Geometric and Number Patterns: Finding Rules and Relationships, Slides of Algebra

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MODULE FOUR

Patterns, Functions and Algebra

a number pattern

Fundamentals in ECD : Mathematics Literacy Learner’s Manual

MODULE FOUR PATTERNS, FUNCTIONS AND ALGEBRA

In this unit you will address the following:

Unit Standard 7448 SO1: Recognize, identify and describe patterns in various contexts. (numeric, geometric, patterns from a variety of contexts.)

SO2: Complete, extend and generate patterns in a variety of contexts. (numeric, geometric, patterns from a variety of contexts.)

Unit Standard 7464 SO1 : Identify geometric shapes and patterns in cultural products. (shapes of and decorations on cultural products such as drums, pots, mats, buildings, and necklaces.)

SO2: Analyze similarities & differences in shapes & patterns, & effect of colour, used by cultures. (analyze similarities and differences in shapes and patterns, and the effect of colour, used by different cultures.)

To do this you will:

• distinguish between geometric and numeric patterns;
• investigate and analyse both geometric and numeric patterns;
• explain and justify patterns observed in both geometric and numeric patterns;
• complete tables of values for numeric patterns;
• use tables of value for numeric patterns to identify rules used to generate numeric patterns;
• answer questions based on information derived from tables of values.

Fundamentals in ECD : Mathematics Literacy Learner’s Manual

UNIT 1

Patterns

MODULE FOUR Unit One: Patterns

1. Geometric patterns A pattern consists of objects arranged in order according to a rule.

Below are several illustrations of African wall patterns. You will almost certainly have seen such patterns in homes, sometimes in the tiles and sometimes in the paintwork. You may even have made such a pattern on the walls of your crèche.

Activity 1:

Investigating geometric patterns

Work alone Look at the patterns above again. Answer these questions about the patterns.

1. Draw a sketch of the object that has been used as the basis for each of these patterns.
2. For each pattern, circle the basic shape then describe, in words, how the shape was moved to create the pattern. Describe both the direction and size of the move. Your description of how to move the shape is called the rule that generates the pattern.
3. Draw a sketch of at least one other pattern that can be generated using the same shape. Describe the rule you used to generate the pattern.

D

A

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D

Time needed 15 minutes

MODULE FOUR Unit One: Patterns

Activity 2:

Extending and generating number patterns

Work alone

1. Write down the next four terms in the pattern above.
2. Work out the number pattern that uses the same rule but starts with 6 as the first number.
3. Complete the number patterns below. In each case write down the rule that was used to generate the pattern. a. … … … 12 15 18 … … … b. … … 12 24 48 … … … … c. … … … … 55 50 45 40 … d. … 81 75 … 63 … … 51 …
4. Create your own patterns using the following rules: a. Subtracting the same number each time. b. Adding the same number each time. c. Multiplying by the same number each time. d. Dividing by the same number each time. e. First multiplying by a number and then adding another each time.
5. Compare your answers for Activity 3 with one of your colleagues.

What have you learned?

In Question 3 you were given at least three of the numbers in the pattern, this is because you will need at least three numbers to establish the rule. Look at the following patterns, if you had only been given the first two numbers you would not have known which of the patterns we wanted to generate. 1 3 5 7 … … … 1 3 9 27 … … …

3. Number patterns that occur in everyday life There are many everyday situations that give rise to number patterns. Think about Palesa who runs the vetkoek stall at the Bantwana Bami ECD Centre’s annual morning market. Palesa’s vetkoek sell for R3,00 each. Instead of calculating how much each order costs she has started to make the following table:

Notice how the numbers in the “Cost” row make a number pattern:

Number of vetkoek 1 2 3 4 5 6 7 8 9 10 11 Cost (in rand) 3 6 9 12

Time needed 20 minutes

Activity 3:

Palesa’s vetkoek

Work alone

1. Write down the rule used to generate the number pattern in the cost row above.
2. Complete Palesa’s table.
3. Rashida wants to buy 12 vetkoek. Describe at least two different ways in which Palesa can work out the cost.
4. Which of the two ways you described above will help Palesa most if Rashida wants to buy 55 vetkoek?

What have you learned?

In describing two ways in Question 3, you almost certainly found the following:

• You can extend the pattern in the second row of the table by adding three to the numbers in the pattern, or
• You can find the number in the second row by multiplying the number in the first row by 3.

Consider the following shapes that children might make on the carpet using rectangular (white) and circular (black) shaped building blocks. Each shape has a number of open squares in the middle.

We can use a table to record the number of squares in each shape as well as the number of each type of building block needed to make the shape.

Shape number 1 2 3 4 5 6 7 8 9 10 11 Number of squares 1 2 3 Number of rectangular blocks 4 7 10 Number of circular blocks 4 6 8

Shape 1 Shape 2 Shape 3 (1 square) (2 squares) (3 Squares)

Fundamentals in ECD : Mathematics Literacy Learner’s Manual

Time needed 15 minutes

3. Use a collection of four different coloured counters, such as plastic bottle tops that are red, blue, green, yellow. Put each colour into a separate container. Children make a row of four counters using a different colour each time. They think about how many different ways they can do this: red blue green yellow blue green yellow red yellow red blue green green yellow red blue
4. With 4 colours and 4 counters in a row there should be 16 possible ways altogether.

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 geometric and numeric patterns? b. Write down one or two questions that you still have about tables of values. c. How will you use what you learned about geometric and numeric patterns and tables of values in your every day life and work?

