Problem Solving: Logically Reason

Here are some guidelines that can assist with logical reasoning:

Read and understand the problem and consider any assumptions being made;
Work out what is mathematically true and what could not possibly be true;
Think about things that might be true and use mathematical reasoning to find out whether they are true or not;
Break a larger problem into smaller steps to see if they can lead to a solution;
Keep a clear record of your work, so that you can use it to justify your solution.

Logical reasoning is particularly helpful where we know about one particular case and need to think about another case. Although to some extent we use logical reasoning with almost all problems.

Worked Examples

First we will read all three examples and have a quick think about them and then we will look at how logical reasoning can help us with each one:

Biology Example

The following diagram shows a magnified cell. Four students have calculated the actual length of the longest side of the cell, but some of them have got the wrong answer. Which of the answers is correct? (a) 5.6µm, (b) 56µm, (c)560µm, (d) 5600µm

Cube Example

The following six pictures all show the same cube. Which symbol is opposite the red cross?

Circle in Square Example

Below is a diagram of a red square with a purple circle drawn through its vertices and a blue square that touches the purple circle on all sides.

What is the relationship between the area of the red square and the blue square and how do you know?

Worked Solutions to Examples

Biology Example

The following diagram shows a magnified cell. Four students have calculated the actual length of the longest side of the cell, but some of them have got the wrong answer.

Solution

Effectively we can use rounding, by noticing that the length of the cell is around three times the marked length, so around 60µm. This leads us to conclude that only one of the answers given is realistically possible.

Cube Example

The following six pictures all show the same cube. Which symbol is opposite the red cross?

Solution

By looking at the different cubes and taking a little information from each of them we can draw out the net of the cube

Circle in Square Example

Below is a diagram of a red square with a purple circle drawn through its vertices and a blue square that touches the purple circle on all sides.

What is the relationship between the area of the red square and the blue square and how do you know?

Solution

So, what do we know at first?

Clearly the blue square is bigger than the red square, but not massively bigger, e.g. we could say it is less than four times bigger. This helps us to establish initial boundaries for our answer.

We don’t know any measurements, so perhaps we will have to name one of our sides of a square x, or would it be better to name the radius of the circle x?

Are there extra lines we could add to the diagram that would help us?

Here is one way that we could solve this problem.

If we ignore the circle and add the following lines onto the diagram, it becomes easier to see the relationship between the two square: