Problem Solving in Mathematics
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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.
Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.
Use Established Procedures
Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.
Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.
Look for Clue Words
Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.
Common clue words for addition problems:
Common clue words for subtraction problems:
- How much more
Common clue words for multiplication problems:
Common clue words for division problems:
Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.
Read the Problem Carefully
This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:
- Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
- What did you need to do in that instance?
- What facts are you given about this problem?
- What facts do you still need to find out about this problem?
Develop a Plan and Review Your Work
Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:
- Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
- If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.
If it seems like you’ve solved the problem, ask yourself the following:
- Does your solution seem probable?
- Does it answer the initial question?
- Did you answer using the language in the question?
- Did you answer using the same units?
If you feel confident that the answer is “yes” to all questions, consider your problem solved.
Tips and Hints
Some key questions to consider as you approach the problem may be:
- What are the keywords in the problem?
- Do I need a data visual, such as a diagram, list, table, chart, or graph?
- Is there a formula or equation that I'll need? If so, which one?
- Will I need to use a calculator? Is there a pattern I can use or follow?
Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.
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120 Math Word Problems To Challenge Students Grades 1 to 8
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- Teaching Tools
- Subtraction
- Multiplication
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- Ordering and number sense
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You sit at your desk, ready to put a math quiz, test or activity together. The questions flow onto the document until you hit a section for word problems.
A jolt of creativity would help. But it doesn’t come.
Whether you’re a 3rd grade teacher or an 8th grade teacher preparing students for high school, translating math concepts into real world examples can certainly be a challenge.
This resource is your jolt of creativity. It provides examples and templates of math word problems for 1st to 8th grade classes.
There are 120 examples in total.
The list of examples is supplemented by tips to create engaging and challenging math word problems.
120 Math word problems, categorized by skill
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?
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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.
Creative Math Prompts
There is a lot of talk in math education circles about the power of noticing and wondering . And (pardon the pun) it's no wonder! When we ask math students to notice and wonder, we shift the focus from teachers’ explanations to students’ ideas. Observing (noticing) and questioning (wondering) are simple, powerful habits that enliven and enrich every aspect of instruction. For the teacher, they support formative assessment, lesson design, classroom discourse, differentiation, and the potential for greater rigor. For the student, they develop self-reliance, confidence, curiosity, perseverance, problem-solving, conceptual understanding, and reasoning.
Does it seem strange that such a simple idea has such profound potential? Try it! As you and your students spend more time observing, questioning, and creating, you develop new habits of mind that transform the learning environment, leading to more productive beliefs about math and how we learn it.
So where to start? A fun and practical approach is to begin with Creative Math Prompts . These are just images—sometimes very simple ones—that you show to students in order to elicit their observations and questions. Images like these open a space for mathematical creativity. The immediate goal is simple—to get students comfortable expressing their thoughts and realizing that their ideas matter. Ultimately, you would like the habits of noticing, wondering, and creating to extend beyond the prompts and become an everyday part of your teaching and your students' thinking.
Please bear in mind that, in keeping with the theme of this website, the Creative Math Prompts below are designed with advanced learners in mind. However, prompts like these are appropriate for all learners, and I encourage you to try them with others! You will discover students whose mathematical insights and creativity blossom even though their strengths may not show up well in traditional tasks and assessments. If you find that some prompts are too complex or advanced, you can usually modify them to suit other needs. All learners benefit from noticing, wondering, and creating!
Read about tips and suggestions for Using Creative Prompts .
See the CMP Content Guide to help you align Creative Math Prompts with the content that you teach.
Click on an image to get more information about a prompt.
EARLY GRADES What do you notice? What do you wonder? What can you create?
more early grades...
MIDDLE GRADES What do you notice? What do you wonder? What can you create?
more middle grades...
LATER GRADES What do you notice? What do you wonder? What can you create?
more later grades...
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3 Problem-Solving Math Activities
Scottie Altland · September 5, 2018 · 1 Comment
A problem is simply a “problem” because there is no immediate, known solution. Problem solving activities in mathematics extend well beyond traditional word problems .
You can provide your student with activities that promote application of math skills while “busting boredom” at the same time! Puzzles and riddles, patterns, and logic problems can all be valuable exercises for students at all levels of mathematics. By engaging in short, fun activities like these, you can help your student become a more skillful, resilient, and successful problem-solver.
When practicing problem-solving skills, be certain to give your student time to explore a problem on her own to see how they might get started. Then discuss their approach together. It is important to provide support during the problem-solving process by showing that you value their ideas and helping them to see that mistakes can be useful. You can do this by asking open-ended questions to help your student gain a starting point, focus on a particular strategy, or help see a pattern or relationship. Questions such as, “What have you done before like this?”, “What can be made from …?” or “What might happen if you change…?” may serve as prompts when they needs inspiration.
Try the activities below to boost your student’s problem-solving skills.
Download the activities here .
1) Toothpick Puzzles
Toothpick puzzles (also referred to as matchstick puzzles) provide students a visualization challenge by applying their knowledge of basic geometric shapes and orientations. The only supplies you need are a box of toothpicks, a workspace, and a puzzle to solve. The goal is for students to transform given geometric figures into others by adding, moving, or removing toothpicks. These puzzles range in complexity and can be found online or in math puzzle books. As an extension, challenge your student to create their own puzzle for someone else to solve.
Sample toothpick puzzles of varying difficulty:
Download solutions to this activity here.
2) Fencing Numbers
The goal of this activity is to create a border or “fence” around each numeral by connecting dots horizontally and vertically so that each digit is bordered by the correct number of line segments.
Print a sheet of dot paper .
Use pencils and scissors to cut the size grid you want to use.
This game can be modified for abilities by adjusting the size of the grid and amount of numerals written. For example, a beginning student might begin with a grid that is 5 x 5 dots with a total of four numerals, while a more advanced student might increase the grid to 7 x 7 dots with six to eight numerals.
Begin by writing the digits 0, 1, 2, and 3 spread repeatedly in between “squares” on the dot paper. Each digit represents the number of line segments that will surround that square. For instance, a square that contains a 3 would have line segments on three sides, and a square that contains a 2 would have line segments on two sides, and so on. See the example boards and solutions for a 5 x 5 grid below.
Beware; there may be multiple solutions for the same problem! Thus, encourage your student to replicate the same problem grid multiple times and look for different solutions. A more advanced student can be challenged to create their own problem. Can they make a grid with only one solution? Is it possible to make a problem with four or more possible solutions?
3) It’s Knot a Problem!
Exercise lateral thinking skills– solving a problem through an indirect and creative approach that is not immediately obvious. You need two people, two pieces of string (or yarn) about one meter long each (or long enough so the person who will wear it can easily step over it), and some empty space to move around. If possible, use two different colored pieces of string. Each person needs a piece of string with a loop tied in both ends so it can be worn like “handcuffs”. Before tying off the loop on the second wrist, the participants loop the string around each other so they are hooked together. The figure below illustrates how the strings should appear when completed.
The goal is to unhook the strings while following these guidelines:
1) The string must remain tied and may not be removed from either participant’s wrists. 2) The string cannot be broken, cut, or damaged in any way.
Caution! This activity not only tests problem-solving skills, but it also promotes positive communication, teamwork, and persistence.
Problem-solving skills are not always taught directly but often learned indirectly through experience and practice. When incorporating problem solving activities aim to make them open-ended and playful to keep your student engaged. Incorporating fun activities like these from time to time foster creative and flexible thinking and can help your student transfer problem solving skills to other subject areas. By providing guidance and helping your student to see a problem from different perspectives, you will help foster a positive disposition towards problem-solving. As your student continues to learn how to effectively solve problems, they increase their understanding of the world around them and develop the tools they need to make decisions about the way they approach a problem.
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February 25, 2020 at 11:13 am
The ideas are very brilliant it encourages critical thinking and also help student think for a solution. Awesome!😍
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26 Good Examples of Problem Solving (Interview Answers)
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Employers like to hire people who can solve problems and work well under pressure. A job rarely goes 100% according to plan, so hiring managers will be more likely to hire you if you seem like you can handle unexpected challenges while staying calm and logical in your approach.
But how do they measure this?
They’re going to ask you interview questions about these problem solving skills, and they might also look for examples of problem solving on your resume and cover letter. So coming up, I’m going to share a list of examples of problem solving, whether you’re an experienced job seeker or recent graduate.
Then I’ll share sample interview answers to, “Give an example of a time you used logic to solve a problem?”
Examples of Problem Solving Scenarios in the Workplace
- Correcting a mistake at work, whether it was made by you or someone else
- Overcoming a delay at work through problem solving and communication
- Resolving an issue with a difficult or upset customer
- Overcoming issues related to a limited budget, and still delivering good work through the use of creative problem solving
- Overcoming a scheduling/staffing shortage in the department to still deliver excellent work
- Troubleshooting and resolving technical issues
- Handling and resolving a conflict with a coworker
- Solving any problems related to money, customer billing, accounting and bookkeeping, etc.
- Taking initiative when another team member overlooked or missed something important
- Taking initiative to meet with your superior to discuss a problem before it became potentially worse
- Solving a safety issue at work or reporting the issue to those who could solve it
- Using problem solving abilities to reduce/eliminate a company expense
- Finding a way to make the company more profitable through new service or product offerings, new pricing ideas, promotion and sale ideas, etc.
- Changing how a process, team, or task is organized to make it more efficient
- Using creative thinking to come up with a solution that the company hasn’t used before
- Performing research to collect data and information to find a new solution to a problem
- Boosting a company or team’s performance by improving some aspect of communication among employees
- Finding a new piece of data that can guide a company’s decisions or strategy better in a certain area
Problem Solving Examples for Recent Grads/Entry Level Job Seekers
- Coordinating work between team members in a class project
- Reassigning a missing team member’s work to other group members in a class project
- Adjusting your workflow on a project to accommodate a tight deadline
- Speaking to your professor to get help when you were struggling or unsure about a project
- Asking classmates, peers, or professors for help in an area of struggle
- Talking to your academic advisor to brainstorm solutions to a problem you were facing
- Researching solutions to an academic problem online, via Google or other methods
- Using problem solving and creative thinking to obtain an internship or other work opportunity during school after struggling at first
You can share all of the examples above when you’re asked questions about problem solving in your interview. As you can see, even if you have no professional work experience, it’s possible to think back to problems and unexpected challenges that you faced in your studies and discuss how you solved them.
Interview Answers to “Give an Example of an Occasion When You Used Logic to Solve a Problem”
Now, let’s look at some sample interview answers to, “Give me an example of a time you used logic to solve a problem,” since you’re likely to hear this interview question in all sorts of industries.
Example Answer 1:
At my current job, I recently solved a problem where a client was upset about our software pricing. They had misunderstood the sales representative who explained pricing originally, and when their package renewed for its second month, they called to complain about the invoice. I apologized for the confusion and then spoke to our billing team to see what type of solution we could come up with. We decided that the best course of action was to offer a long-term pricing package that would provide a discount. This not only solved the problem but got the customer to agree to a longer-term contract, which means we’ll keep their business for at least one year now, and they’re happy with the pricing. I feel I got the best possible outcome and the way I chose to solve the problem was effective.
Example Answer 2:
In my last job, I had to do quite a bit of problem solving related to our shift scheduling. We had four people quit within a week and the department was severely understaffed. I coordinated a ramp-up of our hiring efforts, I got approval from the department head to offer bonuses for overtime work, and then I found eight employees who were willing to do overtime this month. I think the key problem solving skills here were taking initiative, communicating clearly, and reacting quickly to solve this problem before it became an even bigger issue.
Example Answer 3:
In my current marketing role, my manager asked me to come up with a solution to our declining social media engagement. I assessed our current strategy and recent results, analyzed what some of our top competitors were doing, and then came up with an exact blueprint we could follow this year to emulate our best competitors but also stand out and develop a unique voice as a brand. I feel this is a good example of using logic to solve a problem because it was based on analysis and observation of competitors, rather than guessing or quickly reacting to the situation without reliable data. I always use logic and data to solve problems when possible. The project turned out to be a success and we increased our social media engagement by an average of 82% by the end of the year.
Answering Questions About Problem Solving with the STAR Method
When you answer interview questions about problem solving scenarios, or if you decide to demonstrate your problem solving skills in a cover letter (which is a good idea any time the job description mention problem solving as a necessary skill), I recommend using the STAR method to tell your story.
STAR stands for:
It’s a simple way of walking the listener or reader through the story in a way that will make sense to them. So before jumping in and talking about the problem that needed solving, make sure to describe the general situation. What job/company were you working at? When was this? Then, you can describe the task at hand and the problem that needed solving. After this, describe the course of action you chose and why. Ideally, show that you evaluated all the information you could given the time you had, and made a decision based on logic and fact.
Finally, describe a positive result you got.
Whether you’re answering interview questions about problem solving or writing a cover letter, you should only choose examples where you got a positive result and successfully solved the issue.
What Are Good Outcomes of Problem Solving?
Whenever you answer interview questions about problem solving or share examples of problem solving in a cover letter, you want to be sure you’re sharing a positive outcome.
Below are good outcomes of problem solving:
- Saving the company time or money
- Making the company money
- Pleasing/keeping a customer
- Obtaining new customers
- Solving a safety issue
- Solving a staffing/scheduling issue
- Solving a logistical issue
- Solving a company hiring issue
- Solving a technical/software issue
- Making a process more efficient and faster for the company
- Creating a new business process to make the company more profitable
- Improving the company’s brand/image/reputation
- Getting the company positive reviews from customers/clients
Every employer wants to make more money, save money, and save time. If you can assess your problem solving experience and think about how you’ve helped past employers in those three areas, then that’s a great start. That’s where I recommend you begin looking for stories of times you had to solve problems.
Tips to Improve Your Problem Solving Skills
Throughout your career, you’re going to get hired for better jobs and earn more money if you can show employers that you’re a problem solver. So to improve your problem solving skills, I recommend always analyzing a problem and situation before acting. When discussing problem solving with employers, you never want to sound like you rush or make impulsive decisions. They want to see fact-based or data-based decisions when you solve problems. Next, to get better at solving problems, analyze the outcomes of past solutions you came up with. You can recognize what works and what doesn’t. Think about how you can get better at researching and analyzing a situation, but also how you can get better at communicating, deciding the right people in the organization to talk to and “pull in” to help you if needed, etc. Finally, practice staying calm even in stressful situations. Take a few minutes to walk outside if needed. Step away from your phone and computer to clear your head. A work problem is rarely so urgent that you cannot take five minutes to think (with the possible exception of safety problems), and you’ll get better outcomes if you solve problems by acting logically instead of rushing to react in a panic.
