 2 slices and each pizza has 4 slices, how many pizzas should they order?

29. Mixing Multiplication, Addition and Subtraction: Lana has 2 bags with 2 marbles in each bag. Markus has 2 bags with 3 marbles in each bag. How many more marbles does Markus have?

30. Mixing Division, Addition and Subtraction: Lana has 3 bags with the same amount of marbles in them, totaling 12 marbles. Markus has 3 bags with the same amount of marbles in them, totaling 18 marbles. How many more marbles does Markus have in each bag?

## Ordering and number sense word problems

31. Counting to Preview Multiplication: There are 2 chalkboards in your classroom. If each chalkboard needs 2 pieces of chalk, how many pieces do you need in total?

32. Counting to Preview Division: There are 3 chalkboards in your classroom. Each chalkboard has 2 pieces of chalk. This means there are 6 pieces of chalk in total. If you take 1 piece of chalk away from each chalkboard, how many will there be in total?

33. Composing Numbers: What number is 6 tens and 10 ones?

34. Guessing Numbers: I have a 7 in the tens place. I have an even number in the ones place. I am lower than 74. What number am I?

35. Finding the Order: In the hockey game, Mitchell scored more points than William but fewer points than Auston. Who scored the most points? Who scored the fewest points?

## Fractions word problems

36. Finding Fractions of a Group: Julia went to 10 houses on her street for Halloween. 5 of the houses gave her a chocolate bar. What fraction of houses on Julia’s street gave her a chocolate bar?

37. Finding Unit Fractions: Heather is painting a portrait of her best friend, Lisa. To make it easier, she divides the portrait into 6 equal parts. What fraction represents each part of the portrait?

38. Adding Fractions with Like Denominators: Noah walks ⅓ of a kilometre to school each day. He also walks ⅓ of a kilometre to get home after school. How many kilometres does he walk in total?

39. Subtracting Fractions with Like Denominators: Last week, Whitney counted the number of juice boxes she had for school lunches. She had ⅗ of a case. This week, it’s down to ⅕ of a case. How much of the case did Whitney drink?

40. Adding Whole Numbers and Fractions with Like Denominators: At lunchtime, an ice cream parlor served 6 ¼ scoops of chocolate ice cream, 5 ¾ scoops of vanilla and 2 ¾ scoops of strawberry. How many scoops of ice cream did the parlor serve in total?

41. Subtracting Whole Numbers and Fractions with Like Denominators: For a party, Jaime had 5 ⅓ bottles of cola for her friends to drink. She drank ⅓ of a bottle herself. Her friends drank 3 ⅓. How many bottles of cola does Jaime have left?

42. Adding Fractions with Unlike Denominators: Kevin completed ½ of an assignment at school. When he was home that evening, he completed ⅚ of another assignment. How many assignments did Kevin complete?

43. Subtracting Fractions with Unlike Denominators: Packing school lunches for her kids, Patty used ⅞ of a package of ham. She also used ½ of a package of turkey. How much more ham than turkey did Patty use?

44. Multiplying Fractions: During gym class on Wednesday, the students ran for ¼ of a kilometre. On Thursday, they ran ½ as many kilometres as on Wednesday. How many kilometres did the students run on Thursday? Write your answer as a fraction.

45. Dividing Fractions: A clothing manufacturer uses ⅕ of a bottle of colour dye to make one pair of pants. The manufacturer used ⅘ of a bottle yesterday. How many pairs of pants did the manufacturer make?

46. Multiplying Fractions with Whole Numbers: Mark drank ⅚ of a carton of milk this week. Frank drank 7 times more milk than Mark. How many cartons of milk did Frank drink? Write your answer as a fraction, or as a whole or mixed number.

## Decimals word problems

47. Adding Decimals: You have 2.6 grams of yogurt in your bowl and you add another spoonful of 1.3 grams. How much yogurt do you have in total?

48. Subtracting Decimals: Gemma had 25.75 grams of frosting to make a cake. She decided to use only 15.5 grams of the frosting. How much frosting does Gemma have left?

49. Multiplying Decimals with Whole Numbers: Marshall walks a total of 0.9 kilometres to and from school each day. After 4 days, how many kilometres will he have walked?

50. Dividing Decimals by Whole Numbers: To make the Leaning Tower of Pisa from spaghetti, Mrs. Robinson bought 2.5 kilograms of spaghetti. Her students were able to make 10 leaning towers in total. How many kilograms of spaghetti does it take to make 1 leaning tower?

51. Mixing Addition and Subtraction of Decimals: Rocco has 1.5 litres of orange soda and 2.25 litres of grape soda in his fridge. Antonio has 1.15 litres of orange soda and 0.62 litres of grape soda. How much more soda does Rocco have than Angelo?

52. Mixing Multiplication and Division of Decimals: 4 days a week, Laura practices martial arts for 1.5 hours. Considering a week is 7 days, what is her average practice time per day each week?

## Comparing and sequencing word problems

53. Comparing 1-Digit Integers: You have 3 apples and your friend has 5 apples. Who has more?

54. Comparing 2-Digit Integers: You have 50 candies and your friend has 75 candies. Who has more?

55. Comparing Different Variables: There are 5 basketballs on the playground. There are 7 footballs on the playground. Are there more basketballs or footballs?

56. Sequencing 1-Digit Integers: Erik has 0 stickers. Every day he gets 1 more sticker. How many days until he gets 3 stickers?

57. Skip-Counting by Odd Numbers: Natalie began at 5. She skip-counted by fives. Could she have said the number 20?

58. Skip-Counting by Even Numbers: Natasha began at 0. She skip-counted by eights. Could she have said the number 36?

59. Sequencing 2-Digit Numbers: Each month, Jeremy adds the same number of cards to his baseball card collection. In January, he had 36. 48 in February. 60 in March. How many baseball cards will Jeremy have in April?

## Time word problems

66. Converting Hours into Minutes: Jeremy helped his mom for 1 hour. For how many minutes was he helping her?

69. Adding Time: If you wake up at 7:00 a.m. and it takes you 1 hour and 30 minutes to get ready and walk to school, at what time will you get to school?

70. Subtracting Time: If a train departs at 2:00 p.m. and arrives at 4:00 p.m., how long were passengers on the train for?

71. Finding Start and End Times: Rebecca left her dad’s store to go home at twenty to seven in the evening. Forty minutes later, she was home. What time was it when she arrived home?

## Money word problems

60. Adding Money: Thomas and Matthew are saving up money to buy a video game together. Thomas has saved $30. Matthew has saved$35. How much money have they saved up together in total?

61. Subtracting Money: Thomas has $80 saved up. He uses his money to buy a video game. The video game costs$67. How much money does he have left?

62. Multiplying Money: Tim gets $5 for delivering the paper. How much money will he have after delivering the paper 3 times? 63. Dividing Money: Robert spent$184.59 to buy 3 hockey sticks. If each hockey stick was the same price, how much did 1 cost?

64. Adding Money with Decimals: You went to the store and bought gum for $1.25 and a sucker for$0.50. How much was your total?

65. Subtracting Money with Decimals: You went to the store with $5.50. You bought gum for$1.25, a chocolate bar for $1.15 and a sucker for$0.50. How much money do you have left?

67. Applying Proportional Relationships to Money: Jakob wants to invite 20 friends to his birthday, which will cost his parents $250. If he decides to invite 15 friends instead, how much money will it cost his parents? Assume the relationship is directly proportional. 68. Applying Percentages to Money: Retta put$100.00 in a bank account that gains 20% interest annually. How much interest will be accumulated in 1 year? And if she makes no withdrawals, how much money will be in the account after 1 year?

## Physical measurement word problems

72. Comparing Measurements: Cassandra’s ruler is 22 centimetres long. April’s ruler is 30 centimetres long. How many centimetres longer is April’s ruler?

73. Contextualizing Measurements: Picture a school bus. Which unit of measurement would best describe the length of the bus? Centimetres, metres or kilometres?

74. Adding Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Last year, Micha’s dad used 100 litres of gas. This year, her dad used 90 litres of gas. How much gas did he use in total for the two years?

75. Subtracting Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Over the past two years, Micha’s dad used 200 litres of gas. This year, he used 100 litres of gas. How much gas did he use last year?

76. Multiplying Volume and Mass: Kiera wants to make sure she has strong bones, so she drinks 2 litres of milk every week. After 3 weeks, how many litres of milk will Kiera drink?

77. Dividing Volume and Mass: Lillian is doing some gardening, so she bought 1 kilogram of soil. She wants to spread the soil evenly between her 2 plants. How much will each plant get?

78. Converting Mass: Inger goes to the grocery store and buys 3 squashes that each weigh 500 grams. How many kilograms of squash did Inger buy?

79. Converting Volume: Shad has a lemonade stand and sold 20 cups of lemonade. Each cup was 500 millilitres. How many litres did Shad sell in total?

80. Converting Length: Stacy and Milda are comparing their heights. Stacy is 1.5 meters tall. Milda is 10 centimetres taller than Stacy. What is Milda’s height in centimetres?

81. Understanding Distance and Direction: A bus leaves the school to take students on a field trip. The bus travels 10 kilometres south, 10 kilometres west, another 5 kilometres south and 15 kilometres north. To return to the school, in which direction does the bus have to travel? How many kilometres must it travel in that direction?

## Ratios and percentages word problems

82. Finding a Missing Number: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. Jenny has 28 trophies. How many does Meredith have?

83. Finding Missing Numbers: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. The difference between the numbers is 12. What are the numbers?

84. Comparing Ratios: The school’s junior band has 10 saxophone players and 20 trumpet players. The school’s senior band has 18 saxophone players and 29 trumpet players. Which band has the higher ratio of trumpet to saxophone players?

85. Determining Percentages: Mary surveyed students in her school to find out what their favourite sports were. Out of 1,200 students, 455 said hockey was their favourite sport. What percentage of students said hockey was their favourite sport?

86. Determining Percent of Change: A decade ago, Oakville’s population was 67,624 people. Now, it is 190% larger. What is Oakville’s current population?

87. Determining Percents of Numbers: At the ice skate rental stand, 60% of 120 skates are for boys. If the rest of the skates are for girls, how many are there?

88. Calculating Averages: For 4 weeks, William volunteered as a helper for swimming classes. The first week, he volunteered for 8 hours. He volunteered for 12 hours in the second week, and another 12 hours in the third week. The fourth week, he volunteered for 9 hours. For how many hours did he volunteer per week, on average?

## Probability and data relationships word problems

89. Understanding the Premise of Probability: John wants to know his class’s favourite TV show, so he surveys all of the boys. Will the sample be representative or biased?

90. Understanding Tangible Probability: The faces on a fair number die are labelled 1, 2, 3, 4, 5 and 6. You roll the die 12 times. How many times should you expect to roll a 1?

91. Exploring Complementary Events: The numbers 1 to 50 are in a hat. If the probability of drawing an even number is 25/50, what is the probability of NOT drawing an even number? Express this probability as a fraction.

92. Exploring Experimental Probability: A pizza shop has recently sold 15 pizzas. 5 of those pizzas were pepperoni. Answering with a fraction, what is the experimental probability that he next pizza will be pepperoni?

93. Introducing Data Relationships: Maurita and Felice each take 4 tests. Here are the results of Maurita’s 4 tests: 4, 4, 4, 4. Here are the results for 3 of Felice’s 4 tests: 3, 3, 3. If Maurita’s mean for the 4 tests is 1 point higher than Felice’s, what’s the score of Felice’s 4th test?

