Transcript Patterns, Functions and Algebra

Patterns, Functions
and
Algebra
Why do they go together?
Essential Question
Why do Patterns, Functions and
Algebraic Reasoning go together?
What do the three content areas
look like in the elementary grades?
Why teach patterns and functions ?
Looking for patterns trains the mind to
search out and discover the similarities that
bind seemingly unrelated information
together in a whole.
A child who expects things to “make sense”
looks for the sense in things and from this
sense develops understanding.
A child who does not see patterns often does
not expect things to make sense and sees all
events as discrete,separate,and unrelated.
Mary Baratta-Lorton
Vocabulary
Description
Draw a Picture
pattern
Examples
Non-Examples
Patterns
Early experiences should focus on
recognizing regularity.
identifying the same pattern in
different forms.
using patterns to make predictions.
Repeating Patterns
Physical pattern – join in
Using unifix cubes, represent
pattern
For example
Represent pattern using letters
For example
AAB
AAB
Predict 12th element, 20th element…
Recursive (growth) Patterns
Each number is found from the previous
number by repeating some process.
What comes next?
1 , 4 , 7 , 10 , 13 , 16 , …
0 , 2 , 6 , 14 , 30 , 62 , …
2 , 12 , 22 , 32 , 42 , 52 , …
Play Rep-tiles…
Rep –tile Review
For which blocks were you able to
make larger shapes that were
similar to the original block ?
Were there any that could not be
used to make larger,similar shapes?
What pattern,if any, did you notice
in the numbers of blocks used to
build increasingly larger shapes?
Functions
Pairs of numbers that follow a rule
Patterns in the relationship between
two sets of numbers
1
2
3
n
tricycle
tricycles
tricycles
tricycles
3 wheels
6 wheels
? wheels
? wheels
Number of wheels is a function of
the number of tricycles
One way to describe relationship
between number of tricycles and
number of wheels:
(1,3) , (2,6) , (3,9), (x,y)
This set of pairs of numbers forms a
function.
Other Examples
What other patterns can young
children investigate?
wheels on bikes minutes in
hours days in week eyes on faces
pennies in dollars
wheels on cars
wheels on tractor trailers
fingers on hands
Older Children – Formal
Definition: A function is defined as
a set of ordered pairs in which no
two pairs have the same first
number.
The order of numbers is important.
Graphic Organizer
1. Fold a piece of
notebook paper two
times vertically.
The first fold brings
the right edge to
the red line. Then
fold again in half.
2. Now fold vertically
so you have four
sections. The first
fold brings the
bottom edge of the
paper to the top
line. Then fold
again in half.
Function
in words
1.
Wheels to
tricycles
Table
Formula
Graph
Representation
Table
T
W
0
1
0
3
2
6
3
9
…
…
T is number of tricycles
W is number of wheels
Representation
Formula
W = 3T
W is dependent variable
T is independent variable
1’s tell 2’s why ‘dependent’ makes sense.
2’s tell 1’s why ‘independent’ makes sense.
Representation
graph
6
Wheels
5
4
3
This is a linear function.
2
1
0
0
1
2
3
4
Tricycles
5
6
Practice
Using one of the patterns young
children can investigate, with your
partner create a table, formula and
graph. Record on your graphic
organizer.
Share with your table friends,
explaining your work. Record their
example and let them record yours.
Strategies That Most Impact
Achievement
(Marzano and ASCD, 2001 & USDOE:2002)
These are the five strategies that consistently
show the strongest correlation between use
and learning. With your partner, rank them
with 1 being the greatest correlation.We will
come back to this after break…
Vocabulary in context
Non-verbal representations
Extended thinking skills
Advance organizers
Summarizing
RANK
1
2
STRATEGY
Extending Thinking Skills
(Compare/Contrast; Cause/Effect; Classifying;
Analogies/Metaphors)
Summarizing
3
Vocabulary in Context
4
Advance Organizers
5
Non-Verbal Representations
RANK
STRATEGY
%tile
GAIN
1
Extending Thinking Skills
45
2
Summarizing
34
3
Vocabulary in Context
33
4
Advance Organizers
28
5
Non-Verbal
Representations
25