Transcript Slides - Math Professional Development

Patterns and
Pre-Algebra
Upper Elementary
Grades 4–6
1
Agenda
What Is Early Algebra and Why Teach It?
There Are Different Types of Patterns
Algebraic Thinking Is Generalizing Relationships
Your Questions
Break
Algebraic Thinking Is Understanding Change
Pattern Rules Describe Relationships
Patterns, Relations and Functions Can Be Modelled
Your Questions
Lunch
Algebraic Symbols Describe Mathematical Situations—Equality
Your Questions
Break
Algebraic Symbols Describe Mathematical Situations—Variables
Your Questions and Feedback
2
Mathematical Metaphors
When I think of algebra, I feel…
Note: From the University of Alberta/Edmonton Catholic Separate School District No. 7 Collaborative Project, 2003.
3
Why Teach Algebraic Thinking?
Through the study of patterns, students come to
interpret their world mathematically and value
mathematics as a useful tool.
Working with patterns enables students
to make connections both within and
beyond mathematics.
4
Why Teach Algebraic Thinking?
By generalizing patterns, students develop
strategies that can be used to solve a wide
range of problems. Mathematics is seen as
reasoning rather than solving one unrelated
problem after another.
3 ÷ 4 = 3/4
Exploring patterns and pre-algebra in
elementary school lays the foundation for the
study of formal algebra. Rather than a new
topic, algebra becomes a natural extension of
the elementary curriculum and is often defined
as generalized arithmetic and geometry.
5
1
4
9
Concrete Representations
Visual Patterns
Context Problems
16
Numbers
Charts
Tables
6
Patterns can be repeating and made up of a core set of elements –
a core unit that is iterated.
Patterns can be increasing or decreasing and created by orderly change.
9
7
5
32
3
16
8
4
2
7
Learning Task – Last One Standing
No. 1 says “in” and remains standing.
No. 2 says “out” and sits down.
No. 3 says “in” and remains standing.
No. 4 says “out” and sits down.
The contest continues repeatedly around the circle.
The last one standing wins the contest!
Note: Excerpted and reprinted with permission from Green, D. A. (2002). Last One Standing: Creative, Cooperative Problem
Solving. Teaching Children Mathematics, 9(3), 134–139, copyright 2002 by the National Council of Teachers of
Mathematics. All rights reserved.
8
Learning Task – Last One Standing
What patterns were non-numerical? Numerical?
What patterns repeated? How did the patterns grow?
9
Learning Tasks – Charts and Diagrams
What are the Clues?
page 166
Circle Patterns
page 183
Multiple Patterns
page 179
Home Sharing Groups
Expert Groups
10
Learning Tasks – Charts and Diagrams
What are the Clues?
Circle Patterns
Multiple Patterns
Explain the task.
Describe the patterns you found.
What connections are there to other mathematical big ideas?
At what grade level could this task be used?
Share any suggested tips or extensions to the task.
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12
Algebrafying the Curriculum – Solids
Note: From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary Teachers (pp. 26–27), by
K. Willson, L. Gibeau and R. McKay, 2006, Edmonton, AB: Ioncmaste, University of Alberta. Copyright 2006 by the authors.
Reprinted with permission.
13
Algebrafying the Curriculum – Solids
If you know the shape of the base, how can you find the number
of edges, faces and vertices without counting them all?
Shape of Base
No. of Sides
on Base
Edges
Faces
Vertices
Triangle
Rectangle/Square
Pentagon
Hexagon
What patterns are in each of the columns?
How is the base related to the edges, faces and vertices?
How many edges, faces and vertices on a 20-sided pyramid?
14
Algebrafying the Curriculum – Solids
Algebrafying the Curriculum - Combinations
If you had 2 shirts and 3
pants, how many different
outfits could you make?
Algebrafied Version:
What if you had 2 shirts and
4 pants? 5 pants? 6 pants?
How many shirts and pants
will make exactly 21 outfits?
16
Algebrafying the Curriculum - Probability
How many possible
outcomes are there when
you toss two coins?
Algebrafied Version:
How many outcomes
for 3 coins? 4 coins? 5
coins? 10 coins?
heads–tails
tails–tails
tails–heads
heads–heads
Adding one coin doubles the
number of possible outcomes.
17
Learning Task – Pool Tile
What changes? What stays the same?
18
Learning Task – Iguanas
Note: Adapted from Lessons for Algebraic Thinking: Grades K–2, pp. 91–117, by Leyani von Rotz and Marilyn
Burns. Copyright © 2002 by Math Solutions Publications.
Note: Student samples: Edmonton Catholic Schools, 2005.