94. Introducing Proportional Relationships: Store A is selling 7 pounds of bananas for $7.00. Store B is selling 3 pounds of bananas for$6.00. Which store has the better deal?

95. Writing Equations for Proportional Relationships: Lionel loves soccer, but has trouble motivating himself to practice. So, he incentivizes himself through video games. There is a proportional relationship between the amount of drills Lionel completes, in x , and for how many hours he plays video games, in y . When Lionel completes 10 drills, he plays video games for 30 minutes. Write the equation for the relationship between x and y .

## Geometry word problems

96. Introducing Perimeter:  The theatre has 4 chairs in a row. There are 5 rows. Using rows as your unit of measurement, what is the perimeter?

97. Introducing Area: The theatre has 4 chairs in a row. There are 5 rows. How many chairs are there in total?

98. Introducing Volume: Aaron wants to know how much candy his container can hold. The container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. What is the container’s volume?

99. Understanding 2D Shapes: Kevin draws a shape with 4 equal sides. What shape did he draw?

100. Finding the Perimeter of 2D Shapes: Mitchell wrote his homework questions on a piece of square paper. Each side of the paper is 8 centimetres. What is the perimeter?

101. Determining the Area of 2D Shapes: A single trading card is 9 centimetres long by 6 centimetres wide. What is its area?

102. Understanding 3D Shapes: Martha draws a shape that has 6 square faces. What shape did she draw?

103. Determining the Surface Area of 3D Shapes: What is the surface area of a cube that has a width of 2cm, height of 2 cm and length of 2 cm?

104. Determining the Volume of 3D Shapes: Aaron’s candy container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. Bruce’s container is 25 centimetres tall, 9 centimetres long and 9 centimetres wide. Find the volume of each container. Based on volume, whose container can hold more candy?

105. Identifying Right-Angled Triangles: A triangle has the following side lengths: 3 cm, 4 cm and 5 cm. Is this triangle a right-angled triangle?

106. Identifying Equilateral Triangles: A triangle has the following side lengths: 4 cm, 4 cm and 4 cm. What kind of triangle is it?

107. Identifying Isosceles Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 5 cm. What kind of triangle is it?

108. Identifying Scalene Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 6 cm. What kind of triangle is it?

109. Finding the Perimeter of Triangles: Luigi built a tent in the shape of an equilateral triangle. The perimeter is 21 metres. What is the length of each of the tent’s sides?

110. Determining the Area of Triangles: What is the area of a triangle with a base of 2 units and a height of 3 units?

111. Applying Pythagorean Theorem: A right triangle has one non-hypotenuse side length of 3 inches and the hypotenuse measures 5 inches. What is the length of the other non-hypotenuse side?

112. Finding a Circle’s Diameter: Jasmin bought a new round backpack. Its area is 370 square centimetres. What is the round backpack’s diameter?

113. Finding a Circle's Area: Captain America’s circular shield has a diameter of 76.2 centimetres. What is the area of his shield?

114. Finding a Circle’s Radius: Skylar lives on a farm, where his dad keeps a circular corn maze. The corn maze has a diameter of 2 kilometres. What is the maze’s radius?

## Variables word problems

115. Identifying Independent and Dependent Variables: Victoria is baking muffins for her class. The number of muffins she makes is based on how many classmates she has. For this equation, m is the number of muffins and c is the number of classmates. Which variable is independent and which variable is dependent?

116. Writing Variable Expressions for Addition: Last soccer season, Trish scored g goals. Alexa scored 4 more goals than Trish. Write an expression that shows how many goals Alexa scored.

117. Writing Variable Expressions for Subtraction: Elizabeth eats a healthy, balanced breakfast b times a week. Madison sometimes skips breakfast. In total, Madison eats 3 fewer breakfasts a week than Elizabeth. Write an expression that shows how many times a week Madison eats breakfast.

118. Writing Variable Expressions for Multiplication: Last hockey season, Jack scored g goals. Patrik scored twice as many goals than Jack. Write an expression that shows how many goals Patrik scored.

119. Writing Variable Expressions for Division: Amanda has c chocolate bars. She wants to distribute the chocolate bars evenly among 3 friends. Write an expression that shows how many chocolate bars 1 of her friends will receive.

120. Solving Two-Variable Equations: This equation shows how the amount Lucas earns from his after-school job depends on how many hours he works: e = 12h . The variable h represents how many hours he works. The variable e represents how much money he earns. How much money will Lucas earn after working for 6 hours?

## How to easily make your own math word problems & word problems worksheets

Armed with 120 examples to spark ideas, making your own math word problems can engage your students and ensure alignment with lessons. Do:

• Link to Student Interests:  By framing your word problems with student interests, you’ll likely grab attention. For example, if most of your class loves American football, a measurement problem could involve the throwing distance of a famous quarterback.
• Make Questions Topical:  Writing a word problem that reflects current events or issues can engage students by giving them a clear, tangible way to apply their knowledge.
• Include Student Names:  Naming a question’s characters after your students is an easy way make subject matter relatable, helping them work through the problem.
• Be Explicit:  Repeating keywords distills the question, helping students focus on the core problem.
• Test Reading Comprehension:  Flowery word choice and long sentences can hide a question’s key elements. Instead, use concise phrasing and grade-level vocabulary.
• Focus on Similar Interests:  Framing too many questions with related interests -- such as football and basketball -- can alienate or disengage some students.
• Feature Red Herrings:  Including unnecessary information introduces another problem-solving element, overwhelming many elementary students.

A key to differentiated instruction , word problems that students can relate to and contextualize will capture interest more than generic and abstract ones.

## Final thoughts about math word problems

You’ll likely get the most out of this resource by using the problems as templates, slightly modifying them by applying the above tips. In doing so, they’ll be more relevant to -- and engaging for -- your students.

Regardless, having 120 curriculum-aligned math word problems at your fingertips should help you deliver skill-building challenges and thought-provoking assessments.

The result?

A greater understanding of how your students process content and demonstrate understanding, informing your ongoing teaching approach.

## Appendix A: Applications

Apply a problem-solving strategy to basic word problems, learning outcomes.

• Apply a general problem-solving strategy to solve word problems

## Approach Word Problems with a Positive Attitude

The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?

Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student in the cartoon below?

Negative thoughts about word problems can be barriers to success.

When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.

Start with a fresh slate and begin to think positive thoughts like the student in the cartoon below. Read the positive thoughts and say them out loud.

When it comes to word problems, a positive attitude is a big step toward success.

If we take control and believe we can be successful, we will be able to master word problems.

Think of something that you can do now but couldn’t do three years ago. Whether it’s driving a car, snowboarding, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!

## Use a Problem-Solving Strategy for Word Problems

In earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then you’ve increased your math vocabulary as you learned about more algebraic procedures, and you’ve had more practice translating from words into algebra.

You have also translated word sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated.

Now we’ll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We’ll demonstrate the strategy as we solve the following problem.

