Some Non-linear Functions
2. More Geometric Patterns – Triangular and Square numbers
“Triangular numbers” are numbers that correspond to the number of dots needed to make the triangles in the geometric patterns below:
“Square numbers” are numbers that correspond to the number of dots needed to make the squares in the geometric patterns below:
Activity 2:
More Geometric Patterns – Triangular and Square numbers 1. Draw the next two shapes in each of the above geometric patterns.
2. Complete the table below:
3. Describe carefully how you find the values for the shapes with 7, 8 and 9 dots in the side of the shape.
4. In your journal draw a graph showing the relationship between the number of dots in the side of the shape and the total number of dots in the shape for both the triangular numbers and the square numbers.
5. Discuss how the tables of values and the graphs of these relationships are different to the tables of values and the graphs of the linear relationships.
6. Compare your answers to the questions in Activities 1 and 2 with a partner.
Number of dots in the 1 2 3 4 5 6 7 8 9
side of the shape Triangular numbers
(i.e. total number of dots 1 3 6 in triangles)
Square numbers
(i.e. total number of 1 4 9
dots in squares) Time needed
30 minutes
MODULE FOUR Unit Six: Some Non-linear Functions
What have you learned?
You can see that the number patterns made by both the triangular and square shapes are clearly not linear because they are not straight lines on a graph. Now let’s look at a real life example of this.
Activity 3:
Sharing your winnings
Write this activity on separate paper and put it in your portfolio.
Bantwana Bami ECD site has decided to hold a “guess the score” competition for the final match in the soccer championships. Each person who buys a ticket must predict the final score. The person or people who guess the final score correctly will either win all of the prize money or share it evenly among all those who guessed the correct score.
Because this is a fundraising event the prize money is not very large - in fact it is only R16.00! In this example the prize money is small. In real life, the relationships are the same but the numbers are much larger.
1. How different will the amount be if you are the only person to win or if two people win? Express your answer as a fraction of the total prize money.
2. What fraction of the total prize money will you win if you and two other people (i.e. three in total) all guess the correct score?
3. What fraction of the total prize money will you win if you and three other people (i.e. four in total) all guess the correct score?
4. What happens to the fraction of the prize money as the number of people who win increases?
5. In your journal draw a graph to show how the share of the R1600 prize money changes as the number of people who win increases.
6. Compare your answers to the questions in Activity 4 with a partner.
Reflecting on the relationships you have seen in this unit
In this unit you have worked with three different families of non-linear functions.
The table below shows the general equation for each of these families and specific equations that correspond to the examples in the activities in this unit.
Remember in Module 1 you worked with powers of numbers, such as 102. You know that this means 10 ×10. To do the next activity think about the following:
x2means x ×x … i.e. x2= x ×x x3= x ×x ×x and
x4= x ×x ×x ×x etc.
Time needed 30 minutes
Activity 4:
Reflecting on the different relationships in this unit.
Work with a partner:
1. Copy and complete the table:
a. Work out the output values of each of the specific equations for the given input values.
b. Match each of the examples in the activities in this unit to its equation.
What have you learned?
In this unit you have learned that some functions are non-linear. You have seen that the equations, the tables of values and the graphs of these functions are different to the linear functions as a result you can also see more clearly that linear functions have a constant rate of change. Which we can see in a table of values when we notice the following pattern:
You know that linear functions have a constant rate of change and that is why the points of the graphs of the functions lie on a straight line.
+1 +1 +1 +1 +1 +1 +1
Input number 1 2 3 4 5 6 7 8 9 10 11 12 15
Output number
+m +m +m +m +m +m +m
Function type Specific Input number
Examples in
and equation equation 1 2 3 4 5 6 the activities
Exponential
y= a×bx
Quadratic
y= ax2+bx + c
Inverse proportion
y= k x
y= 10× 2x
y= x2
y= 16 x y= 1
2 (x2 + x) y=20
3 × 3x
Time needed 30 minutes
MODULE FOUR Unit Six: Some Non-linear Functions
Non-linear functions do not have constant rates of change and therefore the points of their graphs do not lie on a straight line but rather on a curve with a different shape.
Journal Reflection
Think about what you have learned. Write down all your thoughts, ideas and questions about your learning in your journal. Use these questions to guide you:
a. What did you learn from this unit about the difference between non-linear and linear functions?
b. Write down one or two questions that you still have about the difference between the graphs and equations of exponential, quadratic and inverse proportion functions
Self-assessment Checklist
Reflect on the outcomes that were set for this unit. Think about what you know, what you can do and how you can use what you have learned. Use the key in the table and tick the column next to each outcome to show how well you think you can do these things now.
I can:
Tick as follows: 4=Very well 3=Well 2=Fairly well 1=Not well. 4 3 2 1 1. Recognise and tell the difference between non-linear and
linear functions
2. Recognise and tell the difference between the graphs and equations of exponential, quadratic and inverse
proportion functions
. :
m
In this unit you will address the following:
Unit Standard 7453 SO1:
Form and use algebraic equations and inequalities to represent and solve problems.
(simple linear equations and inequalities.)
SO2:
Manipulate algebraic expressions to find equivalent forms. (common factors, products and grouping using associative, distributive and commutative properties.)
SO3:
Select and use algebraic formulae to solve problems. (Substitution into any formula, solve for one variable, supplied formulae from any context.)
To do this you will:
• identifying the difference between like and unlike terms;
• gathering and adding/subtracting like terms;
• using the meaning of exponents;
• simplifying terms that involve both multiplication and division with exponents;
and
• solve equations using the “trial and improvement” technique;
• solve equations using the “isolating the unknown” technique (together with flow diagrams);
• solve simple inequalities using tables of values;
• factorise expressions in order to show equivalence of expressions using both the
“common factor” and “difference of squares” techniques.