Relationships between variables

In this chapter you will learn about quantities that change, such as the height of a tree. As the tree grows, the height changes. A quantity that changes is called a variable quantity or just a variable. It is often the case that when one quantity changes, another quantity also changes. For example, as you make more and more calls on a phone, the total cost increases. In this case, we say there is a relationship between the amount of money you have to pay and the number of calls you make.

You will learn how to describe a relationship between two quantities in different ways.

Constant and variable quantities

Look for connections between quantities

1. How many fingers does a person who is 14 years old have?

2. How many fingers does a person who is 41 years old have?

3. Does the number of fingers on a person's hand depend on their age? Explain.

There are two quantities in the above situation: age and the number of fingers on a person's hand. The number of fingers remains the same, irrespective of a person's age. So we say the number of fingers is a constant quantity. However, age changes, or varies, so we say age is a variable quantity.

1. Now consider each situation below. For each situation, state whether one quantity influences the other. If it does, try to say how the one quantity will influence the other quantity. Also say whether there is a constant in the situation.

1. The number of calls you make and the amount of airtime left on your cellphone

2. The number of houses to be built and the number of bricks required

3. The number of learners at a school and the duration of the mathematics period

If one variable quantity is influenced by another, we say there is a relationship between the two variables. It is sometimes possible to find out what value of the one quantity, in other words what number, is linked to a specific value of the other variable.

2. Consider the following arrangements:

1. How many yellow squares are there if there is only one red square?

2. How many yellow squares are there if there are two red squares?

3. How many yellow squares are there if there are three red squares?

4. Complete the flow diagram below by filling in the missing numbers.

Can you see the connection between the arrangements above and the flow diagram? We can also describe the relationship between the red and yellow squares in words.

In words:

The number of yellow squares is found by multiplying the number of red squares by 2 and then adding 2 to the answer.

5. How many yellow squares will there be if there are 10 red squares?

6. How many yellow squares will be there if there are 21 red squares?

Different ways to describe relationships

Complete some flow diagrams and tables of values

A relationship between two quantities can be shown with a flow diagram. In a flow diagram we cannot show all the numbers, so we show only some.

1. Calculate the missing input and output numbers for the flow diagram below.

Each input number in a flow diagram has a corresponding output number. The first (top) input number corresponds to the first output number. The second input number corresponds to the second output number, and so on.

We say $$\times 2$$ is the operator.

1. What types of numbers are given as input numbers?

2. In the above flow diagram, the output number 14 corresponds to the input number 7. Complete the following sentences in the same way:

In the relationship shown in the above flow diagram, the output number ______ corresponds to the input number 5.

The input number ______ corresponds to the output number 6.

If more places are added to the flow diagram, the input number ______ will correspond to the output number 40.

2. Complete this flow diagram by writing the appropriate operator, and then write the rule for completing this flow diagram in words.

In words:

3. Complete the flow diagrams below. You have to find out what the operator for (b) is and fill it in yourself.

4. Complete the flow diagram:

A completed flow diagram shows two kinds of information:

• It shows what calculations are done to produce the output numbers.
• It shows which output number is connected to which input number.

The flow diagram that you completed in question 4 shows the following information:

• Each input number is multiplied by 2 and then 3 is added to produce the output numbers.
• It shows which output number is connected to which input number.

The relationship between the input and output numbers can also be shown in a table:

 Input numbers 0 1 5 9 11 Output numbers 3 5 13 21 25
1. Describe in words how the output numbers below can be calculated.

2. Use the table below to show which output numbers are connected to which input numbers in the above flow diagram.

3. Fill in the appropriate operator and complete the flow diagram.

4. The flow diagrams in question 5(a) and 5(c) have different operators, but they produce the same output values for the same input values. Explain.

5. The rule for converting temperature given in degrees Celsius to degrees Fahrenheit is given as: "Multiply the degrees Celsius by 1,8 and then add 32."

1. Check whether the table below was completed correctly. If you find a mistake, correct it.

 Temperature in degrees Celsius 0 5 20 32 100 Temperature in degrees Fahrenheit 32 41 68 212
2. Complete the flow diagram to represent the information in (a).

6. Another rule for converting temperature given in degrees Celsius to degrees Fahrenheit is given as: "Multiply the degrees Celsius by 9, then divide the answer by 5 and then add 32 to the answer."

1. Complete the flow diagram below.

2. Why do you think the flow diagrams in questions 6(b) and 7(a) produce the same output numbers for the same input numbers, even though they have different operators?

3. Will the flow diagram below give the same output values as the flow diagram in question 7(a)? Explain.

7. The rule for calculating the area of a square is: "Multiply the length of a side of the square by itself."

1. Complete the table below.

 Length of side 4 6 10 Area of square 64 144
2. Complete the flow diagram to represent the information in the table.

1. The pattern below shows stacks of building blocks. The number of blocks in each stack is dependent on the number of the stack.

Complete the table below to represent the relationship between the number of blocks and the number of the stack.

 Stack number 1 2 3 4 5 6 7 8 Number of blocks 1 8
2. Describe in words how the output values can be calculated.

Extension: linking dlow diagrams, tables of values, and rules

1. Complete the flow diagrams below.

2. Calculate the differences between the consecutive output numbers and compare them to the differences between the consecutive input numbers. Consider the operator of the flow diagram. What do you notice?

3. Determine the rule to calculate the missing output numbers in this relationship and complete the table:

 Input numbers 1 2 3 4 5 7 10 Output numbers 9 16 23