Pete bought a shirt on sale for $$18$, which is one-half the original price. What was the original price of the shirt? Solution: Step 1. Read the problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don’t understand, look them up in a dictionary or on the Internet. • In this problem, do you understand what is being discussed? Do you understand every word? Step 2. Identify what you are looking for. It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for! • In this problem, the words “what was the original price of the shirt” tell you what you are looking for: the original price of the shirt. Step 3. Name what you are looking for. Choose a variable to represent that quantity. You can use any letter for the variable, but it may help to choose one that helps you remember what it represents. • Let $p=$ the original price of the shirt Step 4. Translate into an equation. It may help to first restate the problem in one sentence, with all the important information. Then translate the sentence into an equation. Step 6. Check the answer in the problem and make sure it makes sense. • We found that $p=36$, which means the original price was $\text{\36}$. Does $\text{\36}$ make sense in the problem? Yes, because $18$ is one-half of $36$, and the shirt was on sale at half the original price. Step 7. Answer the question with a complete sentence. • The problem asked “What was the original price of the shirt?” The answer to the question is: “The original price of the shirt was $\text{\36}$.” If this were a homework exercise, our work might look like this: We list the steps we took to solve the previous example. ## Problem-Solving Strategy • Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don’t understand, look them up in a dictionary or on the internet. • Identify what you are looking for. • Name what you are looking for. Choose a variable to represent that quantity. • Translate into an equation. It may be helpful to first restate the problem in one sentence before translating. • Solve the equation using good algebra techniques. • Check the answer in the problem. Make sure it makes sense. • Answer the question with a complete sentence. For a review of how to translate algebraic statements into words, watch the following video. Let’s use this approach with another example. Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought $11$ apples to the picnic. How many bananas did he bring? In the next example, we will apply our Problem-Solving Strategy to applications of percent. Nga’s car insurance premium increased by $\text{\60}$, which was $\text{8%}$ of the original cost. What was the original cost of the premium? • Write Algebraic Expressions from Statements: Form ax+b and a(x+b). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/Hub7ku7UHT4 . License : CC BY: Attribution • Question ID 142694, 142722, 142735, 142761. Authored by : Lumen Learning. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL • Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected] Teaching support from the UK’s largest provider of in-school maths tuition resources downloaded one to one lessons schools supported Built by teachers for teachers In-school online one to one maths tuition developed by maths teachers and pedagogy experts Hundreds of FREE online maths resources! Daily activities, ready-to-go lesson slides, SATs revision packs, video CPD and more! ## 30 Problem Solving Maths Questions And Answers For GCSE Martin noon. Problem solving maths questions can be challenging for GCSE students as there is no ‘one size fits all’ approach. In this article, we’ve compiled tips for problem solving, example questions, solutions and problem solving strategies for GCSE students. Since the current GCSE specification began, there have been many maths problem solving exam questions which take elements of different areas of maths and combine them to form new maths problems which haven’t been seen before. While learners can be taught to approach simply structured problems by following a process, questions often require students to make sense of lots of new information before they even move on to trying to solve the problem. This is where many learners get stuck. ## How to teach problem solving 6 tips to tackling problem solving maths questions, 10 problem solving maths questions (foundation tier), 10 problem solving maths questions (foundation & higher tier crossover), 10 problem solving maths questions (higher tier). In the Ofsted maths review , published in May 2021, Ofsted set out their findings from the research literature regarding the sort of curriculum and teaching that best supports all pupils to make good progress in maths throughout their time in school. Regarding the teaching of problem solving skills, these were their recommendations: • Teachers could use a curricular approach that better engineers success in problem-solving by teaching the useful combinations of facts and methods, how to recognise the problem types and the deep structures that these strategies pair to. • Strategies for problem-solving should be topic specific and can therefore be planned into the sequence of lessons as part of the wider curriculum. Pupils who are already confident with the foundational skills may benefit from a more generalised process involving identifying relationships and weighing up features of the problem to process the information. • Worked examples, careful questioning and constructing visual representations can help pupils to convert information embedded in a problem into mathematical notation. • Open-ended problem solving tasks do not necessarily mean that the activity is the ‘ideal means of acquiring proficiency’. While enjoyable, open ended problem-solving activities may not necessarily lead to improved results. Are you a KS2 teacher needing more support teaching reasoning, problem solving & planning for depth ? See this article for FREE downloadable CPD There is no ‘one size fits all’ approach to successfully tackling problem solving maths questions however, here are 6 general tips for students facing a problem solving question: • Read the whole question, underline important mathematical words, phrases or values. • Annotate any diagrams, graphs or charts with any missing information that is easy to fill in. • Think of what a sensible answer may look like. E.g. Will the angle be acute or obtuse? Is £30,000 likely to be the price of a coat? • Tick off information as you use it. • Draw extra diagrams if needed. • Look at the final sentence of the question. Make sure you refer back to that at the end to ensure you have answered the question fully. There are many online sources of mathematical puzzles and questions that can help learners improve their problem-solving skills. Websites such as NRICH and our blog on SSDD problems have some great examples of KS2, KS3 and KS4 mathematical problems. ## Read more: KS2 problem solving and KS3 maths problem solving In this article, we’ve focussed on GCSE questions and compiled 30 problem solving maths questions and solutions suitable for Foundation and Higher tier students. Additionally, we have provided problem solving strategies to support your students for some questions to encourage critical mathematical thinking . For the full set of questions, solutions and strategies in a printable format, please download our 30 Problem Solving Maths Questions, Solutions & Strategies. ## 30 Problem Solving Maths Questions, Solutions & Strategies Help your students prepare for their maths GCSE with these free problem solving maths questions, solutions and strategies These first 10 questions and solutions are similar to Foundation questions. For the first three, we’ve provided some additional strategies. In our downloadable resource, you can find strategies for all 10 Foundation questions . ## 1) L-shape perimeter Here is a shape: Sarah says, “There is not enough information to find the perimeter.” Is she correct? What about finding the area? • Try adding more information – giving some missing sides measurements that are valid. • Change these measurements to see if the answer changes. • Imagine walking around the shape if the edges were paths. Could any of those paths be moved to another position but still give the same total distance? The perimeter of the shape does not depend on the lengths of the unlabelled edges. Edge A and edge B can be moved to form a rectangle, meaning the perimeter will be 22 cm. Therefore, Sarah is wrong. The area, however, will depend on those missing side length measurements, so we would need more information to be able to calculate it. ## 2) Find the missing point Here is a coordinate grid with three points plotted. A fourth point is to be plotted to form a parallelogram. Find all possible coordinates of the fourth point. • What are the properties of a parallelogram? • Can we count squares to see how we can get from one vertex of the parallelogram to another? Can we use this to find the fourth vertex? There are 3 possible positions. ## 3) That rating was a bit mean! The vertical line graph shows the ratings a product received on an online shopping website. The vertical line for 4 stars is missing. If the mean rating is 2.65, use the information to complete the vertical line graph. ## Strategies • Can the information be put into a different format, either a list or a table? • Would it help to give the missing frequency an algebraic label, x ? • If we had the data in a frequency table, how would we calculate the mean? • Is there an equation we could form? Letting the frequency of 4 star ratings be x , we can form the equation \frac{45+4x}{18+x} =2.65 Giving x=2 ## 4) Changing angles The diagram shows two angles around a point. The sum of the two angles around a point is 360°. Peter says “If we increase the small angle by 10% and decrease the reflex angle by 10%, they will still add to 360°.” Explain why Peter might be wrong. Are there two angles where he would be correct? Peter is wrong, for example, if the two angles are 40° and 320°, increasing 40° by 10% gives 44°, decreasing 320° by 10% gives 288°. These sum to 332°. 10% of the larger angle will be more than 10% of the smaller angle so the sum will only ever be 360° if the two original angles are the same, therefore, 180°. ## 5) Base and power The integers 1, 2, 3, 4, 5, 6, 7, 8 and 9 can be used to fill in the boxes. How many different solutions can be found so that no digit is used more than once? There are 8 solutions. ## 6) Just an average problem Place six single digit numbers into the boxes to satisfy the rules. The mean in maths is 5 \frac{1}{3} The median is 5 The mode is 3. How many different solutions are possible? There are 4 solutions. 2, 3, 3, 7, 8, 9 3, 3, 4, 6, 7, 9 3, 3, 3, 7, 7, 9 3, 3, 3, 7, 8, 8 ## 7) Square and rectangle The square has an area of 81 cm 2 . The rectangle has the same perimeter as the square. Its length and width are in the ratio 2:1. Find the area of the rectangle. The sides of the square are 9 cm giving a perimeter of 36 cm. We can then either form an equation using a length 2x and width x . Or, we could use the fact that the length and width add to half of the perimeter and share 18 in the ratio 2:1. The length is 12 cm and the width is 6 cm, giving an area of 72 cm 2 . 8) It’s all prime The sum of three prime numbers is equal to another prime number. If the sum is less than 30, how many different solutions are possible? There are 5 solutions. 2 can never be used as it would force two more odd primes into the sum to make the total even. ## 9) Unequal share Bob and Jane have £10 altogether. Jane has £1.60 more than Bob. Bob spends one third of his money. How much money have Bob and Jane now got in total? Initially Bob has £4.20 and Jane has £5.80. Bob spends £1.40, meaning the total £10 has been reduced by £1.40, leaving £8.60 after the subtraction. ## 10) Somewhere between Fred says, “An easy way to find any fraction which is between two other fractions is to just add the numerators and add the denominators.” Is Fred correct? ## Solution Fred is correct. His method does work and can be shown algebraically which could be a good problem for higher tier learners to try. If we use these two fractions \frac{3}{8} and \frac{5}{12} , Fred’s method gives us \frac{8}{20} = \frac{2}{5} \frac{3}{8} = \frac{45}{120} , \frac{2}{5} = \frac{48}{120} , \frac{5}{12} = \frac{50}{120} . So \frac{3}{8} < \frac{2}{5} < \frac{5}{12} The next 10 questions are crossover questions which could appear on both Foundation and Higher tier exam papers. We have provided solutions for each and, for the first three questions, problem solving strategies to support learners. ## 11) What’s the difference? An arithmetic sequence has an nth term in the form an+b . 4 is in the sequence. 16 is in the sequence. 8 is not in the sequence. -2 is the first term of the sequence. What are the possible values of a and b ? • We know that the first number in the sequence is -2 and 4 is in the sequence. Can we try making a sequence to fit? Would using a number line help? • Try looking at the difference between the numbers we know are in the sequence. If we try forming a sequence from the information, we get this: We can now try to fill in the missing numbers, making sure 8 is not in the sequence. Going up by 2 would give us 8, so that won’t work. The only solutions are 6 n -8 and 3 n -5. 12) Equation of the hypotenuse The diagram shows a straight line passing through the axes at point P and Q . Q has coordinate (8, 0). M is the midpoint of PQ and MQ has a length of 5 units. Find the equation of the line PQ . • We know MQ is 5 units, what is PQ and OQ ? • What type of triangle is OPQ ? • Can we find OP if we know PQ and OQ ? • A line has an equation in the form y=mx+c . How can we find m ? Do we already know c ? PQ is 10 units. Using Pythagoras’ Theorem OP = 6 The gradient of the line will be \frac{-6}{8} = -\frac{3}{4} and P gives the intercept as 6. ## 13) What a waste Harry wants to cut a sector of radius 30 cm from a piece of paper measuring 30 cm by 20 cm. What percentage of the paper will be wasted? • What information do we need to calculate the area of a sector? Do we have it all? • Would drawing another line on the diagram help find the angle of the sector? The angle of the sector can be found using right angle triangle trigonometry. The angle is 41.81°. This gives us the area of the sector as 328.37 cm 2 . The area of the paper is 600 cm 2 . The area of paper wasted would be 600 – 328.37 = 271.62 cm 2 . The wasted area is 45.27% of the paper. ## 14) Tri-polygonometry The diagram shows part of a regular polygon and a right angled triangle. ABC is a straight line. Find the sum of the interior angles of the polygon. Finding the angle in the triangle at point B gives 30°. This is the exterior angle of the polygon. Dividing 360° by 30° tells us the polygon has 12 sides. Therefore, the sum of the interior angles is 1800°. ## 15) That’s a lot of Pi A block of ready made pastry is a cuboid measuring 3 cm by 10 cm by 15 cm. Anne is making 12 pies for a charity event. For each pie, she needs to cut a circle of pastry with a diameter of 18 cm from a sheet of pastry 0.5 cm thick. How many blocks of pastry will Anne need to buy? The volume of one block of pastry is 450 cm 3 . The volume of one cylinder of pastry is 127.23 cm 3 . 