introduction to unit 1: patterns & sequences mathematics 12 foundations

Introduction to Unit 1Introduction to Unit 1: : Patterns & Sequences Patterns & Sequences

Mathematics 12 FoundationsMathematics 12 Foundations

DefinitionsDefinitions::1. Sequence:

An arrangement of numbers, symbols, or pictures in order.

Each item or term follows another according to a rule.

Example: {a, b, c, d, e} is the sequence of the first 5 letters

alphabetically {0, 1, 0, 1, 0, 1, ...} {0, 1, 0, 1, 0, 1, ...} {20, 25, 30, 35, ...} {20, 25, 30, 35, ...}

DefinitionsDefinitions::2. Term:

Each item in a sequence.

3. Finite Sequence: A sequence which eventually ends.Example: {1, 3, 5, 7}

4. Infinite Sequence: A sequence which continues endlessly.

Example: {1, 2, 3, 4 ,...}

DefinitionsDefinitions::5. Fibonacci Sequence:

A sequence which is determined by calculating the sum of the previous two terms.

Terms one and two have the value of 1. Example: {1, 1, 2, 3, 5, 8, …}

6. Fibonacci Number: A term in the Fibonacci sequence.

Fibonacci Sequence:Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding the

two numbers before it together The 2 is found by adding the two

numbers before it (1+1) The 21 is found by adding the two

numbers before it (8+13)

Investigation #1Investigation #1 – Properties of – Properties of Designs (page 2)Designs (page 2)

In most types of construction, it is important to make and extend patterns. A bricklayer builds towers using an odd number of bricks in each row.

Each new tower has one more row than the previous tower had.

Tiffany, a bricklayer, wants to know the number of bricks needed to build a tower with 10 rows.

Investigation #1Investigation #1 – Properties of – Properties of Designs (page 2)Designs (page 2)

She writes the number of bricks for each tower as a term in a sequence: {1, 4, 9 …}.

Purpose Describe a patterning rule for extending a

sequence of numbers symbols, or pictures.

Procedure Use a geometrical pattern to draw below

the next three towers in the bricklayer’s project above.

What is the Pattern or Sequence?What is the Pattern or Sequence? What numbers will we write below for this sequence?

{___, ___, ___, ___, ___, ___, ___, ___, ___, ___ …}

Draw a graph showing the number of bricksversus the number of rows in the tower.

Questions:Questions:1. Choose ONE of the following which best

describes the shape of the graph you drew above. Line Parabola Sine Cosine Absolute Value

2. Explain using the idea of slope, why the graph you plotted can’t be a linear relationship.

3. Describe a rule for finding the number of bricks in a tower, if you know the number of rows in the tower.

4. Use the rule from question #3 to determine the number of bricks in a tower with the following number of rows.

5. Fill in the table below with the missing information..