You can use all of the ideas above to describe your problem solving skills when asked interview questions about the topic. If you say that you do the things above, employers will be impressed when they assess your problem solving ability.
If you practice the tips above, you’ll be ready to share detailed, impressive stories and problem solving examples that will make hiring managers want to offer you the job. Every employer appreciates a problem solver, whether solving problems is a requirement listed on the job description or not. And you never know which hiring manager or interviewer will ask you about a time you solved a problem, so you should always be ready to discuss this when applying for a job.
Related interview questions & answers:
- How do you handle stress?
- How do you handle conflict?
- Tell me about a time when you failed
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Creative Problem Solving in School Mathematics
Creative Problem Solving in School Mathematics is an outstanding resource for introducing problem solving to beginning students in grades 4-8. The text uses nearly 400 challenging nonroutine problems to extend elementary and middle school mathematics into such topics as sequences, series, principles of divisibility, geometric configurations, and logic.
The text is written by the creator of the popular Mathematics Olympiads in the Elementary and Middle Schools program, and is intended to be a broad and deep resource for teachers and homeschoolers who wish to challenge their students to think creatively about mathematics.
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Cultivating Creativity
Age 5 to 18.
Published 2011
"The joy of confronting a novel situation and trying to make sense of it - the joy of banging your head against a mathematical wall, and then discovering that there may be ways of either going around or over that wall"
- how we present content
- how we model good practice
- how we encourage our students to be creative
Presenting content
- some have made progress in understanding that squares do not have to be constructed with sides parallel to the edges of the paper they are drawn on;
- some have begun to identify relationships between the amount of tilt and the areas of squares;
- others have been able to generalise and offer a justification of Pythagoras' Theorem for right-angled triangles with two short sides of integer length.
Posing problems
Questioning.
- some students used an algebraic approach (calling the original three numbers a, b and c);
- others used trial and improvement: one of these groups starting by saying that the smallest number has to be 5 or less in order to make a total of 11...;
- another group of students found a solution almost "by accident" and at this point they were given a similar problem to encourage them to explore a generalisable method.
The curriculum
- engage in problem solving and problem posing;
- have access to experimental opportunities (environments) to explore which have the potential to lead to particular mathematical ideas;
- are mathematising (identifying the mathematics in situations);
- make connections with other mathematical experiences;
- engage in and examine other people's mathematics;
- are not constrained by the content of the previous lessons but supported by them;
- value individuality and multiple outcomes;
- value creative representation of findings.
Concluding remarks
- Boaler, J. (1997). Experiencing School Mathematics . Buckingham, Open University Press.
- Gardner, M. (1965). Mathematical Puzzles and Diversions , Penguin.
- Mason, J., L. Burton, et al. (1982). Thinking Mathematically , Prentice Hall.
- Olkin, I. and A. Schoenfeld, H. (1994). A Discussion of Bruce Reznick's Chapter [Some Thoughts on Writing for the Putnam]. Mathematical Thinking and Problem Solving Schoenfeld, A, H. Hillside NJ, Lawrence Erlbaum: 39-51.
- Piggott, J. S. and E. M. Pumfrey (2005). Mathematics Trails - Generalising , CUP.
- Watson, A. and J. Mason (1998). Questions and Prompts for Mathematical Thinking , Association of Teachers of Mathematics.
This article first appeared in Mathematics Teaching, Vol 202, p3-7 in 2007.
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Creative problem solving: basics, techniques, activities
Why is creative problem solving so important.
Problem-solving is a part of almost every person's daily life at home and in the workplace. Creative problem solving helps us understand our environment, identify the things we want or need to change, and find a solution to improve the environment's performance.
Creative problem solving is essential for individuals and organizations because it helps us control what's happening in our environment.
Humans have learned to observe the environment and identify risks that may lead to specific outcomes in the future. Anticipating is helpful not only for fixing broken things but also for influencing the performance of items.
Creative problem solving is not just about fixing broken things; it's about innovating and creating something new. Observing and analyzing the environment, we identify opportunities for new ideas that will improve our environment in the future.
The 7-step creative problem-solving process
The creative problem-solving process usually consists of seven steps.
1. Define the problem.
The very first step in the CPS process is understanding the problem itself. You may think that it's the most natural step, but sometimes what we consider a problem is not a problem. We are very often mistaken about the real issue and misunderstood them. You need to analyze the situation. Otherwise, the wrong question will bring your CPS process in the wrong direction. Take the time to understand the problem and clear up any doubts or confusion.
2. Research the problem.
Once you identify the problem, you need to gather all possible data to find the best workable solution. Use various data sources for research. Start with collecting data from search engines, but don't forget about traditional sources like libraries. You can also ask your friends or colleagues who can share additional thoughts on your issue. Asking questions on forums is a good option, too.
3. Make challenge questions.
After you've researched the problem and collected all the necessary details about it, formulate challenge questions. They should encourage you to generate ideas and be short and focused only on one issue. You may start your challenge questions with "How might I…?" or "In what way could I…?" Then try to answer them.
4. Generate ideas.
Now you are ready to brainstorm ideas. Here it is the stage where the creativity starts. You must note each idea you brainstorm, even if it seems crazy, not inefficient from your first point of view. You can fix your thoughts on a sheet of paper or use any up-to-date tools developed for these needs.
5. Test and review the ideas.
Then you need to evaluate your ideas and choose the one you believe is the perfect solution. Think whether the possible solutions are workable and implementing them will solve the problem. If the result doesn't fix the issue, test the next idea. Repeat your tests until the best solution is found.
6. Create an action plan.
Once you've found the perfect solution, you need to work out the implementation steps. Think about what you need to implement the solution and how it will take.
7. Implement the plan.
Now it's time to implement your solution and resolve the issue.
Top 5 Easy creative thinking techniques to use at work
1. brainstorming.
Brainstorming is one of the most glaring CPS techniques, and it's beneficial. You can practice it in a group or individually.
Define the problem you need to resolve and take notes of every idea you generate. Don't judge your thoughts, even if you think they are strange. After you create a list of ideas, let your colleagues vote for the best idea.
2. Drawing techniques
It's very convenient to visualize concepts and ideas by drawing techniques such as mind mapping or creating concept maps. They are used for organizing thoughts and building connections between ideas. These techniques have a lot in common, but still, they have some differences.
When starting a mind map, you need to put the key concept in the center and add new connections. You can discover as many joints as you can.
Concept maps represent the structure of knowledge stored in our minds about a particular topic. One of the key characteristics of a concept map is its hierarchical structure, which means placing specific concepts under more general ones.
3. SWOT Analysis
The SWOT technique is used during the strategic planning stage before the actual brainstorming of ideas. It helps you identify strengths, weaknesses, opportunities, and threats of your project, idea, or business. Once you analyze these characteristics, you are ready to generate possible solutions to your problem.
4. Random words
This technique is one of the simplest to use for generating ideas. It's often applied by people who need to create a new product, for example. You need to prepare a list of random words, expressions, or stories and put them on the desk or board or write them down on a large sheet of paper.
Once you have a list of random words, you should think of associations with them and analyze how they work with the problem. Since our brain is good at making connections, the associations will stimulate brainstorming of new ideas.
5. Storyboarding
This CPS method is popular because it tells a story visually. This technique is based on a step-creation process. Follow this instruction to see the storyboarding process in progress:
- Set a problem and write down the steps you need to reach your goal.
- Put the actions in the right order.
- Make sub-steps for some steps if necessary. This will help you see the process in detail.
- Evaluate your moves and try to identify problems in it. It's necessary for predicting possible negative scenarios.
7 Ways to improve your creative problem-solving skills
1. play brain games.
It's considered that brain games are an excellent way to stimulate human brain function. They develop a lot of thinking skills that are crucial for creative problem-solving.
You can solve puzzles or play math games, for example. These activities will bring you many benefits, including strong logical, critical, and analytical thinking skills.
If you are keen on playing fun math games and solving complicated logic tasks, try LogicLike online.
We created 3500+ puzzles, mathematical games, and brain exercises. Our website and mobile app, developed for adults and kids, help to make pastime more productive just in one place.
2. Practice asking questions
Reasoning stimulates you to generate new ideas and solutions. To make the CPS process more accessible, ask questions about different things. By developing curiosity, you get more information that broadens your background. The more you know about a specific topic, the more solutions you will be able to generate. Make it your useful habit to ask questions. You can research on your own. Alternatively, you can ask someone who is an expert in the field. Anyway, this will help you improve your CPS skills.
3. Challenge yourself with new opportunities
After you've gained a certain level of creativity, you shouldn't stop developing your skills. Try something new, and don't be afraid of challenging yourself with more complicated methods and techniques. Don't use the same tools and solutions for similar problems. Learn from your experience and make another step to move to the next level.
4. Master your expertise
If you want to keep on generating creative ideas, you need to master your skills in the industry you are working in. The better you understand your industry vertical, the more comfortable you identify problems, find connections between them, and create actionable solutions.
Once you are satisfied with your professional life, you shouldn't stop learning new things and get additional knowledge in your field. It's vital if you want to be creative both in professional and daily life. Broaden your background to brainstorm more innovative solutions.
5. Develop persistence
If you understand why you go through this CPS challenge and why you need to come up with a resolution to your problem, you are more motivated to go through the obstacles you face. By doing this, you develop persistence that enables you to move forward toward a goal.
Practice persistence in daily routine or at work. For example, you can minimize the time you need to implement your action plan. Alternatively, some problems require a long-term period to accomplish a goal. That's why you need to follow the steps or try different solutions until you find what works for solving your problem. Don't forget about the reason why you need to find a solution to motivate yourself to be persistent.
6. Improve emotional intelligence
Empathy is a critical element of emotional intelligence. It means that you can view the issues from the perspective of other people. By practicing compassion, you can understand your colleagues that work on the project together with you. Understanding will help you implement the solutions that are beneficial for you and others.
7. Use a thinking strategy
You are mistaken if you think that creative thinking is an unstructured process. Any thinking process is a multi-step procedure, and creative thinking isn't an exclusion. Always follow a particular strategy framework while finding a solution. It will make your thinking activity more efficient and result-oriented.
Develop your logic and mathematical skills. 3500+ fun math problems and brain games with answers and explanations.
About Creative Problem Solving
Welcome! We're excited that you are interested in teaching Creative Problem Solving at BEAM Discovery.
Why Creative Problem Solving?
The Creative Problem Solving courses are designed to introduce students to contest problem solving. There are two reasons we believe this can be so important for students.
First, contest-style problems challenge students to apply their mathematical knowledge in new ways. They cannot be solved using procedures alone, and so students must genuinely think about their mathematics knowledge to correctly solve them. It is our hope that students will make this transition in their mathematical thinking.
Second, math contests are a way for students to stay involved with mathematics beyond BEAM. After the summer, we can help students' schools register for math competitions for them. Moreover, there are numerous online tools that students can continue to use after the summer, and by starting them off on those tools now, they will be able to further their educations on their own.
Different Classes
Each Creative Problem Solving course is designed to equip students to do well in the MATHCOUNTS contest. The courses take shape by identifying common problem types and learning the mathematics to do well on those problems.
We believe that interesting content makes learning problem solving strategies more exciting. We furthermore believe that depth is more valuable than breadth to learn mathematics and how to do mathematics. Hence, Creative Problem Solving courses teach problem solving through the lens of a particular content area.
We have prepackaged courses available on several content areas, which have been taught at BEAM Discovery before successfully. These courses are:
Counting Without Counting (with a focus on combinatorics and the multiplication principle)
Primes, Powers and Solving Problems (with a focus on divisibility and prime factorization)
Unruly Patterns, Sequences, and The Rules That Govern Them (with a focus on sequences)
Words Meet Numbers: An Algebra Story (with a focus on algebraic work problems)
Of course, if you’d like to design your own curriculum, feel free to propose another area! For example, we’ve thought that geometry (with a focus on deriving angles and side lengths) might make a good course.
If you’re developing your own course, we do have many years of archived math contest problems and copies of Art of Problem Solving textbooks that you can use to plan your course. You will have the flexibility (and responsibility) for developing your day-to-day lesson plans, selecting problems, and creating handouts.
Other Important Notes
Remember that the goal of this course is to give students a start that they can build off of. Sometimes class will move slowly and might not cover as much as you hoped, especially because students might be missing important basic skills. If they feel empowered to continue learning on their own, then the class has been a huge success.
While we believe that competition can be healthy, it is not for everyone, and competitions often turn young people off from mathematics. To prevent this, it is important to emphasize how interesting the problems are and the learning that students are doing independent of winning or losing. The class should always maintain a positive atmosphere for all students and should not spotlight students who are particularly quick or solve more problems; everyone's progress should be celebrated.
Additionally, contests often emphasize speed to a much greater extent than speed is valuable when doing mathematics. While preparing students for the speed required on contests, your course should also emphasize that this is an artificial element of most math contests and that they should strive to do and enjoy the problems rather than to rush through. Please don't let students walk out thinking that speed is a measure of their mathematical skill!
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To read more about our hiring process and submit the initial application, read this website:
Once we have received your application, our hiring committee will review it and then get back to you if we feel you're a good fit.
IF YOU’RE APPLYING TO CREATE YOUR OWN CURRICULUM
If you’re applying to create your own curriculum rather than teach a pre-packaged course, the second step in the process will be to create a course description about how you would teach your course.
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10 Strategies for Problem Solving in Math
Jessica Kaminski
8 minutes read
June 19, 2022
Kids often get stuck when it comes to problem solving. They become confused when you offer them word problems or include an unknown variable like x in their math question. In such cases, teachers have to guide kids through this problem-solving maze, which is why this article covers the strategies for problem solving in math and the ways your students can leverage them.
What Are Problem Solving Strategies in Math?
To solve an issue, one must have a reliable strategy. Strategies for problem solving in math refer to methods of approaching math questions to ensure accurate results and increased efficiency. Such strategies simplify math for kids with no experience in problem solving and those already familiar with it.
