Transcript number patterns

Algebraic Patterning
Autauga County Inservice Day
January 4, 2010
Deanna McKinley
DPES
Introduction
AL COS #3 states Solve problems using
numeric and geometric patterns.
Bullet states determine a verbal rule for a
function given the input and output
The study of patterns is a key part of algebraic
thinking. They involve relationships and
generalizations.
NCTM Standards show the
progression
PRIMARY CLASSROOMS (K-5)
• Recognize, describe, and extend patterns such as
sequences of sounds and shapes or simple numeric
patterns and translate from one representation to
another. Analyze how both repeating and growing
patterns are generated
• Describe, extend, and make generalizations about
geometric and numeric patterns. Represent and analyze
patterns and functions, using words, tables, and graphs
Grades 6–8
• Represent, analyze, and generalize a variety of patterns
with tables, graphs, words, and, when possible, symbolic
rules
Why Patterns?
Patterning is critical to the abstraction of mathematical
ideas and relationships, and the development of
mathematical
reasoning
in
young
children.
(English, 2004; Mulligan, Prescott & Mitchelmore, 2004; Waters, 2004)
The integration of patterning in early mathematics
learning can promote the development of mathematical
modelling, representation and abstraction of
mathematical ideas.
(Papic & Mulligan, Preschoolers’ Mathematical Patterning)
Patterns are everywhere; we just need to learn to
notice them.
Patterns progression
1. Copy a pattern and create the next element
2. Predict relationship values by continuing the pattern with
systematic counting
3. Predict relationship values using recursive methods e.g.
table of values, numeric expression
4. Predict relationship values using direct rules e.g. ? x 3 + 1
5. Express a relationship using algebraic symbols with
structural understanding e.g. m = 6f + 2 or m = 8 + 6(f – 1)
Wright (1998). The learning and Teaching of Algebra: Patterns, Problems and Possibilities.
Patterns Jargon
Number sequences
-
Number patterns
Sequential
-
Spatial
Linear
Geometric
-
Sequential rules
-
Repeating patterns
Trends
Functional rules
-
-
Growing pattern
Values
Activities for Exploring Number Patterns
Paper Folding
Fold a piece of paper in half, and then in half again, and
again, until you make six folds. When you open it up, how
many sections will there be? Make a chart. Continue to fold
as you look for a pattern.
Paper Tearing
Tear a piece of paper in half and give half to someone else.
Each person then tears the piece of paper in half and passes
half on to another person. How many people will have a
piece of paper after 10 rounds of tearing paper like this?
Continue tearing paper and record on a chart. Look for
patterns as you complete each round.
What do we do?
Spatial repeating patterns
What do we do?
Repeating patterns with beads
Set
1
2
3
4
..
10
..
25
..
n
Blue
Green
Total
What do we do?
Spatial growing patterns
What do we do?
Spatial and number patterns
Shape
1
2
3
..
10
..
n
# sticks
What do we do?
Spatial growing patterns
Squares
Sticks
3n + 1
4n + 2
3n + 1
4n + 4
What do we do?
Number Machines
Some student work...
Pre-repeating patterns
Post-repeating patterns
Pre-Spatial & Number patterns
Post-Number patterns
Post-Spatial patterns
Pre-Number Machines
Post-Number
Machines
And … Post-Functions
Pre-Number Machines
Post-Number Machines
Some points
Lots of hands on material based exploration followed by group
discussion. Materials can get in the way and we have to move on.
Develop understanding by decomposing spatial shapes in a pattern (i.e.,
finding what is different and similar)
We found beads very helpful to elicit discussion leading to functional rules
between the colors
Some students preferred to work with the numbers than the spatial
patterns (they could see patterns easier), therefore keep using the
numbers and spatial patterns together.
Don't always put the values of a number pattern table in order.
Take the number machines to the next level and then require students to
connect it to the functional rule for a number pattern.
Bellringers and Journal Ideas
Start to use all numbers (rational, irrational,
weird, negative) and get students to experiment
with calculators.
(Stacey and MacGregor, Building foundations for Algebra, 1997)
Connecting patterns – tables – graphs.
