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Assessing and Enhancing Mathematical Thinking in Primary Education
The problem requires solving this scenario: Seven people gather for lunch while the first
person shakes hands with all seven others. The second person greeted all participants except the
first who already received a handshake. The third person exchanged handshakes with everyone
except the first two individuals while the remaining participants followed suit until everyone
shook hands with everyone else. The solution process requires multiple steps to handle this
problem through various problem representations.
Solution 1: Numerical Calculation
One of the ways to approach the problem is to divide it into several sub-problems. The
first person shakes hand with all the other five people while the second person shakes hand with
four other people since he has already shaken hand with the first person and so on. This goes on
until the seventh person does not have anyone to greet, since he has already greeted everyone
else. To find the total number of handshakes, we can just sum up the handshakes of each person:
6 + 5 + 4 + 3 + 2 + 1 = 21
This means that there was a total of twenty-one handshakes. Every figure in the sum
indicates the number of new acquaintances that the given individual makes.
Solution 2: Visual Representation
To enhance the understanding of the solution, it is possible to provide a diagram that
would describe the situation under consideration. For the purpose of this task, let every
individual be represented by a point and the interaction between two individuals by a line
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connecting these points (Bartolini & Martignone, 2020). The first person extends his hand to the
other six people, which is depicted by 6 lines. The second person has already greeted the first by
shaking hands, so he only shakes hand with the other five, depicted by 5 lines. This continues
until all handshakes are counted in the process of determining the total number of handshakes
needed for the group.
This can be represented in a graph in which each point (person) is connected by lines
(handshakes). The number of lines denote the number of new handshakes each person forms.
This way of presenting the information also strengthens the notion that the first person shakes
hands with 6 other people, the second – with 5, and so on.
Solution 3: Abstract Representation Using a Table
Another way to formulate the problem abstractly is to construct a table that describes the number
of new handshakes that each person makes:
Person Handshakes with Already Shaken Hands New Handshakes
1 6 None 6
2 5 1 (Person 1) 5
3 4 1 (Person 1) & 2 4
4 3 1 (Person 1), 2, 3 3
5 2 1 (Person 1), 2, 3, 4 2
6 1 1 (Person 1), 2, 3, 4, 5 1
7 0 All others 0
Assessing and Enhancing Mathematical Thinking in Primary Education
The problem requires solving this scenario: Seven people gather for lunch while the first
person shakes hands with all seven others. The second person greeted all participants except the
first who already received a handshake. The third person exchanged handshakes with everyone
except the first two individuals while the remaining participants followed suit until everyone
shook hands with everyone else. The solution process requires multiple steps to handle this
problem through various problem representations.
Solution 1: Numerical Calculation
One of the ways to approach the problem is to divide it into several sub-problems. The
first person shakes hand with all the other five people while the second person shakes hand with
four other people since he has already shaken hand with the first person and so on. This goes on
until the seventh person does not have anyone to greet, since he has already greeted everyone
else. To find the total number of handshakes, we can just sum up the handshakes of each person:
6 + 5 + 4 + 3 + 2 + 1 = 21
This means that there was a total of twenty-one handshakes. Every figure in the sum
indicates the number of new acquaintances that the given individual makes.
Solution 2: Visual Representation
To enhance the understanding of the solution, it is possible to provide a diagram that
would describe the situation under consideration. For the purpose of this task, let every
individual be represented by a point and the interaction between two individuals by a line
, 2
connecting these points (Bartolini & Martignone, 2020). The first person extends his hand to the
other six people, which is depicted by 6 lines. The second person has already greeted the first by
shaking hands, so he only shakes hand with the other five, depicted by 5 lines. This continues
until all handshakes are counted in the process of determining the total number of handshakes
needed for the group.
This can be represented in a graph in which each point (person) is connected by lines
(handshakes). The number of lines denote the number of new handshakes each person forms.
This way of presenting the information also strengthens the notion that the first person shakes
hands with 6 other people, the second – with 5, and so on.
Solution 3: Abstract Representation Using a Table
Another way to formulate the problem abstractly is to construct a table that describes the number
of new handshakes that each person makes:
Person Handshakes with Already Shaken Hands New Handshakes
1 6 None 6
2 5 1 (Person 1) 5
3 4 1 (Person 1) & 2 4
4 3 1 (Person 1), 2, 3 3
5 2 1 (Person 1), 2, 3, 4 2
6 1 1 (Person 1), 2, 3, 4, 5 1
7 0 All others 0