There are various ways to implement problem solving strategies in math, and each method is different. While none is foolproof, they can improve your student’s problem-solving skills, especially with exercises and examples. The keyword here is practice — the more problems students solve, the more strategies and methods they pick up.
Strategies for Problem Solving in Math
Even if a student is not a math whiz, appropriate strategies for problem-solving in math can help them find solutions. Students may solve math issues in many ways, but here are ten math strategies for problem solving with high success rates. Depending on usage and preference, the strategies give kids renewed confidence as they work through difficulties.
Understand the Problem
Before solving a math problem, kids need to know and understand their nature. They should identify if the question is a fraction problem , a word problem, a quadratic equation, etc. An excellent way to boost their understanding is to look for keywords in the problem, revisit other similar questions, or check online. This step keeps the student on track.
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Guess and Check
The guess and check approach is one of the time-intensive strategies for problem solving in math. Students are to keep guessing until they find the proper answer.
After assuming a solution, kids need to put it back into the math problem to determine its accuracy. The procedure may seem laborious, but it often uncovers patterns in a child’s thought process.
Work It Out
When kids are working on a math problem, please encourage them to write down every step. This strategy is a self-monitoring method for math students since it demands that they first understand the problem. If they immediately start solving the problem, they risk making mistakes.
Using this strategy, students will keep track of their ideas and correct mistakes before arriving at a final answer. Even after working out their math problems in the supplementary sheet, a child may still ask you to explain the processes. This confirmation stage etches the steps they took to solve the problem in their minds.
Work Backwards
There are times when math problems may be best solved by looking at them differently. Kids need to understand that recreating math problems will be handy for project management and engineering careers.
Using the “Work Backwards” strategy, students anticipate challenges in real-world situations and prepare for them. They can start with the final result and reverse engineer it to arrive at the initial problem.
A math problem that may seem confusing to kids can generally become simpler once you represent it visually. Having kids visualize and act out the math problem are some of the most effective math strategies for problem solving.
Drawing a picture or making tally marks on a sheet of working-out paper is a visualization option. You could also model the process on the whiteboard and give students a marker to doodle before writing down the solution.
Find a Pattern
Pattern recognition strategies help kids understand math fundamentals and remember formulas. The best way to uncover patterns in a math problem is to teach pupils to extract and list relevant details. They can use the strategy when learning shapes and repetitive concepts, which makes the approach one of the most effective elementary math strategies for problem solving.
Using this method, students will recognize similar information and find the missing details. Over time, this approach will help students solve math problems faster.
One of the best problem solving strategies for math word problems is asking oneself, “what are some possible solutions to this issue?” It helps you consider the question more carefully, think outside the box, and avoid tunnel vision when facing challenges. So, encourage kids to muse over math problems and not settle for the first answer that enters their minds.
Draw a Picture or Diagram
Like visualization, creation of a diagram of a math problem will help kids figure out the best ways to approach it. Use shapes or numbers to represent the forms to keep things basic. Depending on the situation, patterns and graphs may also be valuable, and you can encourage kids to use dots or letters to represent the items.
Diagrams are even beneficial in many non-geometrical situations. After studying, students can create sketches of the concepts they read about for later revision. The approach will help kids determine what kind of math problem they are dealing with and the steps needed whenever they encounter a similar idea.
Trial and error method
Trial and error approach may be one of the most common strategies for solving math problems. However, the efficiency of this strategy depends on its application. If students blindly try solving math questions without specific formulas or directions, the chances of success will be low.
On the other hand, if they start by making a list of possible solutions based on preset guidelines and then attempting each one, they increase their odds of finding the correct answer. So, don’t be quick to discourage kids from using the trial and error strategy.
Review answers with peers
Strategies for problem solving in math that involve reviewing solutions with peers are enjoyable. If students come up with different answers to the same question, encourage them to share their thought processes with the rest of the class.
You could also have a session with the class to compare children’s working techniques. This way, students can discover loopholes in their ideas and make the necessary adjustments.
Check out the Printable Math Worksheets for Your Kids!
Many strategies for problem solving in math influence students’ speed and efficiency in tests. That is why they need to learn the most reliable approaches. By following the problem solving strategies for math listed in this article, students will have better experiences dealing with math problems.
Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master's degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.
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Creative problem solving examples
Example: I find that I'm most productive when I have some time to
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10 Examples of Creative Problem Solving
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8 Creative Solutions to Your Most Challenging Problems
Creative problem solving: process, techniques, examples
1. Balance Divergent and Convergent Thinking 2. Reframe Problems as Questions 3. Defer Judgment of Ideas 4. Focus on Yes, And Instead of
Do math equations
Homework is a necessary part of school that helps students review and practice what they have learned in class.
Answers in 5 seconds
You can use math to determine all sorts of things, like how much money you'll need to save for a rainy day.
Do my homework
I love spending time with my friends when I have free time.
Decide math problem
I can help you with your homework if you need it.
Deal with mathematic question
There's more to your application than just filling out the forms.
To determine what the math problem is, you will need to take a close look at the information given and use your problem-solving skills. Once you have determined what the problem is, you can begin to work on finding the solution.
Creative Problem Solving
What Are Good Outcomes of Problem Solving? Saving the company time or money Making the company money Pleasing/keeping a customer Obtaining new customers
Find the right method
There is no one-size-fits-all method for goal setting and achieving success. The best method for you depends on your individual goals and circumstances.
Free time to spend with your friends
In just 5 seconds, you can get the answer to your question.
More than just an application
Knowing is half the battle.
The Basics of Creative Problem Solving
Creative problem solving is not just about fixing broken things it's about innovating and creating something new. Observing and analyzing the environment, we
Determine mathematic problem
Determine mathematic
What is mathematical creativity, how do we develop it, and should we try to measure it? PART 2
By Keith Devlin @profkeithdevlin
“What is creative mathematical thinking?” That’s the question I set out to answer last month. The discussion got this far: Creative mathematical thinking is non-algorithmic mathematical thinking.
The question arose when a long-time friend (and former teacher) from the ed tech world and I had an email exchange, prompted in part by the publication of a LinkedIn survey of industry leaders which ranked creativity as the number one skill they look for in employees.
The online magazine EdSurge picked up on the LinkedIn survey results to conduct its own (informal) survey of various thought leaders in different domains (film, writing, teaching, museums, and technology companies of different sizes), asking, “Is creativity a skill (that can be developed through practice and repetition)?” They published the results in the January 21 issue .
The answers given ranged all over. An associated Twitter poll EdSurge conducted came down slightly in favor of “yes.” None of this is scientific, of course. The relevant takeaway is that professionals in different areas for whom creativity is a relevant notion do not agree as to what it is. (Nor did my ed tech friend and I.)
Moreover, the EdSurge survey was by no means specific to mathematics. Indeed, the only responses that came close having particular relevance to mathematics or mathematics learning were acclaimed teacher and Moonshots author Esther Wojcicki’s view that creativity is not a skill but a mindset, and Google Education Evangelist Jaine Casap’s observation that:
“[Creativity is] embedded in problem-solving, for example. You must use creativity to think of new ways to define and solve problems. Creativity also separates us from machines or robots. For example, an algorithm is a prescribed process, a pattern of commands a machine (or technology) follows. A human can look at issues from a variety of angles—in a nonlinear way! Creativity can be the ‘how’ part of problem-solving.”
