Problem Solving

NIE!

QCM 501

Name:

Class:

Date:

How Many Handshakes?

There are 5 people in a group.

They shook each other’s hands once and only once. (a) How many handshakes were there altogether?

Page 1 of 2

Tutorial Group 8, Mrs. Chang Suo Hui Christine Lee Puay Joo, Lee Wei Loong

NIE!

QCM 501

How many handshakes would there be if there were (b) 10 people in the group? (c) 20 people in the group? (d) 40 people in the group?

Page 2 of 2 Tutorial Group 8, Mrs. Chang Suo Hui Christine Lee Puay Joo, Lee Wei Loong

NIE!

QCM 501

Suggested Solution

A

B

C

D

E

(a) A shakes hands with 4 people

- 4 handshakes

B has already shaken hands with A, and now shakes the remaining three

- 3 handshakes

C has shaken hands with A and B, now shakes with D and E

- 2 handshakes

D has shaken hands with A, B and C, now shakes with E.

- 1 handshake

E has shaken hands with all people

- 0 handshakes Total

- 10 handshakes

(b)

Similarly, A shakes hand with 9 people, B shakes hands with 8 people, C shakes hands with 7 people and so on...

Total number of handshakes is therefore: 9 + 8 +7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 Solution to finding out sum of 1 to 9: 9 + 1 = 10 8 + 2 = 10 7 +3 = 10

4 x 10 = 40

6 + 4 = 10 remaining 5

40 + 5 = 45

1 of 3 Tutorial Group 8, Mrs. Chang Suo Hui Christine Lee Puay Joo, Lee Wei Loong

NIE!

QCM 501

(c) Again, A shakes hands with 19 people, B with 18 people, C with 17 people and so on...

Total number of handshakes is therefore: 19 + 18 + 17 + ... + 3 + 2 + 1 = 190 Solution to finding out sum of 1 to 19: 19 + 1 = 20 18 + 2 = 20 17 +3 = 20 “

9 x 20 = 180

“ “ 11 + 9 = 20

180 + 10 = 190

remaining 10

(d) After attempting the handshake questions 3 times, students should be able to visualize the handshaking process mentally otherwise, illustrating the process of 40 people exchanging handshakes is discouraged. Students are prompted to utilize their prior solutions to discover a pattern, i.e. for 5 people exchanging handshakes, total handshakes = 4 + 3 + 2 + 1 for 10 people exchanging handshakes, total handshakes = 9 + 8 + 7 + ... + 2 + 1 for 20 people exchanging handshakes, total handshakes = 19 + 18 + 17 + ... + 2 + 1 and therefore, for 40 people exchanging handshakes, total handshakes = 39 + 38 + 37 + ... + 2 + 1 Again, students might face difficulty summing 1 to 39. Prompt students to search for a pattern and utilize the pattern to aid them in solving the sum, i.e.

39 and 1 make 40, 38 and 2 make 40, 37 and 3 make 40 and so on....

39 + 1 = 40 38 + 2 = 40 “ “

19 x 40 = 760

21 + 19 = 40 remaining 20

760 + 20 = 780

2 of 3 Tutorial Group 8, Mrs. Chang Suo Hui Christine Lee Puay Joo, Lee Wei Loong

NIE!

QCM 501

Alternative Solution (All parts) If there are 40 people, each person has a total of 39 handshakes. If there are 40 people, there would be: 40 people x 39 handshakes = 1560 handshakes But, this involves each pair handshaking twice, therefore, number of handshakes, if each person shakes hands with another once and only once, would be: 1560 ÷ 2 = 780 There would be 780 handshakes if there were 40 people and they shook each other’s hands once and only once.

Similarly, for 20 people, the total number of handshakes would be: 20 x 19 = 380 380 ÷ 2 = 190 For 10 people, the total number of handshakes would be: 10 x 9 = 90 90 ÷ 2 = 45 For 5 people, the total number of handshakes would be: 5 x 4 = 20 20 ÷ 2 = 10

3 of 3 Tutorial Group 8, Mrs. Chang Suo Hui Christine Lee Puay Joo, Lee Wei Loong

Before (Understand the Problem) 15 minutes 1) Teacher introduces the problem and tasks pupils to read the problem carefully. ‣ What does the problem require you to find out? (the total number of handshakes if there are 5 people for part (a), 10 people for part (b) and 40 people for part (c)) ‣ Teacher will allow pupils to pose questions regarding parts which they find ambiguous.

2) Teacher directs students to highlight the important points present in the problem. ‣ What information do you think you must pay close attention to when you are solving the problem? (handshake once and only once) ‣ What do you think this means?