12 pies will require 1526.81 cm 3 . Dividing the volume needed by 450 gives 3.39(…). Rounding this up tells us that 4 pastry blocks will be needed. ## 16) Is it right? A triangle has sides of (x+4) cm, (2x+6) cm and (3x-2) cm. Its perimeter is 80 cm. Show that the triangle is right angled and find its area. Forming an equation gives 6x+8=80 This gives us x=12 and side lengths of 16 cm, 30 cm and 34 cm. Using Pythagoras’ Theorem 16 2 +30 2 =1156 Therefore, the triangle is right angled. The area of the triangle is (16 x 30) ÷ 2 = 240 cm 2 . ## 17) Pie chart ratio The pie chart shows sectors for red, blue and green. The ratio of the angles of the red sector to the blue sector is 2:7. The ratio of the angles of the red sector to the green sector is 1:3. Find the angles of each sector of the pie chart. Multiplying the ratio of red : green by 2, it can be written as 2:6. Now the colour each ratio has in common, red, has equal parts in each ratio. The ratio of red:blue is 2:7, this means red:blue:green = 2:7:6. Sharing 360° in this ratio gives red:blue:green = 48°:168°:144°. ## 18) DIY Simultaneously Mr Jones buys 5 tins of paint and 4 rolls of decorating tape. The total cost was £167. The next day he returns 1 unused tin of paint and 1 unused roll of tape. The refund amount is exactly the amount needed to buy a fan heater that has been reduced by 10% in a sale. The fan heater normally costs £37.50. Find the cost of 1 tin of paint. The sale price of the fan heater is £33.75. This gives the simultaneous equations p+t = 33.75 and 5 p +4 t = 167. We only need the price of a tin of paint so multiplying the first equation by 4 and then subtracting from the second equation gives p =32. Therefore, 1 tin of paint costs £32. ## 19) Triathlon pace Jodie is competing in a Triathlon. A triathlon consists of a 5 km swim, a 40 km cycle and a 10 km run. Jodie wants to complete the triathlon in 5 hours. She knows she can swim at an average speed of 2.5 km/h and cycle at an average speed of 25 km/h. There are also two transition stages, in between events, which normally take 4 minutes each. What speed must Jodie average on the final run to finish the triathlon in 5 hours? Dividing the distances by the average speeds for each section gives times of 2 hours for the swim and 1.6 hours for the cycle, 216 minutes in total. Adding 8 minutes for the transition stages gives 224 minutes. To complete the triathlon in 5 hours, that would be 300 minutes. 300 – 224 = 76 minutes. Jodie needs to complete her 10 km run in 76 minutes, or \frac{19}{15} hours. This gives an average speed of 7.89 km/h. 20) Indices a 2x × a y =a 3 (a 3 ) x ÷ a 4y =a 32 Find x and y . Forming the simultaneous equations Solving these gives This final set of 10 questions would appear on the Higher tier only. Here we have just provided the solutions. Try asking your learners to discuss their strategies for each question. ## 21) Angles in a polygon The diagram shows part of a regular polygon. A , B and C are vertices of the polygon. The size of the reflex angle ABC is 360° minus the interior angle. Show that the sum of all of these reflex angles of the polygon will be 720° more than the sum of its interior angles. Each of the reflex angles is 180 degrees more than the exterior angle: 180 + \frac{360}{n} The sum of all of these angles is n (180 + \frac{360}{n} ). This simplifies to 180 n + 360 The sum of the interior angles is 180( n – 2) = 180 n – 360 The difference is 180 n + 360 – (180 n -360) = 720° ## 22) Prism and force (Non-calculator) The diagram shows a prism with an equilateral triangle cross-section. When the prism is placed so that its triangular face touches the surface, the prism applies a force of 12 Newtons resulting in a pressure of \frac{ \sqrt{3} }{4} N/m^{2} Given that the prism has a volume of 384 m 3 , find the length of the prism. Pressure = \frac{Force}{Area} Area = 12÷ \frac{ \sqrt{3} }{4} = 16\sqrt{3} m 2 Therefore, the length of the prism is 384 ÷ 16\sqrt{3} = 8\sqrt{3} m ## 23) Geometric sequences (Non-calculator) A geometric sequence has a third term of 6 and a sixth term of 14 \frac{2}{9} Find the first term of the sequence. The third term is ar 2 = 6 The sixth term is ar 5 = \frac{128}{9} Diving these terms gives r 3 = \frac{64}{27} Giving r = \frac{4}{3} Dividing the third term twice by \frac{4}{3} gives the first term a = \frac{27}{8} ## 24) Printing factory A printing factory is producing exam papers. When all 10 of its printers are working, it can produce all of the exam papers in 12 days. For the first two days of printing, 3 of the printers are broken. At the beginning of the third day it is discovered that 2 more printers have broken down, so the factory continues to print with the reduced amount of printers for 3 days. The broken printers are repaired and now all printers are available to print the remaining exams. How many days in total does it take the factory to produce all of the exam papers? If we assume one printer prints 1 exam paper per day, 10 printers would print 120 exam papers in 12 days. Listing the number printed each day for the first 5 days gives: Day 5: 5 This is a total of 29 exam papers. 91 exam papers are remaining with 10 printers now able to produce a total of 10 exam papers each day. 10 more days would be required to complete the job. Therefore, 15 days in total are required. ## 25) Circles The diagram shows a circle with equation x^{2}+{y}^{2}=13 . A tangent touches the circle at point P when x=3 and y is negative. The tangent intercepts the coordinate axes at A and B . Find the length AB . Using the equation x^{2}+y^{2}=13 to find the y value for P gives y=-2 . The gradient of the radius at this point is - \frac{2}{3} , giving a tangent gradient of \frac{3}{2} . Using the point (3,-2) in y = \frac {3}{2} x+c gives the equation of the tangent as y = \frac {3}{2} x – \frac{13}{2} Substituting x=0 and y=0 gives A and B as (0 , -\frac {13}{2}) and ( \frac{13}{3} , 0) Using Pythagoras’ Theorem gives the length of AB as ( \frac{ 13\sqrt{13} }{6} ) = 7.812. ## 26) Circle theorems The diagram shows a circle with centre O . Points A, B, C and D are on the circumference of the circle. EF is a tangent to the circle at A . Angle EAD = 46° Angle FAB = 48° Angle ADC = 78° Find the area of ABCD to the nearest integer. The Alternate Segment Theorem gives angle ACD as 46° and angle ACB as 48°. Opposite angles in a cyclic quadrilateral summing to 180° gives angle ABC as 102°. Using the sine rule to find AC will give a length of 5.899. Using the sine rule again to find BC will give a length of 3.016cm. We can now use the area of a triangle formula to find the area of both triangles. 0.5 × 5 × 5.899 × sin (46) + 0.5 × 3.016 × 5.899 × sin (48) = 17 units 2 (to the nearest integer). ## 27) Quadratic function The quadratic function f(x) = -2x^{2} + 8x +11 has a turning point at P . Find the coordinate of the turning point after the transformation -f(x-3) . There are two methods that could be used. We could apply the transformation to the function and then complete the square, or, we could complete the square and then apply the transformation. Here we will do the latter. This gives a turning point for f(x) as (2,19). Applying -f(x-3) gives the new turning point as (5,-19). ## 28) Probability with fruit A fruit bowl contains only 5 grapes and n strawberries. A fruit is taken, eaten and then another is selected. The probability of taking two strawberries is \frac{7}{22} . Find the probability of taking one of each fruit. There are n+5 fruits altogether. P(Strawberry then strawberry)= \frac{n}{n+5} × \frac{n-1}{n+4} = \frac{7}{22} This gives the quadratic equation 15n^{2} - 85n - 140 = 0 This can be divided through by 5 to give 3n^{2} - 17n- 28 = 0 This factorises to (n-7)(3n + 4) = 0 n must be positive so n = 7. The probability of taking one of each fruit is therefore, \frac{5}{12} × \frac{7}{11} + \frac {7}{12} × \frac {5}{11} = \frac {70}{132} ## 29) Ice cream tub volume An ice cream tub in the shape of a prism with a trapezium cross-section has the dimensions shown. These measurements are accurate to the nearest cm. An ice cream scoop has a diameter of 4.5 cm to the nearest millimetre and will be used to scoop out spheres of ice cream from the tub. Using bounds find a suitable approximation to the number of ice cream scoops that can be removed from a tub that is full. We need to find the upper and lower bounds of the two volumes. Upper bound tub volume = 5665.625 cm 3 Lower bound tub volume = 4729.375 cm 3 Upper bound scoop volume = 49.32 cm 3 Lower bound scoop volume = 46.14 cm 3 We can divide the upper bound of the ice cream tub by the lower bound of the scoop to get the maximum possible number of scoops. Maximum number of scoops = 122.79 Then divide the lower bound of the ice cream tub by the upper bound of the scoop to get the minimum possible number of scoops. Minimum number of scoops = 95.89 These both round to 100 to 1 significant figure, Therefore, 100 scoops is a suitable approximation the the number of scoops. ## 30) Translating graphs The diagram shows the graph of y = a+tan(x-b ). The graph goes through the points (75, 3) and Q (60, q). Find exact values of a , b and q . The asymptote has been translated to the right by 30°. Therefore, b=30 So the point (45,1) has been translated to the point (75,3). Therefore, a=2 We hope these problem solving maths questions will support your GCSE teaching. To get all the solutions and strategies in a printable form, please download the complete resource . Looking for additional support and resources? You are welcome to download any of the secondary maths resources from Third Space Learning’s resource library for free. There is a section devoted to GCSE maths revision with plenty of maths worksheets and GCSE maths questions . There are also maths tests for KS3, including a Year 7 maths test , a Year 8 maths test and a Year 9 maths test Other valuable maths practice and ideas particularly around reasoning and problem solving at secondary can be found in our KS3 and KS4 maths blog articles. Try these fun maths problems for KS2 and KS3, SSDD problems , KS3 maths games and 30 problem solving maths questions . For children who need more support, our maths intervention programmes for KS3 achieve outstanding results through a personalised one to one tuition approach. Do you have students who need extra support in maths? Every week Third Space Learning’s maths specialist tutors support thousands of students across hundreds of schools with weekly online 1-to-1 lessons and maths interventions designed to plug gaps and boost progress. Since 2013 we’ve helped over 150,000 primary and secondary students become more confident, able mathematicians. Find out more about our GCSE Maths tuition or request a personalised quote for your school to speak to us about your school’s needs and how we can help. Subsidised one to one maths tutoring from the UK’s most affordable DfE-approved one to one tutoring provider. Related Articles ## FREE Guide to Maths Mastery All you need to know to successfully implement a mastery approach to mathematics in your primary school, at whatever stage of your journey. Ideal for running staff meetings on mastery or sense checking your own approach to mastery. ## Privacy Overview ## Study Smarter 17 maths problem solving strategies to boost your learning. Worded problems getting the best of you? With this list of maths problem-solving strategies , you'll overcome any maths hurdle that comes your way. Friday, 3rd June 2022 • What are strategies? ## Understand the problem Devise a plan, carry out the plan, look back and reflect, practise makes progress. Problem-solving is a critical life skill that everyone needs. Whether you're dealing with everyday issues or complex challenges, being able to solve problems effectively can make a big difference to your quality of life. While there is no one 'right' way to solve a problem, having a toolkit of different techniques that you can draw upon will give you the best chance of success. In this article, we'll explore 17 different math problem-solving strategies you can start using immediately to deepen your learning and improve your skills. ## What are maths problem-solving strategies? Before we get into the strategies themselves, let's take a step back and answer the question: what are these strategies? In simple terms, these are methods we use to solve mathematical problems—essential for anyone learning how to study maths . These can be anything from asking open-ended questions to more complex concepts like the use of algebraic equations. The beauty of these techniques is they go beyond strictly mathematical application. It's more about understanding a given problem, thinking critically about it and using a variety of methods to find a solution. ## Polya's 4-step process for solving problems We're going to use Polya's 4-step model as the framework for our discussion of problem-solving activities . This was developed by Hungarian mathematician George Polya and outlined in his 1945 book How to Solve It. The steps are as follows: We'll go into more detail on each of these steps as well as take a look at some specific problem-solving strategies that can be used at each stage. This may seem like an obvious one, but it's crucial that you take the time to understand what the problem is asking before trying to solve it. Especially with a math word problem , in which the question is often disguised in language, it's easy for children to misinterpret what's being asked. Here are some questions you can ask to help you understand the problem: Do I understand all the words used in the problem? What am I asked to find or show? Can I restate the problem in my own words? Can I think of a picture or diagram that might help me understand the problem? Is there enough information to enable me to find a solution? Is there anything I need to find out first in order to find the answer? What information is extra or irrelevant? Once you've gone through these questions, you should have a good understanding of what the problem is asking. Now let's take a look at some specific strategies that can be used at this stage. ## 1. Read the problem aloud This is a great strategy for younger students who are still learning to read. By reading the problem aloud, they can help to clarify any confusion and better understand what's being asked. Teaching older students to read aloud slowly is also beneficial as it encourages them to internalise each word carefully. ## 2. Summarise the information Using dot points or a short sentence, list out all the information given in the problem. You can even underline the keywords to focus on the important information. This will help to organise your thoughts and make it easier to see what's given, what's missing, what's relevant and what isn't. ## 3. Create a picture or diagram This is a no-brainer for visual learners. By drawing a picture,let's say with division problems, you can better understand what's being asked and identify any information that's missing. It could be a simple sketch or a more detailed picture, depending on the problem. ## 4. Act it out Visualising a scenario can also be helpful. It can enable students to see the problem in a different way and develop a more intuitive understanding of it. This is especially useful for math word problems that are set in a particular context. For example, if a problem is about two friends sharing candy, kids can act out the problem with real candy to help them understand what's happening. ## 5. Use keyword analysis What does this word tell me? Which operations do I need to use? Keyword analysis involves asking questions about the words in a problem in order to work out what needs to be done. There are certain key words that can hint at what operation you need to use. How many more? How many left? Equal parts Once you understand the problem, it's time to start thinking about how you're going to solve it. This is where having a plan is vital. By taking the time to think about your approach, you can save yourself a lot of time and frustration later on. There are many methods that can be used to figure out a pathway forward, but the key is choosing an appropriate one that will work for the specific problem you're trying to solve. Not all students understand what it means to plan a problem so we've outlined some popular problem-solving techniques during this stage. ## 6. Look for a pattern Sometimes, the best way to solve a problem is to look for a pattern. This could be a number, a shape pattern or even just a general trend that you can see in the information given. Once you've found it, you can use it to help you solve the problem. ## 7. Guess and check While not the most efficient method, guess and check can be helpful when you're struggling to think of an answer or when you're dealing with multiple possible solutions. To do this, you simply make a guess at the answer and then check to see if it works. If it doesn't, you make another systematic guess and keep going until you find a solution that works. ## 8. Working backwards Regressive reasoning, or working backwards, involves starting with a potential answer and working your way back to figure out how you would get there. This is often used when trying to solve problems that have multiple steps. By starting with the end in mind, you can work out what each previous step would need to be in order to arrive at the answer. ## 9. Use a formula There will be some problems where a specific formula needs to be used in order to solve it. Let's say we're calculating the cost of flooring panels in a rectangular room (6m x 9m) and we know that the panels cost$15 per sq. metre.