None of those asked gave a definitive answer to the question as to whether creativity could be objectively measured. For my ed tech friend and I, however, leaving the question unanswered was not a viable option. We wanted to know if it were possible, in principle, to develop digital tools that developed creative mathematical thinking and measured it. We needed a definition. It did not have to be “the correct definition.” That seems out of reach given where we all are today, if indeed there is a definitive, clean, concise answer. But is there a notion of “mathematical creativity” that (1) makes a reasonable claim on being referred to by that name, (2) can be implemented in a digital math learning tool, (3) is developed by engaging with the tool, and (4) permits automated assessment by the tool? As long as the notion is easy to understand and clearly specified, such tools could be built. Everyone would know exactly what skill or ability (or mindset, etc.) is being developed and measured, and researchers could take on the task of determining how the defined notion and its implementation compare with other learning outcomes and metrics.
As it turns out, there is such a notion, which had been doing the rounds since the early 1990s. Before I say what it is, it’s probably a good idea to watch (or, re-watch) two excellent TED talk videos on creativity by Sir Kenneth Robinson: His talk Do schools kill creativity? given in Monterey, CA, in 2006 [SPOILER: The answer is “yes”] and the sequel Bring on the learning revolution! , given at the same venue in 2015.
Most people I have talked to about creativity have already seen those videos, and agree that Robinson is absolutely right in saying that creative thought comes naturally to humans, with young children exhibiting seemingly endless creativity in all manner of domains. Anyone who has spent any time with young children, as parents, teachers, or whatever, has surely observed that. But as Robinson correctly, and eloquently, observes, systemic education tends to drive the creativity out of them.
In the case of mathematics education, creativity is suppressed by the adoption of an excessive focus on the mastery of basic algorithmic skills. To be sure, mathematics educators could, until recently, defend that emphasis by pointing to the crucial need to master calculation—a need that lasted throughout the three millennia period up until the 1990s, when calculation was a crucial life skill but there were no machines to do it for us.
ASIDE: While that defense has some merit, I find it hard to accept that the need for calculation “drill” meant the almost total suppression of creative mathematics. “Drill of skill” turned into “drill and kill”—the precious commodity killed being any interest in mathematics as a pleasurable mental activity. There was never an either-or choice; time could have been devoted to engagement with creative mathematical thinking.
Be that as it may, with Robinson’s talks fresh in my mind from an N’th re-watch, I went back and looked at the one notion that, by and large, mathematicians had agreed was a reasonable first definition of mathematical creativity. (At least, the relatively few mathematicians who had spent some time trying to come to grips with the elusive concept so agreed.)
That notion has a history going back to the 1940s, which seems to be when mathematicians, mathematics educators, and philosophers first started to reflect on the issue, of particular note among them being Henri Poincaré (1948), Jacques Hadamard (1945), and George Pólya (1962).
Mathematical creativity – a definition
The definition mathematicians and mathematics educators settled on is very much along the lines of the
mathematical creativity is non-algorithmic decision making
we eventually arrived at in Part 1 of this post.
Taking that general idea as a starting point, Gontran Ervynck, an educator in the Faculty of Science at the Katholieke Universiteit Leuven, in Belgium, came up with a definition (Ervynck 1991) of mathematical creativity that I personally find productive (as do many others).
I’ll elaborate a bit about the background to Ervynck’s contribution later, but first let me cut to the chase and present his definition. I should, however, preface it by noting that he was trying to define creativity in advanced mathematical thinking. What I find attractive, however, is that his definition distills mathematical creativity to an essence that works equally well for learners of all ability levels, both for learning and assessment. Moreover, that notion could be implemented in digital learning tools.
Ervynck approached mathematical creativity in terms of three stages of mathematical competence (Ervynck 1991, pp.42-43):
The first stage (Stage 0) is referred to as the preliminary technical stage , which consists of “some kind of technical or practical application of mathematical rules and procedures, without the user having any awareness of the theoretical foundation.”
The second stage (Stage 1) is that of algorithmic activity , which consists primarily of performing mathematical techniques, such as explicitly applying an algorithm repeatedly.
The third stage (Stage 2) is referred to as creative ( conceptual, constructive ) activity . This is the stage in which true mathematical creativity occurs, and consists of non-algorithmic decision making. Ervynck comments that “The decisions that have to be taken may be of a widely divergent nature and always involve a choice.”
Although Ervynck describes the process by which a mathematician arrives at the creative thinking stage after going through two earlier stages, his description of mathematical creativity nevertheless ends up very similar to those of others who have considered the topic of mathematical creativity, such as Poincaré and Hadamard.
I should point out that, in accepting Ervynck’s concept as a working definition of mathematical creativity, mathematicians and mathematics educators are really taking the word “creativity” and giving it a specific meaning within mathematics. (Mathematicians do this with everyday words all the time.) In this case, the result is a notion that (1) makes sense within mathematics, (2) makes sense within mathematics education, (3) can be applied to all mathematics learners, regardless of experience or ability, and (4) can be applied to mathematics learners in a graded fashion, based on the nature of the choices they make. In addition, it accords very well with the kind of creativity Ken Robinson talked about in his talks. That’s why I like it so much.
What the definition does not capture, however—at least not directly—is the notion of mathematical creativity that is tacitly assumed when we talk about highly creative people. That kind of population was the focus of Einav Aizikovitsh-Udi’s 2014 study The Extent of Mathematical Creativity and Aesthetics in Solving Problems among Students Attending the Mathematically Talented Youth Program .
While Ervynck’s three-stages concept still applies to exceptional individuals, the essence of creativity that Aizikovitsh-Udi studied involves making highly unusual choices that lead to unusual results that stand out from most others. The mathematics community as a whole has very little difficulty recognizing that kind of creativity when we see it, just as is the case for exceptional creativity in all other domains. But do we understand it? Do we know how to develop it? Do we know how to measure it?
Regardless of any progress we may one day obtain on those questions, the Aizikovitsh-Udi is interesting as it stands as a study of exceptional mathematical creativity as it exists. Certainly, the goal of the study was not to figure out if that kind of creativity could be effectively assessed algorithmically, by technology or by hand. To do so would presumably require analyzing the sequences of choices that lead to the desired result, but such an approach seems highly unlikely to be successful. Algorithms can identify unusual sequences of steps, but as any research mathematician knows from long and frustrating experience, the vast majority of those unusual sequences don’t work—even if they seem like wise choices at the time.
In contrast, the thought experiment my ed tech friend and I were having was the degree to which technology could develop and measure the (mathematical) creativity in regular children that Ken Robinson was talking about. Such a technology, it one were possible, would clearly be a significant benefit to the mathematics education community. I don’t think that is necessarily out of reach. In fact, starting with the Ervynck notion of mathematical creativity, I see real potential to make useful progress. But time alone will tell.
Finally, I promised I’d say something about the history of studies of mathematical creativity that led to the Ervynck definition.
The earliest attempt I am aware of to study mathematical creativity was a fairly extensive questionnaire published in the French periodical L’Enseigement Mathematique in 1902. This questionnaire, and a lecture on creativity by Henri Poincaré to the Societé de Psychologie, inspired his colleague Jacques Hadamard to investigate the psychology of mathematical creativity (Hadamard, 1945). Hadamard based his study on informal inquiries among prominent mathematicians and scientists in America, including George Birkhoff, George Pólya, and Albert Einstein, about the mental images they used in doing mathematics.
