maths problem solving ideas

120 Math Word Problems To Challenge Students Grades 1 to 8

Written by Marcus Guido

• 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 . ( See our entire list of back to school resources for teachers here .)

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

Addition word problems.

Best for: 1st grade, 2nd grade

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?

5. Adding to 1,000: How many cookies did you sell if you sold 320 chocolate cookies and 270 vanilla cookies?

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

Best for: 1st grade, second grade

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

Best for: 2nd grade, 3rd grade

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

Best for: 3rd grade, 4th grade, 5th grade, 6th grade

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

Best for: 4th grade, 5th grade

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

Best for: Kindergarten, 1st grade, 2nd grade

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

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade, 5th grade

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

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade

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

Best for: 4th grade, 5th grade, 6th grade

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

Best for: 4th grade, 5th grade, 6th grade, 7th grade

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

Best for: 4th grade, 5th grade, 6th grade, 7th grade, 8th grade

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

Best for: 6th grade, 7th grade, 8th grade

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.

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Hey teachers! 👋

Use Prodigy to spark a love for math in your students – including when solving word problems!

Problem Solving Activities: 7 Strategies

• Critical Thinking

Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.

In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough. 

I was so excited!

We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies. 

It was a proud moment for me!

Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy. 

Genius right? 

After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name. 

What a crushing blow! Just when I thought I did something special, I find out I did it all wrong. 

I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom. 

When I Finally Saw the Light

To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students. 

Problem Solving Activities

Here are seven ways to strategically reinforce problem solving skills in your classroom. 

Seasonal Problem Solving

Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!

Cooperative Problem Solving Tasks

Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.

Notice and Wonder

Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it. 

Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.

Math Starters

Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here ! 

Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.

Calculators

Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!

Three-Act Math Tasks

Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons . 

Getting the Most from Each of the Problem Solving Activities

When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next. 

Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below. 

Which of the problem solving activities will you try first? Respond in the comments below.

Shametria Routt Banks

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2 responses.

This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.

Thank you, Scott! Best wishes to you and your pre-service teachers this year!

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Free Math Worksheets — Over 100k free practice problems on Khan Academy

Looking for free math worksheets.

You’ve found something even better!

That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun!

Just choose your grade level or topic to get access to 100% free practice questions:

Kindergarten, basic geometry, pre-algebra, algebra basics, high school geometry.

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Statistics and probability

High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

• Addition and subtraction
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• Addition and subtraction within 20
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• Measurement and data
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• Place value
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Frequently Asked Questions about Khan Academy and Math Worksheets

Why is khan academy even better than traditional math worksheets.

Khan Academy’s 100,000+ free practice questions give instant feedback, don’t need to be graded, and don’t require a printer.

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Khan Academy is a nonprofit with a mission to provide a free, world-class education to anyone, anywhere.

Want to get even more out of Khan Academy?

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5 Strategies for Successful Problem Solving

• Powerful teaching strategies
• December 26, 2023
• Michaela Epstein

Blog > 5 Strategies for Successful Problem Solving

Problem solving can change the way students see maths – and how they see themselves as maths learners.

But, it's tough to help all students get the most out of a task.

To help, here are  5 Strategies for Problem Solving Success.

These are 5 valuable lessons I've learned from working with teachers across the globe .  You can use these strategies with all your students, no matter their level.

5 Strategies for Problem Solving Success

Your own enthusiasm is quickly picked up by your students. So, choose a problem, puzzle or game that you’re excited and curious about.

How do you know what will spark your curiosity? Do the task yourself!

(That’s why,  in the workshops I run , we spend a lot of time  actually  exploring problems. It’s a chance to step into students' shoes and experience maths from their perspective.)

Often, curriculum content becomes the goal of problem solving. For example, adding fractions, calculating areas or solving quadratic equations.

But, this is a mistake! Here's why-

Low floor, high ceiling tasks give students choices. Choices about what strategies to use, tools to draw on – and even what end-points to get to.

The most valuable goals focus on building confidence and capability in problem solving. For example:

• To make and break conjectures
• To use and evaluate different strategies
• To organise data in meaningful ways
• To explain and justify their conclusions.

The start of a task is what will get your students curious and hungry to get underway.

Consider: What's the least information your students will need?

At  ​our Members' online PL sessions​ , we look at one of four possibilities for launching a problem:

• Present a mystery to explore
• Present an example and non-example
• Run a demonstration game
• Show how to use a tool.

Keep the launch short – under 5 minutes. This is just enough to keep students’ attention AND share essential information.

Let’s face it, problem solving is hard, no matter your age or mathematical skill set.

Students aren’t afraid of hard work – they’re afraid of feeling or looking stupid. And, when those tricky maths moments do come, you can help.

Using questions, tools and other prompts can bring clarity and boost confidence.

(Here's a  free question catalogue  you might find handy to have in your back pocket.)

This careful support will help your students find problem solving far less daunting. Instead, it can become a chance for wonderous mathematical exploration.

Picture this: Your students are elbows deep in a problem, there’s a buzz in the air – oh, and only a minute until the bell.

The  most important  stage of a problem solving task – right at the end – is often the one that gets dropped off.

Why does ‘wrapping up’ matter?

In the last 10 minutes of a problem, students can share conjectures, strategies and solutions. It's also a chance to consider new questions that may open up further exploration.

In wrapping up, important learning will happen. Your students will observe patterns, make connections and clarify conjectures. You might even notice ‘aha’ moments.

Five strategies for problem solving success:

• Choose a task that YOU'RE keen on,
• Set a goal for strengthening problem solving skills,
• Plan a short launch to make the task widely accessible,
• Use questions, tools and prompts to support productive exploration, and
• Wrap up to create space for pivotal learning.

Join the Conversation

Dear Michaela, Greetings !! Thank you for sharing the strategies for problem solving task. These strategies will definitely enhance the skill in the mindset of young learners. In India ,Students of Grade 9 and Grade 10 have to learn and solve lot of theorems of triangle, Quadrilateral, Circle etc. Being an educator I have noticed that most of the students learn the theorems and it’s derivation by heart as a result they lack in understanding the application of these theorems.

I will appreciate if you can share your insights as how to make these topics interesting and easy to grasp.

Once again thanks for sharing such informative ideas.

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• DIGITAL GAMES

Problem-Solving Strategies

October 16, 2019

There are many different ways to solve a math problem, and equipping students with problem-solving strategies is just as important as teaching computation and algorithms. Problem-solving strategies help students visualize the problem or present the given information in a way that can lead them to the solution. Solving word problems using strategies works great as a number talks activity and helps to revise many skills.

Problem-solving strategies

1. create a diagram/picture, 2. guess and check., 3. make a table or a list., 4. logical reasoning., 5. find a pattern, 6. work backward, 1. create a diagram/draw a picture.

Creating a diagram helps students visualize the problem and reach the solution. A diagram can be a picture with labels, or a representation of the problem with objects that can be manipulated. Role-playing and acting out the problem like a story can help get to the solution.

Alice spent 3/4 of her babysitting money on comic books. She is left with $6. How much money did she make from babysitting?

2. Guess and check

Teach students the same strategy research mathematicians use.

With this strategy, students solve problems by making a reasonable guess depending on the information given. Then they check to see if the answer is correct and they improve it accordingly.  By repeating this process, a student can arrive at a correct answer that has been checked. It is recommended that the students keep a record of their guesses by making a chart, a table or a list. This is a flexible strategy that works for many types of problems. When students are stuck, guessing and checking helps them start and explore the problem. However, there is a trap. Exactly because it is such a simple strategy to use, some students find it difficult to consider other strategies. As problems get more complicated, other strategies become more important and more effective.

Find two numbers that have sum 11 and product 24.

Try/guess  5 and 6  the product is 30 too high

  adjust  to 4 and 7 with product 28 still high

  adjust  again 3 and 8 product 24

3. Make a table or a list

Carefully organize the information on a table or list according to the problem information. It might be a table of numbers, a table with ticks and crosses to solve a logic problem or a list of possible answers. Seeing the given information sorted out on a table or a list will help find patterns and lead to the correct solution.

To make sure you are listing all the information correctly read the problem carefully.

Find the common factors of 24, 30 and 18

Logical reasoning is the process of using logical, systemic steps to arrive at a conclusion based on given facts and mathematic principles. Read and understand the problem. Then find the information that helps you start solving the problem. Continue with each piece of information and write possible answers.

Thomas, Helen, Bill, and Mary have cats that are black, brown, white, or gray. The cats’ names are Buddy, Lucky, Fifi, and Moo. Buddy is brown. Thoma’s cat, Lucky, is not gray. Helen’s cat is white but is not named Moo. The gray cat belongs to Bill. Which cat belongs to each student, and what is its color?

A table or list is useful in solving logic problems.

Since Lucky is not gray it can be black or brown. However, Buddy is brown so Lucky has to be black.

Buddy is brown so it cannot be Helen’s cat. Helen’s cat cannot be Moo, Buddy or Lucky, so it is Fifi.

Therefore, Moo is Bill’s cat and Buddy is Mary’s cat.

5. Find a pattern.

Finding a pattern is a strategy in which students look for patterns in the given information in order to solve the problem. When the problem consists of data like numbers or events that are repeated then it can be solved using the “find a pattern” problem-solving strategy. Data can be organized in a table or a list to reveal the pattern and help discover the “rule” of the pattern.

 The “rule” can then be used to find the answer to the question and complete the table/list.

Shannon’s Pizzeria made 5 pizzas on Sunday, 10 pizzas on Monday, 20 pizzas on Tuesday, and 40 pizzas on Wednesday. If this pattern continues, how many pizzas will the pizzeria make on Saturday?

6. Working backward

Problems that can be solved with this strategy are the ones that  list a series of events or a sequence of steps .

