Transcript Patterns - mathsleadteachers

Having Fun with Algebra
[Level 1,2 and 3]
Look at this growing pattern of fish.
Describe how a child, from each curriculum level,
would solve how many sticks would be needed to
make 5 fish.
How do you know what have you
done?
Purpose:
• Explore Algebraic thinking
• Discuss key characteristics of pre-algebra
• Recognise the link between the multiplicative
strategies and the development of algebraic
patterning
• Using a word rule to explain the algebra in a
problem.
• Finding the number rule to predict relationships in
a pattern
What is Algebra?
• Discuss in your groups.
The New Zealand Curriculum
Level One
Equations and Expressions
• Communicate and explain counting, grouping, and equal
sharing strategies using words, numbers and pictures.
Patterns and Relationships
• Generalize that the next counting number gives the result of
adding one object to a set and that counting in a set tell how
many.
• Create and continue sequential patterns.
Level Two
Equations and Expressions
• Communicate and interpret simple additive structures using
words, diagrams,and symbols.
Patterns and Relationships
• Generalize that whole numbers can be partitioned in many
ways.
• Find rules for the next member of a sequential pattern.
What is a pattern?
Discuss with a partner.
What is a repeating pattern?
What is a growing pattern?
Growing Patterns
“You just keep adding a row every time to what you had
before … that’s one bigger than before” (4 year old)
Difference between repeating
and growing pattern:
Patterns:
Use of vocabulary
– Repeat
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Predict
Explain
Plan
Growing
Equal
Same as
different equal
not equal
More
Less
Balanced
unbalanced
So… What can we do?
• Use patterns to infer one thing from another and make
predictions
• Represent aspects of a situation with numbers to make it
easier to see patterns in a situation
• Describing number patterns means providing a precise rule
that produces a pattern
• Finding strategies that help us become better at recognising
common types of patterns
• The Number system has a lot of special patterns to make
working with numbers easier
• Some numbers have interesting or useful properties
First Steps in Mathematics
Understanding equality:
• Discuss: What do we understand by?
=
The equals sign
What does the equals sign mean in each of these
situations?
7+8+9=
X=3
4+5=?+3
How many ways can you make 16?
• Jamie has to write many names for the
number sixteen. He organizes his expressions
using patterns:
• 0 + 16
• 1 + 15
• 2 + 14
16 – 0
17 – 1
18 – 2
• 14 + 2
• 15 + 1
• 0 + 16
30 -14
31 - 15
32 - 16
8+8
4+4+4+4
2+2+2+2+2+2+2+2
2 + 14 = 3 + ?
You have a minute to think about
this!!
• What is the total of all the numbers
from 1-100?
Staircases:
One block is needed to make a 1-step upand-down staircase. It takes one step to
get up and one step to get down.
This is a called a 2-step staircase as it
takes two steps to go up and two steps
to go down.
How many cubes they think would be
needed to make a 5-step staircase?
Make this pattern:
• Can you visualize a pattern that would help
you find a rule?
• Share your patterns with a partner: What is
your rule?
• If the pattern had 100 steps how many blocks
would you need altogether?
What do you notice?
How would you explain in words?
What rule can you come up with?
Matchstick Patterns
T = 2n + 2(n - 2)
T = 4(n - 2) + 4
T = 4(n - 1)
T = 4n - 4
T = 2n + 2(n - 2)
T = 4(n - 2) + 4
T = 4(n - 1)
T = 4n - 4
How do we use letters in mathematics?
1) A letter can be used to represent a number
In the formula for the area of a rectangle, the length
base of the rectangle is often represented by b
2) A letter can be used to stand for a specific unknown
number that needs to be found
In a triangle, x is often used for the size of the angle
students need to find
3) A letter can be used as a variable that can take a
variety of possible values
4) In the sequence, (n, 2n+1), n takes on the values of
the natural numbers – sequentially
Solve:
• 3n + 2 =14
This is an equation. What word problem could
you think of?
Fish pattern
Neil writes Tn = 6 + 4 (n-1)
Iris writes Tn = 3n + 4 (n-1)
Fish pattern
6
6+4
6+4+4
6 + 2 lots of 4
6 + 4 (n-1)
Fish pattern
1st
pattern
2nd
pattern
3rd
pattern
2 lots of 3 parallel sticks
4 lots of mountains with 2 sticks
2 x n + 4 (n-1)
2x3 + 4 (n-1)
6+8
Materials
Imaging
Property of
numbers
Generalisation
from numbers
In summary
What is Algebra?
Results of Research
Students who are highly numerate in terms of the
measures used in the Numeracy Project; start
with and maintain a clear advantage in algebra
Solving these kind of problems by
patterns in tables, if done by itself, is
ineffective.
Dynamic dictionary to record.
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Reference words
Meanings (associated language)
Diagrams
Symbols
Representations
Resources:
• Resources needed
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Book 9
Dice
Counters
Toothpicks
Straws
A3 paper
NZC
Standards
Sharpies
Whiteborad pens
Book 1