Hadamard’s study was influenced by the Gestalt psychology popular at the time. He hypothesized that mathematicians’ creative process followed the four-stage Gestalt model of preparation–incubation–illumination–verification (Wallas, 1926). That model provides a characterization of the mathematician’s creative process, but it does not define creativity per se .
Many years later, in 1976, a number of scholars interested in the notion of mathematical creativity came together to form the International Group for the Psychology of Mathematics (PME), which began to meet annually at different venues around the world to share research ideas. In 1985, a Working Group of PME was formed to look at creativity in advanced mathematical thinking. The volume Advanced Mathematical Thinking , edited by mathematics educator David Tall at the University of Warwick in the UK (Tall 1991), resulted from the work of that group. In Chapter 3 of that book, Ervynck presents his analysis of mathematical creativity.
The PME volume is a mammoth, comprehensive work, full of powerful insights, that I have done no more than delve into from time to time. From what I’ve read (and from what Tall says in his Preface), at the end of the day, we really don’t know how the logically-sequenced solutions and proofs mathematicians write out relate to the mental processes by which they arrive at those arguments. Tall writes (p.xiv):
“[T]here is a huge gulf between the way in which ideas are built cognitively and the way in which they are arranged and presented in deductive order. This warns us that simply presenting a mathematical theory as a sequence of definitions, theorems and proofs (as happens in a typical university course) may show the logical structure of the mathematics, but it fails to allow for the psychological growth of the developing human mind.”
Salutary advice for teachers and students alike.
Final thoughts
My take-home conclusions from my discussion with my ed tech friend? With today’s technologies having eliminated the need for humans to master computation (of any kind), learning and assessment have to focus on creative mathematics (as defined above).
Teaching computational skills was relatively easy—albeit too often done in a way that turned people off the subject—and assessment could be done with automation. In contrast, developing and assessing creative mathematics are much more problematic.
Technology may help for the early school grades, say through to middle school, but even then it is likely to be a challenging task to develop systems that work really well, and in my view it’s highly likely that if they do work well it will as supplementary tools dispensed as and when appropriate by an experienced teacher.
As to higher grade levels, I’d look to the considered opinions of experienced mathematics educators and developmental clinical psychologists. They, perhaps informed by conclusions generated by machine-learning algorithms, can certainly have (some) value in terms of identifying creative mathematical talent. Such an approach could be useful in deciding who should be given the benefit of a focused mathematical education and when to conduct an educational intervention for a particular student. Decisions about resources allocation have to be made, and it’s always better to make them with as much information as possible. And from society’s perspective, technology can surely help develop creativity and provide useful measurements of an individual’s creative potential. But at the end of the day, each individual decision is at best an educated bet.
In particular, the most dramatic forms of creativity are often missed as such at the time. Georg Cantor’s theory of infinite sets was initially regarded as the wild mental ramblings of a deranged mind; only later was it recognized as a work of creative genius. In earth science, it took fifty years before the scientific community recognized that Alfred Wegener’s theory that the surface of the earth consisted of separate plates, whose drifting led to the formation of today’s continents and were the cause of earthquakes, was a creative explanation having scientific validity–supported by evidence not available in Wegener’s time. And in music, Stravinsky’s Right of Spring met a similar fate. Etc.
Leaving creative genius aside, however, I should conclude by acknowledging that these Final Thoughts about the potential for ed tech in the development and assessment of creative mathematical ability, are at present no more than a considered (and somewhat informed) opinion from an experienced mathematics educator. Pass the salt.
Aizikovitsh-Udi, E. (2014). The Extent of Mathematical Creativity and Aesthetics in Solving Problems among Students Attending the Mathematically Talented Youth Program. In Creative Education 5 , pp.228-241
Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), Advanced mathematical thinking (pp. 42-53). Dordrecht: Kluwer.
Hadamard, J. (1945). Essay on the psychology of invention in the mathematical field . Princeton, NJ: Princeton University Press.
Poincaré, H. (1948). Science and method . New York: Dover.
Pólya, G. (1962) Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving .New York: Wiley
D. Tall (Ed.) (1991). Advanced mathematical thinking . Dordrecht: Kluwer (2002 edition available on Google Books)
Wallas, G. (1926). The art of thought . New York: Harcourt, Brace & Jovanovich.
Table of Contents
26 January 2021
Reading Time: 2 minutes
Do your children have trouble solving their Maths homework?
Or, do they struggle to maintain friendships at school?
If your answer is ‘Yes,’ the issue might be related to your child’s problem-solving abilities. Whether your child often forgets his/her lunch at school or is lagging in his/her class, good problem-solving skills can be a major tool to help them manage their lives better.
Children need to learn to solve problems on their own. Whether it is about dealing with academic difficulties or peer issues when children are equipped with necessary problem-solving skills they gain confidence and learn to make healthy decisions for themselves. So let us look at what is problem-solving, its benefits, and how to encourage your child to inculcate problem-solving abilities
Problem-solving skills can be defined as the ability to identify a problem, determine its cause, and figure out all possible solutions to solve the problem.
- Trigonometric Problems
Problem Solving Skills: Meaning, Examples & Techniques
What is problem-solving, then? Problem-solving is the ability to use appropriate methods to tackle unexpected challenges in an organized manner. The ability to solve problems is considered a soft skill, meaning that it’s more of a personality trait than a skill you’ve learned at school, on-the-job, or through technical training. While your natural ability to tackle problems and solve them is something you were born with or began to hone early on, it doesn’t mean that you can’t work on it. This is a skill that can be cultivated and nurtured so you can become better at dealing with problems over time.
Problem Solving Skills: Meaning, Examples & Techniques are mentioned below in the Downloadable PDF.
Benefits of learning problem-solving skills
Promotes creative thinking and thinking outside the box.
Improves decision-making abilities.
Builds solid communication skills.
Develop the ability to learn from mistakes and avoid the repetition of mistakes.
Problem Solving as an ability is a life skill desired by everyone, as it is essential to manage our day-to-day lives. Whether you are at home, school, or work, life throws us curve balls at every single step of the way. And how do we resolve those? You guessed it right – Problem Solving.
Strengthening and nurturing problem-solving skills helps children cope with challenges and obstacles as they come. They can face and resolve a wide variety of problems efficiently and effectively without having a breakdown. Nurturing good problem-solving skills develop your child’s independence, allowing them to grow into confident, responsible adults.
Children enjoy experimenting with a wide variety of situations as they develop their problem-solving skills through trial and error. A child’s action of sprinkling and pouring sand on their hands while playing in the ground, then finally mixing it all to eliminate the stickiness shows how fast their little minds work.
Sometimes children become frustrated when an idea doesn't work according to their expectations, they may even walk away from their project. They often become focused on one particular solution, which may or may not work.
However, they can be encouraged to try other methods of problem-solving when given support by an adult. The adult may give hints or ask questions in ways that help the kids to formulate their solutions.
Encouraging Problem-Solving Skills in Kids
Practice problem solving through games.