In this strategy, the students must start with the solution and work back to the beginning. Each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backward they will need to subtract. And if they multiply working forwards, they must divide when working backward.

Mom bought a box of candy. Mary took 5 of them, Nick took 4 of them and 31 were given out on Halloween night. The next morning they found 8 pieces of candy in the box. How many candy pieces were in the box when mom bought it.

For this problem, we know that the final number of candy was 8, so if we work backward to “put back” the candy that was taken from the box we can reach the number of candy pieces that were in the box, to begin with.

The candy was taken away so we will normally subtract them. However, to get back to the original number of candy we need to work backward and do the opposite, which is to add them.

8 candy pieces were left + the 31 given out + plus the ones Mary took + the ones Nick took

8+31+5+4= 48   Answer: The box came with 48 pieces of candy.

Selecting the best strategy for a problem comes with practice and often problems will require the use of more than one strategies.

Print and digital activities

I have created a collection of print and digital activity cards and worksheets with word problems (print and google slides) to solve using the strategies above. The collection includes 70 problems (5 challenge ones) and their solution s and explanations.

sample below

How to use the activity cards

Allow the students to use manipulatives to solve the problems. (counters, shapes, lego blocks, Cuisenaire blocks, base 10 blocks, clocks) They can use manipulatives to create a picture and visualize the problem. They can use counters for the guess and check strategy. Discuss which strategy/strategies are better for solving each problem. Discuss the different ways. Use the activities as warm-ups, number talks, initiate discussions, group work, challenge, escape rooms, and more.

Ask your students to write their own problems using the problems in this resource, and more, as examples. Start with a simple type. Students learn a lot when trying to compose a problem. They can share the problem with their partner or the whole class. Make a collection of problems to share with another class.

For the google slides the students can use text boxes to explain their thinking with words, add shapes and lines to create diagrams, and add (insert) tables and diagrams.

Many of the problems can be solved faster by using algebraic expressions. However, since I created this resource for grades 4 and up I chose to show simple conceptual ways of solving the problems using the strategies above. You can suggest different ways of solving the problems based on the grade level.

Find the free and premium versions of the resource below. The premium version includes 70 problems (challenge problems included) and their solutions

There are 2 versions of the resource

70 google slides with explanations + 70 printable task cards

70 google slides with explanations + 11 worksheets

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K-5 Math Centers

K-5 math ideas, 3rd grade math, need help organizing your k-5 math block, 5 ways to include math problem solving activities in your classroom.

Are you looking for math problem solving activities that are not too easy and not too hard, but juuust right? I’ve got something just for you and your students.

Solve and Explain Problem Solving Tasks are open-ended math tasks that provide just the right amount of challenge for your kids. Here’s a little more about them.

Open-ended math problem solving tasks:

• promote multiple solution paths and/or multiple solutions
• boost critical thinking and math reasoning skills
• increase opportunities for developing perseverance
• provide opportunities to justify answer choices
• strengthen kids written and oral communication skills

What Makes These So Great?

• All Common Core Standards are covered for your grade level
• 180+ Quality questions that are rigorous yet engaging
• They are SUPER easy to assemble
• Provide opportunities for meaningful math discussions
• Perfect for developing a growth mindset
• Easily identify student misconceptions so you can provide assistance
• Very versatile (check out the different ways to use them below)

You can find out more details for your grade level by clicking on the buttons below.

I’m sure you really want to know how can you use these with your kids. Check out the top 5 ideas on how to use Solve and Explain Problem Solving Tasks in your classroom.

How and When Can I Use Them?

Solve and Explain Tasks Cards are very versatile. You can use them for:

• Math Centers  – This is my favorite way to use these! Depending on your grade level, there are at least two (Kinder – 2nd) or three (3rd-5th) tasks types per Common Core standard. And each task type has 6 different questions. Print out each of the different tasks types on different color paper. Then, let students choose which one question from each task type they want to solve.

• Problem of the Day  – Use them as a daily math journal prompt. Print out the recording sheet and project one of the problems on your white board or wall.  Students solve the problem and then glue it in their spiral or composition notebooks.

• Early Finisher Activities  -No more wondering what to do next!Create an early finishers notebook where students can grab a task and a recording sheet. Place the cards in sheet protectors and make copies of the Early Finisher Activity Check-Off card for your kids to fill out BEFORE they pull a card out to work on. We want to make sure kids are not rushing through there first assignment before moving on to an early finisher activity.

• Weekly Math Challenges  – Kids LOVE challenges! Give students copies of one of the problems for homework. Then give them a week to complete it. Since many of the questions have multiple solutions and students have to explain how they got their answers, you can have a rich whole group discussion at the end of the week (even with your kindergarten and 1st grade students).

Shop Recommended Resources

• Formative Assessments  – Give your students a problem to solve. Then use the Teacher Scoring Rubric to see how your kids are doing with each standard. Since they have to explain their thinking, this is a great way to catch any misconceptions and give feedback to individual students.

So this wraps up the top 5 ways that you can use problem solving tasks in your classroom.  Click your grade level below to get Solve and Explain problem solving tasks for your classroom.

• Read more about: K-5 Math Ideas

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25 Exciting Hands on Math Activities for Elementary

Engaging elementary students in hands on math activities is a fantastic way to foster a deeper understanding of mathematical concepts while making learning enjoyable.

These activities not only make math come alive but also encourage active participation and critical thinking.

By incorporating hands-on experiences, educators can create an enriching learning environment that sets a strong foundation for future mathematical success.

We will explore a variety of creative and effective hands on math activities for elementary students, ensuring that the learning process is both fun and educational.

Math Manipulative Creations: Building Geometric Shapes

One exciting hands-on math activity for elementary students involves using math manipulatives like building blocks or interlocking cubes to create various geometric shapes. Start by introducing basic shapes like squares, rectangles, triangles, and circles. Then, challenge students to use the manipulatives to construct these shapes on their own. As they become more comfortable, encourage them to combine shapes to form more complex figures, such as making a hexagon from triangles or a parallelogram from rectangles. This activity not only reinforces shape recognition but also enhances spatial awareness and problem-solving skills.

Fraction Beach: Sandbox Exploration of Fractions

Create a tactile learning experience with a Fraction Beach activity. Fill a large sandbox with sand and divide it into sections representing different fractions. Students can use shovels, buckets, and other tools to manipulate the sand, creating visual representations of fractions. This hands-on approach offers a unique way for students to grasp the concept of fractions as parts of a whole.

Fraction Pizza Party: Understanding Fractions through Play

Fractions can be a challenging concept for elementary students to grasp, but a Fraction Pizza Party activity can make it more accessible and enjoyable. Provide students with circular paper cutouts resembling pizza slices. Ask them to divide the slice into different fractions, such as halves, thirds, and quarters, using markers or colored pencils. Then, they can decorate each slice to represent the fraction they’ve created. This hands-on approach allows students to visualize fractions and comprehend their relationship to a whole, transforming abstract ideas into tangible concepts. Related: 20 Creative Math Door Decoration Ideas

Shape Hunt Scavenger Hunt: Exploring Geometry in the Environment

Take math learning beyond the classroom with a Shape Hunt Scavenger Hunt. This activity involves going outdoors or exploring different areas of the school to identify and categorize various shapes in the environment. Provide students with a list of shapes to find, such as squares, circles, rectangles, and triangles. They can document their discoveries by taking photos or drawing sketches of the objects they find. Not only does this activity reinforce shape recognition, but it also encourages students to see the relevance of math in their surroundings.

Money Math Market: Applying Real-life Math Skills

Teaching elementary students about money can be both practical and enjoyable through a Money Math Market activity. Set up a pretend market with various items labeled with price tags. Give students play money and have them shop for items, making calculations of their purchases and finding the total cost. This hands-on experience helps students understand the value of different coins and bills, as well as practice addition and subtraction in a real-world context. It also promotes financial literacy from an early age.

Measurement Olympics: Exploring Length, Weight, and Volume

Host a Measurement Olympics to help elementary students grasp the concepts of length, weight, and volume in a playful way. Set up various stations where students can measure objects using rulers, scales, and measuring cups. They can compare the lengths of different items, weigh objects, and pour liquids into containers to understand volume. This activity not only enhances their measurement skills but also encourages them to apply mathematical reasoning to real-world scenarios.

Tangram Teasers: Puzzling with Shapes and Spatial Relationships

Tangrams are a fantastic tool to develop spatial reasoning and geometry skills in elementary students. Provide each student with a set of tangram pieces – seven flat shapes that can be combined to form various figures. Challenge students to recreate specific shapes, animals, or objects using the tangram pieces. This activity promotes critical thinking, problem-solving, and spatial visualization while engaging students in a creative puzzle-solving adventure.

Math Board Games: Learning Through Play

Engage elementary students in the world of board games that are designed to teach and reinforce mathematical concepts. Games like “Monopoly,” “Chutes and Ladders,” and “Uno” can be adapted to incorporate math challenges. Students roll dice, draw cards, or spin spinners, answering math questions or performing calculations as part of the game mechanics. This approach blends learning with leisure, making math more enjoyable and interactive. Related: 20 Helpful Grief Activities for Elementary Students

Pattern Play: Unveiling the Magic of Patterns

Patterns are everywhere, and the Pattern Play activity helps elementary students discover the magic behind them. Begin with simple patterns of colors, shapes, or numbers, and let students continue the sequence. Gradually increase the complexity of the patterns as they become more confident. This activity hones pattern recognition skills and lays the foundation for understanding more intricate mathematical concepts, such as algebraic patterns and sequences.

Geometry in Art: Creating Symmetrical Masterpieces

Merge math and art with the Geometry in Art activity, where students explore symmetry and geometric shapes to design their own masterpieces. Provide them with drawing materials and guide them to create symmetrical images by drawing on one side and reflecting the design on the other. This activity not only encourages creativity but also reinforces concepts like line symmetry and geometric transformations, showcasing the beautiful intersection of math and artistic expression.