19
Learning Task – Iguanas
Fish
Caterpillars
Note: Adapted from Lessons for Algebraic Thinking: Grades K–2, pp. 2–11, by Leyani von Rotz and Marilyn Burns.
Copyright © 2002 by Math Solutions Publications.
20
Learning Task – What Changes?
What Stays the Same?
Frame 1
Frame 2
Frame 3
Frame 4
Frame
number
1
2
3
4
Number of
Tiles
4
8
12
16
21
Learning Task – What Changes?
What Stays the Same?
2, 4, 6, 8,10,…
1, 3, 6, 10, 15,…
Create a visual pattern for your
number sequence.
Can you find others who had
the same sequence?
20, 16, 12, 8, 4
2, 4, 8, 16, 32,…
22
Understanding Change
in Real-life Contexts
Heartbeat ↔ Age
Time in seconds ↔ Level of water
Distance driven ↔ Gas in tank
23
Input
2
4+
Note: Adapted from Lessons for Algebraic Thinking: Grades K–2, pp. 91–117, by Leyani von Rotz and Marilyn
Burns. Copyright © 2002 by Math Solutions Publications.
Note: Student samples: Edmonton Catholic Schools, 2005.
Output
6
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 A pattern rule must account for all elements of a pattern,
including the first one.
 Pattern rules can describe recursive or functional relationships.
+4
+6
+8
start
at 2
Recursive rule: from step to step or frame to frame
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 A pattern rule must account for all elements of a pattern,
including the first one.
 Pattern rules can describe recursive or functional relationships.
(1 × 1) + 1
(2 × 2) + 2
(3 × 3) + 3
(4 × 4) + 4
“x2 + x”
Functional rule: from step no. to step or frame no. to frame
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Frame
Number
Frame
1
2
3
4
5
6
2
6
12
20
30
42
Looking at relationships
down columns
Recursive
Looking at relationships
across columns
Functional
27
Learning Task – Twelve Days of Christmas
Analyzing Recursive Patterns
Note: From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary Teachers (p. 34),
by K. Willson, L. Gibeau and R. Mckay, 2006, Edmonton, AB: Ioncmaste, University of Alberta. Copyright 2006 by the
authors. Reprinted with permission.
28
Learning Task – Twelve Days of Christmas
Analyzing Recursive Patterns
How many gifts were given on the 12th day?
Note: From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary Teachers (p. 36),
by K. Willson, L. Gibeau and R. Mckay, 2006, Edmonton, AB: Ioncmaste, University of Alberta. Copyright 2006 by the
authors. Reprinted with permission.
29
Learning Task – Twelve Days of Christmas
Analyzing Recursive Patterns
I can’t do
this
anymore!
How many gifts were given on the 12th day?
Note: From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary Teachers (p. 40),
by K. Willson, L. Gibeau and R. Mckay, 2006, Edmonton, AB: Ioncmaste, University of Alberta. Copyright 2006 by the
authors. Reprinted with permission.
30
Learning Task – Twelve Days of Christmas
Analyzing Recursive Patterns
How many gifts were given on the 12th day?
Note: From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary
Teachers (pp. 37–39), by K. Willson, L. Gibeau and R. Mckay, 2006, Edmonton, AB: Ioncmaste, University of
Alberta. Copyright 2006 by the authors. Reprinted with permission.
31
Learning Task – Twelve Days of Christmas
Analyzing Recursive Patterns
What is the total
amount of each gift
given?
Using multiplication sentences helped these
students to discover a new pattern. As the
number of days each gift was given
decreases by one, the quantity of each gift
increases by one.
12 × 1
11 × 2
10 × 3
9×4
8×5
7×6
12 days with 1 partridge
11 days with 2 turtle doves
10 days with 3 French hens
9 days with 4 calling birds
8 days with 5 golden rings
7 days with 6 geese
Note: From Analyzing Students’ Thinking on Mathematical Tasks: Professional Development for Elementary Teachers (p. 41),
by K. Willson, L. Gibeau and R. Mckay, 2006, Edmonton, AB: Ioncmaste, University of Alberta. Copyright 2006 by the
authors. Reprinted with permission.
