patterns and algebra in stages 3 and 4

7/30/2019 Patterns and Algebra in Stages 3 and 4

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Patterns and Algebra in

Stages 3 and 4

Judy Anderson

The University of Sydney

[email protected]

AIS Conference 2008


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AIS Conference 2008 2

Patterns and Algebra in K10

Expressing generality (x+ y= y+x)

Using and interpreting functions ( y= 3x)

Solving equations ( 3x 1 = 5)


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How can we begin? Some magic

A story

An everyday situation

A puzzle

An investigation


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The Magic of Numbers

Think of a number between 1 and 64

Write down everything you can about your number

Share this with a friend can you think of any other

interesting things about these two numbers?

Finding your number


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A long time ago A mathematician invented the game of chess and presented it to

the king. The king was so pleased with the game that he asked

the mathematician to name a reward.

The mathematician looked at the chessboard, consisting of 64squares, and asked for some rice according to the rule:

One grain of rice on the first square, 2 grains of rice on the

second square, 4 grains on the third square, 8 on the fourth and

so on until the last square.

The king thought that the mathematician was a bit simple, so he

readily agreed and sent for the rice from the royal warehouse.

How much rice did the king need?


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I noticed the other morning when hanging some

clothes on the line that I hang each item separatelywith two pegs per item.

How many pegs would I need to hang 1 item, 2 items,

5 items, 30 items, and so on? How would you describe

this pattern in words?

Everyday number patterns


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The clothesline A peggy problem!

When I was a kid in the country, I used two pegs forthe first towel, and then one new peg for each

additional towel.How many pegs are required for 1 item, 2 items, 5items, 30 items, and so on?

Mum used to overlap the towels but she would put an

extra peg in the middle of each towel. What does thepattern look like now?



Think of a number

Add 2

Multiply by 4

Subtract 4

Divide by 4

Subtract the number your started with

What is the answer?

Use Algebra to show how this works?



Another think of a number

Think of a number between 2 and 10

Multiply it by nine

Add the two digits of your answer

Subtract five

Choose a letter of the alphabetcorresponding to the number



Calendar Patterns

June 2008

Sun Mon Tues Wed Thurs Fri Sat

1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31



Calendar Patterns

June 2008


1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31


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Algebraic thinking

Patterns and Algebra

Generating and investigating

patterns

Observing, predicting and proving

Describing relationships

Making generalisations and

proving results

Using and applying algebraic

symbolism to solve problems

Working

Mathematically Questioning

Applying Strategies

Communicating

Reasoning

Reflecting


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The developmental sequence

Early Stage 1

Stage 1

Stage 2

Stage 3

Stage 4


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Building foundations for Algebra

in K 6 Mathematics

Number Patterns pattern work leads to expressinggenerality

eg continue the pattern

3, 6, 9, 12, .

1, 4, 9, 16, ..

Number Relationshipsbuilding understandings of

number and operations is also very important


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Building foundations includes:

Understanding the properties of numbers andoperations

Using allnumbers, not just whole numbers

Seeing the operations, not just the answers

125 5 = x 5 = 125

4 + 1 = + 2


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Whats my rule?


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ActivityGenerate three different number patterns

that include the number 12.

Try to use different kinds of numbers and

different operations.

Look at one of your neighbours patterns and

find the next three numbers in the pattern.


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Questions to pose: What number comes next?

How do you know?

What number will be 10th?

How do you know?

Can you predict the 20th number?

How could you check if you are correct?

Does the number xbelong to this pattern?


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Consider investigating other patterns

16 - 9=

26 - 9 =

36 - 9 =continue, predict other cases and explain

The aim of pattern work is to:

develop facility and flexibility with numbers

build intuitive understanding of properties.


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Stage 3 Build simple geometric patterns involving multiples

Complete a table of values for geometric and

number patterns

Number ofTriangles 1

Number ofSides 3

Number ofTriangles 1 2 3 4 5 6 7

Number ofSides 3 6 9 12 15 - -


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Describe a pattern in words in more than one

way

(determining a rule to describe the pattern from the table)

Number ofTriangles 1 2 3 4 5 6 7

Number of

Sides 3 6 9 12 15 - -

It looks like the 3 times

tables.

You multiply the top

number by three to get the

bottom number.


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Construct, verify and complete number sentences

involving the four operations with a variety of numbers

completing number sentences:

5 + = 124

7 = 7.7

constructing number sentences to match a word

problem checking solutions and describing strategies


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Learning about using inverse operations

I think of a number, multiply it by 3, take away 9 andthen divide by 5. The answer is 3. What was the

number I thought of?

( ( 3 - 9) 5 ) = 3

Answer: 8


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Example of backtracking

I think of a number, multiply it by 3, take away 9 and then divide

by 5. The answer is 3. What was the number I thought of?

3 3 -9 5


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Stage 4 - Introducing Pronumerals

Kis the number of letters in your name - so always stands

for a number

Ktakes multiple values (unknown or variable)


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2 (g+ 4) = g+ 4 + g+ 4

= g+ g+ 8

= 2g+ 8


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2 (n+n+6)

(2 + n) 2 + 6


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Describe this pattern in words?

Describe this pattern using symbols?

What is the value of the 10

th

, 20

th

, 50

th

, 100

th

terms?

Compare your answers.

Discuss any differences to clarify who has predicted correctly.

x 1 2 3 4 5 6 7

y 1 5 9 13 17 21 25


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Functional Thinking:Students have difficulty connecting the top number with the

bottom number

Common errors:

x goes up by 1 and y goes up by 4

x+1= y + 4

ystarts at one and you keep adding 4

x= 1 + 4y

Algebra is not a personal shor thand to jot th ings down

Algebra canno t be used to express al l patterns


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Students need to develop mature operations

Counting Adding Multiplying


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AIS C f 2008 32

Algebra resources

Syllabus and Sample Units of Work

DET Patterns and Algebra

RIC pattern books (Paul Swan)

Origo algebra books (Elizabeth Warren)

AAMT

Others???

patterns and algebra in stages 3 and 4

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