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Patterns and Algebra in Stages 3 and 4
Judy AndersonThe University of Sydney
AIS Conference 2008
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Patterns and Algebra in K–10
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 64 squares, 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 separately with 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 patternsEveryday number patterns
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The clothesline – A peggy problem!
When I was a kid in the country, I used two pegs for the first towel, and then one new peg for each additional towel.
How many pegs are required for 1 item, 2 items, 5 items, 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 the pattern look like now?
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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?
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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 alphabet
corresponding to the number …
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Calendar PatternsJune 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
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Calendar PatternsJune 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
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Calendar PatternsJune 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
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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 Algebrain K – 6 Mathematics
Number Patterns – pattern work leads to expressing generality
eg continue the pattern
3, 6, 9, 12, ….
1, 4, 9, 16, …..
Number Relationships –building understandings of number and operations is also very important
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Building foundations includes:
Understanding the properties of numbers and operations
Using all numbers, not just whole numbers
Seeing the operations, not just the answers
125 5 = � � x 5 = 125
4 + 1 = � + 2
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What’s my rule?
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Activity
Generate three different number patterns that include the number 12.
Try to use different kinds of numbers and different operations.
Look at one of your neighbour’s 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 ‘x’ belong 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 of Triangles 1
Number of Sides 3
Number of Triangles 1 2 3 4 5 6 7
Number of Sides 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 of Triangles 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 + � = 12 – 4
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 and then 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
K is the number of letters in your name - so always stands for a number
K takes 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 10th, 20th, 50th, 100th 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
“ y starts at one and you keep adding 4”x = 1 + 4y
Algebra is not a personal shorthand to jot things down
Algebra cannot be used to express all patterns
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Students need to develop mature operations
Counting Adding Multiplying
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Algebra resources …
Syllabus and Sample Units of Work
DET Patterns and Algebra
RIC pattern books (Paul Swan)
Origo algebra books (Elizabeth Warren)
AAMT
Others???