Exposing kids to various riddles, mysteries, and treasure hunts, puzzles, and games not only enhances their critical thinking but is also an excellent bonding experience to create a lifetime of memories.
Create a safe environment for brainstorming
Welcome, all the ideas your child brings up to you. Children learn how to work together to solve a problem collectively when given the freedom and flexibility to come up with their solutions. This bout of encouragement instills in them the confidence to face obstacles bravely.
Invite children to expand their Learning capabilities
Whenever children experiment with an idea or problem, they test out their solutions in different settings. They apply their teachings to new situations and effectively receive and communicate ideas. They learn the ability to think abstractly and can learn to tackle any obstacle whether it is finding solutions to a math problem or navigating social interactions.
Problem-solving is the act of finding answers and solutions to complicated problems.
Developing problem-solving skills from an early age helps kids to navigate their life problems, whether academic or social more effectively and avoid mental and emotional turmoil.
Children learn to develop a future-oriented approach and view problems as challenges that can be easily overcome by exploring solutions.
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Frequently Asked Questions (FAQs)
How do you teach problem-solving skills.
Model a useful problem-solving method. Problem solving can be difficult and sometimes tedious. ... 1. Teach within a specific context. ... 2. Help students understand the problem. ... 3. Take enough time. ... 4. Ask questions and make suggestions. ... 5. Link errors to misconceptions.
What makes a good problem solver?
Excellent problem solvers build networks and know how to collaborate with other people and teams. They are skilled in bringing people together and sharing knowledge and information. A key skill for great problem solvers is that they are trusted by others.
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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 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.
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How Did You Solve It? Metacognition in Mathematics
The role of metacognition in problem-solving, metacognitive awareness and regulation, creating opportunities for metacognition to flourish, recording student thinking, math is meta.
Charlie has a giant bag of gumballs and wants to share them with his friends. He gives half of what he has to his buddy, Jaysen. He gives half of what's left after that to Marinda. Then he gives half of what's left now to Zack. His mom makes him give 5 gumballs to his sister. Now he has 10 gumballs left. How many gumballs did Charlie have to begin with?
Boaler, J. (2016). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. San Francisco, CA: Jossey-Bass.
Boaler, J. (2019). Limitless mind: Learn, lead, and live without barriers. New York, NY: HarperOne.
Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L. B. Resnick (Ed.), The nature of intelligence. Hillsdale, NJ: Erlbaum.
Pólya, G. (1945/2014). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press.
Sam Rhodes is a visiting assistant professor of mathematics education at James Madison University in Harrisonburg, VA. His research focus is on developing students' abilities to problem-solve through student discourse.
ASCD is a community dedicated to educators' professional growth and well-being.
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Although creative problem solving traditionally deals with problems that have multiple solutions, such as those found in management, math usually involves only one solution. But, Geometry and other math units often pose problems where there are multiple ways of coming to the same solution. Here is an example:
Brain teasers, logic puzzles and math riddles give students challenges that encourage problem-solving and logical thinking. They can be used in classroom gamification, and to inspire students to tackle problems they might have previously seen as too difficult. If you want to get your students excited about math class, this post is for you.
Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.
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.
Creative Math Prompts There is a lot of talk in math education circles about the power of noticing and wondering. And (pardon the pun) it's no wonder! When we ask math students to notice and wonder, we shift the focus from teachers' explanations to students' ideas.
For example, a beginning student might begin with a grid that is 5 x 5 dots with a total of four numerals, while a more advanced student might increase the grid to 7 x 7 dots with six to eight numerals. Begin by writing the digits 0, 1, 2, and 3 spread repeatedly in between "squares" on the dot paper.
The following are illustrative examples of creative problem solving. Counterfactual Thinking Solving current problems by considering what would have happened if the past had been slightly different. For example, considering a current career choice by thinking about your choices up to this point and the universe of paths not taken.
Answer: 9. To solve this fun maths question, you need to understand how the area of a parallelogram works. If you already know how the area of a parallelogram and the area of a triangle are related, then adding 79 and 10 and subsequently subtracting 72 and 8 to get 9 should make sense.
Creative Problem Solving Examples Example #1: Adapting Customer Service to Evolving Customer Expectations and Needs The Complex Problem: Customer service always has room for...
I think the key problem solving skills here were taking initiative, communicating clearly, and reacting quickly to solve this problem before it became an even bigger issue. Example Answer 3: In my current marketing role, my manager asked me to come up with a solution to our declining social media engagement.
Overview. Creative Problem Solving in School Mathematics is an outstanding resource for introducing problem solving to beginning students in grades 4-8. The text uses nearly 400 challenging nonroutine problems to extend elementary and middle school mathematics into such topics as sequences, series, principles of divisibility, geometric ...
Creative problem solving examples - An 8 step approach to inside-out problem solving 1. Define the (right) problem 2. Check your mindset 3. Empathize with the. Math Index. ... To solve a math problem, you need to figure out what information you have. 4. Get the Most useful Homework explanation ...
A wealth of NRICH activities based on plastic Geoboards, number cards or plastic cubes, for example, give students room to manipulate objects as they think through ideas. Modelling As teachers, we rarely model problem solving in the mathematics classroom.
Now it's time to implement your solution and resolve the issue. Top 5 Easy creative thinking techniques to use at work 1. Brainstorming Brainstorming is one of the most glaring CPS techniques, and it's beneficial. You can practice it in a group or individually. Define the problem you need to resolve and take notes of every idea you generate.
The Creative Problem Solving courses are designed to introduce students to contest problem solving. There are two reasons we believe this can be so important for students. First, contest-style problems challenge students to apply their mathematical knowledge in new ways. They cannot be solved using procedures alone, and so students must ...
The guess and check approach is one of the time-intensive strategies for problem solving in math. Students are to keep guessing until they find the proper answer. After assuming a solution, kids need to put it back into the math problem to determine its accuracy. The procedure may seem laborious, but it often uncovers patterns in a child's ...
Creative problem solving examples - A classic trick is to open a book or dictionary and pick out a random word. Then generate ideas that somehow incorporate. ... Figure out mathematic problem. I enjoy working on math problems because they provide a challenge and a chance to use my problem-solving skills.
"[Creativity is] embedded in problem-solving, for example. You must use creativity to think of new ways to define and solve problems. Creativity also separates us from machines or robots. For example, an algorithm is a prescribed process, a pattern of commands a machine (or technology) follows.
Problem solving can be difficult and sometimes tedious. ... 1. Teach within a specific context. ... 2. Help students understand the problem. ... 3. Take enough time. ... 4. Ask questions and make suggestions. ... 5. Link errors to misconceptions. What makes a good problem solver?
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.
The Role of Metacognition in Problem-Solving. Despite the popular belief that mathematics is about memorizing and precisely following algorithms and procedures, mathematics is actually a subject of critical thinking, problem-solving, and creativity (Boaler, 2016, 2019). Given that problem-solving is, by definition, the process of working ...
When teaching mathematics, critical thinking skills can be used, practiced and enhanced by effective cognitive methods. Critical thinking can enhance creative problem solving options by encouraging students to seek new strategies when solving mathematical problems. Mathematics teachers know the importance of mathematical
Jan 15, 2016 - Explore MathPlus's board "Creative Problem Solving", followed by 270 people on Pinterest. See more ideas about creative problem solving, problem solving, solving.