Time Travel Adventure: Learning with Elapsed Time

Help elementary students grasp the concept of elapsed time through a Time Travel Adventure. Present them with scenarios involving different start and end times, and ask them to calculate the duration in hours and minutes. You can use real-life scenarios like planning a day’s activities or calculating travel times between destinations. This activity strengthens students’ time-telling skills and develops their understanding of time intervals.

Math Storytelling: Numbers and Narratives

Combine math and literacy with the Math Storytelling activity, where students create their own mathematical stories. Encourage them to invent characters, settings, and situations that involve mathematical concepts. They can use addition, subtraction, multiplication, or division to solve problems within their stories. Sharing their stories not only nurtures their creative skills but also reinforces their understanding of mathematical operations in a meaningful context.

Math Puzzles and Riddles: Brain-Teasing Challenges

Engage elementary students’ critical thinking skills with Math Puzzles and Riddles. Provide them with age-appropriate puzzles that require logical reasoning and mathematical insights to solve. These puzzles can range from number sequences and logic puzzles to classic riddles with mathematical twists. The challenge of cracking these brain-teasers makes math enjoyable and encourages students to think outside the box.

Fractional Kitchen: Exploring Fractions in Cooking

Bringing fractions into the kitchen creates an interactive and delicious learning experience. Organize a Fractional Kitchen activity where students follow recipes and use measuring cups and spoons to prepare dishes. As they measure ingredients like flour, sugar, and liquids, they gain a tangible understanding of fractions in a practical context. This activity bridges the gap between abstract fractions and real-life applications.

Math Around the World: Exploring Cultural Numerical Systems

Expose elementary students to the diversity of numerical systems with the Math Around the World activity. Introduce them to various counting systems used in different cultures, such as Roman numerals or the Mayan number system. Allow them to explore these systems by representing numbers and performing basic calculations. This activity fosters cultural awareness while deepening students’ appreciation for the universality of mathematical principles.

Math Art Gallery: Graphing Coordinate Plane Creations

Transform the concept of graphing on a coordinate plane into an artistic endeavor with a Math Art Gallery activity. Provide students with a set of coordinates and guide them to plot points on graph paper. As they connect the dots, intricate designs and images will emerge. This activity allows students to explore the relationship between numbers, coordinates, and visual representations, enhancing their graphing skills in an engaging way.

Math Jeopardy: Interactive Review Game

Elevate the excitement of reviewing math concepts by playing Math Jeopardy. Create a game board with different categories and point values related to the curriculum. Divide students into teams and take turns choosing questions from the board. Students must solve the presented problems within a specified time frame to earn points. Math Jeopardy not only reinforces learning but also promotes friendly competition and collaboration among students.

Math-Inspired Building: Exploring Geometry with Construction

Combine engineering and mathematics with a Math-Inspired Building activity. Provide students with materials like toothpicks, marshmallows, or straws, and challenge them to create geometric shapes or structures using their imagination. This hands-on experience helps them understand concepts like angles, symmetry, and stability, all while fostering creativity and hands-on problem-solving skills.

Math Mysteries: Solving Interactive Math Whodunits

Engage elementary students in mathematical detective work with Math Mysteries. Present them with scenarios where they must solve mathematical clues to uncover the answers. These mysteries can involve a range of math topics, from solving equations to deciphering patterns. As students work through the clues, they not only sharpen their math skills but also enjoy the thrill of solving puzzles.

Math in Nature: Outdoor Exploration of Patterns and Shapes

Nature is a treasure trove of mathematical inspiration. Take students on an outdoor exploration to discover Math in Nature. Observe patterns in leaves, petals, and seeds. Look for symmetry in butterfly wings or geometric shapes in rock formations. Encourage students to document their findings through sketches or photographs. This activity fosters a deeper connection between math and the world around us while encouraging curiosity and observation skills. Related: 100 Fun Questions Of The Day for Elementary Students

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• 23 Fun Shark Activities for Preschoolers

Sohaib Hasan Shah

Sohaib's journey includes 10+ years of teaching and counseling experience at BCSS School in elementary and middle schools, coupled with a BBA (Hons) with a minor in Educational Psychology from Curtin University (Australia) . In his free time, he cherishes quality moments with his family, reveling in the joys and challenges of parenthood. His three daughters have not only enriched his personal life but also deepened his understanding of the importance of effective education and communication, spurring him to make a meaningful impact in the world of education.

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Building Problem-Solvers: Engaging Maths Challenges & Playful Games for Primary Students

Table of Contents

Building Problem-Solvers: Incorporating challenging maths puzzles and games into primary education is a powerful strategy for developing problem-solving skills in young learners. We recognise the importance of engaging pupils with interactive tasks that not only stimulate their mathematical abilities but also build their confidence in tackling complex challenges. By designing diverse mathematical exercises that range from simple to complex, we create opportunities for children to develop fluency as mathematical thinkers and problem solvers, a skill set that is essential in today’s fast-paced, ever-evolving world.

It’s crucial to foster a supportive maths culture within the classroom, one that encourages risk-taking and persistence in overcoming obstacles. Education in primary mathematics should not only focus on procedural fluency but also on the understanding of fundamental mathematical concepts . Tools and strategies such as visual representations and heuristic approaches can be invaluable in promoting this kind of deep comprehension. By integrating these techniques with a varied set of maths puzzles and games, we make learning mathematics an engaging and dynamic experience.

Key Takeaways

• Developing problem-solving skills in maths is essential for primary education.
• Diverse and challenging puzzles enhance mathematical thinking.
• A supportive learning environment encourages perseverance and growth in maths.

Understanding the Fundamentals of Maths

In our pursuit of developing proficient problem solvers, it’s essential to start with a strong foundation in mathematics. We focus on building an understanding of numbers and operations, as well as recognising the significance of patterns and relationships. These components are vital in nurturing a comprehensive mathematical mindset.

Number and Operations

At the core of mathematical understanding are numbers and operations . This involves grasping how numbers work and interact with one another through basic operations : addition, subtraction, multiplication, and division. For example:

Addition: It’s combining quantities to increase the total.

Subtraction: It’s determining how much one quantity is greater than another

Multiplication : It simplifies repeated addition.

Division: It’s finding out how many times a number can be subtracted from another in equal parts.

Understanding these operations enables children to solve real-world problems effectively.

Patterns and Relationships

Patterns and relationships form the building blocks for higher-level mathematical thinking. They involve seeing connections and predicting what comes next, based on a sequence or rule.

Patterns: These may be numerical or geometric, allowing us to anticipate the next element in a sequence.

Relationships: Understanding that certain pieces of information are related, and using this to solve problems, is key to mathematical reasoning . An example might be in understanding that if a car covers more distance in an hour at a faster speed, there’s a direct relationship between speed and distance.

In fostering these fundamental skills, we offer our learners not just theory but engaging challenges through mathematical puzzles and games that reinforce these concepts. By participating in such activities, children can apply these foundational maths skills, making connections between theory and practice in a dynamic and enjoyable way.

Teaching Mathematics Effectively

In our quest to develop sharp problem solvers and creative thinkers, we must focus on how mathematics is taught in primary schools. Our approach bridges the gap between traditional teaching and innovative, engaging methods .

Role of Teachers

We, as educators, play a pivotal role in moulding students into proficient problem-solvers. By incorporating a variety of maths puzzles and games into our classrooms, we make abstract concepts tangible and accessible. We believe in adapting our teaching styles to meet individual learning needs, devising strategies that make mathematics more than just a subject – instead, a captivating journey of discovery.

Curriculum Integration

The curriculum should not solely dictate our teaching, but serve as a dynamic framework that integrates real-world problem-solving and logical reasoning through mathematical activities . It is essential we weave challenging exercises with relevant contexts into lessons, thus aligning with our commitment to a broad and balanced education where every child flourishes.

The Importance of Problem-Solving Skills

Problem-solving skills are integral to learning mathematics, as they enable children to approach complex problems strategically and with confidence. Our goal is to harness these skills through puzzles and games that challenge and engage primary school children.

Developing Critical Thinking

We understand that critical thinking is the foundation of effective problem-solving. Our puzzles and games are designed to stimulate students’ thinking and reasoning processes , encouraging them to make connections and derive solutions based on logical deduction.

Cultivating Patience and Persistence

Throughout the problem-solving journey, we foster a sense of patience and persistence in our young learners. Recognising that solutions are not always immediate or straightforward, our resources teach children the value of perseverance and resilience in the face of challenging tasks.

Engaging Pupils with Interactive Maths Games

Interactive maths games are a brilliant way for us to make mathematics appealing and accessible to primary school pupils. By selecting the right games and integrating 21st-century technology, we can transform the learning experience into an interactive adventure that captivates our young learners.

Selecting Appropriate Games

When choosing maths games, we must ensure that they are appropriately challenging and aligned with the curriculum. The games should be designed to encourage pupils to think critically and develop their problem-solving skills. We often look for options that offer multiple levels of difficulty, which enables us to cater to the diverse abilities within a classroom. The aim is to select games that not only educate but also genuinely engage the students, making the learning process both enjoyable and effective.

Integrating ICT in Maths Games

In today’s digital age, integrating Information and Communication Technology (ICT) in maths games is not just innovative; it’s essential. We leverage ICT to provide simulations and virtual environments where pupils can explore mathematical concepts. This interactive technology helps to create immersive experiences that make abstract ideas more concrete. Using ICT, we can give pupils the chance to practise and hone their maths skills through fun, tech-driven games, fostering a more dynamic and interactive learning environment.

Incorporating interactive maths games within the classroom is a testament to our commitment to engaging and educating our pupils in a way that resonates with their experiences and interests. Through careful selection and the use of modern ICT, we are able to provide a learning experience that is both enriching and exciting.

