MATHTRAKresources.mathtrak.com.au/manuals/mathtraksetC.pdf · MATHTRAK - online training in Mathematical problem solving with hints, fully worked solutions, and certificates. SET

MATHTRAK - online training in Mathematical problem solving with hints, fully worked solutions, and certificates.
SET C"Solving problems is a practical art, like swimming, or skiing, or playing the piano: youcan learn it only by imitation and practice. . . . if you wish to learn swimming you haveto go in the water, and if you wish to become a problem solver you have to solveproblems." - George Polya

Contract for: Class: Date commenced:

Rules ofconduct:

1. You are responsible for your own good conduct.

2. Use your time effectively.

3. If you are doing these in class, do not disturb your teacher's lesson. Wait until theteacher is available, then raise your hand. If working with others, do so quietly.

Contract: 1. Complete all questions.

2. Initially try to solve the problems by yourself. Only after you have tried severalideas and when you are completely lost for ideas should you talk the problem overwith a friend, or look for hints at MATHTRAK.

3. Full working must be shown for each question. As a guide, your working should besuch that you can give your solution to another student who hasn't been able tosolve the problem, and that student should be able to follow your ideas without anyadditional assistance from you.

4. Do not tell another student the answer, no matter how desperate they claim to be.The answers to these questions are only valuable if they have been arrived at by thesolver and the solution fully explained. Remember "If you give someone a fish,they will eat today only, but if you teach them to fish, they will eat for a lifetime."

5. Your teacher will want to record your progress. Show your work to your teacher ona regular basis.

Progress: 1T 1C 2T 2C 3T 3C 4T 4C 5T 5CInitial anddate whencompletedProgress: 6T 6C 7T 7C 8T 8C 9T 9C 10T 10CInitial anddate whencompleted

Hints, and solutions for each of these problems are available online at MATHTRAK athttp://www.mathtrak.com.au .

MATHTRAKMathematical problem solving at www.mathtrak.com.au

Hints to help when solving Mathematical Problems

Hi! It's Coach Cal here. Over the years when I have been solving mathematical problems, Ihave stopped to think about what it is that I do or think about when I am solving a problem.What follows is a list of the different stages my mind goes through when I am solving problems.Maybe it might help you to get started and have some success. The following is my thinkingprocess when I solve problems.

Getting Started I try to understand the problem as it is presented. I draw a diagram to help me read thequestion closely. I often try to "guess" an acceptable answer.

Engaging I get interested in the problem as one worth solving. It is important to me to get asolution.

Mulling I begin to feel comfortable with the problem, no longer completely lost, and start workingon the problem. Often I simplify the problem - using easier numbers.

Gaining Insight I am convinced that my efforts are worth it and that I can solve it.

Keeping Going I have a pretty strong feeling that I am onto a good idea and that the end appears to bein sight. I test my idea on a simpler but easier problem.

Solving I use my tested idea to get a solution.

Checking I question how valid my answer is.

Looking Back I think about how I solved this problem and how I might use the ideas in a futureproblem.

Here is another way of looking at mathematical problem solving. This is a summary of the ideas inGeorge Polya's book "How to solve it", you may already have seen his ideas - they are bestdescribed as:

SEE , PLAN , DO , CHECK.It might be an idea to read this book if you are keen to improve your problem solving skills. Itshould be in your school library, if not ask your maths teacher. Anyway, here is the summary:

Understand the Problem - (SEE)• Carefully read the problem.• Decide what you are trying to do.• Identify the important data.

Devise a plan - (PLAN)• Gather together all available information.• Consider some possible actions:• look for a pattern;• draw a sketch;• make an organised list;• simplify the problem;• quess and check;

• make a table;• write a number sentence;• act out the problem;• identify a sub-task; and• check the validity of given information.

Carry out the plan - (DO)• Implement a particular plan of attack.• Revise and modify the plan as needed.• Create a new plan if necessary.

Check the answer - (CHECK)• Ensure you have used all the important information.• Decide whether or not the answer makes sense.• Check that all of the given conditions of the problem are met by the answer.• Put your answer in a complete sentence.