There is no mention of the word 'area', and yet that is exactly what we need to calculate. The problem requires us to use the formula for the area of a rectangle (A = l x w) in order to find the total cost of the flooring panels.

## 10. Eliminate the possibilities

When there are a lot of possibilities, one approach could be to start by eliminating the answers that don't work. This can be done by using a process of elimination or by plugging in different values to see what works and what doesn't.

## 11. Use direct reasoning

Direct reasoning, also known as top-down or forward reasoning, involves starting with what you know and then using that information to try and solve the problem . This is often used when there is a lot of information given in the problem.

By breaking the problem down into smaller chunks, you can start to see how the different pieces fit together and eventually work out a solution.

## 12. Solve a simpler problem

One of the most effective methods for solving a difficult problem is to start by solving a simpler version of it. For example, in order to solve a 4-step linear equation with variables on both sides, you could start by solving a 2-step one. Or if you're struggling with the addition of algebraic fractions, go back to solving regular fraction addition first.

Once you've mastered the easier problem, you can then apply the same knowledge to the challenging one and see if it works.

## 13. Solve an equation

Another common problem-solving technique is setting up and solving an equation. For instance, let's say we need to find a number. We know that after it was doubled, subtracted from 32, and then divided by 4, it gave us an answer of 6. One method could be to assign this number a variable, set up an equation, and solve the equation by 'backtracking and balancing the equation'.

Now that you have a plan, it's time to implement it. This is where you'll put your problem-solving skills to the test and see if your solution actually works. There are a few things to keep in mind as you execute your plan:

## 14. Be systematic

When trying different methods or strategies, it's important to be systematic in your approach. This means trying one problem-solving strategy at a time and not moving on until you've exhausted all possibilities with that particular approach.

Once you think you've found a solution, it's important to check your work to make sure that it actually works. This could involve plugging in different values or doing a test run to see if your solution works in all cases.

## 16. Be flexible

If your initial plan isn't working, don't be afraid to change it. There is no one 'right' way to solve a problem, so feel free to try different things, seek help from different resources and continue until you find a more efficient strategy or one that works.

## 17. Don't give up

It's important to persevere when trying to solve a difficult problem. Just because you can't see a solution right away doesn't mean that there isn't one. If you get stuck, take a break and come back to the problem later with fresh eyes. You might be surprised at what you're able to see after taking some time away from it.

Once you've solved the problem, take a step back and reflect on the process that you went through. Most middle school students forget this fundamental step. This will help you to understand what worked well and what could be improved upon next time.

Whether you do this after a math test or after an individual problem, here are some questions to ask yourself:

What was the most challenging part of the problem?

Was one method more effective than another?

Would you do something differently next time?

What have you learned from this experience?

By taking the time to reflect on your process you'll be able to improve upon it in future and become an even better problem solver. Make sure you write down any insights so that you can refer back to them later.

There is never only one way to solve math problems. But the best way to become a better problem solver is to practise, practise, practise! The more you do it, the better you'll become at identifying different strategies, and the more confident you'll feel when faced with a challenging problem.

The list we've covered is by no means exhaustive, but it's a good starting point for you to begin your journey. When you get stuck, remember to keep an open mind. Experiment with different approaches. Different word problems. Be prepared to go back and try something new. And most importantly, don't forget to have fun!

The essence and beauty of mathematics lies in its freedom. So while these strategies provide nice frameworks, the best work is done by those who are comfortable with exploration outside the rules, and of course, failure! So go forth, make mistakes and learn from them. After all, that's how we improve our problem-solving skills and ability.

Lastly, don't be afraid to ask for help. If you're struggling to solve math word problems, there's no shame in seeking assistance from a certified Melbourne maths tutor . In every lesson at Math Minds, our expert teachers encourage students to think creatively, confidently and courageously.

If you're looking for a mentor who can guide you through these methods, introduce you to other problem-solving activities and help you to understand Mathematics in a deeper way - get in touch with our team today. Sign up for your free online maths assessment and discover a world of new possibilities.

From our blog.

## How to Get Better at Maths — 9 Tips to Improve your Grades

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## 73 Crazy Riddles for Kids [with Answers] — Can you do them all?

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## 5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems  includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving  focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

• The problem has important, useful mathematics embedded in it.
• The problem requires high-level thinking and problem solving.
• The problem contributes to the conceptual development of students.
• The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
• The problem can be approached by students in multiple ways using different solution strategies.
• The problem has various solutions or allows different decisions or positions to be taken and defended.
• The problem encourages student engagement and discourse.
• The problem connects to other important mathematical ideas.
• The problem promotes the skillful use of mathematics.
• The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

• It must begin where the students are mathematically.
• The feature of the problem must be the mathematics that students are to learn.
• It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a  neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

## Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

• Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
• What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
• Can the activity accomplish your learning objective/goals?

## Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

• Allows students to show what they can do, not what they can’t.
• Provides differentiation to all students.
• Promotes a positive classroom environment.
• Advances a growth mindset in students
• Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

• YouCubed – under grades choose Low Floor High Ceiling
• NRICH Creating a Low Threshold High Ceiling Classroom
• Inside Mathematics Problems of the Month

## Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

• Dan Meyer’s Three-Act Math Tasks
• Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

## Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

• The teacher presents a problem for students to solve mentally.
• Provide adequate “ wait time .”
• The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
• For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
• Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

## Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

• “Everyone else understands and I don’t. I can’t do this!”
• Students may just give up and surrender the mathematics to their classmates.
• Students may shut down.

• “I think I can do this.”
• “I have an idea I want to try.”
• “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

## Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

• Provide your students a bridge between the concrete and abstract
• Serve as models that support students’ thinking
• Provide another representation
• Support student engagement
• Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

The home of mathematics education in New Zealand.

• Teaching material
• Problem solving activities

## Problem Solving

This section of the nzmaths website has problem-solving lessons that you can use in your maths programme. The lessons provide coverage of Levels 1 to 6 of The New Zealand Curriculum. The lessons are organised by level and curriculum strand.  Accompanying each lesson is a copymaster of the problem in English and in Māori.

Choose a problem that involves your students in applying current learning. Remember that the context of most problems can be adapted to suit your students and your current class inquiry. Customise the problems for your class.

• Level 1 Problems
• Level 2 Problems
• Level 3 Problems
• Level 4 Problems
• Level 5 Problems
• Level 6 Problems

The site also includes Problem Solving Information . This provides you with practical information about how to implement problem solving in your maths programme as well as some of the philosophical ideas behind problem solving. We also have a collection of problems and solutions for students to use independently.

## Solving Word Problems in Mathematics

Steps of solving a word problem.

## Search form

You are here, problem solving in mathematics.

Not all teachers are comfortable with using the technique of problem solving, which is an attempt to find the answer to a problem when the method of solution is not known. This research shows how problem solving can be used in secondary school mathematics classes.

## PROBLEM SOLVING In Mathematics

Derek Holton Department of Mathematics and Statistics, University of Otago

Jim Neyland Mathematics and Science Education Centre, Victoria University of Wellington

Julie Anderson Department of Mathematics and Statistics, University of Otago

## WHERE ARE WE?

In 1993, a new national mathematics curriculum which emphasised the importance of problem solving was introduced in New Zealand. Three years later there was evidence that a significant proportion of teachers believe problem solving to be important and many are making some effort to incorporate problem solving into their teaching. However, despite this evident commitment, problem solving does not seem to occupy a regular place in most classrooms.

By problem solving we mean the attempt to find the answer to a problem when the method of solution is not known. In problem-solving situations the solver has to use strategic skills to find appropriate mathematical techniques which will settle the question. Neyland (1995) discusses the similarities and differences between the problem-solving approach and seven other approaches to teaching mathematics.

It is worth noting that what is a problem to one person may not be a problem to another. For a five-year-old determining the number of legs three sheep have will almost certainly be a problem. However, it should not be a problem for most 15-year-olds. The five-year-old may need to draw, use equipment, or employ some other method to solve the problem. The 15-year-old will just say “3 times 4 equals 12”.

Word problems do not necessarily involve problem solving. If a word problem is introduced to practise a technique which has just been acquired, it should be reasonably obvious what technique has to be used. On the other hand, problem-solving situations may be presented as wordy problems, though this is not always the case.

According to the National Council of Teachers of Mathematics:

Problem Solving should be the central focus of the mathematics curriculum. As such it is a primary goal of all mathematics instruction and an integral part of all mathematical activity. Problem Solving is not a distinct topic but a process that should permeate the entire program and provide the context in which concepts and skills can be learned.

We would like to underline the last sentence of the above quote as we find it difficult to differentiate between problem solving on the one hand and mathematics, or its essential nature, on the other.

In Australia, these views are reflected in A National Statement on Mathematics for Australian Schools (1990), where problem solving is one of four sub-headings in the Mathematical Inquiry strand:

The mathematical processes described within this strand cannot be developed in isolation from the work of other strands. They should pervade the whole curriculum.

In New Zealand, the importance of problem solving has been underscored by the Ministry of Education publication Implementing Mathematical Processes in Mathematics in the New Zealand Curriculum (1995):

Problem solving is the first heading in the Mathematical Processes strand; unless the ability to solve problems is developed, there is little point in studying mathematics.

The Mathematical Processes strand in the Mathematics in the New Zealand Curriculum document provides the umbrella for learning in the other five strands of Number, Measurement, Geometry, Algebra, and Statistics. It incorporates three areas: problem solving, developing logic and reasoning, and communicating mathematical ideas. We have decided to focus our attention on problem solving because it can provide the medium for both developing mathematical processes and interlacing the learning of specific knowledge and skills. However, is problem solving valued in our classrooms?

## THE STUDIES

The work in this paper is based upon two pieces of research. The first was a questionnaire that was given to 11 secondary school mathematics departments in a provincial city of New Zealand. Some 53 teachers responded to the questionnaire (representing a 100 percent response rate from the schools) and the issues raised were discussed with seven experienced mathematics teachers.

The second study involved observations of problem-solving sessions in five secondary classrooms at the form 3 and 4 level [year 9 and 10 in New Zealand; year 8 and 9 in Australia].