Designing Challenging Maths Puzzles

In our pursuit to build effective problem-solvers, we focus on introducing maths puzzles that are engaging and non-routine, designed to foster natural curiosity in learners.

Creating Non-Routine Tasks

We believe that the essence of problem-solving lies in the ability to tackle non-routine tasks. These are not your everyday textbook problems, but instead, they present a scenario that requires students to apply concepts in ways they might not have anticipated. In constructing these tasks, it’s vital to strike a balance—too easy and they won’t push the envelope; too hard and they might discourage learners. For example, a puzzle involving pattern recognition might ask pupils to identify the underlying rule of a sequence and predict the next set of numbers.

Fostering Curiosity Through Puzzles

Curiosity drives us to explore the unknown, and with maths puzzles, it translates into learners venturing beyond their comfort zones. We craft puzzles that inherently provoke students’ interest and pique their natural inquisitiveness. A well-thought-out puzzle can act as an open inquiry, where the journey to the solution is as valuable as the solution itself. Consider a jigsaw arithmetic puzzle that requires learners to not only solve for missing pieces but also to understand why those pieces fit together as they do.

Incorporating Diverse Mathematical Tasks

When we introduce a variety of mathematical tasks to primary students, we lay the foundation for robust problem-solving skills. By engaging in an assortment of challenges , children can develop a deeper understanding of numbers, spatial awareness, and logical reasoning.

Exploration and Investigation

We believe that learning should be an adventure, where exploration and investigation play pivotal roles. By offering tasks that encourage students to explore, we open up opportunities for hands-on learning and inquisitive thinking. For instance, we might ask them to investigate the number of ways to reach a total of 10 using only red and blue counters. This simple task invites them to explore addition and the concept of combinations.

Measurement and Geometry

Measurement and geometry are two areas where children can apply maths to real-world scenarios. We often ask our students to measure lengths and widths of classroom objects, using rulers and other tools to relate the numbers to physical attributes. Then, we might move on to explore geometric shapes by examining and constructing models , allowing for a tangible understanding of edges, faces, and vertices. This hands-on experience is invaluable for cementing their conceptual knowledge.

Overcoming Obstacles in Problem Solving

Engaging with maths puzzles and games in primary education can sometimes present challenges. It’s essential we understand these stumbling blocks and employ effective strategies to help our children become confident problem solvers.

Identifying Common Mistakes

Often, obstacles in problem solving arise from common misunderstandings or repeated mistakes . We might see children rush through a problem without fully understanding it, or they may become fixed on one approach and not consider alternative methods. Documenting these mistakes offers us a chance to address them directly.

• Rushing : Not taking enough time to understand the problem.
• Fixation on one strategy : Failing to consider different angles.
• Overlooking details : Missing out on crucial information within the problem.

Strategies to Overcome Challenges

After pinpointing the typical mistakes, our strategies should focus on overcoming these barriers and reinforcing effective problem solving habits.

• Encourage thorough reading : Urge pupils to read problems several times.
• Promote multiple approaches : Introduce a variety of methods to tackle a single problem.
• Detail orientation : Teach children to pay attention to all the information given.

Remember, practice makes perfect, and providing children with a mixture of puzzles and games can build their resilience and adaptability in maths.

Creating a Supportive Maths Culture

At the heart of nurturing future problem-solvers is the establishment of a supportive maths culture . This means creating an environment where every child feels valued and capable of mastering mathematical challenges .

Encouraging Open Communication

We recognise the importance of open communication in the classroom. It’s crucial to foster an atmosphere where pupils feel comfortable to express their ideas, ask questions, and share their experiences. In our classroom, we encourage learners to articulate their thought processes and reasoning. This not only clarifies their understanding but also enriches peer learning, as students learn from each other’s perspectives. Such a culture not only enhances their communication skills but also demystifies complex concepts, making maths more approachable .

Building Confidence and Growth Mindset

We strive to infuse our pupils with confidence and a growth mindset . To do this, we emphasise the belief that abilities can be developed through dedication and hard work. This contrasts with a fixed mindset, where children might believe their skills are static and unchangeable. By celebrating effort rather than innate ability , we inspire our students to embrace challenges and learn from mistakes—an approach that is pivotal for fostering resilient problem solvers . Our aim is to show that mathematics is not about being ‘right’ all the time; instead, it’s about the adventure of learning and improving.

Assessing and Enhancing Problem-Solving Capabilities

In our quest to build adept problem-solvers through maths puzzles and games , it’s vital that we effectively assess and enhance primary students’ problem-solving capabilities. This not only involves gauging their current skill levels but also providing constructive feedback to aid their growth.

Conducting Meaningful Assessments

When it comes to assessing problem-solving activities, our approach is to create evaluations that are as engaging as the learning experiences themselves. We believe that a student’s problem-solving skills are best understood by observing them during actual problem-solving activities. Such assessments might involve practical tasks where pupils apply heuristics or use visual representations to tackle mathematical problems, as highlighted by Developing Mathematical Problem-Solving Skills . These tasks are designed to mirror ‘real life’ scenarios, requiring students to think critically and creatively.

Providing Constructive Feedback

Feedback is a powerful tool in our educational arsenal—it informs students about their performance and provides guidance on how they can improve. Our feedback is specific, timely, and always focused on strategies that students can use to enhance their problem-solving abilities. Whether it’s encouraging greater representational fluency or exploring diverse strategies, we ensure that our feedback helps students reflect and grow as problem solvers. By drawing on resources such as the Real Engagement in Active Problem Solving (REAPS) model, we support teachers in offering feedback that fosters creative problem solving in mathematics.

In all of our endeavours at LearningMole, we strive to provide learning experiences that not only educate but also excite. We’re dedicated to fostering environments where every student can become a confident and proficient problem solver.

Resources for Primary Maths Education

We know that the right resources can make all the difference in empowering primary maths education. So, let’s explore some excellent materials that you can access for free, as well as delve into the treasures offered by NRICH to enrich our young learners’ mathematical journey.

Free Teaching Materials

We pride ourselves on providing a variety of free teaching materials designed to make maths engaging and fun . You’ll find interactive tutorials , activity sheets , and articles, all tailored for the curious minds of primary children. These resources are not only educational, but they also allow children to absorb key maths concepts in a way that feels like play.

• Interactive Tutorials : Engage with our step-by-step guides that bring clarity to complex maths problems.
• Articles : Discover insights and tips targeted at enhancing the teaching experience.
• Activity Sheets : Download and print these for hands-on practice that reinforces mathematical understanding.

One platform we admire is LearningMole , which offers a wealth of content to help children discover the joy of maths through various fun and creative resources.

Utilising NRICH Resources

When it comes to using NRICH resources , we’re looking at a treasure trove of maths puzzles and activities that are perfect for primary pupils. NRICH aims to challenge and excite young minds with games that are both intriguing and highly educational.

• Challenging Puzzles : They stimulate strategic thinking and offer varying degrees of difficulty to suit all levels.
• Classroom Activities : Carefully crafted to promote collaborative problem-solving among students.

These resources from NRICH not only complement our teaching but also bring a new dimension to primary maths, making our lessons more dynamic and effective.

Frequently Asked Questions

In this section, we’ll address some of the most common inquiries related to enhancing problem-solving skills through mathematics puzzles and games in primary education.

What are some engaging activities that can enhance problem-solving abilities in primary school children?

Children in primary school can greatly benefit from activities such as building structures using different shapes or collaborative tasks where they devise solutions to “real life” scenarios. These activities encourage them to apply mathematical concepts in practical ways.

Can you suggest some games that improve problem-solving skills for young learners?

Certainly! Games like chess, Sudoku, and even certain board games that require strategic thinking can improve problem-solving skills . These games challenge young minds to think ahead and plan their moves carefully.

How can puzzles be effectively used to develop mathematical problem-solving competencies in children?

Puzzles can be used in teaching by posing them as interesting challenges that are mathematically meaningful . Puzzles such as tangrams, magic squares, and logic puzzles encourage children to use mathematical reasoning and pattern recognition.

In what ways can teachers incorporate problem-solving exercises into their primary classroom curriculum?

Teachers can introduce problem-solving in the classroom by integrating puzzles and games into lesson plans. They might also ask thought-provoking questions that lead to problem-solving discussions or use differentiated instruction to cater to various learning styles.

Which hands-on group activities can help children build teamwork and problem-solving skills concurrently?

Group activities that involve building projects from common objects like blocks or recycled materials allow children to work together. They can also participate in team-based challenges that require collective problem-solving and decision-making.

What types of maths challenges are suitable for primary students to promote critical thinking and reasoning?

Suitable challenges include pattern identification , sequencing tasks, basic arithmetic puzzles, and solving mathematical puzzles . These can be tailored to match the students’ age and proficiency level, ensuring the tasks remain engaging and appropriately challenging.

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32 Mathematical Ideas: Problem-Solving Techniques

Jenna Lehmann

Solving Problems by Inductive Reasoning

Before we can talk about how to use inductive reasoning, we need to define it and distinguish it from deductive reasoning.

Inductive reasoning is when one makes generalizations based on repeated observations of specific examples. For instance, if I have only ever had mean math teachers, I might draw the conclusion that all math teachers are mean. Because I witnessed multiple instances of mean math teachers and only mean math teachers, I’ve drawn this conclusion. That being said, one of the downfalls of inductive reasoning is that it only takes meeting one nice math teacher for my original conclusion to be proven false. This is called a counterexample . Since inductive reasoning can so easily be proven false with one counterexample, we don’t say that a conclusion drawn from inductive reasoning is the absolute truth unless we can also prove it using deductive reasoning. With inductive reasoning, we can never be sure that what is true in a specific case will be true in general, but it is a way of making an educated guess.