(Teacher will elicit responses from pupils, and list them down on the whiteboard. E.g. shaking with right and left hand would be considered as two handshakes)

‣ ‣ ‣ ‣ ‣ ‣ ‣

Allow pupils to act-it-out: Call for 5 pupil volunteers to come forward, to act the problem out with their understanding of ‘handshake once and only once’.

Teacher will get the class to pin point the errors that the pupil volunteers might have done in their demonstration.

Get the first pupil, Pupil A to demonstrate shaking the other four pupils’ hands. T will ask the class to observe and count along.

How many handshakes did Pupil A make with his other 4 friends?

Teacher follows the count on the whiteboard.

Teacher will ask Pupil B to continue the demonstration.

The class will observe and count along as they did for Pupil A. Probable error: Pupil B will shake the hands of the rest of the pupils in the group, including Pupil A when it comes to Pupil B’s turn to act it out.

3) Teacher helps pupils achieve a better understanding of the given condition “Handshake once and only once”: ‣ Teacher will remind pupils about the condition stated in the question and ask them to observe if Pupil B has obeyed the instructions by shaking Pupil A’s hand again. ‣ Whose hands must Pupil B shake now? ( Pupils C, D and E) ‣ Must Pupil B shake Pupil A’s hand again? Why? (No, Pupil B would then be shaking Pupil A’s hand twice)

‣ ‣ ‣ ‣ ‣

Act-it-out continuation: Teacher will get Pupil B to act his part out correctly the 2nd time.

Teacher will ask the class the number of handshakes that Pupil B has made, and continue the count on the whiteboard.

Teacher will subsequently get Pupils C, D and E to act it out.

Teacher will take the opportunity to affirm the pupils’ understanding of the given condition in the problem.

After pupil volunteers have completed acting-it-out, teacher will direct the class to think of the appropriate solution method and allow student volunteers to model the the problem solution on the board.

Page 1 of 4 Tutorial Group 8, Ms. Chang Suo Hui Christine Lee Puay Joo, Lee Wei Loong

During (Plan and Do) 30 minutes 1)

Individual seat work ‣ Teacher tasks pupils to complete the other parts of the problem individually. ‣ Teacher walks around the class to monitor pupils’ work and provide assistance to pupils when needed. ‣ Encourage pupils to illustrate the problem: Without acting the problem out, what can you do to help you better visualize the problem?

Scaffold students’ responses to get them to identify the chosen heuristic: draw representations

2)

Difficulties pupils may face while solving the problem ‣ Without any form of physical act-it-out, pupils may be unable to illustrate the problem correctly on paper and hence be inclined to think that for instance, the total number of handshakes is 4 x 5 (4 handshakes per person multiplied by 5 people in the group) for part (a) due to pupils repeating counting of handshakes. ‣ For part (c), pupils may be stuck as they find it infeasible or too tedious to illustrate the handshakes made by 20 people. ‣ They are also likely to make careless mistakes in their illustrations. ‣ For parts (c) and (d), pupils may face complications having to add up all the relevant digits to find the total number of handshakes. ‣ E.g. Part (c): pupils may not be able to derive the answer to 19 + 18 + 17 + ... + 3 + 2 + 1 = ___.

3)

Steps to address difficulty: Pupils may be stuck as they find it infeasible or too tedious to illustrate the handshakes made by 20 people. ‣ Teacher will provide hints or cues: Is there any pattern that you can observe from parts (a) and (b)? ‣ Teacher gets pupils to relook at their working solutions for parts (a) and (b), and asks them what is the common pattern that they see in the working. ‣ Pupils should be able to recognize that the digits involved in the repeated addition solution are (number of people – 1 ), the second digit is 1 less than the first, third digit is 1 less than the second, and so on... until they add the final digit 1, in the case of 5 people in one group, it would be 4 + 3 + 2 + 1. ‣ Therefore, pupils will be encouraged to apply this heuristic skill: finding a pattern, to derive the solution equation instead of drawing representations.

4)

Steps to address difficulty: For parts (c) and (d), pupils may face complications having to add up all the relevant digits to find the total number of handshakes.

Teacher scaffolds by providing hints or cues: ‣ Is there another way of adding up this long string of digits together instead of adding them one by one? ‣ Can you find a pattern in these digits that could possibly allow you to obtain the answer easier and faster? ‣ E.g. Part (c): Look at the first digit and the last digit in your solution equation, look at the 2nd digit and the 2nd last digit from the back, the 3rd digit and the 3rd last digit and so on..what do you realize? How many pairs can you form? Is there a number left out? Can you find the sum now?

