PROBLEM SOLVING What is problem solving? 

In its broadest sense it is the process of applying previously acquired knowledge to new and unfamiliar situations.

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Rather than mindlessly applying rules, you need to explore different approaches until you find the one that is appropriate.

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When problems are given in words you need to extract the information you need so that you can work out what is to be done mathematically.

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Start with a positive attitude. We all developed a wide range of problem solving skills as we grew up. To each new situation we all bring strengths, skills and experience, drawing on these resources to solve problems. In maths it is similar; we use maths skills we already have to explore ways of solving new problems.

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There are many ways to reach a solution, none is necessarily best, but some are better than others for particular types of problems.

Fundamentals of problem solving Understanding the problem     

Read the problem carefully to make sure you understand what it is asking you to find Check that you understand all the words and symbols used Write down a summary of the problem using your own words or symbols Pinpoint relevant information and jot it down Write down what you want to find

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Solving the problem This involves exploring different strategies and selecting one - or a combination - which successfully solves the problem.      

Draw a diagram Use hands on materials Guess and check Look for patterns Make a list, table or chart Make it simpler - break the problem down into manageable sub-problems or replace the hard numbers with easier ones.

Checking the answer   

Re-read the problem Make sure that you have answered the question Check that your answer is reasonable.

Exercises 1.

Using the numbers 1 to 6, fill in the circles so that each row of three numbers adds up to 10. Each number can be used only once.

2.

A kennel contains 40 dogs. Among them there are 12 German Shepherds, 5 Cocker Spaniels, 7 Collies and 4 terriers. How many dogs are there in the kennel?

3.

I have in my hand two New Zealand coins, which total $1.20 in value. One of them is not a 20C piece. What are the coins

2 Updated Feb 2013

4.

Chris, Mike and Nathan live in adjacent houses. Mike lives in the middle house. They work as a chemist, a computer programmer and a mechanic, but not necessarily in that order. The computer programmer waters Nathan's garden when he goes away. The chemist goes next door to collect Chris when he wants to go jogging. What occupation does each person have? Chris

Mike

Nathan

Chemist Computer Programmer Mechanic

5.

There is a pattern in the numbers in this triangle. Fill in the next two rows of numbers. 1 1 1 1 1 _ _

2 3

4 _

_

1

3 6

_ _

1 1 4 _

_

1 _

_

_ _

_

6.

On an athletics circular running track there are six flags spaced evenly around it. If it took Fred 30 seconds to get to the third flag, how long would it take him to get all the way around the track?

7.

Three people are in a restaurant deciding who gets which of three desserts. They all want to choose something different. The choices are Death by Chocolate, Strawberry Cheesecake, and Banana Split. Allan is allergic to strawberries. John likes Strawberry Cheesecake and Banana Split. Peter has a hard time deciding but finally chooses Strawberry Cheesecake. Which dessert does each person eat?

3 Updated Feb 2013

8.

Sandy has a rather large collection of books. When a friend asked him what kind of books he liked he said: "I have six times as many novels as biographies. I have 16 more non-fiction books than biographies. All together I have 96 books." How many of each type of book does Sandy have?

9.

Frank has three more T-shirts than sweatshirts. He has a total of 17 T-shirts and sweatshirts. How many T-shirts does he have?

10. In a room there are several owners with their dogs. On counting the heads there were 10. On counting the feet there were 34. How many dogs were there and how many people? 11. Paul is having a party. The first time the doorbell rings, one guest enters. On each successive ring a group enters that has two more people than the group just entered on the previous ring. How many guests arrive together on the 10th ring? How many on the 20th ring? 12. Tom is working out a roster for the staff on a weeklong training programme. Each person will attend two half-day sessions on the programme. There are five staff members, Anne, Mario, Phillip, Kirsten and Chris. Each staff member has other work commitments, which limit his/her availability. Tom has a list of these commitments:       

Anne cannot attend on Monday,Thursday or Friday Mario wants to attend all day Thursday Chris can attend on a Monday morning Anne prefers two half days to one whole day Kirsten wants to attend two afternoons in a row Chris and Phillip can both attend on Friday Phillip can only attend in the mornings

Can you complete the roster to everyone's satisfaction? Mon

Tue

Wed

a.m. p.m.

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Thur

Fri

13. Bill stood on the middle rung of a ladder while painting the outside of his house. When he had finished the section he was working on he went up three rungs. When he ran out of paint he had to go down five rungs to be able to reach the paint tin on the ground. Then he climbed up seven rungs to the top and worked until he had finished the job. How many rungs did the ladder have?

14. The difference between two numbers is 75. If the larger number is divided by the smaller number, the answer is 6 and the remainder 5. What are the two numbers? 15. How many triangles are there in this design?

16. Tony is investing $1000 in a very special bank account in which the money doubles every year.(We should be so lucky!) For instance, at the end of the first year he has $2000, at the end of the second year he has $4000. How much does he have at the end of five years? How much at the end of ten years?

17. A rock group has four members: Bob, Diana, Gill and Paul. They play bass, drums, guitar and piano. No one plays an instrument, which begins with the same letter as his or her first name. Bob shares a flat with the piano player. Paul and the drummer are best friends. Gill plays bass, which she borrowed, from Bob. Which instrument does each person play. 18. Two planes leave Sydney at 9am. One flies west to Perth, the other flies east to Auckland. The westbound plane travels at 425 km per hour and the other at 475 km per hour. How long will it take for the two planes to be 3000 km apart?

5 Updated Feb 2013

19. Matt arranged to have a design printed on sweatshirts to raise money for his rugby club. It cost $75 plus $3.50 for each sweatshirt printed. He received an account for $425 from the printing firm, but the invoice did not state how many sweatshirts were printed. He needed to do some calculations in order to check the account. How many sweatshirts were there? 20. Using the numbers 1 to 7, fill in the circles so that each row of three numbers adds up to 10. Each number can be used only once.

21. Shane decided to lay paving bricks along the side of his garden. He started off slowly, but gradually became more efficient, noticing that each hour he laid eight more bricks than in the previous hour. Over a period of six hours of hard work he laid 240 bricks. How many bricks did he lay in the first hour? 22. A group of 42 people book into a resort hotel for the weekend. When they arrive they are pleased to find that there are four times as many single rooms as double rooms. How many couples and how many single people make up the group? 23. When landscaping his garden, Danny built a set of steps using railway sleepers. the first five steps are shown below:

Without actually drawing the steps and counting each sleeper, can you work out how many sleepers he needed to make a set of ten steps? 6 Updated Feb 2013

24. If everyone in a class of thirty shook hands with everyone else, how many handshakes would there be? 25. Sally and Richard were mixing a jug of orange cordial and decided to try an experiment. Each of them had a one litre jug: one contained 600mL water and the other 600mL orange cordial. Sally poured 200mL of water from his jug into Richard's jug of orange cordial. Richard did the same: he poured 200 mL of the mixture from his jug into Sally's jug. They then asked themselves if there was more water in the orange cordial or more orange cordial in the water? 26. Two groups of flatmates were discussing the amount of shampoo their respective flats used. The boys flat said; "we use up to three bottles of shampoo in two weeks". While the girls flat said; "we use up to four bottles of shampoo in three weeks". Who uses the most shampoo in a week, one of the boys or one of the girls? How much more? 27. At Christmas if each family member gives a gift to every other family member, how many presents would be given in a family of ten?

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Solutions 1.

1 6

4

3 2.

40 dogs

3.

$1 and 20 cents

4.

Chris – Computer Programmer Mike - Chemist Nathan - Mechanic

5.

1 1

5 6

5

2

10 15

10 20

5

1

15

6

1

6.

90 seconds

7.

John - Banana Split Allan - Death by Chocolate Peter - Strawberry Cheesecake

8.

60 novels, 10 biographies, 26 non-fiction books

9.

10 T-shirts and 7 sweatshirts

10. 7 dogs and 3 people 11. 19 guests, 39 guests 12. Mon

Tue

Wed

Thur

Fri

a.m.

Chris

Anne

Phillip

Mario

Phillip

p.m.

Kirsten

Kirsten

Anne

Mario

Chris

13. 11 rungs 14. 14, 89 15. 27 triangles 8 Updated Feb 2013

16. $32 000, $1 024 000 17. Bob plays drum Diana plays piano

Gill plays bass Paul plays guitar

18. 3 hours 20 minutes 19. 100 sweatshirts 20.

2 4

6

1 1

3 7 21. 20 bricks

6

5 5

7

22. 7 couples and 28 single people 23. 55 sleepers 24. No. of students

No. of handshakes

Total

2

1

1

3

1+2

3

4

1+2+3

6

5

1+2+3+4

10

6

1+2+3+4+5

15

n

1 + 2 +….+ (n-1)

n = 30

n(n  1) 2 435

25. There is exactly the same amount of water in the orange cordial as there is orange cordial in the water. 26. A boy uses 1/6 more than a girl 27. 90 presents

9 Updated Feb 2013