Deductive reasoning depends on a hypothesis that is considered to be true. In other words, if X = Y and Y = Z, then we can deduce that X = Z. An example of this might be that if we know for a fact that all dogs are good, and Lucky is a dog, then we can deduce that Lucky is good.

Strategies for Problem Solving

No matter what tool you use to solve a problem, there is a method for going about solving the problem.

• Understand the Problem: You may need to read a problem several times before you can conceptualize it. Don’t become frustrated, and take a walk if you need to. It might take some time to click.
• Devise a Plan: There may be more than one way to solve the problem. Find the way which is most comfortable for you or the most practical.
• Carry Out the Plan: Try it out. You may need to adjust your plan if you run into roadblocks or dead ends.
• Look Back and Check: Make sure your answer gives sense given the context.

There are several different ways one might go about solving a problem. Here are a few:

• Tables and Charts: Sometimes you’ll be working with a lot of data or computing a problem with a lot of different steps. It may be best to keep it organized in a table or chart so you can refer back to previous work later.
• Working Backward: Sometimes you’ll be given a word problem where they describe a series of algebraic functions that took place and then what the end result is. Sometimes you’ll have to work backward chronologically.
• Using Trial and Error: Sometimes you’ll know what mathematical function you need to use but not what number to start with. You may need to use trial and error to get the exact right number.
• Guessing and Checking: Sometimes it will appear that a math problem will have more than one correct answer. Be sure to go back and check your work to determine if some of the answers don’t actually work out.
• Considering a Similar, Simpler Problem: Sometimes you can use the strategy you think you would like to use on a simpler, hypothetical problem first to see if you can find a pattern and apply it to the harder problem.
• Drawing a Sketch: Sometimes—especially with geometrical problems—it’s more helpful to draw a sketch of what is being asked of you.
• Using Common Sense: Be sure to read questions very carefully. Sometimes it will seem like the answer to a question is either too obvious or impossible. There is usually a phrasing of the problem which would lead you to believe that the rules are one way when really it’s describing something else. Pay attention to literal language.

This chapter was originally posted to the Math Support Center blog at the University of Baltimore on November 6, 2019.

Math and Statistics Guides from UB's Math & Statistics Center Copyright © by Jenna Lehmann is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution. 

1. Link problem-solving to reading

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it. 

3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process. 

4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks. 

5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.” 

6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process. 

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

Outdoor Math Games – 50 Ideas That Really Work

Getting maths into the outdoors is one of the biggest concerns many practitioners have. It is probably the thing I am asked about the most where maths is concerned.

This post is by far the most comprehensive quantity of high-quality outdoor math activities for children aged 3 to 6 that you will be able to find all in one place.

Why do maths outside?

The benefits of maths outdoors are many, which include:

• You can do things on a bigger scale
• Maths can become really active and physical
• You can use a wider range of found objects
• You have the space for large games
• Some children prefer being outdoors and will access maths opportunities more freely than indoors

Outdoors offer both an opportunity to learn new things, as well as practice basic math concepts they already understand with bigger and more active resources.

Here we go – let’s get stuck into some of the best and tried and tested outdoor math activities that will get your children loving maths outside.

Table of Contents

Number Stones

I have created quite an array of different types of number stones. These are actually great for use either inside or out. Lots of the math games I will demonstrate below can be played with these, although you can use many other things such as sticks, leaves, or conkers.

I just bought lots of white pebbles from a hardware shop and some acrylic paint.

If you’re looking to buy similar pebbles, something on Amazon like this will work.

For acrylic paint, some bog-standard primary colors like these will do the job.

I painted the stones with acrylic paint in different ways, and sometimes used a Sharpie pen if I needed to put an outline on them.

The number stones I have created include:

Sets of 1-10 Stones

These are alternately colored, yellow, white, yellow, white, to begin the idea of odd and even in a visual way. Great for ordering or using to label.

If you don’t want to create your own stones, you can also buy beautiful number stones, thus saving you time. Yellow Door painted number stones are a good choice (check out the latest price on Amazon here.)

Matching Numeral To Quantity Stones

Have some stones with numbers on, and some with quantities. It could be dots, or it could be something like insects.

These are brilliant for one to one correspondence in particular. If you want to find out what one to one correspondence is, and the best ways to teach it, then check this out.

Animal Stones

I have some ladybird stones, some leopard ones, as well as some zebra and bee stones. Any animal that has a multi-colored pattern can be copied. These stones are great for games.

2s, 5s And 10s Stones

Great for higher-ability children in the early years, and for challenging younger kids who are starting to count in different ways.

These are perfect for ordering numbers in a range of ways.

Outdoor Models and Images

Mathematicians think in pictures.

Models and images seem to work really well outside because you can make them big, and they can be the focus of active games.

The benefits of using models and images include:

• They present maths as pictures
• They are visual
• They allow for problem-solving
• They allow for the manipulation of physical objects
• They can be used again and again, often in different ways

Chalkboard Paint

A lot of the resources that I have created are using black chalkboard paint . This is one of the ultimate resources in the early years. It is great for painting large surfaces for mark-making, such as fences, sheds, pallets, or old furniture.

I imagine any can of chalkboard paint would work, but I use American Crafts DIY Chalkboard Paint (check out the current price on Amazon) . It does the job great.

I have painted several pieces of MDF from a hardware shop and then used these for games. However, you could just as easily chalk the games on the floor. I also once painted tree stumps with lots of games using acrylic paint. They lasted for at least a year, even out in the rain.

Addition Triangles

These are great for exploring number bonds. They represent the part-part-whole model used in Singapore Maths.

I have painted my addition triangles onto a piece of MDF. Using the animal stones (or another resource such as leaves and sticks), you can explore number bonds.

For example, start with five stones in the top circle. Split them up into different quantities, and put the stones in the two lower circles. For example, it could be 1+4, or 2+3.

Higher-ability children could try to record what they have found.

2 Sets For Addition

This math game is another good one when children are beginning to add. You can use painted stones, or just found objects from the area.

Place some objects in either circle and then simply count how many you have got altogether. This is an excellent way of adding two sets and finding the total.

Trellis Models And Images

One of the best things I ever tried was sawing up a big trellis that I found at a garden center. I am pretty sure I saw this on Pinterest but cannot remember where. I cut it up into many different models, such as:

• A ten frame
• A five frame
• A two-part addition frame
• A four-part frame

There are so many games that you can play using these. Children really love finding objects to experiment with in the frames.

Ten frames are a key resource to introduce to young children both inside and out . To find out more about what ten frames are and the many ways to use them, take a look at this .

Board Games Outside

Board games are fantastic for a range of mathematical skills. They encourage:

• Counting for a purpose
• Recognizing numbers
• Problem-solving
• Find more/less

Some examples of fantastic outdoor board games include:

Noughts And Crosses

This seems to be known more by the modern generation as tic tac toe.

It is a brilliant game for problem-solving and simple counting, as well as turn-taking. You can easily create simple boards on wood or logs, and use painted stones or objects such as pine cones and conkers.

I have recently seen a board made out of rope.

It is good for simple counting, doing mental math and subitizing, as well as thinking ahead.

Connect Four

Another classic game from the old days.

If you play this game on the floor it is actually easier to get four in a row than if you play the standard plastic version.

Create a large grid board, and then use different objects for each player. It could be stones vs. conkers. Put one on at a time, and try to get four in a row. You can actually have more than 2 players if you want as well. Great for counting, subitizing numbers, tactics and problem solving.

Racetrack 1-6 Game

Have a dice with numerals on it and a board that is a simple grid, with numerals 1 to 6, and some other boxes. Pick which number you think will win, and put your colored stone at the top of the grid for that number. Then roll the dice multiple times. Every time you roll it, put a stone on the right numeral on the grid. The first number is the number that fills up its grid.

This is a fantastic activity for number recognition. If you want to find out my favorite 16 games to teach recognizing numerals, then take a look at this article.

Racetrack 1-12 Game

This is quite a bit harder, and has a lot of probability and calculating involved. But this game makes math problems and math equations more enjoyable to solve.

Have a racetrack that is a 12 x 4 grid. Have the numbers 1-12 written on the bottom of the grid.

You will need two dice, ideally a dots dice and a numeral dice.

To start with every player predicts which number they think will win. They use something such as a colored stone to signify this by putting the stone at the top of the grid in the number column.

Then you take turns to roll a dice. For example, you might get 3+4. Put a stone on the first box in the 7 column. Keep going, adding stones to the number columns. The ‘winner’ is the first number to come out 3 times and make it to the top of the grid.

This game is excellent for multiple things:

• Calculating
• Counting on
• Probability – Numbers 6 and 7 are the two most likely to win. They have the most combinations of numbers possible. Number 1 can never win as you can never throw a 1 with two dice. Children may start to get a bit of an idea of this when they play.
• Thinking about ‘more’. I.e. how many more a number needs to win.

Number Line Race

This is a great way of exploring a number line.

Have a long number line either chalked on the floor (or painted on a piece of wood). Each child has one ‘counter’ such as a colored stone.

The idea is that everyone starts at the beginning of the line. The first child rolls a 1-6 dice. They move their stone along the line to that point (e.g. 5). Then the other children go. For the next go, you simply keep moving along the number line. The winner is the person that gets to the end first.

This is good for number recognition, and also problem solving (such as how far your opponent is ahead of you)

Blank Number Line Race

I also have a number line that has blank spaces on it. Children can add their own numbers to it.

They could be 1, 2, 3, etc., or they could mix it up. They could go backwards. Or they could start from a number that’s not 1. You could also do a game counting in tens.

Active Outdoor Math Games

Here are some physical outdoor games that can get children really engaged.

Stepping Stones

This is an excellent game of cooperation and teamwork.

I use rubber spots for this game with numbers on them. You can find similar spots to these at Amazon here.

You need one more spot than you have children, so if you have a team of six children you will need 7 spots (with numbers 1 to 7 on).