SET C

TRAINING PROBLEM 1:Find a two-digit number that is twice the sum of its digits.



SET C

CHALLENGE PROBLEM 1:

Find the largest two-digit number, which is seven timesthe sum of its digits.



SET C

TRAINING PROBLEM 2:

When diplomatic functions are held in Asria it ismandatory that everyone shake hands with everyoneelse. At one function there were 30 diplomats.How many handshakes were there altogether?



SET C


At a Sales Conference every participant makes a pointof shaking hands with everyone else. At the dinnerfunction it was announced that there had been 3160handshakes that day. How many sales peopleparticipated in the conference?



SET C

TRAINING PROBLEM 3:

If 41 =+x

x , then =+ 22 1

xx ?



SET C


If x2 + y2 = 17 and xy = 4, what is the largest value ofx+y?



SET C

TRAINING PROBLEM 4:

PQRS is a square of length 8 cm. Semicircles are drawnon QR and SR as shown. Find the area of the shadedregion (correct to the nearest whole number).


MATTRAK - online training in Mathematical problem solving with hints, fully worked solutions, and certificates.

SET C


A "Gothic" window is constructed on a base BC oflength 3 metres so that two arcs with radius length of 3metres are drawn from B and C and meet at A. What isthe area of the window (correct to one decimal place)?



SET C

TRAINING PROBLEM 5:

Two towns A and B are separated by 540 km ofstraight road. A car starts off at town A and drives totown B at 100 km/h. At precisely the same time aplane leaves town B and flies directly above the roadtowards town A at a speed of 300 km/h. How far away(in kilometres) from town A will they be when theplane is directly above the car?



SET C


John can run 4 times faster than his little brother Pat.On the circular track below, both start running frompoint A at the same time. If John runs anti-clockwiseand Pat runs clockwise, they meet at B. What is the sizeof the smaller angle AOB in degrees?



SET C

TRAINING PROBLEM 6:

Julie and her son Mark celebrate their birthdays on thesame day. Julie is today three times as old as Mark willbe in a year's time. In ten years time she will be twiceas old as Mark is then. How old is Mark today?



SET C


If 2x+3y = 7 and 2y+4z =21, and 3x+z =17 what isx+y+z?



SET C

TRAINING PROBLEM 7:

A train climbed slowly up a hill at an average speed of20 km/h. The following downhill run was three timesas long. What was the average downhill speed, if thetrain averages 30 km/h for the whole uphill anddownhill trip?



SET C


Town A is 500 km uphill from town B. A car travelsfrom A to B at 100 km/h and then back from B to Aat 50 km/h. What is the average speed of the wholetrip? (Answer to the closest km/h).



SET C

TRAINING PROBLEM 8:

The base of a cubic peg sits on the base of a cylindricalhole whose cross-sectional radius is 10 cm. What is thelength of one edge of the largest cube to fit (correct to2 decimal places)?



SET C


A new confectionary product called "Child's Play" isabout to enter the market. The concept is similar toothers on the market where a chocolate has a plasticcontainer inside which holds a toy. The plasticcontainer in "Child's Play" is to be spherical in shapeand has to be big enough to just contain a cube of side5cm. What is the diameter of the smallest plasticcontainer that will hold this cube? (Answer correct toone decimal place.)



SET C

TRAINING PROBLEM 9:

Pam the photographer wants to stand five people in aline so that she can take a group photograph. Howmany different ways can she put these people in a line?


MATTRAK - online training in Mathematical problem solving with hints, fully worked solutions, and certificates.

SET C


There are ten competitors in a swimmingchampionship final. In how many different ways canthe Gold, Silver and Bronze medals be awarded to theswimmers?



SET C

TRAINING PROBLEM 10:

I toss a red and a green die. What is the probability(correct to two decimal places) that both die will landso that they have the same face up?



SET C


I toss the red and green die again. What is theprobability that the face on the red die is higher thanthe face on the green die?


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MATHTRAKresources.mathtrak.com.au/manuals/mathtraksetC.pdf · MATHTRAK - online training in Mathematical problem solving with hints, fully worked solutions, and certificates. SET