32
Learning Task – Patterns with Perimeters
Analyzing Functional Patterns
33
Learning Task – Patterns with Perimeters
Analyzing Functional Patterns
Grade 4: add 2 to the frame number/number of tiles
Grade 5: t + 2 or  + 2 where t &  are the number of triangles in the string
Grade 6: p = t + 2 where p is perimeter and t is the number of triangles
3 × 5 = 15
(3 × 5) – (2 x 4) = 7
Perimeter = number of sides x number of tiles – [2 × (number of tiles – 1)]
(square):
P = 4 × number of tiles – [2 × (number of tiles – 1)]
(pentagon):
P = 5 × number of tiles – [2 × (number of tiles – 1)]
(hexagon):
P = 6 × number of tiles – [2 × (number of tiles – 1)]
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Learning Task – Patterns with Perimeters
Analyzing Functional Patterns
Across the rows,
the perimeter
increases by
Number
of tiles
Perimeter
Equilateral
triangle
Square
Pentagon
Hexagon
1 unit
1
3 units
4 units
5 units
6 units
2 units
2
4 units
6 units
8 units
10 units
3 units
3
5 units
8 units
11 units
14 units
4 units
4
6 units
10 units
14 units
18 units
5 units
5
7 units
12 units
17 units
22 units
10 units
10
12 units
22 units
32 units
42 units
20 units
20
22 units
42 units
62 units
82 units
100 units
100
102 units
202 units
302 units
402 units
Triangles
Squares
Pentagon
Hexagon
Perimeter
increases by
1 unit each
frame
Perimeter
increases by
2 units each
frame
Perimeter
increases by
3 units each
frame
Perimeter
increases by
4 units each
frame
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Typical Pattern Lesson
Developing relational thinking
1. Build the first three frames and extend:
How many in the next frame?
2. Keep extending the pattern, frame by frame:
The next frame after that?
3. Identify a pattern:
What do you notice? Why is this happening?
4. Push students to generalize:
How many in the 10th frame? 100th frame?
How can you find the answer for any given frame?
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37
Learning Task – Handshake Problem
Ten students arrive at a special gathering for
students taking part in a Mathematics Fair. As the
students are all from different schools, the teacher
wants each to get to know the others. The teacher
asks each student to shake hands with each of
the other students and introduce themselves. How
many handshakes took place?
38
Learning Task – Handshake Problem
Note: Student samples: Edmonton Catholic Schools, 2005
39
Learning Task – Handshake Problem
Models become tools for thinking
40
What Is Early Algebra?
Representing and analyzing mathematical problems
using algebraic symbols
Understanding the meaning
of equality
Understanding the
conventions and multiple
meanings of variables
41
What do
elementary
students think
the equal sign
means?
42
8+4=
+5
7
12
17
12 and
17
Other
Grades 1 and 2
5%
58%
13%
8%
16%
Grades 3 and 4
9%
49%
25%
10%
7%
Grades 5 and 6
2%
76%
21%
1%
0%
• The answer comes next: 8 + 4 = 12 + 5
• Use all the numbers (overgeneralizing associative property): 8 + 4 = 17 + 5
• Extending the problem: 8 + 4 = 12 + 5 = 17
Note: From Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (p. 4), by T. P. Carpenter, M. L. Franke and
L. Levi, 2003, Portsmouth, NH: Heinemann. Copyright 2003 by the authors. Reprinted with permission.
43
Robin: Second-grade student
18 + 27 =  + 29
“Twenty-nine is 2 more than 27, so the number in the box has to be
2 less than 18 to make the 2 sides equal. So it’s 16.”
Note: From Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (p. 4), by T. P. Carpenter, M. L. Franke
and L. Levi, 2003, Portsmouth, NH: Heinemann. Copyright 2003 by the authors. Reprinted with permission.
44
Equality and inequality between quantities can be considered as:
•
•
•
•
whole to whole relationships (5 = 5)
part–part to whole relationships (3 + 5 = 8)
whole to part–part relationships (8 = 5 + 3)
part–part to part–part relationships (4 + 4 = 3 + 5).
45
Learning Tasks – True or False Mini-lessons
Determine if these equations are
true or false without calculating the
actual sum or difference. Use
relational thinking!
37 + 56 = 39 + 54
33 – 27 = 34 – 26
471 – 382 = 474 – 385
674 – 389 = 664 – 379
583 – 529 = 83 – 29
46
Learning Task – Virtual Pan Balance ~ Shapes
Note: Excerpted and reprinted with permission from National Council of Teachers of Mathematics. (2006). Pan Balance—Shapes.
Illuminations. Retrieved November 29, 2006, from http://illuminations.nctm.org/ActivityDetail.aspx?ID=33, copyright 2006 by
the National Council of Teachers of Mathematics. All rights reserved.
47
Learning Task – Virtual Pan Balance ~ Numbers
Note: Excerpted and reprinted with permission from National Council of Teachers of Mathematics. (2006). Pan Balance—Numbers.
Illuminations. Retrieved November 29, 2006, from http://illuminations.nctm.org/ActivityDetail.aspx?ID=26, copyright 2006 by
the National Council of Teachers of Mathematics. All rights reserved.