The children are in a line. The child at the front has all the spots (they have a hard job). The person at the back of the line actually has the hardest job, as they have to bend down and pick the spots up.

The idea is that the floor is now lava. You cannot stand on it. However, you can stand on the spots that are the stepping stones over the lava.

The child at the front finds number 1 and puts it on the floor. They stand on it. Then they put down number 2 in front of them, and step on that. Then number 3 etc. As there is a free stepping stone behind them, the others start to step onto the stones.

When number 7 is on the floor, the children should be standing on stones 2 to 7. Then it is the turn of the person at the back to pick up stone number 1 and pass it down the line.

They help each other to pass it down and the process starts again. The person at the back keeps on bending down and picking up the spare stepping stone.

I like to get them to say the number as they stand on it. This game is good for number recognition and ordering.

Using water spray-bottles

This is dead easy. Have water spray bottles and a dice. (A spray bottle lo oks a bit like this if you weren’t sure!)

Roll the dice and spray the bottle that number of times. It could be on the fence or on wallpaper. If you are brave you add a bit of paint to the water in the bottles.

Sticks Activities

There are so many learning opportunities can you perform with sticks including:

• Making 2D shapes.
• Ordering through size
• Making pictures – such as houses, vehicles, rockets, etc.
• Counting them
• Making ten-frames, or number lines

Balloon Games

Balloons are great for physical development, cooperation, and turn taking. Some great outdoor maths games with balloons include:

• Have a small group of children in a circle (approximately 4 to 6 children). Write numbers on a balloon in pen. Maybe put about 6 numbers on the balloon. One child hits the balloon up into the air, then another hits it up and continues. When you hit the balloon, say the number that you hit.
• Have lots of balloons, each with one number on them. Start with one balloon, keep it up, and when you hit it say the number. Then add another balloon to the circle, then another. Keep saying the number as you hit each balloon.
• Have one balloon again (this time with no numbers written on it). The first person says ‘One’ as they hit the balloon. Then the next person says, ‘two’. Continue like this.
• The harder version is counting back from 10 to 0. The first person hits the balloon and says ‘ten’. Then the next person says ‘nine’ as they hit it, and so on.

Maths Ball Games

All sorts of counting and number activities can really be brought to life with ball games . Here are some great examples:

Have Bibs With Numbers

This is one of the most fun outdoor math activities on this list!

Have shirts or bibs with numbers on that the children wear. The child-size version of these number sports vests would work well, as an example.

If you don’t have anything like this, you can write numbers on stickers and they stick them on their tops.

Stand in a circle.

The first child says a number that someone else is wearing and throws the ball to them. That person says ‘thank you’. Then that person chooses someone else, says their number and throws it to them.

This game can be done as a rolling game if they are struggling with catching.

Great for numeral recognition in a fun context. 

Write Numbers On The Balls

Either put stickers with numbers on onto balls, or write directly onto them if you don’t mind having permanent number balls. We use large rubber balls that look a bit like this (check them out on Amazon).

Again this could be a rolling game or a throwing game, depending on the skills of the children.

Start with one ball, but you want to quickly extend it to at least two and hopefully more.

One child says someone’s name and throws a ball to them. That child catches it and says the number.

Then they say someone else’s name and throw it to them. Repeat.

If you have two or three balls going at once there is lots of communication and teamwork required.

Pass The Number Balls Around The Circle

Using the same balls that you created for the last game, play a simple numeral recognition game.

The children sit in a circle and pass the balls around. When they are holding a ball they say the number that is written on it, before passing the ball to the next person.

This game is good because even if you don’t recognize all the numbers, you can listen to the person next to you and copy what they have said. Hopefully, in this way, you are absorbing what the numbers are in some way.

For those who know the numbers already, it is great for speed and quick recognition practice.

Symmetry – Building Half A Giant Outside

This is a great provocation for using outdoors with natural loose parts – things like sticks, leaves, conkers, reels, and building blocks. Just whatever you can find in the outdoor space. To find out the many resources you can use for loose parts play, take a look at this list of at least 100 ideas .

There is a simple way to do it, and a trickier way.

The simple way is for the adult to draw the silhouette of a huge giant on the floor with chalk. Draw a central line down the center of the body, from the top of the head down to the feet.

Now children ‘build’ the giant. Whatever they place on one side – for example, 4 conkers for hair – they try to copy on the other side.

There are harder ways of trying this game out. One harder variation is just draw a central line on the floor with chalk and nothing else. Then, once again, they try to build the giant on one side, and copy it on the other. This will create a much more random creation, and really get them thinking

Washing Lines

There are so many things you can do with washing lines , and it really is so simple to set up. Pretty much all you need is a piece of string.

You can put the string:

• Between two fences
• Between two walls
• I know several teachers who have put wooden broomsticks into pots and then filled the pot with concrete so that the brooms stand up. These are great for creating a washing line between two brooms

If you put a washing line up outside, the big thing to be careful of is that children do not run into it by mistake. The big threat is things like neck injuries.

Some ways to combat this include:

• Having the string next to a wall or fence
• Have it above their head height, maybe with a platform for them to stand on to peg things on to it
• Have it in an area where running is not possible

Anyway, here are some fantastic outdoor washing line activities:

• Hanging up the giant’s clothes. The trick is to bring some large adult clothes in, and you will be amazed at how the children are convinced they are giant clothes. Hang them next to baby clothes, and talk about the differences in size
• Hanging up pairs of socks
• Finding objects outside, such as sticks and leaves and pegging them up. You can peg found objects up in patterns, or you can add or subtract from them. For example, you could have three red leaves and add two brown leaves. How many have you got altogether?
• Ordering numbers. Ideally use something natural, like pegs with numbers on them or wood slices with numbers on them.

Parachute Games

Parachutes are another fantastic resource that can be used either adult-led outside or can be used independently if you model the games and how to use them to the children.

The parachute that I use looks like this:

These are really resilient and can create years of fun and learning. (Check out the latest price of this parachute on Amazon).

Here are some excellent maths parachute games:

• Number dive. Have some numbers underneath the parachute. Have several children holding the parachute. Pick a child to go first, and say ‘1,2,3 – lift!’ Everyone lifts the parachute high into the air. Then shout a number. That child dives under the parachute, grabs the number, and brings it out to safety. Repeat for other children. If the children are doing it by themselves, then the child that has just gone picks the person to go next, and also what number they will get.
• Shape Dive! This is the same as number dive but with shapes.
• Singing counting songs . This works really well if you have some toys to go with it. For example, five monkey toys so you can sing 5 Cheeky Monkeys Jumping On The Bed. Great for singing the song, and taking off one monkey at a time. (Check out these gorgeous monkey finger puppets you could use for this activity here.)
• Bean bag bounce. This is an excellent number bonds game. Have ten bean-bags on the parachute. You have a few seconds where everyone tries to bounce the bean-bags off the parachute. Count back by going 5-4-3-2-1-0. When you get to zero, see now many bean bags are left on the parachute. You will always have a number bond. If there are two on the chute, then there will be three on the ground. If there are four on the chute, there will be six on the ground.
• Hit The Number! Have some rubber numbers (or something similar) and balance them on top of the parachute as everyone holds it together. Have a rubber ball for this game as well. The adult says a number, and everyone tries to roll the ball onto that number. This is a game of cooperation as well as number recognition. You can potentially make it harder by saying things like ‘one more than three’ (and trying to roll the ball onto number four).

Outdoor Games Top-Tips

• Make everything big!
• Make it active
• Use lots of models and images for games
• Use found objects like stones, sticks and leaves
• Use balls, and floor games, and things like balloons

Outdoor math games have loads of fantastic benefits.

It makes maths fun!

It consolidates prior learning in an active and concrete way.

It helps children to link maths learning to the real world.

It is also extremely cheap and quite simple to set up. Pretty much all of these outdoor math activities use resources that are easy to source, and cost next to nothing or are free.

So why not give outdoor math games a go today?

If you are reading this and are based in the UK, then you may well be interested in my Practical Early Years Maths Training Courses that I run around the UK. Check them out to find a venue near you. All sessions are hands-on, practical, and based on numerous ideas to help you find success in teaching maths.

Outdoor Loose Parts Ideas – The Essential Guide

21 Circle Time Games – That Actually Work!

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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

Choosing the Right Task

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
• Graham Fletcher3-Act 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.

Instead, you and your students could say the following:

• “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

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Maths Problem Solving: Engaging Your Students And Strengthening Their Mathematical Skills

Meriel Willatt

Maths problem solving can be challenging for pupils. There’s no ‘one size fits all’ approach or strategy and questions often combine different topic areas. Pupils often don’t know where to start. It’s no surprise that problem solving is a common topic teachers struggle to teach effectively to their pupils.

In this blog, we consider the importance of problem solving and share with you some ideas and resources for you to tackle problem solving in your maths classroom, from KS2 up to GCSE.

What is maths problem solving?

Why is maths problem solving so difficult, how to develop problem solving skills in maths, maths problem solving ks2, maths problem solving ks3, maths problem solving gcse.

Maths problem solving is when a mathematical task challenges pupils to apply their knowledge, logic and reasoning in unfamiliar contexts. Problem solving questions often combine several elements of maths.

We know from talking to the hundreds of school leaders and maths teachers that we work with as one to one online maths tutoring providers that this is one of their biggest challenges: equipping pupils with the skills and confidence necessary to approach problem solving questions.

The Ultimate Guide to Problem Solving Techniques

Download these 9 ready-to-go problem solving techniques that every pupil should know

The challenge with problem solving in maths is that there is no generic problem solving skill that can be taught in an isolated maths lesson. It’s a skill that teachers must explicitly teach to pupils, embed into their learning and revisit often.