The main questions in the questionnaire were:

## How is problem solving being used and how often?

There are a variety of approaches to problem solving in the classroom. The questionnaire avoided questions about specific types of activities (open-ended tasks, investigations, short challenging tasks, projects, or mathematical modelling) and concentrated on broader areas of usage that had been observed in classrooms. The six categories considered were homework tasks, lesson starters not directly related to the content of the lesson, lesson starters related to the content of the lesson, throughout lesson, at the end of a unit of work, and catering for the needs of more able students. See table 1 for results.

From table 1, we see that problem-solving activities are being used as homework tasks more regularly than any of the other five categories surveyed. However, of those teachers using homework tasks at least weekly, under half (48 percent) used problem solving through all of a lesson at least weekly. This suggests that homework tasks are often independent activities which are not supported by problem solving in the classroom.

## Lesson starters

One of the teaching approaches identified by Sigurdson et al. (1994) is the problem-process approach , which involves using simple problems related to the mathematical content of the lesson, solving these problems through student-teacher interaction in a whole class setting and focusing on the processes being used. While using problems as lesson starters seems a good starting-off point for the implementation of a problemsolving approach, over half of the teachers surveyed never used problemsolving starters or used them only sometimes. Less than four percent use lesson starters related to content daily. But, more surprisingly, stand-alone problems are used as lesson starters at least once a week by only 21 percent of respondents. This suggests that teachers should be encouraged to use short challenging tasks to initiate problem solving in their classroom. This might be a useful point to begin teacher professional development in this area.

## Throughout lesson

Over one-third of teachers (40 percent) stated that they were using a problem-solving approach throughout all of their teaching of a lesson, at least weekly. This suggests that a regular integrated approach to problem solving is occurring in a number of classrooms. However, evidence from later written responses suggested there may have been an overstatement of the actual situation.

The reason for the relatively small percentage of teachers using a problem-solving approach regularly through all lessons may be found in Burkhardt (1988). He believes that standard teaching methods are largely single track and depend on the method being explained by the teacher. On the other hand, problem solving is multi-track and is led by the methods proposed by individual students. Burkhardt feels that problem solving is difficult for the teacher. He identifies three areas giving rise to this difficulty— mathematical, pedagogical, and personal .

Mathematically the teacher must scan the different approaches that students are using and assess how useful each of these may be. The teacher then has to decide how best to complete the mathematical task from each of these starting points.

Pedagogically the teacher must decide when to help and when not to help, and what support and questioning to provide for each student or each group of students.

Personally there is the problem of teacher confidence. Because teachers may be confronted with approaches that they have not considered before, they need to believe that they can determine, possibly with student help, which approaches lead to solutions and which don’t. This is a potentially non-trivial barrier to tackling problem solving in class.

## End of unit

In the past it has been the practice to include word problems at the end of a section in order for students to practise newly learned skills in potentially novel situations. Some 27 percent of the respondents are using problem solving in this way at least monthly. It is worth noting though that all but one of the teachers who used problem solving at the end of a unit of work was also using problem-solving lesson starters or was also using problem solving throughout all of their teaching weekly. Applying skills to problems at the end of a unit of work seems to be only one component of a teacher’s problem-solving repertoire.

## Extension students

Many schools encourage their capable students to experience independent problem solving in a variety of ways. These include mathematics or computer clubs, small groups working apart from the normal class, through mathematics competitions, or via individual extension material. In a survey of talented mathematics students, Curran, Holton, Daniel, and Pek (1992) found that such students enjoyed the challenge and the open approach of problemsolving tasks. Burkhardt (1988) highlights a common belief in the need for problem solving for the gifted child but is reserved about how best to balance this need in the gifted students’ mathematical diet.

The survey shows that, at least monthly, individual students are being given specific extension opportunities in the area of problem solving by 43 percent of teachers. Other able students will benefit from being exposed to regular classroom problem-solving activities. However, of the 57 percent of classrooms which are not providing specific problem-solving experiences regularly for their capable students, some 41 percent do not provide opportunities for problem solving in the regular classroom at least monthly. Hence, in over 23 percent of classrooms, bright students are only occasionally being exposed to any form of problem solving.

## What strategies are being promoted?

Pólya (1973) stresses the importance of strategies for solving problems. Strategies are just means to discover a solution. They are almost always not the method of solution nor a justification of the answer. They are, however, a mechanism by which the answer may be found and then justified, if necessary, by some other means.

The suggested learning experiences in the problem-solving section of Mathematics in the New Zealand Curriculum include the “devising, using, and modifying of problem solving strategies”. The questionnaire in our study referred to 12 strategies from Mathematics in the New Zealand Curriculum . These were the strategies that we thought would be commonly used by teachers. The results are summarised in table 2.

The predominance of Find a pattern, etc is perhaps to be expected. It is one of the easiest techniques to promote because of the natural occurrence of patterning in students’ environments. In addition, children come to secondary school with a history of patterning experience from primary school for teachers to draw on. Further, the emphasis on patterning is encouraged by Mathematics in the New Zealand Curriculum . Exploring patterns and relationships is an achievement objective for all levels.

We believe that Make a list and Guess and check are also popular because they are easy to teach and can be applied to a range of problems. Equally it may be that these strategies are well suited to the types of problems that are frequently used in the early years of high school.

Surprisingly though, Draw a diagram was only rated as a commonly-used strategy by just over half of the teachers. It is not clear whether this is because teachers feel that this is not “proper” mathematics at the secondary level or whether its value as a tool is not appreciated. In the research project we have seen diagrams being of great value, especially to weaker students. It provides a closer representation of the problem than, for instance, algebra does, and hence may make some problems more accessible to students.

A similar accessibility can be provided through the strategies Make a model or Act out . These heuristics (or the means by which a solution is sought) give more control of the problem to the students and can provide motivation. Lovitt and Clarke (1988) advocate these approaches, referring to them as “kinesthetic”. It is now acknowledged that children learn in different ways and the tactile, physical, pictorial thinkers could well benefit if these strategies and Draw a diagram were used more often. It may be that some teachers avoid these strategies because they view them as being more appropriate for primary schools. They may also find the strategies too difficult to implement because of the resources needed and the time constraints on their lessons. Only a quarter of respondents regularly discuss Thinking creatively . It is difficult for a teacher to model with students that “flash of brilliance” which seems to come from nowhere. However, the fostering of such insights by discussion and brain storming is vital if we are to extend the horizons of students’ thinking and to enable them to strike out in new directions.

## Strategies need to be taught

Our research supports the conclusion that strategies need to be taught. They also need to be practised otherwise they become forgotten. This seems to be especially true for young students and students who are weak mathematically.

Generally a number of strategies can be used in one problem. For example:

Greedygrimes Charlie ate a total of 100 jellybeans in 5 days, each day eating 6 more than the previous day. How many jellybeans did he eat on the third day?

This problem was solved by students in form 4 [year 10 in New Zealand, year 9 in Australia] using:

The Australian National Statement on Mathematics says that the methods of good problem solvers are likely to be “idiosyncratic” and that students need to discuss a variety of strategies to increase their “awareness of the range of techniques available”. This is supported by Beagle (1979), Pólya (1973), and Schoenfeld (1992). The first of these authors states:

… problem solving strategies are both problem and student specific often enough to suggest that finding one (or few) strategies which should be taught to all, or most students, is far too simplistic.

It is not clear, though, how heuristics should be taught. Holton (1994) and Neyland (1994) share a concern that teaching should not lead to strategies being employed like another set of algorithms or rules. They also say that students should be encouraged to see how to translate strategies to other problems and situations. Strategy usage and teaching is not straightforward and much more research is required in these areas.

## How would you like to use problem solving in the future?

Just under 60 percent of the teachers surveyed want to use problem solving more often, suggesting that a receptive environment for future development does exist. They do identify a number of common difficulties though. Among these are:

Our experience is that problem solving does take more time, especially in the early stages. However, time can be saved later. One of the teachers in the research project commented that using a group investigative approach to construction tasks and locus problems enabled her students to develop a number of concepts in one lesson which usually took her two or three lessons of more structured teaching. This approach also freed her to help individuals who were struggling, resulting in all students achieving more than usual in the time.

There is a real concern for the needs of less-able students. This relates in particular to their reading ability. One of the classes in our research project is a low-ability class. They work effectively at problem solving. This may be because all the students are of comparable ability, because the teacher uses problems that can be completed in ten to fifteen minutes, because the reading level required by the problems is not too high, or because the teacher chooses problems in a single lesson that require the same strategies.

## What help do you need with problem solving?

The teachers in the survey overwhelmingly requested pre-prepared resources and problems that would fit in with content strands. Some 74 percent of the respondents asked for resources of this kind. Some typical comments regarding teacher needs were:

As Neyland (1994) and Lovitt (1995) say, problem-solving activities by themselves are not enough to guarantee good problem solving. However, a start in this direction cannot be made without these activities. There is an urgent need to produce them in a form which makes them easily used.

Other comments by teachers included the provision of appropriate physical environments for problem solving. This meant appropriate equipment as well as space. Teachers also felt they needed help and training in scaffolding, heuristics, and metacognition (thinking about thinking—an important aspect of problem solving). Some assistance in extension work for bright students would also be appreciated.

Two valuable suggestions were made which could usefully be explored. These are:

With the change of the political environment, teachers may find the Ministry of Education in New Zealand will no longer provide the continuing services it has done in the past. Resources for teachers are less likely to be seen as a central responsibility and if not contracted out may be left to private enterprise to produce. Until these appear on the scene the next phase of teacher development will be largely left in the hands of the teachers themselves.

## A TEACHING PERSPECTIVE

In the short period since the implementation of the new curriculum, it appears that most teachers are making at least some effort to incorporate problem-solving approaches into their mathematics classrooms. At present, problem solving is not a regular feature of most classrooms but it is clear that teachers are prepared to increase their use of problem solving. This increased use can be facilitated by the production of appropriate content-related problems. Indeed the teachers surveyed see this as the first priority to the further implementation of a problem-solving approach to mathematics.

The advantages proposed for problem solving in school are that it:

Unfortunately simply giving teachers good reasons for teaching problem solving, and giving them good problems to pass on to their students, does not produce good problem solving. As Lovitt (1995) says:

All the early problem-solving efforts were mostly devoted to the creation of suitable problems in the belief that teachers could present these in classrooms and generate effective learning with the same maths they used for expository teaching. It has taken some time to recognise that this is not the case …

Neyland (1994) expresses the same view:

Some well designed [problem-solving activities] do not result in the active learning event intended. There appear to be two main reasons for this. Firstly, the way the activity was originally presented to the class diminished its potential. And secondly, the interactive component of the teaching–learning process was not adequately prepared and unanticipated problems arose. [emphasis in original]

Before starting to teach problem solving, teachers should be clear why, apart from the fact that it is in the curriculum, they intend teaching it. It is important too, to note that the problem-solving approach to teaching mathematics is different from what has become the traditional approach. However, much of the problem-solving approach is not new. Scaffolding, which is not limited to problem solving, has its origins in the Socratic method. Heuristics too have a long pedigree. So how is problem solving different from what has become the standard “chalk and talk” method for teaching mathematics?

The main difference is one of attitude or philosophy on the part of the teacher. The shift is from the so-called “sage on the stage” to the “guide by the side”. Philosophically, the teacher needs to change from a giving role to an encouraging role, from “here is how to solve a linear equation” to “how might we solve this linear equation? ”

Naturally the students will not be able to invent all the mathematics they need for themselves. There will still be things that they need to be told. They will still need to practise both skills and problemsolving processes. However, more time needs to be spent by the students exploring mathematics with the teacher as a guide. During this exploration, seeds will be planted and students will develop connections between various parts of their learning which will increase learning, understanding and retention later ( see Hiebert and Carpenter, 1992).