48
Generalizing The Meaning of the Operations
Part-part-whole
or Collection
Comparison
Whole Unknown
Part Unknown
Connie has 15 red marbles and 28
blue marbles. How many marbles
does she have?
Connie has 43 marbles. 15 are red
and the rest are blue. How many
blue marbles does Connie have?
Difference
Unknown
Unknown Big
Quantity
Unknown Small
Quantity
Connie has 15 red
marbles and 28 blue
marbles. How many
more blue marbles
than red marbles does
Connie have?
Connie has 15 red
marbles and some
blue marbles. She has
13 more blue marbles
than red ones. How
many blue marbles
does Connie have?
Connie has 28 blue
marbles. She has 13
more blue marbles
than red ones. How
many red marbles
does Connie have?
whole
Big Quantity
Small Quantity
part
Differenc
e
part
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Multiplication
Givens: number of
groups and number of
objects in each group
Problem Type
Grouping/
Partitioning
Givens: total number of
objects and the number
of objects in each group
Partitive Division
Givens: total number
of objects and the
number of groups
Gene has 4 tomato
plants. There are 6
tomatoes on each plant.
How many tomatoes are
there altogether?
Gene has some tomato
plants. There are 6
tomatoes on each plant.
Altogether there are 24
tomatoes. How many
tomato plants does Gene
have?
Gene has 4 tomato
plants. There is the
same number of
tomatoes on each plant.
Altogether there are 24
tomatoes. How many
tomatoes are on each
tomato plant?
The giraffe in the zoo is 3
times as tall as the
kangaroo. The kangaroo
is 6 feet tall. How tall is
the giraffe?
The giraffe is 18 feet tall.
The kangaroo is 6 feet
tall. The giraffe is how
many times taller than the
kangaroo?
The giraffe is 18 feet
tall. She is 3 times as
tall as the kangaroo.
How tall is the
kangaroo?
Equal Groups
Multiplicative
Comparison
Measurement
Division
2
K
4
4×
G
6
6
6
6
6
6
6
6
50
Symmetric Problems
A farmer plants a rectangular vegetable garden that measures 2 m along one
side and 5 m along an adjacent side. How many m 2 of garden did the farmer
plant?
Area and Array
A baker has a pan of fudge that measures 8 inches on one side and 9 inches on
another side. If the fudge is cut into square pieces 1 inch on a side, how many
pieces of fudge does the pan hold?
A farmer plants a rectangular garden. She has enough room to make the garden
5 m on one side. How long does she have to make the adjacent side in order to
have 10 m2 of garden?
Combinations
The Friendly Old Ice Cream Shop has 2 types of cones I(waffle and plain). They
have 5 flavours of ice cream (chocolate, vanilla, strawberry, rainbow and tiger).
How many one-scoop combinations of an ice cream flavour and cone type can
you get at the Friendly Old Ice Cream Shop?
5
2
5
2
C
S
V
R
T
W
Wc
Ws
Wv
Wr
Wt
P
Pc
Ps
Pv
Pr
Pt
51
Write an algebraic equation to represent this situation:
“There are twenty-four students
for every teacher in Grade 5.”
52
Write an algebraic equation to represent this situation:
“There are twenty-four students
for every teacher in Grade 5.”
In various studies, from 1/3 to 1/2 of adults
answered this question incorrectly.
53
Common Student Misconceptions About Variables
Variables Are
Objects
n + 3 = 12
x + 3 = 12
+ 3 = 12
+ 3 = 12
have different
answers
Variables Always Have
One Value
a=a
is always true, but
c=a
is never true
x–x=0
is a solvable equation
54
Variables as a Specific Unknown
a × 9 = 36
Mathematician’s Rule
×
=
36
×
=
36
55
Variables as Dynamic Quantities
Pattern Rules:
Expressions and
Equations
p+4
s = 24p
e=s+c
Formulas,
Conversions and
Rates
cm = 2.54 × i
p = $2.10 × d
A=l×w
56
Variables as Generalizers
a+b=c
b+a=c
c–a=b
c–b=a
a+b=b+a
(x + y) + z = x + (y + z)
a+0=a
a×b=c
b×a=c
c÷a=b
c÷b=a
a–b≠b–a
57
Learning Tasks – Variables
Algebra
Islands
a
b
The Variable
Machine
c
58
Understanding the Meaning of Variables
Note: From Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School, by T. P. Carpenter, L. M. Franke and
L. Levi, 2003, Portsmouth, NH: Heinemann. Reproduced with permission.
59
Each problem that I solved
became a rule which served afterwards
to solve other problems.
René Descartes
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