The ‘handshakes’ problem

This assignment centres around the following problem.

The ‘handshakes’ problem

Seven mathematicians met up one week.

The first mathematician shook hands with all the others.

The second one shook hands with all the others apart from the first one since they had already shaken hands.

The third one shook hands with all of the others apart from the first and the second mathematicians, and so on, until everyone had shaken hands with everyone else.

How many handshakes were there altogether?

There are 3 (three) tasks that need to be completed. Read through the following steps to understand and meet the requirements of each task:

Task A: Solving the problem

Step 1: Solve The ‘handshakes’ problem yourself.

Step 2: Provide at least two solutions to the problem, and include both concrete and abstract representations.

Step 3: Use photos, drawings or diagrams, and algorithms to show your abstract and concrete solutions.

Step 4: Include an explanation in words about how you solved both representations. 

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Task B: Teaching a student

Step 1: Select one student from Year 3 to 6 and have him/her attempt The ‘handshakes’ problem. 

You should make available a range of materials such as paper, pens, pencils, rulers, or any other appropriate concrete materials you think the student might use. Allow the student to use their own methods and choose the materials you have provided. Only scaffold when necessary and do not teach the student how to complete the activity.

Please note:

• The identity of the child should be protected and participation should be voluntary and with parental/caregiver permission.
• You are allowed to conduct the task if you have a child who falls into the appropriate year bracket. 
• You should not approach a school to complete this assignment, but if you struggle to find a child to teach, reach out to your OLA for assistance.

Step 2: Once you find a child to teach, parents/caregivers must sign a Consent form (DOCX 43 KB) Download Consent form (DOCX 43 KB). You must submit this note with your assignment (even if you are teaching your own child). This should be included in the appendices (refer to Task C). 

Step 3: If your student finds ‘The handshakes problem’ too easy or too difficult, administer the following problems instead.

Easier option: The ‘colourful jerseys’ problem

In a class, there are 20 students. They are wearing jerseys of 3 different colours: blue, green, and red.

The class was lined up in a way that the pattern of jerseys was: blue, blue, green, green, red, blue, blue, green, green, red…

1. What colour jersey is the last child wearing?
2. How many students are wearing a red jersey?

Harder option: The ‘lollies, lollies, lollies’ problem

On Monday, Sam, Sonny and Sylvia share some lollies that they have been given.

Sonny was given twice as many lollies as Sam.

Sylvia was given three times as many lollies as Sam.

They are given the same number of lollies each day up to and including Friday. 

1. If Sylvia got a total of 18 lollies on Tuesday and Wednesday, how many lollies did Sonny get for the whole five days?
2. How many lollies would Sam need to get on Saturday if he wanted to have 39 lollies altogether?

Step 4: Interview the student to gather further information for your report. Code the questions you ask the student and the student responses (SI, 01, etc.) so that this information can be used in your analysis.

Step 5: Submit a transcript of this interview as part of the Appendix for the assignment. During the activity, record any observations that would be relevant to assessing the student’s thinking and achievement. The observations recorded must be submitted with the assignment. Code these observations (OS1 etc.) so that they can be used to justify the achievement or non-achievement of the mathematical outcomes addressed in the analysis.

Step 6: Collect, organise, annotate and submit the student’s works samples that show their mathematical thinking as they investigate the problem. Code these work samples (WS01 etc.) and submit these work samples in the Appendix of the assignment.

Step 7: Using all of the evidence you have collected, write a report (approx. 800-1000 words) analysing the work samples, your observations and interview notes to determine the student’s knowledge and thinking according to the current NSW K-10 Mathematics Syllabus (NSW Government Education Standards Authority, 2024). Include direct links to the evidence by using your coding of the interview transcript, student observations and work samples, as well as the NSW K-10 Mathematics Syllabus, to indicate the multiple levels of achievement of this student.

Step 8: Justify your analysis discussion with references from at least five relevant sources and include current mathematics literature, as well as your eText: Booker, G., Bond, D., & Seah, R. (2021). Teaching Primary Mathematics (6th ed.). Pearson Australia.

Child safety note:

You may take photos of your work and the activity station, but you must not take pictures of the child you are teaching.

Task C: Suggestions for improvement (Approx. 400 words)

Step 1: Describe three activities or pedagogical strategies that you would use to further develop your student’s mathematical thinking. Include a rationale, based on your reading, for the activities or strategies that you describe.

Step 2: Ensure your recommendations address the issues reported in the analysis and clearly link to the identified needs of your particular student.

Step 3: If your suggestions are pedagogical strategies, provide a brief example of an activity that exemplifies how you would implement your recommendation/s.

Step 4: Include the following appendices:

• Consent form (DOCX 43 KB).Download Consent form (DOCX 43 KB).
• Coded interview transcript.
• Annotated and coded work sample.
• Coded observations of student behaviour and student comments/explanations.
• Examples of alternate problems and/or activities to support your discussion of suggested improvements.

Useful wensites:

• NSW Government Education Standards Authority. (2024). Mathematics K-10 Syllabus (2024). https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-k-10-2022/content
• Western Sydney University Library. (2020). American Psychological Association (APA) referencing style guide. https://library.westernsydney.edu.au/__data/assets/pdf_file/0009/1943487/cite_APA.pdf

Supporting resources

• Consent form (DOCX 43 KB) Download Consent form (DOCX 43 KB).
• Assignment 2 Template (PDF 360 KB) Download Assignment 2 Template (PDF 360 KB).
• NSW K-10 Mathematics Syllabus (NSW Government Education Standards Authority, 2024).
• American Psychological Association (APA) referencing style guide (PDF 756 KB) (Western Sydney University Library, 2020).
• Library Study Smart—a library resource that assists you in completing your assignments (Western Sydney University Library, 2021). 
• Western Sydney University Library has further information about referencing on their Referencing and citation page.

 

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