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. Distinguish between geometric and numeric patterns
2. Investigate and analyse both geometric and numeric patterns
3. Explain and justify patterns observed in both geometric and numeric patterns
4. Complete tables of values for numeric patterns
5. Use tables of value for numeric patterns to identify rules used to generate numeric patterns
6. Answer questions based on information derived from tables of values

Fundamentals in ECD : Mathematics Literacy Learner’s Manual

MODULE FOUR Unit One: Patterns

MODULE FOUR Unit Two: Finding rules for number patterns

Relational approach In the relational approach we try to establish a relationship or rule between the position of the term in the pattern (i.e. the term number) and the term itself. You can see this in the case of Rashida’s vetkoek:

Cost = number of vetkoek multiplied by the cost of one vetkoek

You can write this in a shorter way:

Cost = number of vetkoek × 3

The relational approach has the advantage that you do not have to repeat a large number of simple operations. But it can be difficult to find the relationship or rule.

Our challenge is to study patterns and find the rule or relationship from structure of the pattern. You need to learn to look at the patterns in different ways. Here you will explore how to use tables and graphs to show the structure of patterns.

2. Graphs Think back to Palesa and her vetkoek in the previous unit. Felicity runs the cold drink stall next door to Palesa. Felicity has also decided to find a quick way to calculate the price of an order but instead of a table she has prepared a graph. This is the graph.

How much to charge?

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Cold drinks

Activity 1:

Using graphs to represent number patterns

Work alone

1. Why do you think that Felicity has not joined the points on her graph?
2. Reproduce Felicity’s graph on the squared grid paper in your journal. On the same graph add a graph for the costs of Palesa’s vetkoek (from unit 1) to this graph.
3. How are the graphs for Palesa’s vetkoek and Felicity’s cold drinks the same? How are they different?
4. Make a table similar to Palesa’s vetkoek table for Felicity’s cold drinks.
5. In the notes on the relational approach above you read a rule for finding the cost of any number of Palesa’s vetkoek. Find a similar rule for working out the cost of any number of Felicity’s cold drinks.
6. Compare your answers with a partner.
7. Make a list of ways in which the tables, the graphs and the rules for Palesa’s vetkoek and Felicity’s cold drink are the same and different. Look for a connection between these similarities and differences.

Activity 2:

Building block shapes again

Work alone Look at these shapes again:

In unit 1 you copied and completed this table. This is what it looks like.

1. Draw graphs from the table, for the number of squares, the number of rectangles and the number of circles required for the different shape numbers. The first three points for each pattern have already been marked on this graph for you to see.

Shape number 1 2 3 4 5 6 7 8 9 10 11 Number of squares 1 2 3 Number of rectangular blocks 4 7 10 Number of circular blocks 4 6 8

Shape 1 Shape 2 Shape 3 (1 square) (2 squares) (3 Squares)

Fundamentals in ECD : Mathematics Literacy Learner’s Manual

Time needed 25 minutes

Time needed 25 minutes

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 graphs? b. Write down one or two questions that you still have about different representations of numeric patterns. c. How will you use what you learned about graphs in your every day life and work?

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. Draw graphs to describe situations
2. Convert between different representations of a numeric pattern including: tables, graphs and verbal descriptions of the rules
3. Answer questions about a situation based on the graphs, tables of values and/or verbal descriptions of the situation
4. Describe situations using graphs, table and verbal descriptions
5. Distinguish between the recursive and the relational approach to finding given terms in a numeric pattern.

Fundamentals in ECD : Mathematics Literacy Learner’s Manual

MODULE FOUR Unit Two: Finding rules for number patterns

MODULE FOUR Unit Three: More number patterns

You have to use your imagination a bit because plants do not grow as perfectly as the graph suggests.

1. You can see that the teacher has joined the points on her graph. Is this correct? Explain your answer.
2. Make a table of values showing the days and the corresponding height of the plant.
3. Think about Rashida’s vetkoek, Felicity’s cold drinks and the building block patterns. Which of those patterns is most similar to the sunflower’s growth pattern? Explain how the patterns are similar and also how they are different.
4. Work out a rule for finding the height of the plant for any given number of days.

Activity 2:

The cost of a phone call

Work alone Not all of the teachers at the Bantwana Bami ECD Centre own a cell phone and the Centre’s telephone may not be used for private calls. Shaheeda has decided to make her phone available for anybody but has decided to charge for the calls. She charges per second and has made the following table to give people an idea of the cost of their call.

Sunflower growth pattern

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0 1 2 3 4 5 6 7 8 9 10 Day

Length of call in minutes 1 1 2 2 3 3 4 4 5 Length of the call in seconds 60 90 120 150 180 Cost of the phone call (in rands) 1,20 1,80 2,40 3,00 3,

(^112) 2 1 2 1 2

Time needed 30 minutes

1. Copy and complete the table.
2. In your journal draw a graph for the cost of a call for a given number of seconds. Did you join the points on your graph? Why?
3. Think about Rashida’s vetkoek, Felicity’s cold drinks, the building block patterns and the sunflower’s growth. Which of those patterns is most similar to the phone call cost pattern? Explain how the patterns are similar and also how they are different.
4. Work out a rule for determining the cost of a phone call for any given number of seconds.
5. Use your rule, or use another way, to find the following: (a) The cost of a 480 second phone call (b) The cost of a 68 second phone call (c) The cost of a 13 minute phone call (d) The number of seconds that you can speak for if you have R7, (e) The number of minutes that you can talk for if you have R12, 2. Flow diagrams We sometimes use flow diagrams to summarise a rule. In a flow diagram the input number is your starting number. The operator tells what you have to do to the input number to produce the output number. You can have more than one operator in a flow diagram.

The diagram below shows the flow diagram for the rule: “multiply the input number by 3 and add 4 to the answer”

Input Numbers

Operators

Output Numbers 1

x 3 + 4

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3

5

8

7

10

a

b

c

Fundamentals in ECD : Mathematics Literacy Learner’s Manual