When pupils are first introduced to a topic, they cannot start problem solving straight away using it. Problem solving relies on deep knowledge of concepts. Pupils need to become familiar with it and practice using it in different contexts before they can make connections, reason and problem solve with it. In fact, some researchers suggest that it could take up to two years to do this (Burkhardt, 2017). 

At Third Space Learning, we specialise in online one to one maths tutoring for schools, from KS1 all the way up to GCSE. Our lessons are designed by maths teachers and pedagogy experts to break down complex problems into their constituent parts. Our specialist tutors then carefully scaffold learning to build students’ confidence in key skills before combining them to tackle problem solving questions.

In order to develop problem solving skills in maths, pupils need lots of different contexts and word problems in which to practise them and the opportunity to engage in mathematical talk that draws on their metacognitive skills. 

The EEF suggests that to develop problem solving skills in maths, teachers need to teach pupils:

• To use different approaches to problem solving
• Use worked examples
• To use metacognition to plan, monitor and reflect on their approaches to problem solving

Below, we take a closer look at problem solving at each stage, from primary school all the way to GCSEs. We’ve also included links to maths resources and CPD to support you and your team’s classroom teaching.

At lower KS2, the National Curriculum states that pupils should develop their ability to solve a range of problems. However, these will involve simple calculations as pupils develop their numeracy skills. As pupils progress to upper KS2, the demand for problem solving skills increases. 

“At this stage, pupils should develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of problems.” National curriculum in England: mathematics programmes of study (Upper key stage 2 – years 5 and 6)

KS2 problem solving can often fall into the trap of relying on acronyms, such as RICE, RIDE or even QUACK. The most popular is RUCSAC (Read, Underline, Calculate, Solve, Answer, Check). While these do aim to simplify the process for young minds, it encourages a superficial, formulaic approach to problem solving, rather than deep mathematical thinking. Also, consider how much is wrapped up within the word ‘solve’ – is this helpful?

We teach thousands of pupils KS2 maths problem solving skills every week through our one to one online tutoring programme for maths. In our interventions, we encourage deep mathematical thinking by using a simplified version of George Polya’s four stages of problem solving. Here are the four stages:

Understand the problem

• Devise a strategy for solving it
• Carry out the problem solving strategy
• Check the result

We use UCR as a simplified model: Understand, Communicate & Reflect. You may choose to adapt this depending on the age and ability of your class.

For example:

Maisy, Heidi and Freddie are children in the same family. The product of their ages is a score. How old might they be?

There are three people.

There are three numbers that multiply together to make twenty (a score is equal to 20). There will be lots of answers, but no ‘right’ answer.

Communicate

To solve the word problem we need to find the numbers that will go into 20 without a remainder (the factors).

The factors of 20 are 1, 2, 4, 5, 10 and 20.

Combinations of numbers that could work are: 1, 1, 20 1, 2, 10 1, 4, 5 2, 2, 5.

The question says children, which means ‘under 18 years’, so that would mean we could remove 1, 1, 20 from our list of possibilities. 

In our sessions, we create a nurturing learning environment where pupils feel safe to make mistakes. This is so important in the context of problem solving as the best problem solvers will be resilient and able to overcome challenges in the ‘Reflect’ stage. Read more: What is a growth mindset

Looking for more support teaching KS2 problem solving? We’ve developed a powerpoint on problem solving, reasoning and planning for depth that is designed to be used as CPD by primary school teachers, maths leads and SLT. 

The resource reflects on how metacognition can enhance reasoning and problem solving abilities, the ‘curse’ of real life maths (think ‘Carl buys 60 watermelons…) and how teachers can practically implement and teach strategies in the classroom.

You may also be interested in: 

• Developing Thinking Skills At KS2
• KS2 Maths Investigations
• Word problems for Year 6

At KS3, the importance of seeing mathematical concepts as interconnected with other skills, including problem solving, is foregrounded. The National Curriculum also stresses the importance of a strong foundation in maths before moving on to complex problem solving.

“Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programme of study for key stage 3 is organised into apparently distinct domains, but pupils should build on key stage 2 and connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems” National curriculum in England: mathematics programmes of study (Key stage 3)

“Decisions about progression should be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content in preparation for key stage 4.” National curriculum in England: mathematics programmes of study (Key stage 3)

For many students, the transition from primary to secondary school can be a huge challenge.

Especially in the aftermath of the Covid-19 pandemic and the resultant school closures, students may arrive into Year 7 with various learning gaps and misconceptions that will hold them. Some students may need focused support to plug these gaps and grow in confidence.

You can give pupils a smoother transition from KS2 to KS3 with personalised one to one online tuition with specialist tutors with Third Space Learning. Our lessons cover content from Years 5-7 and build a solid foundation for pupils to develop their problem solving skills. Pupils are supported towards independent practice through worked examples, questioning and support slides.

The challenge for KS3 maths problem solving activities is that learners may struggle to get invested unless you start with a convincing hook. Engage your young mathematicians on topics you know well or you know they’ll be invested in and try your hand at designing your own mathematical problems. Alternatively, get some inspiration from our crossover ability and fun maths problems .

Since the new GCSE specification began in 2015, there has been an increased focus on non-routine problem solving questions. These questions demand students to make sense of lots of new information at once before they even move on to selecting the strategies they’ll use to find the correct answer. This is where many learners get stuck.

In recent years, teachers and researchers in pedagogy (including Ofsted) have recognised that open ended problem solving tasks do not in fact lead to improved student understanding. While they may be enjoyable and engage learners, they may not lead to improved results.

SSDD problems (Same Surface Different Depth) can offer a solution that develops students’ critical thinking skills, while ensuring they engage fully with the information they’re provided. The idea behind them is to provide a set of questions that look the same and use the same mathematical hook but each question requires a different mathematical process to be solved.

Read more about SSDD problems , tips on writing your own questions and download free printable examples. There are also plenty of more examples on the NRICH website.

Worked examples, careful questioning and constructing visual representations can help students to convert the information embedded in a maths challenge into mathematical notations. Read our blog on problem solving maths questions for Foundation, Crossover & Higher examples, worked solutions and strategies.

Remember that students can only move on to mathematics problem solving once they have secure knowledge in a topic. If you know there are areas your students need extra support, check our Secondary Maths Resources library for revision guides, teaching resources and worksheets for KS3 and GCSE topics.

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10 Math Tasks for the Beginning of the Year

I have a rule about the first day of school: always do some math. No, that doesn't mean you have to start Lesson 1.1 as soon as students walk through the door, but it does mean that you should give your students a preview of the kind of thinking, reasoning, puzzling, and sense-making that they'll be doing in your class this year. Ideally, students will be so highly engaged that they barely even recognize they're doing math -- and certainly not the kind of math they're used to in school.

I also have an inordinate appreciation for what I call "interesting problems". These are tasks that use mathematical thinking and strategy, but don't require specific content knowledge like the formula for the equation of a circle or knowing what a composite function does. They are highly accessible, highly engaging, and have multiple solution strategies. The task itself can be explained in a few sentences and students can work on them for 20 minutes or 2 hours, depending on how far they want to take it. I scour the internet for tasks like these and have been collecting them for YEARS on my computer. I decided this was the year to bring them to the light and share them with the Math Medic community.

These tasks don't require formal content knowledge, but they do help students engage in the mathematical practices and develop mathematical habits of mind, such as:

Looking for and making use of structure

Representing one's thinking

Working systematically

Visualizing

Developing a convincing argument

Conjecturing and generalizing

While I've curated this list with high school students in mind, many of these tasks could be done with middle schoolers or even with adults. The inspiration for these questions came from all over this great big internet, but have been adapted and reformatted for classroom use. So, without further ado, here are my (current) top 10 "interesting problems" to do on the first day of school.

10 Interesting Problems

The Shopping Cart Task

A linear context in a LOT of disguise. Many solution strategies and great opportunities for representing one's thinking with a model or visual.

The Locker Problem

This one is set up with multiple parts providing lots of natural extensions. Thinking about a number's properties is key to this task! Make sure to print the 100s chart that is on page two on a separate sheet of paper. You can offer it to everyone or as an optional support.

A Leg to Stand On

Loads of solution strategies on this one as well. Your teacher brain might scream system of equations with 4 variables, but you'll be surprised at the intuitive solutions your students find to solve this problem.

To Run or Not to Run?

Perfect after a summer of olympics. Students deal with rates in this problem, which is an important concept for any age group and relevant for any math course.

Four 4's

This one's been famous for a long time but I'm sharing it anyway because students do great with it!

Page Turner

This one and the next two all encourage students to think systematically. There's a brute force solution but making use of structure will illuminate an easier way.

How Many 7's?

This is a good intro to thinking systematically and has a nice extension. I would use this in an Algebra 1 or Geometry course.

Oddball Numbers

This one is very difficult, so we recommend saving it for your Precalc or above courses.

This one is the most recent in my collection and I'm still thinking about the extension part!

Toothpick Challenge

I've often done this one with Geometry students because of the shapes and visual reasoning components.

Editable versions of these tasks can be found in this Google Drive folder .

How to use these in your classroom:

Pick ONE task for students to work on. We don't recommend giving multiple tasks back-to-back because it can start to feel like a worksheet, rather than a puzzle.

Solve the problem yourself first! We are purposefully not giving solutions here , so make sure you've wrestled with the problem yourself before handing it out to students.

Have students work on these in groups of 2, 3, or 4. Make sure they have enough materials available to hash out their ideas and represent their strategies. These are great to do on vertical non-permanent surfaces or poster paper.

Decide how long you will let students work. If doing this on the first day of school, we recommend about 20 minutes. If students don't have a solution by then, that is totally fine. A surprising number of them will keep thinking about the problem throughout the day or even at home.

Be ready with some extensions for groups who finish early, but make sure they understand what "done" means. Have they clearly communicated their strategy? Have they convinced themselves and others that their strategy will hold up? If giving an extension, make sure it's related to the given task, not just a different task. It's important that students are challenged in the depth of their reasoning, not in the quantity of problems.

