Patterns and Algebra in Stages 3 and 4

Report
AIS Conference 2008
Patterns and Algebra in
Stages 3 and 4
Judy Anderson
The University of Sydney
[email protected]
Patterns and Algebra in K–10

Expressing generality (x + y = y + x)
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Using and interpreting functions ( y = 3x )
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Solving equations ( 3x – 1 = 5)
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How can we begin?
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Some magic …
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A story …
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An everyday situation …
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A puzzle …
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An investigation …
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The Magic of Numbers

Think of a number between 1 and 64
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Write down everything you can about your number
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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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Everyday number patterns

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?
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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
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Multiply by 4
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Subtract 4
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Divide by 4
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Subtract the number your started with
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What is the answer?
Use Algebra to show how this works?
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Another think of a number …
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Think of a number between 2 and 10
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Multiply it by nine
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Add the two digits of your answer
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Subtract five
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Choose a letter of the alphabet
corresponding to the number …
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Calendar Patterns
June 2008
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Calendar Patterns
June 2008
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Calendar Patterns
June 2008
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Algebraic thinking …
Patterns and Algebra

Generating and investigating
patterns
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Observing, predicting and proving
Working
Mathematically
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Describing relationships
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Questioning

Making generalisations and
proving results
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Applying Strategies
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Communicating
Using and applying algebraic
symbolism to solve problems
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Reasoning
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Reflecting

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The developmental sequence
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Early Stage 1
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Stage 1
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Stage 2
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Stage 3
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Stage 4
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Building foundations for Algebra
in K – 6 Mathematics
 Number Patterns – pattern work leads to expressing
generality
eg continue the pattern
3, 6, 9, 12, ….
1, 4, 9, 16, …..
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Number Relationships –building understandings of
number and operations is also very important
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Building foundations includes:
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Understanding the properties of numbers and
operations
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Using all numbers, not just whole numbers
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Seeing the operations, not just the answers
125  5 = •
•
4 + 1 =•
x 5 = 125
+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:
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What number comes next?
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How do you know?
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What number will be 10th?
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How do you know?
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Can you predict the 20th number?
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How could you check if you are correct?
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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
2
3
5
6
7
Number of
Sides
3
6
9 12 15
-
-
4
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
Describe a pattern in words in more than one
way
Number of
Triangles
1
2
3
5
6
7
Number of
Sides
3
6
9 12 15
-
-
4
(determining a rule to describe the pattern from the table)
‘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
-9
5
3
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Stage 4 - Introducing Pronumerals

K is the number of letters in your name - so always stands
for a number
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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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x
1
2
3
4
5
6
7
y
1
5
9
13
17
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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.
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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
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Multiplying
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Algebra resources …
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Syllabus and Sample Units of Work
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DET Patterns and Algebra
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RIC pattern books (Paul Swan)
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Origo algebra books (Elizabeth Warren)
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AAMT
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Others???
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