So, instead of teachers taking students down well-trodden paths in a sequential unfolding of mathematical structures, there should be more emphasis on the students themselves structuring mathematical knowledge for specific problem contexts. Certainly the structures that are formed this way have to be justified but not necessarily in the one way shown in the textbook. Where possible, students should be given the opportunity to provide their own justifications. Many of these will be correct and may well be different from the text-book proof.

This is not to say that the development of mathematics during the year is anarchic. Teachers should know from the start what material is to be covered. However, there should be some flexibility in the manner and order in which the content arises. Of course, in highstakes years, when there are external examinations, the course may have to be slightly more structured. Nevertheless, it is important even on these occasions for teachers’ questioning to be open rather than closed in order to stimulate students’ thinking.

In the problem-solving approach then, there is less emphasis on students applying rules to problems which have been carefully chosen to fit those rules. There is greater emphasis on the creative construction of mathematical structures and solutions in non-routine situations and over a range of contexts.

## Incorporating process and model

In order to give the following five-part overview of the problem-solving process we combine the model of problem solving given in Holton and Neyland (1996) with the heuristics and metacognition discussed in that paper ( see figure 1).

We recall that heuristics are means by which a solution is sought. Most of the heuristics of Begg (1994) are covered in this overview of the problem-solving process. We also show that certain heuristics tend naturally to appear at certain stages. In what follows, heuristics are emphasised in italics.

## I Getting started

This is the first part of problem solving and includes “problem” and “experiment” from figure 1. It also covers the first two phases of Póya’s four-phase model.

## 1. Understand the problem

The problem will either be posed by the students themselves, presented in writing, or given verbally. If the problem is not one posed by the students, a first and crucial step is for them to make the problem their own; to become familiar with its conditions, characteristics, and variables. If it is in writing the solver has to read the problem and read it carefully. Many a solver has begun by solving a problem they thought was there but was actually not.

As the solvers read the problem they should be asking the various questions posed by Pólya. They should also be looking for key words . These are words which, if changed, will lead to a quite different problem. They are words or phrases which are essential in the solution of the problem. For novice problem solvers, underlining key words is a useful strategy. It is also a good idea for the problem solver to restate the problem for themselves . This helps to emphasise the key aspects of the problem. Other strategies such as experimenting with special cases also help the student to understand the problem.

Consider the problem below.

Peter is 18 years older than his daughter Sue who is 7 years old. How old will Peter be when he is twice as old as Sue?

In this problem, “18”, “7”, and “twice as old” all appear to be key words or phrases. The question, and more importantly the answer, is altered if key words are changed. (In actual fact, “7” is an extraneous piece of information here.) On the other hand, “Peter” could be “Peta” and “Sue”, “Sam” without changing the answer or the method of solution.

In this particular problem it is obvious that both Peter and Sue age at the same rate, one year at a time. This is nowhere explicitly stated but it is an hidden assumption . Many problems have implicit conditions that turn out to be important in finding a solution. So it is vital to make sure that hidden assumptions have been noted.

One of the metacognitive aspects of problem solving is to know when to come back to the original problem and read it again. This may be necessary to see if you are on the right track of a solution or if you have inadvertently started to solve another problem. It may also help you to try another approach if you have made no progress with a particular method of attack or change the point of view . Once an answer has been obtained it is important to reread the question to ensure that the solution you have obtained does indeed solve the original problem.

Apart from the metacognition (or “thinking about thinking”) at the start of a problem—which largely relates to monitoring and controlling the problem-solving processes—it is necessary for solvers to go through a mental list of heuristics in an effort to find a few which will get them started.

It is worth asking the following Pólya questions:

In addition it is worth considering:

## 3. Experiment

This is often an early stage in the problem-solving process. It has two main functions. The first is to get a feel for the problem. The second is to start to produce some evidence for a conjecture. The experiments may be calculations or measurements or a variety of things depending on the problem in hand.

The experiments should be systematic . The aim is to get central, useful information that can be collated in some way, rather than an unconnected jumble of, say, numbers. And the results of experiments should be recorded in some logical fashion— in a list, table , or diagram . When appropriate, consider the case n = 1, then n = 2 and so on. It is worth keeping these experiments until the problem has been completely solved. They may well be useful to provide or inspire a counterexample later.

Another problem that is useful to illustrate some ideas here is the frog problem.

Four spotted frogs (S) and four green frogs (G) are sitting on lilypads as in figure 2. Frogs can move to the next lilypad if it is free or they can jump over another frog if there is an empty lilypad on the other side. What is the smallest number of moves which can interchange the green and the spotted frogs?

There is a certain amount of symmetry in this problem. Whatever happens next, certainly it doesn’t matter whether a spotted or a green frog starts first.

This is a good problem too, to illustrate solving a simpler problem first . It’s easier to do the problem above once you’ve tried moving one frog of each type, two frogs of each type, and so on.

This is a common feeling at the start of a new problem. Success and experience will give the solver confidence to continue. However, even expert problem solvers face panic on occasions. Be prepared to think “there is no way I will ever do this problem” and then get on and realise you can.

## II Conjecturing

The next process in problem solving involves conjecture.

This can be the most enjoyable and interesting part of the problemsolving process. In more difficult problems it is necessary to find a pattern, rule , or relation in order to produce a conjecture.

Some problems give up the correct conjecture immediately, other problems take more time. In harder problems it may be necessary to discard a number of conjectures before obtaining the right one. Some people find some problems easier to make conjectures about than others. And conjectures require practice. So it is necessary to start with simple problems and work up.

Does the conjecture make sense? Is it consistent with previous knowledge and the data that has been assembled in the experimental phase? Does the conjecture imply anything that feels wrong? Intuition and common sense can and should be used here. A conjecture which implies that the height of a mountain is five centimetres must surely be wrong. Any conjecture should be justifiable . Even though it cannot be proved at this stage, there should be good reasons for choosing one conjecture over another.

## III Proof/Counter example

The more difficult problems will require the solver to trade off conjecture against proofs against counterexamples. The process is a dynamic one which is only completed when the final step in the solution is written down. Frequently, trying to prove a conjecture gives an idea for a counterexample and looking for a counterexample can give an idea for a proof. Whether starting down the proof or counterexample trail the solver may see that the conjecture needs adjusting and the process starts again.

It is difficult to see whether to first try to justify a conjecture or show it is wrong. The decision as to which to try first depends on the solver and the problem. If testing a few cases will cover all possible counterexamples, then the solver should go that route. This will either produce a counterexample or strongly confirm the conjecture. In the latter case, of course, a proof will still be required.

Some problems look like others and this may suggest a way to proceed to a proof. A slight change of a known proof might work. Or the problem may be one which suggests a well-known proof technique such as proof by contradiction or mathematical induction .

## 1. Extreme cases

In trying to find counterexamples it is often valuable to try extreme examples or extreme cases . These examples are somehow at the edge of the spectrum of values that are being used. For instance, what happens if a number is very small or what happens as “n” approaches infinity? What happens if the triangle is isosceles? Extreme cases are usually easier to handle than more general situations so it is worth testing a few before trying a proof. Even if they do not provide a counterexample they may well give the solver some useful information which can be used later. Here is a problem which can be quickly solved by looking at an extreme case: Do all one-litre milk containers have the same surface area? The answer, no, can be easily obtained by imagining a one-litre box-shaped milk container with a base the size of a tennis court and a height a fraction of a millimetre. Clearly this one-litre container has a surface area much larger than the one in the fridge.

## 2. Special cases

A conjecture can also be tested against special cases rather than extreme cases. Special cases are typical cases such as the ones during the experimental phase. Where possible, cases would be tested because they are straight forward or easy to test. But sometimes more complicated cases are forced on the solver.

## 3. Simpler problems

Sometimes problems are far too difficult to solve the first time. They may involve far too many variables to be able to understand easily. In that case reduce the problem to a simpler problem in some way. For instance, in the frog problem using fewer frogs is a good way to start. Similarly a gambling problem involving five dice might be reduced at first to one involving only two. Insight gained from the two dice case may well lead to the solution of the original problem. Here the aim is to keep as many of the essentials of the problem as possible while reducing the problem to a manageable size. Using simpler problems is also something that is useful at the experimental stage, as we saw earlier. Sometimes problems can be simplified in more than one way and students should learn to choose the best from the range of available simplified situations.

## 4. Exhaust all possibilities

One of the very simple methods of proof is to exhaust all possibilities . If there are sufficiently few cases to handle (under 50, say), then by looking at each one in turn, the solver can learn about the entire problem. This way any conjecture about the situation is immediately verified or disproved. Simple combinatorial problems such as the behaviour of two dice are often open to this approach. For instance, how many different ways are there for two dice to give a sum of four?

When a solution has been obtained by older children by the exhaustive approach, it is worth asking the Pólya phase four question “Can you derive the result differently?” They should be looking for more sophisticated approaches. There are times, however, when no other method is available. Certainly this will be the case with younger children. And finding a justification, any justification, is better than no justification at all.

## 5. Guess and check

One of the simplest ways of tackling a problem is guess and check . This is sometimes also called trial and error . In the problem of Peter and Sue, an answer can be found by guessing and checking. It is therefore a good strategy for simple problems. There are drawbacks, however. First, if the problem involved has a large number of cases, for instance the five dice gambling game we alluded to earlier, then it may not be easy to guess the correct answer in a reasonable time. Second, guessing and checking will give an answer. However, the method cannot tell you whether the answer is unique. It may well be that the problem has a number of answers. Only an exhaustive search will be able to determine all answers in such a problem. Finally, the guess and check method in complicated situations may only reveal the conjecture required; it may not justify that conjecture.

The efficiency of guess and check can be increased with metacognition by using guess and improve . Here subsequent guesses are improved using the data of past guesses. Again in simple problems where a unique answer is almost certain, this is a useful strategy. In Peter and Sue’s problem, we could first guess that Peter was 30, in which case Sue would be 12. The check 2 × 12≤ 30, shows that we have guessed incorrectly. We could then guess, say 40. Now 2 × 22 = 44 which is more than 30. The next guess should then be directed between 30 and 40. Guessing and thinking allows us to hone in on the correct answer.

To make sure that we don’t keep making the same guesses, the results could be recorded in a table such as that in figure 3. Another advantage of the table is that it enables us to move our guesses in the right direction; it enables us to see how best to use guess and improve.

We actually came across a nice solution of this problem where the student had made a model . On the edges of two pieces of paper he wrote the numbers 1 to 30. He then put the edges of the paper side by side and moved one of them until corresponding numbers were 18 higher on one edge ( see figure 3). He then looked along the edges to try to find where one number was twice the other. (Actually he didn’t put down enough numbers, but what he had was enough to enable him to see the right answer.)

## 6. Work backwards

Work backwards is often useful in problems which involve a sequence of moves, events, or operations leading to a known end point. By reversing the process the starting configuration can be obtained; this is a bit like running a video in reverse. It is useful for investigating two-person games. Start from the last move and work back to the starting position. Work backwards is the basis of many commonly used methods for solving equations. This strategy is also useful when dealing with proofs of trigonometric identities.