Page 2 of 4 Tutorial Group 8, Ms. Chang Suo Hui Christine Lee Puay Joo, Lee Wei Loong

During (Plan and Do) 30 minutes 5)

Questions to assist weaker pupils ‣ Can you tell me what you are thinking? ‣ What do you understand from the problem? ‣ If pupils could not understand parts (c) and (d) for example, teacher provides hints and cues by directing them to relate to the earlier scaffolded parts of the problem, i.e. parts (a) and (b): ‣ Don’t you think that parts (c) and (d) are similar to parts (a) and (b) that you have solved? (to find the total number of handshakes) ‣ Can you tell me how do you go about writing the problem solution for (c)? In part (b), there were 10 people involved, now you have 20 people. ‣ Don’t you think it is very difficult to draw out the handshakes for 20 people? ‣ So, is there a pattern that you can find in the problem equation that you have written for parts (a) and (b)? ‣ Teacher guides pupils to extend their solutions to parts (c) and (d).

6)

For pupils who approach the teacher to check if their answers are correct ‣ Teacher does not confirm with them if their answers are correct or wrong. ‣ Teacher encourages pupils to verbally explain why do they think their answers are right. ‣ Have you considered all the handshakes? ‣ Did you double count the handshakes? ‣ How do you know that you have summed up the numbers correctly?

7)

Teacher to get pupils to share their methods and solutions with their shoulder partners ‣ Teacher walks around to listen in on pupil discussion and picks out good and bad methods to be shared with the class.

8)

Extension for pupils who finished earlier ‣ Teacher tasks pupils who have completed the heuristic questions earlier to attempt an extension question to challenge them further (Appendix A).

Page 3 of 4 Tutorial Group 8, Ms. Chang Suo Hui Christine Lee Puay Joo, Lee Wei Loong

After (Looking Back) 15 minutes 1)

Sharing of Solution ‣ Teacher will share the solutions to the other parts of the problem and takes the opportunity to share the alternative solution. ‣ Teacher to clarify doubts on the alternative solution

2)

Reflection ‣ Teacher will call for pupil volunteers to share their difficulties faced when doing the questions, and how they managed to overcome them.

3)

Revision and Summary ‣ Teacher will revise and summarize key learning points ‣ Can you tell me what are the heuristic skills that you have applied to solve this problem? ‣ How can you apply these heuristic skills that you have learnt today to other types of problems?

4)

Extension ‣ Task extension question (Appendix A) as homework ‣ Students encouraged to try alternative solution along with the solutions they worked out over Problem Solving lesson

Page 4 of 4 Tutorial Group 8, Ms. Chang Suo Hui Christine Lee Puay Joo, Lee Wei Loong

NIE!

QCM 501

Reference Delta Sigma Theta Sorority, Inc., and the Delta Research and Educational Foundation. (n.d.) The Handshake Problem. Retrieved October 20, 2011 from: http://www.deltasee.org/communities/edManual/section3d.pdf

Parent Educator Manual v.1

THE HANDSHAKE PROBLEM (Number Theory) Mathematics Standard — Number and Operations Audience: Grades 4— Adult Materials: None — strictly “Minds-On” Directions: Purpose the Question: If 6 people are in attendance at a meeting, and each person shakes hands with every other person only one time, what is the total number of handshakes will occur? …………………………….Allow time for exploration…………………………….

(Encourage participants to solve the problem using any method or strategy they feel comfortable with) Possibilities: Demonstration, drawings, narrative, discussion Ask each group to be prepared to share their respective solutions in writing. This helps students to realize the importance of writing in mathematics as well as recording data in an organized manner.

Move around the room to listen to the group discussions. You may need to guide and probe but do not give answers. If any group has difficulty getting started, ask a question or two to point them in a solid direction but no answers please!

Demonstrating solutions: One suggestions is to issue a large poster sheet (from a flip chart) to each group for recording their result. The richness of this activity lies in the fact that in most cases, each group has a different approach to the problem. Students then begin to realize that there is more than one way to solve a problem and that ‘informal’ methods are a acceptable as formal methods. At the least, they begin to see the background and ‘why’ things work versus the isolated rote memorization of a formula.

Extensions: Change the number of people at the meeting to 8, then 10, then 20, etc., to see if students begin to see a pattern. Students should eventually be able to make a conjecture. I have had students as early as 4 (with some guidance) learn to form a general rule.

www.deltasee.org

Section III – Page 27

Tutorial Group 8, Mrs. Chang Suo Hui Christine Lee Puay Joo, Lee Wei Loong

NIE!

QCM 501

Name:

Class:

Date:

How Many Chess G ames?

There are 100 chess-players in a chess competition.

Each player can compete with another player once and only once.

How many games were played?

Appendix A Tutorial Group 8, Mrs. Chang Suo Hui Christine Lee Puay Joo, Lee Wei Loong