Transcript CH 14

CHAPTER 14
Algebraic Thinking: Generalizations,
Patterns, and Functions
Elementary and Middle School
Mathematics Teaching Developmentally
Ninth Edition
Van de Walle, Karp and Bay-Williams
Developed by E. Todd Brown /Professor
Emeritus University of Louisville
Big Ideas
1. Algebra is a useful tool for generalizing arithmetic
and representing patterns in our world.
2. Methods we use to compute and structures in our
number system can and should be generalized.
3. Symbols, especially involving equality and
variables, must be well understood conceptually
for students to be successful in mathematics.
4. Understanding of functions is strengthened when
they are explored across representations (e.g.
equations, tables, and graphs).Each mathematical
model provides a different view of the same
relationship.
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-2
Strands of Algebraic Thinking
Blanton, 2008; James Kaput- 2008
Research suggests three strands of algebraic
reasoning, all infusing the central notions of
generalization and symbolization.
1. Study of structures in the number system
2. Study of patterns, relations, and functions
3. Process of mathematical modeling, including the
meaningful use of symbols
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-3
Connecting Number and Algebra
Number Combinations- looking for generalizations
begins early with decomposition of numbers.
For example
Generalization can be analyzed when data is
recorded in a table.
Place Value- Sum of 49 + 18 can be found with the
use of the hundreds chart. Moves on the chart
can be recorded with arrows for up and down.
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-4
Connecting Number and Algebra
cont.
Algorithms- Sets of problems are good ways for
students to look for and describe patterns across
problems
For example
1 x 12 = 1 x 14 =
1 x 12 =
3 x 12 = 3 x 12 =
2
8
4
4
8
Solve and focus on what they notice. How are the
problems alike and different? How does the
difference in the problem affect the answer?
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-5
Structure in Number System: Properties
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-6
Applying the Properties of Addition and
Multiplication
At the heart of what it means to do mathematics is
•
Noticing generalizable properties 1+6 = 6+1
•
Distributive property central to basic facts 6 x 8
can be split up (5 x 8) + (1 x 8)
•
Justifying a conjecture
provides exploration
of the number system
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-7
Try this one
Activity 14.2 Five Ways to Zero
Materials- deck of cards, counters and number lines
Directions- Draw a card and record five different
ways to get to 0 using number sentences.
For example if you have a 7
7–5–2=0
7 + 3 – 10 = 0
Follow- up with record five different ways to get
your number
Discuss what was true about all of the problems
they wrote.
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-8
Study of Patterns and Functions
Repeating patterns
— identifying
patterns that have
a core that
repeats
— looking for
patterns in
number (placevalue)
_ repeating patterns
in seasons, days
of week, simple
singing do, mi, mi,
do, mi,mi
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-9
Study of Patterns and Functions
cont.
Growing patterns
Explore patterns that involve a
progression from step to
step (sequences)
Extend or identify the core
Look for generalization or
algebraic relationship
Geometric good exemplars to
see the pattern and students
can manipulate the objects.
Include fractions and decimals
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-10
Study of Patterns and Functions cont.
Three types of patterns
Recursive- how a pattern
changes from step to step, x
to y
Covariational thinking- how
two quantities vary in relation,
as step increases by 1,
number of tiles in T goes up
by 3
Correspondence relationshiplook across the table to see
how to use the input to
generate the output
3x + 1
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-11
Study of Patterns and Functions cont.
Growing patterns
represented by
1.Physical
models
2.Tables
3.Words
4.Symbols
5.Graphs
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-12
Study of Patterns and Functions cont.
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-13
Try this one
Activity 14.9 Sketch a Graph
Materials- Graph paper
Directions- sketch a graph to match the following situations.
A.
The temperature of a frozen dinner 30 minutes before
removed from the freezer until removed from the microwave
and placed on the table.
B.
The value of a 1970 Volkswagen Beetle from the time it was
purchased to the present.
C.
The level of water in the bathtub from the time you begin to
fill it to the time it is completely empty after your bath.
D.
Profit in terms of number of items sold
E.
The height of a thrown baseball from when it is released to
the time it hits the ground
F.
Speed of the same baseball
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-14
Linear Functions
Linear functions a subset of function
Rate of Change and Slope
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-15
Linear Functions cont.
Proportional
Represented in these
growing patterns
Non-proportional
One value is constant
For example; pattern block
perimeter problem no matter
what step number you are on
there are two units (one on
each end) that must be added.
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-16
Meaningful Use of Symbols
In algebra symbols represent real situations and are
useful tools in representing situations
Equivalence and equal and inequality signs
— understand and symbolize relationships in our
number system
— how we mathematically represent quantitative
relationships
_ conceptualize as a balance
_ Reinforce that equal means “is the same as”
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-17
Try this one
Activity 14.15 True or False
Materials- Equations
Directions-
More challenging problems use fractions, decimals,
and larger numbers.
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
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Relational Thinking
Operational view- equal sign means “do something”
6 + 7 = ____
Relational- computational view- equal sign
symbolizes a relation between two calculations
7+n=6+9
6 + 9 is 15 so I have to figure out 7 plus what
equals 15
Relational-structural view- equal sign signifies a
numeric relationship 7 + n = 6 + 9
7 is one more that 6 on the other side, so n should
be one less than 9
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-19
Meaning of variables
Variables can be used to represent
•
A unique but unknown quantity
•
A quantity that varies
Example of a variable used as an unknown:
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-20
Meaning of variables cont.
TRY THIS ONE
Examples of problems with
multiple scales with quantities
and variables that may vary.
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
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Using Expressions and Variables
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
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Meaning of Variables cont.
Number line used
to build understanding
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
Table used to guide
thinking and recording
variables
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Mathematical Modeling
Creating equations to describe a situation is
an important skill.
Van de Walle, Karp, Bay-Williams, Elementary and Middle School
Mathematics: Teaching Developmentally, 9/e , © 2016, 2013, 2010, 2007,
2004 by Pearson Education, Inc. All Rights Reserved.
14-24