If you're looking for more tasks like these, I highly recommend the NRICH site from the University of Cambridge.

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Maths problem-solving – Activities for Early Years settings

• Written By: Judith Dancer
• Subject: Maths

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Critical thinking doesn’t have to be a daunting prospect. There are simple, effective and exciting ways to encourage children’s maths problem-solving skills, says Judith Dancer…

Maths is a subject many adults lack confidence in. Having struggled with it at school they often avoid it, wherever possible, when grown up.

But if maths seems scary for some people, then maths problem-solving can cause even more anxiety. There is no ‘safety net’ of knowing the ‘correct answer’ beforehand. This is because maths problem-solving lends itself to investigation and exploration with lots of possible tangents.

Understandably this is often the area of maths where many practitioners feel least confident. However, young children, who are not restrained by right answers, feel the most enthused and animated.

The non-statutory  Development Matters Guidance , as part of ‘creating and thinking critically’ in the Characteristics of Effective Learning, identifies that practitioners need to observe how a child is learning, noting how a child is:

• thinking of ideas;
• finding ways to solve problems;
• finding new ways to do things;
• making links and noticing patterns in their experience;
• making predictions;
• testing their ideas;
• developing ideas of grouping, sequences, cause and effect;
• planning, making decisions about how to approach a task, solve a problem and reach a goal;
• checking how well their activities are going;
• changing strategy as needed;
• reviewing how well the approach worked.

All of these elements are, at one time or another, part of the problem-identifying and solving process – although not at the same time and in the same problem.

Role of the adult

Maths problem-solving for young children involves them understanding and using two kinds of maths:

• Maths knowledge – learning and applying an aspect of maths such as counting, calculating or measuring.
• Maths thinking skills – reasoning, predicting, talking the problem through, making connections, generalising, identifying patterns and finding solutions.

The best maths problems for children are the ones that they identify themselves. They will be enthused, fascinated and more engaged in these ‘real’, meaningful problems.

Children need opportunities to problem-solve together. As they play, they will often find their own mathematical problems.

One of the key roles of practitioners is to provide time, space and support for children. We need to develop situations and provide opportunities in which children can refine their maths problem-solving skills and apply their mathematical knowledge.

Supporting maths problem-solving

You can effectively support children’s developing maths problem-solving strategies through:

• Modelling maths talk and discussion – language is part of maths learning because talking problems through is vital. Children need to hear specific mathematical vocabulary in context. You can promote discussion through the use of comments, enabling statements and open-ended questions.
• Providing hands-on maths problem-solving activities across all areas of the setting. Children learn maths through all their experiences and need frequent opportunities to take part in creative and engaging experiences. Maths doesn’t just happen in the maths learning zone!
• Identifying potential maths learning indoors and outdoors. Provide rich and diverse open-ended resources that children can use in a number of different ways to support their own learning. It is important to include natural and everyday objects and items that have captured children’s imaginations, including popular culture.

Maths problem-solving possibilities

Spell it out.

This experience gives children lots of opportunities to explore calculating, mark making, categorising and decisions about how to approach a task.

What you need to provide:

• Assorted containers filled with natural materials. This includes leaves, pebbles, gravel, conkers, twigs, shells, fir cones, mud and sand. Include some ‘treasure’ – sequins, gold nuggets, jewels and glitter.
• Bottles and jugs of water, large mixing bowls, cups, a ‘cauldron’, small bottles, spoons and ladles.
• Cloaks and wizard hats.
• Laminated ‘spells’ – e.g. “To make a disappearing spell, mix 2 smooth pebbles, 2 gold nuggets, 4 fir cones, a pinch of sparkle dust, 3 cups of water”.
• Writing frameworks for children’s own spell recipes and a shiny ‘Spell Book’ to stick these in.
• Temporary mark-making opportunities such as chalk on slate.

The important thing with open-ended maths problem-solving experiences like this is to observe, wait and listen. Then, if appropriate, join in as a co-player with children, following their play themes.

So if children are mixing potions, note how children sort or categorise the objects. What strategies do they use to solve problems? What happens if they want eight pebbles and they run out? Observe what they do next.

When supporting children’s maths problem-solving, you need to develop a wide range of strategies and ‘dip into’ these appropriately. Rather than asking questions, it is often more effective to make comments about what you can see. For example, say, “Wow, it looks as though there is too much potion for that bottle”.

Acting as a co-player offers lots of opportunities to model mathematical behaviours. This might include reading recipes for potions and spells out loud, focusing on the numbers – one feather, three shells…

Going, going, gone

We all know that children will engage more fully when involved in experiences that fascinate them. If a particular group has a real passion for cars and trucks , consider introducing maths problem-solving opportunities that extend this interest.

This activity offers opportunities for classifying, sorting, counting, adding and subtracting, among many other things.

• Some unfamiliar trucks and cars and some old favourites. Ensure these include metal, plastic and wooden vehicles that can be sorted in different ways.
• Masking tape and scissors.
• Sticky labels and markers.

Mark out some parking lots on a smooth floor, or huge piece of paper using masking tape. Lining paper is great for this. Line the vehicles up around the edge of the floor area.

Encourage one child to select two vehicles that have something the same about them. Ask the child, “What is the same about them?”.

When the children have agreed on what is the same – e.g. size, materials, colour, lorries or racing cars – the child selects a ‘parking lot’ to put the vehicles in. So this first parking lot could be for ‘red vehicles’.

Another child chooses two more vehicles that have something the same. Do they belong in the same ‘parking lot’, or a different parking lot? E.g. these vehicles could both be racing cars.

What happens when a specific vehicle could belong in both lots? E.g. it could belong in the set of red vehicles and also belongs in the set of racing cars.

Support the children as they discuss the vehicle. Make new ‘parking lots’ with masking tape and create labels for the groups, if you choose.

Observe children’s strategies

It’s really important to observe the strategies the children use. Where appropriate, ask the children to explain what they are doing and why.

If necessary, introduce and model the use of the vocabulary ‘the same as’ and ‘different from’. Follow children’s discussions and interests. If they start talking about registration plates, consider making car number plates for all the wheeled toys outdoors.

Do the children know the format of registration plates? Can you take photos of cars you can see in the local environment?

Camping out

Constructing camps and dens outdoors is a good way to give children the opportunity to be involved in lots of maths problem-solving experiences and construction skills learning. This experience offers opportunities for using the language of position, shape and space, and finding solutions to practical problems.

• Materials to construct a tent or den such as sheets, curtains, poles, clips and string.
• Rucksacks, water bottles, compasses and maps.
• Oven shelf and bricks to build a campfire or barbecue.
• Buckets and bowls and water for washing up.

Encourage the children to explore the resources and decide which materials they need to build the camp. Suggest they source extra resources as they are needed.

Talk with the children about the best place to make a den or erect a tent and barbecue. During the discussion, model the use of positional words and phrases.

Follow children’s play themes. This could include going on a scavenger hunt collecting stones, twigs and leaves and going back to the campsite to sort them out.

Encourage children to try different solutions to the practical problems they identify. Use a running commentary on what is happening without providing the solution to the problem.

Look for opportunities to develop children’s mathematical reasoning skills by making comments such as, “I wonder why Rafit chose that box to go on the top of his den.”

If the children are familiar with traditional tales, you could extend this activity by laying a crumb trail round the outdoor area for children to follow. Make sure that there is something exciting at the end of the trail. It could be a large dinosaur sitting in a puddle, or a bear in a ‘cave’.

Children rarely have opportunities to investigate objects that are really heavy. Sometimes they have two objects and are asked the question, “Which one is heavy?” when both objects are actually light.

This experience gives children the chance to explore really heavy things and measures (weight). They also need to cooperate and find new ways to do things.

• A ‘building site’ in the outdoor area. Include hard hats, builders’ buckets, small buckets, shovels, spades, water, sand, pebbles, gravel, guttering, building blocks, huge cardboard boxes and fabric (this could be on a tarpaulin).
• Some distance away, builders’ buckets filled with damp sand and large gravel.
• Bucket balances and bathroom scales.

With an open-ended activity such as this, it is even more important to observe, wait and listen as the children explore the building site and the buckets full of sand and gravel.

Listen to the discussions the children have about moving the sand and the gravel to the building site. What language do they use?

Note the strategies they use when they can’t lift the large buckets. Who empties some of the sand into smaller buckets? Who works together collaboratively to move the full bucket? Does anyone introduce another strategy, for example, finding a wheelbarrow or pull-along truck?

Where and when appropriate, join in the children’s play as a co-player. You could act in role as a customer or new builder. Ask, “How can I get all this sand into my car?”. “How much sand and gravel do we need to make the cement for the foundations?”.

Extend children’s learning by modelling the language of weight:

• heavy/heavier than/heaviest
• light/lighter than/lightest
• about the same weight as/as heavy as
• balance/weigh

Judith Dancer is an author, consultant and trainer specialising in communication and language and mathematics. She is co-author, with Carole Skinner, of  Foundations of Mathematics – An active approach to number, shape and measures in the Early Years .

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• Consider changing the context, the names of characters, or the numbers in the problem to make it more suitable for your class. Click to download a collection of templates to support you to create your own versions of problems.  
• Captivate your students (hook them into the problem) with a creative introduction.  
• Having shared the problem, have your students retell in their own words what the problem is asking them to find out. To experience success a student must first fully understand the problem.  
• Make suitable materials available. Many students solve a problem and demonstrate their solution using materials.  
• Be clear yourself about exactly what you want your students to learn from their problem solving experience.  Make this explicit to your students. (Finding a solution is only part of the learning).  

• Conclude a problem solving session with your students by identifying important learning and together generalize this to other situations.  
• The problems are set within full lessons which outline a suggested teaching sequence. These are guidelines only. 

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