Purpose
This problem solving activity has an algebra focus.
Achievement Objectives
NA3-8: Connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns.
Student Activity
Six business people meet for lunch and shake hands with each other.
How many handshakes are there?
Specific Learning Outcomes
Describe in words the rules for the pattern.
Identify the pattern of triangular numbers.
Devise and use problem solving strategies to explore situations mathematically (systematic list, draw a picture, use equipment).
Description of Mathematics
The maths that is involved in this problem depends on the approach that is used to solve it. If the students look for patterns starting with the simpler cases (2 people etc) the problem involves triangular numbers.
Required Resource Materials
Activity
The Problem
Six business people meet for lunch and shake hands with each other. How many handshakes are there?
Teaching Sequence
- Introduce the problem by getting 3 students to role-play people meeting and shaking hands.
- Count and record the number of handshakes. Discuss other ways of convincing others that there are 3 handshakes (eg, draw a picture).
- Pose the problem for the students to work on in pairs or small groups.
- Brainstorm ways to solve the larger problem (act it out, make a list and look for a pattern). List these on the board for the students to consider.
- As the students work ask questions that focus their thinking on working systematically and looking for patterns.
How are you keeping track of the handshakes? (diagram, list)
How many handshakes do you think that there would be if you added another person?
What do you notice about the number of handshakes and the number of people?
How could you record your work so that you could look for a pattern? - Share results.
Extension
How many handshakes are there at the meeting if people come in pairs and shake hands with everyone except their own partners.
Solution
If two people shake hands there is one handshake.
If three people shake hands there are 3 handshakes.
If four people shake hands there are 3 more handshakes so 3 + 3 = 6 in total.
If five people shake hands there are another 4 handshakes so 6 + 4 = 10.
For 6 people there are another 5 handshakes so 10 + 5 = 15.
A second pattern that may be described is that each person has to shake hands with all the others. If there are 6 people each person has 5 handshakes to make. But each time a handshake occurs there are 2 people involved. This means that you only need ½ (6 x 5 ) = 15.
Solution to the Extension
6 people = 12 handshakes (15 – 3 = 12, subtract 3 for the shakes that are between partners).
Attachments
ShakingHands.pdf154.06 KB
TeHariru.pdf231.01 KB
Add to plan
Level Three
Certainly! I'm well-versed in mathematics and problem-solving strategies, especially those involving sequential patterns, algebraic reasoning, and exploring mathematical situations. In this context, the problem revolves around determining the number of handshakes when a specific number of people meet and shake hands in a group setting. This problem typically involves exploring triangular numbers and recognizing patterns to derive solutions.
The problem of calculating handshakes among a group of individuals follows a predictable pattern. For instance, when two people meet, there's one handshake. With three people, there are three handshakes. As the number of individuals increases, the number of handshakes follows a pattern related to triangular numbers: 1, 3, 6, 10, 15, etc. This pattern can be deduced by systematically listing the handshakes or by drawing diagrams to visualize the interactions.
Moreover, there's an extension to this problem where individuals come in pairs and shake hands with everyone except their own partners. This scenario requires subtracting the handshakes between partners from the total number of handshakes to find the final count accurately.
To solve this problem efficiently, students can use various problem-solving strategies, such as drawing diagrams, creating systematic lists, or using mathematical equations to establish the relationship between the number of people and the resulting handshakes.
The critical concepts involved here include:
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Sequential Patterns: Understanding how the number of handshakes increases as the number of people in a group increases, following a triangular number pattern.
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Algebraic Reasoning: Recognizing the relationship between the number of people and the total handshakes, possibly expressed in terms of triangular numbers or through an algebraic formula.
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Problem-Solving Strategies: Encouraging students to use systematic approaches, draw diagrams, and look for patterns to explore and solve mathematical problems effectively.
By comprehensively applying these concepts, students can describe the rules governing the pattern of handshakes, identify triangular number patterns, and devise strategies to explore and solve mathematical scenarios related to this problem.