SEE HANDOUTS
Basic fact patterns
Instant recognition of series
5x
0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
2x
0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
4x
0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
3x
0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
9x
0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
6x
0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
8x
0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
7x
0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70
Fractions - Decimals - Percentages
Halves, quarters, and eighths
1/2
0.5
50%
1/4
0.25
25%
1/8
0.125
12.5%
1/2 x table 0.5
1.0
1.5
2.0
2.5
…
5x table
1/4 x table
0.25
0.50
0.75
1.00
1.25
…
25x table
1/8 x table
0.125 0.250 0.375 0.500 0.625 … 125x table
Patterns
Internal patterns
10x
0,
10, 20, 30, 40, 50, 60, 70, 80, 90, 100
5x
0,
5, 10,
2x
0,
2, 4, 6, 8, 10,
12, 14, 16, 18, 20
8x
0,
8, 16, 24, 32, 40,
48, 56, 64, 72, 80
9x
0,
9, 18, 27, 36, 45, 54, 63, 72, 81, 90
15, 20,
25, 30,
35, 40,
45, 50
Fractions – Decimals - Percentages
Thirds, ninths, and sixths times table
1/3
0.333
33.3%
1/9
0.111
11.1%
1/6
0.166
16.6%
0.666 0.999 (=1!)
…
1/3 x table
0.333
1/9 x table
0.111 0.222 0.333 0.444 …0.999 (=1) 11x table
1/6 x table
0.166 0.333, 0.500, 0.666, 0.833, 1.000
More basic facts Patterns
Instant recognition of series
Instant recognition of membership
Power series
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
Square numbers
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Triangular numbers
1, 3, 6, 10, 15, 21, 28, 36, 45
Cubic numbers
1, 8, 27, 81, 125
A pattern is a pattern is a pattern
Kindergarteners are asked to make patterns
and to continue a pattern by coloring
So by 3rd grade we better be asking a little
more… Making the steps toward generalizing
Stage 1
– create, represent and continue a variety of
number patterns and supply missing elements
– build
number
(addition and subtraction facts to at least 20)
relationships
make
generalizations
relationships
‘when I count by
fives the last
number goes
five, zero, five,
zero, …’
about
number
‘When I add zero it
does not change the
number’
An odd number
plus an odd
number always
equals an even
number’
– use the equals sign to record equivalent
number relationships
Stage 2
– generate, describe and record number
patterns using a variety of strategies
1, 4, 7, 10, …
2.2, 2.0, 1.8, 1.6, …
1 2 3 4 5 6 7
4, 4, 4, 4, 4, 4, 4, …
(generate using calculator, materials or mental strategies)
– build
number
relationships
(relating multiplication and division facts to at least 10 x 10)
2x4
=
4x2
(applying the commutative property)
‘6x4 = 24; so 24÷4 = 6
and 24÷6 = 4.
(relating multiplication and division)
‘The multiplication facts for
6 are double the
multiplication facts for 3’
(describing the relationships)
–
complete simple number sentences by
calculating the value of a missing number
Find
Find
so that 5+
= 13,
so that 28 =
x7
Stage 3 - algebra without symbols
– build simple geometric patterns involving
multiples
– complete a table of values for geometric and
number patterns
Number of
Triangles
Number of
Triangles
Number of
Number of
SidesSides
1 2 3 4 5 6 7
1
3 3 6 9 12 15 -
-
– 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.’
Construct, verify and complete number sentences
completing number sentences:
5 + • = 12 – 4
7 x • = 7.7
constructing number sentences to match a
word problem
checking solutions and describing strategies
Stage 4 - algebra with symbols
Five outcomes at this Stage:
*use letters to represent numbers
*describe number patterns with symbols
*simplify, expand
expressions
and
factorise
algebraic
*solve equations and simple inequalities
*graph on the number plane
A Row of Triangles, Squares, or Pentagons
A Row of Triangles
If you line up 100 equilateral triangles (like the green ones in Pattern
Blocks) in a row, what will the perimeter measure? If you think of this as
a long banquet table, how many people can be seated? Create a chart to
record the data and look for patterns as you add triangles.
A Row of Squares
What will the perimeter measure if you line up a row of 100 squares? If
this is a long table, how many people can be seated? Create a chart for
your data like the one in the Row of Triangles problem.
