Shake Hands

Introduction

If all the students in this room shook hands with each other, how many handshakes would there be altogether?

Investigate this problem on your own, if you need some prompts, take a look at these:

  • Let's start small, what if there are only 2 students? 3, 4, or 5?

  • How would you represent this on a diagram? Look at the circle on the right, how is this relevant?

  • You may wish to organise your results in a table. What should you have as headings of the two rows?

  • How is the number of handshakes increasing as the number of students increase (term to term rule)?

  • Can you describe a way of working out the number of handshakes for any number of students (position to term rule)?

  • Verify that your rules work by:

    • First, finding an answer for 6 students with your rules,

    • Then drawing a diagram/ showing your calculation to check that your answer is correct.

  • Here is what Sam thinks for 7 students: He says that since each mathematician shakes hands 6 times, there must be 7Ă—6 handshakes altogether. Helen disagrees; she worked out 7+6+5+...+2+1 and got a different answer.

    • Who is right? For the person who got this wrong, what's wrong with the reasoning?

    • How can the method be modified?

  • Now use your rule to find out the following:

    • One day, 161 students met. How many handshakes took place this time?

    • There are 4851 handshakes at a gathering where everyone shakes hands? How many students would there be?

    • What about the following numbers of handshakes?

      • 6214

      • 3655

      • 7626

      • 8656

Further Questions and Challenges

What if now there are 2 classes? Da Vinci has 30 students and Einstein has 32?

  • How many handshakes are there within each class?

  • How many handshakes are there between each class?

  • How many handshakes are there in total?

  • What if there are "d" students in Da Vinci and "e" students in Einstein? Can you come up with a general rule for:

    • Within;

    • Between and

    • Total?

  • What if we include Fleming with "f" students?

Further Practice

The essential skill introduced in this lesson is "triangular numbers". Other skills you may also wish to recap/ learn about are "square root", "cube numbers" and "cube root". Practise these relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.


Transum

Here is a game to practise recognising the different types of numbers including triangular numbers quickly. Start from level 2 and work up to level 5.

Extension

Transum

Look at this Letters in Numbers starter activity to consolidate understanding of the different types of numbers.

Look at this Satisfaction puzzle to classify the different types of numbers.

For more goodies on different types of numbers, look at these activities on transum.


nRich

Look at this Mystic Rose (a beautiful image created by joining together points that are equally spaced around a circle). Can you describe how to construct a Mystic Rose?


In a chess tournament every contestant is supposed to play exactly one game against every other contestant. However, contestant A withdrew from the tournament after playing only ten games, and contestant B withdrew after just one game. A total of 55 games were played. Did A and B play each other?


Add together all the numbers from 1 to 100, how long do you think this will take you? 5050. Read how Clever Carl (a world famous mathematician) solved this problem in minutes!