## 6. All information

If the solver is unable to make any headway it is often worth re-reading the question in an attempt to ensure that all of the information given in the problem has been used. Sometimes there is an implicit detail that is not mentioned specifically but which is vital to the solution. Knowing general facts such as the sum of the dots on opposite sides of a dice is seven may well be required to solve a problem. (See hidden information in the experimental stage.)

Having obtained a complete solution the solver should go through the justification and check every step. Even if the proof passes the check test, sometimes the solver still has a nagging doubt. In all cases it is a good idea for the solver to get someone else to check their solutions. At this stage too, sometimes a solver may see a quicker, more efficient, way to solve the problem. The new method needs to be written down and checked too.

## IV GENERALISATION/EXTENSION

1. generalisation.

A generalisation is a problem which contains the original problem as a subset or special case. Generalisations can often be found by increasing the number of digits (in some number problems), stepping up the dimension (in a geometry problem), or by increasing the number of variables in some way. For instance, you may want to do the frog problem with an arbitrary number of frogs (n, say) on either side. This is a generalisation of the four frog case.

The reason for considering generalisations is that they give a result which is true for a much wider class of objects. One difficulty with generalisations is that they may require a quite different justification from the justification used in the original problem.

## 2. Extension

An extension of a problem is one which is related to the original problem but not by way of a generalisation. These can be found by changing one of the conditions of the original problem in some way. It may be that a problem can be extended by changing addition to multiplication. As with generalisations, the proof of an extension may not be linked in any way to the proof of the original problem.

The final part of the problem-solving process may in fact be to give it up!

Some problems are too difficult to be solved right now. So, at times, students will have to abandon some problems. If they are forced to abandon all problems they tackle then the problems are too difficult.

This will depend on the teacher and the student. It becomes a matter of priorities. However, any reasonable problem will take more than 10 minutes and less than two hours (unless it is an extended investigation). In general, we suspect that students can go further than we usually expect. This is especially true of the better students. And it should be remembered that expert problem solvers like to sleep on a problem if no solution is at first forthcoming.

In cases where the problem is too hard for both the teacher and the student, outside help should be summoned. This may be from another teacher, a book, or a local university.

## Implementing problem solving

If you are a teacher who has been convinced philosophically of the importance of problem solving in mathematics, or if you see it in the curriculum and feel a responsibility to teach problem solving, the next question to be resolved is how ? The first thing to point out is that there is no unique way to teach problem solving. Teachers and schools will need to develop their own styles. Each teacher will need to develop an individual style which will no doubt develop and be a function of the school, the students, and many other things. An easy way to start is by using a series of one-off problems that may not necessarily be related to the content of the remainder of the lesson. This allows the teacher to gain confidence and gather together a string of useful problems. However, the danger is that students will not realise the relevance and importance of problem solving that stands alone, not integrated with the rest of the curriculum.

Our research suggests that problem-solving lessons work well around a three-stage format. The first stage is a whole-class format, where the problem is discussed by teachers and students, and students may suggest possible heuristics.

In the second stage, students work in small groups. The aim of this stage is to enable students to become involved with the problem and attempt to solve it for themselves. During this period, the teacher is able to go from group to group to provide scaffolding. However, our research suggests that most scaffolding during group work is undertaken by the students themselves as they give peer tuition to their group members.

The third stage is a reporting back stage which gives the students an opportunity to say what they have done and how they did it. Here the spotlight is again on the students who are reporting on their work. Different methods of solution are to be encouraged during this stage. It is important that student thinking and learning takes place in this final stage. One reason for the reporting is to improve students’ ability to communicate but even more important is the chance for students to see alternative approaches to a problem. These alternative approaches should be useful on future occasions. In this stage too, there is an opportunity for students to see links between different parts of mathematics which is again another aid towards understanding.

We have observed these three stages in a variety of problem-solving classes. They seem to work equally well with weak and strong students and with young and old students. A general rule of thumb is that the three-stage cycle should be repeated with young students and weak students. For these children the three stages should take no more than 15 minutes. Very good students may only go around the cycle once in a one-hour lesson.

Problem solving is not easy for many teachers as it involves a rethinking of their whole approach to teaching mathematics. The difficulties arise for many reasons. Some relate to potential loss of control in going from a closed to an open environment where it is never clear in which direction the students will head next. Some relate to the wider range of teaching skills required to handle the situation. Other difficulties are due to social phenomena, including student expectation. But it is our experience that most of these difficulties are surmountable by many teachers, even though at times they may be close to giving up. In our current research, one teacher was in despair after what she thought was a bad lesson. The next problem-solving lesson she taught was extremely successful, with the students seeing links that she herself had not previously seen. From that moment she has been sold on problem solving.

Another teacher had worked solidly doing a competent job for over two terms with a weak-streamed class. In a regular third-term exam (not based on problem solving) her students were 10 percent above similar students in another class. Her students had gained considerably on word problems. The practice they had had in problem-solving situations had given them confidence to successfully tackle the word problems in the exam.

DEREK HOLTON is Professor in the Department of Mathematics and Statistics, University of Otago. E-mail: [email protected]

JIM NEYLAND is Lecturer of Mathematics Education in the Mathematics and Science Education Centre, Victoria University of Wellington. E-mail: [email protected]

JULIE ANDERSON is in the Department of Mathematics and Statistics, University of Otago, and St Hilda’s College, Dunedin.

We would like to thank the Ministry of Education, Research Section, for the financial support which allowed us to engage in the research on which this paper is based. Our thanks also go to the many teachers and students who allowed us into their classrooms.

For more details of this research, see:

Holton D., Spicer, T., & Thomas, G. (1995). Is problem solving too hard? Proceedings of Mathematics Education Research Group of Australasia , 345–351.

Holton D., Spicer T., Thomas G., & Young, S. (1996). The benefits of problem solving in the learning of mathematics (Report to the Ministry of Education). Dunedin: Otago University, Department of Mathematics and Statistics.

The New Zealand curriculum documents are:

Ministry of Education. (1992). Mathematics in the New Zealand curriculum . Wellington: Learning Media Ltd.

Ministry of Education. (1995). Implementing mathematical processes in the New Zealand curriculum . Wellington: Learning Media Ltd.

The Australian curriculum documents are:

Australian Education Council. (1990). A national statement on mathematics for Australian schools . Carlton, Vic: Curriculum Corporation.

That problem solving should be the central focus of the mathematics curriculum is from page 23 of:

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematic . Reston, VA: Author.

The problem-process approach is described by:

Sigurdson, S. E., Olson, A. T., & Mason, R. (1994). Problem solving and mathematics learning. Journal of Mathematical Behaviour 13, 361–388.

That talented mathematics students enjoyed the challenge and the open approach of problem-solving tasks is noted by:

Curran, J., Holton, D., Daniel, C., & Pek, W. H. (1992). A survey of talented secondary mathematics students . Dunedin: Department of Mathematics and Statistics, University of Otago.

A common belief in the need for problem solving for the gifted child is highlighted in:

Burkhardt, H. (1988). Teaching problem solving. In H. Burkhardt, S. Groves, A. Schoenfeld & K. Stacey (Eds.), Problem solving: A world view (pp. 17–42). Nottingham, UK: Shell Centre.

The importance of strategies for solving problems is noted in:

Pólya, G. (1973). How to solve it (3rd ed.). Princeton, NJ: Princeton University Press.

The kinesthetic approaches are advocated by:

Lovitt, C., & Clarke, D. J. (1988). Mathematics curriculum and teaching program, activity bank (Vols. 1 and 2). Canberra, ACT: Curriculum Development Centre.

That the methods of good problem solvers are likely to be “idiosyncratic” is supported by:

Beagle (1979) pp. 145–146, see above.

Pólya (1973), see above.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, meta-cognition and sense making in mathematics. In Douglas A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan.

That students should be encouraged to see how to translate strategies to other problems and situations is stated by:

Holton, D. (1994). Problem solving. In J. Neyland (Ed.), Mathematics education: A handbook for teachers (Vol. 1, pp. 18–30). Wellington: Wellington College of Education.

Neyland, J. (1994). Designing rich mathematical activities. In J. Neyland (Ed.), Mathematics education: A handbook for teachers (Vol. 1, pp.106–122). Wellington: Wellington College of Education.

That problem-solving activities by themselves are not enough to guarantee good problem solving is stated by:

Lovitt, C. (1995). Personal communication.

That time needs to be spent by the students exploring mathematics with the teacher as a guide is noted by:

Hiebert, J. & Carpenter, J. P. (1996). The benefits of problem solving to the learning of mathematics (Report to the Ministry of Education, Wellington). Dunedin: University of Otago, Department of Mathematics and Statistics.

For more details of the heuristics used in the problemsolving process see:

Begg, A. (1994). Mathematics: Content and process. In J. Neyland (Ed.), Mathematics education: A handbook for teacher (Vol. 1, pp. 183–192). Wellington: Wellington College of Education.

Ministry of Education. (1995). Implementing mathematical process in mathematics in the New Zealand curriculum . Wellington: Learning Media, Ltd.

National Council of Teachers of Mathematics. (1980). An agenda for action . Reston, VA: Author.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics . Reston, VA: Author.

Neyland, J. (1995). Eight approaches to teaching mathematics. In J. Neyland (Ed.), Mathematical education: A handbook for teachers (Vol. 2, pp. 34–48). Wellington: Wellington College of Education.

For a good example of problem-solving teachers include:

Stacey, K. & Groves, S. (1985). Strategies for problem solving . Camberwell, Vic: Lattude Publications.

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## How to Solve Math Problems

Last Updated: May 16, 2023 Fact Checked

This article was co-authored by Daron Cam . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 574,162 times.

Although math problems may be solved in different ways, there is a general method of visualizing, approaching and solving math problems that may help you to solve even the most difficult problem. Using these strategies can also help you to improve your math skills overall. Keep reading to learn about some of these math problem solving strategies.

## Understanding the Problem

• Draw a Venn diagram. A Venn diagram shows the relationships among the numbers in your problem. Venn diagrams can be especially helpful with word problems.
• Draw a graph or chart.
• Arrange the components of the problem on a line.
• Draw simple shapes to represent more complex features of the problem.

## Expert Q&A

• Seek help from your teacher or a math tutor if you get stuck or if you have tried multiple strategies without success. Your teacher or a math tutor may be able to easily identify what is wrong and help you to understand how to correct it. Thanks Helpful 1 Not Helpful 1
• Keep practicing sums and diagrams. Go through the concept your class notes regularly. Write down your understanding of the methods and utilize it. Thanks Helpful 1 Not Helpful 0

## You Might Also Like

• ↑ Daron Cam. Math Tutor. Expert Interview. 29 May 2020.
• ↑ http://tutorial.math.lamar.edu/Extras/StudyMath/ProblemSolving.aspx
• ↑ https://math.berkeley.edu/~gmelvin/polya.pdf

To solve a math problem, try rewriting the problem in your own words so it's easier to solve. You can also make a drawing of the problem to help you figure out what it's asking you to do. If you're still completely stuck, try solving a different problem that's similar but easier and then use the same steps to solve the harder problem. Even if you can't figure out how to solve it, try to make an educated guess instead of leaving the question blank. To learn how to come up with a solid plan to use to help you solve a math problem, scroll down! Did this summary help you? Yes No

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