A Row of Pentagons
What will the perimeter measure if you line up a row of 100 pentagons?
If this is a long table, how many people can be seated? Create a chart for
your data like the one in the Row of Triangles and Row of Squares
problem.
Handshakes
Suppose everyone in the room shakes hands with
every other person in the room. How many
handshakes will that be? (With one person, there
will be no handshake. With two people, there will
be one handshake. How many handshakes will
there be with three people? Four? Continue by
creating a chart of the data and look for patterns.
Hundred Board Wonders
Select a rule from the list below to explore number patterns using a hundred board
1.
Numbers with a two in them
2.
Numbers whose digits have a difference of 1 (Be sure the students always select
numbers whose tens-place digit is 1 greater than the ones-place digit.)
3.
Numbers with a 4 in them
4.
Numbers that are multiples of 3
5.
Numbers with a 7 in them
6.
Numbers that are multiples of 5
7.
Numbers with a 0 in them
8.
Numbers that are divisible by 6
9.
Numbers with a 5 in the tens place
10.
Numbers that are multiples of 4
11.
Numbers having both digits the same
12.
Numbers that are both multiples of 2 and 3
13.
Numbers that are divisible by 8
14.
Numbers whose digits add to 9 (example: 63)
Tiling a Patio
You are designing square patios. Each patio has a
square garden area in the center. You use brown
tiles to represent the soil of the garden. Around
each garden, you design a border of white tiles.
Build the three smallest square patios you can
design with brown tiles and white tiles for the
border.
Record the number of each color tile
needed for the patios in a table. Continue filling in
the table as you design the next two patios.
What’s the Best Deal?
Your boss at the video game company where you work has given you two
choices of salary schemes. You have to decide which of the choices will
permit you to reach your goal of $1000 the fastest. You will need to support
your choice by showing the data your collected and by describing the graphs
for each situation.
Choice 1: Your salary will be doubled each day. You will earn $1 the first
day, $2 the second day, $4 the third day, $8 the fourth day, and so on.
Choice 2: Your salary will increase $3 each day. You will make $3 the
first day, $6 the second day, $9 the third day, $12 the fourth day, and so
on.
Which of these two ways to earn your salary will get you to $1000 the fastest?
Complete a table for each plan. On graph paper, draw a graph for the total
earnings for each salary plan.
Amazing Calendars
Calendars provide examples of number patterns to explore.
Consider the following:
1. How does the calendar change as you look across a
week?
2. How does it change as you go up? Down? Across?
Diagonal?
3. What is the sum of all the numbers in a square? Find
other sums? Do you notice any patterns?
4. What other patterns do you notice?
Let’s Go Fishing!
Ten people are fishing in a boat that has 11 seats. Five
people are on one side and five are on the other side with
an empty seat between them. What is the minimum number
of moves it would take for the five people in the front of
the boat to exchange places with the five people in the back
of the boat? (You may only jump over one person at a time
and one person can only be in a seat at a time.)
Table Problem
For a special dinner, the McKinley’s are using
square tables that can be put together to make
rectangles.
One table can seat 4 people
Two tables can seat 6 people
Three tables can seat 8 people
Table problem
How many people can be seated at 4 tables?
How could we help a student stuck at this point?
How many tables would be needed to seat 16
people? How do you know?
What are we looking for in an explanation?
Table problem
What are some possible numbers of people
where there would be seats left over? How do
you know?
This is the first step in generalizing, noticing what is
and isn’t in the pattern.
What description might a 3rd grader give? A 6th
grader?
Table problem
Mr. Barrett says that he will tell you how many
tables he needs, and then he will ask you how
many people are coming. How would you
figure out how many people are coming?
Possible Strategies?
Table problem
Each table adds 2
more people
The number of
people is always
even
tables people
1
4
2
6
3
8
4
10
Summary
TEACH STANDARDS AND
DOCUMENT WHAT YOU TEACH.
RETEACH IN SMALL GROUPS
PER STANDARD.
QUESTIONS AND CONCERNS?
THANK YOU!
CONTACT INFO
DEANNA MCKINLEY
DANIEL PRATT ELEMENTARY SCHOOL
[email protected]
[email protected]
www.dpeseagles.com