Download content area: patterns, functions and algebra

INVESTIGATING NUMBER PATTERNS FROM GEOMETRIC
DIAGRAMS
INTERMEDIATE: Lynette Dreyer
Algebra
Patterns
Functions
RATIONALE
This workshop pack is designed to engage Intermediate Phase educators in investigating
number patterns from geometric diagrams.
CONTENT AREA: PATTERNS, FUNCTIONS AND ALGEBRA
The learner will be able to recognize, describe and represent patterns and relationships, as
well as to solve problems using algebraic language and skills.
Topics and Skills:
o Investigates and extents numeric and geometric patterns looking for a relationship or
rules, including patterns.
o Describes, explains and justifies observed relationships or rules in own words.
o Represents and uses relationships between variables in order to determine input and
/or output values in a variety of ways using:
 Verbal descriptions;
 Flow diagrams;
 Tables
 Equations
 Graphs
OBJECTIVES
At the end of the workshop educators will know:
o How to investigate number patterns from geometric diagrams.
o How to derive rules/equations from the identified patterns.
RESOURSES
Flip chart, flip chart pens, and boxes of matches, counters, rulers, pair of scissors, pencils and
erasers.
Investigating number patterns from geometric diagrams
Activity 1
The sketches below represent shapes, which are made of matchsticks.
Stage 1
Stage 2
Stage 3
1. Use a box of matches provided to build 4, 5, 6 and 7 triangles.
2. Is there a pattern? Describe the pattern.
3. What is the relationship between the number of triangles and the number of
matchsticks used to form triangles?
4. Can you formulate a rule?
5. Use the rule you discovered in question 4 to find out how many matches are needed
to make:
(i)
11 triangles
(ii)
15 triangles
(iii) 80 triangles
Activity 2
This diagram shows how a pattern of squares is built up in a row using matches:
2.1 Count the number of matchsticks needed to make up one square, two squares, up to four
squares.
2.2 How many matches will be needed for 5 squares?
2.3 Write down the rule you think will work for linking the number of matches to the number
of squares.
Activity 3
Mzitho Tile Company produces mosaics (patterns made of small tiles) in the form of squares
of differing sizes as shown in the diagram below. Each small tile is a square of side 1cm. A
size 1 mosaic has one black tile in the middle and one black tile per arm; a size 2 has two
black tiles per arm and so on. The other tiles are white.
Size 1
Size 2
Size 3
3.1 Copy and complete this table:
Size
number
Number
of black
tiles
1
2
3
4
5
9
13
17
5
6
7
3.2 What rule could you use to find out the number of black tiles from the size number of the
pattern? Write your rule down.
3.3 How many black tiles will be there in each case?
(i)
size 20
(ii)
size 40
(iii) size 100
Activity 4
This sequence is generated from a pattern of dots, which form a triangle.
By adding another row of dots and counting all the dots we can find the next number of the
sequence.
4.1 How many dots will be used for stages 6 and 7?
4.2 Can you formulate a rule?
Activity 5
Chairs may be arranged around tables as shown in the following diagrams:
Chairs
Table
5.1 How many chairs can be arranged around 3 tables?
5.2 Copy the following table into your book and complete the table by filling in the missing
values.
Number 1
2
3
4
5
……
16
…….
of
tables
(t)
Number 8
10
…….
……. 80
of
chairs
( c)
5.3 Write down an equation to represent the relationship between the number of tables (t) and
the number of chairs (c) that can be arranged around those tables.
5.4 (i) Use the equation to determine the number of chairs that can be arranged around 19
tables.
(ii) If you arrange 62 chairs around tables, how many tables are there?
5.5 Use the table to draw a graph to represent the relationship between the number of tables
and the number of chairs that can be arranged around those tables.
5.6 By referring to the patterns above, explain why the graph that you have drawn is a
straight line.
Activity 6
PASCAL TRIANGLE
1
1
1
1
1
1
2
3
4
1
3
6
1
4
1
6.1 Extend the triangle by two rows and explain how you got the numbers you filled in.
6.2 Colour in boxes containing triangular numbers.