Download Algebra Tiles . . . Get Them out Dust Them Off

Algebra Tiles. . .
Get Them out
Dust Them Off
Connie Schlimme
Sanderson High
Wake Co.
[email protected]
1
Algebra Tiles. . .
Get Them out
Dust Them Off
Adapted from REL HYBIRD ALGEBRA
RESEARCH PROJECT which was adapted
from David McReynolds, AIMS PreK-16
Project and Noel Villarreal, South Texas
Rural Systemic Initiative
November , 2007
Algebra Tiles
• Manipulatives used to
enhance student
understanding of concepts
traditionally taught at
symbolic level.
• Provide access to symbol
manipulation for students
with weak number sense.
• Provide geometric
interpretation of symbol
manipulation.
3
Algebra Tiles
• Support cooperative
learning, improve discourse
in classroom by giving
students objects to think
with and talk about.
• When I listen, I hear.
• When I see, I remember.
• But when I do, I
understand.
4
Algebra Tiles
• Algebra tiles can be used to
model operations involving
integers.
• Let the small yellow square
represent +1 and the small
red square (the flip-side)
represent -1.

The yellow and red squares are
additive inverses of each other.
5
Algebra Tiles
• Algebra tiles can be used to model
operations involving variables.
• Let the green rectangle represent
+1x or x and the red rectangle (the
flip-side) represent -1 x or -x .

The green and red rods are additive
inverses of each other.
6
Algebra Tiles
• Let the blue square represent x2. The red
square (flip-side of blue) represents -x2.
• As with integers, the red shapes and their
corresponding flip-sides form a zero pair.
7
YOU MAY HAVE AN EARLIER
VERSION THAT LOOK LIKE
THIS:
These will work just as well. . .
with a little alteration
-
-
8
Zero Pairs
• Called zero pairs because they are
additive inverses of each other.
• When put together, they model zero.
• Don’t say “cancel out” for zeroes!
Instead say “they form a zero pair” or
“they add up to zero.”
9
10
Addition of Integers
• Addition is ―combining.‖
• Combining involves the
forming and removing zero
pairs.
• For each of the given
examples, use algebra tiles
to model the addition.
• Draw pictorial diagrams
which show the modeling.
• Write the manipulation
performed.
11
Addition of Integers
(+3) + (+1) =
• Combined like objects to get four
positives
(-2) + (-1) =
• Combined like objects to get
three negatives
12
Addition of Integers
(+3) + (-1) = +2
• Make a zero pair, two
positives remain.
(+3) + (-4) = -1
• Make three zero pairs, one
negative remains.
• Have students practice
applying the concept with
larger integers. Fade away
the manipulatives.
13
Subtraction of Integers
• Subtraction is ―taking
away.‖
• If we do not have enough
positive or negative tiles to
take away, we represent
the number in a different
form by adding zero pairs.
• For each of the given
examples, use algebra tiles
to model the subtraction.
• Draw pictorial diagrams
which show the modeling
process.
• Write a description of the
actions taken.
14
Subtraction of Integers
(+5) – (+2) = +3
• Take away two positives,
three positives remain
(-4) – (-3) = -1
• Take away three
negatives, one negative
remains
15
Subtracting Integers
(+3) – (-5) = +8
Add some zero pairs, take away five
negatives, eight positives remain
(-4) – (+1)= -5
Add some zero pairs, take away one
positive, five negatives remain
16
Subtracting Integers
(+3) – (-3)=
• After students have seen many
examples, students will begin to see
the pattern that subtracting gives the
same result as adding the opposite.
(+3) – (-3) has the same value as 3 + 3
17
Addition/Subtraction of Integers
Try These:
• (+1) + (+ 3)
• (-3) + (-4)
• (+4) + (-6)
• (-7) – (-2)
18
Multiplication of Integers
• Multiplication is repeated addition – we
are combining sets of tiles.
• The first factor tells us the number of sets
needed.
• The second factor tells the size of the set
(how many tiles) and the kind of
numbers/tiles to use (positive or
negative).
• If the first factor is negative,
we interpret that as ―the opposite of.‖
19
Multiplication of Integers
number
of sets
size and
kind of set
(+2)(+3) = +6
(+3)(-4) = +12
Two sets of three
positives
Three sets of
four negatives
20
Multiplication of Integers
• If the first factor is negative it means ―the
opposite of.‖
(-2)(+3) = -6
The opposite of two
sets of three positives
(-3)(-1) = +3
The opposite of three
sets of one negative
21
Multiplication of Integers
Let’s Practice
• (+4)(+2)
• (+5)(-3)
• (-2)(+2)
• (-3)(-3)
22
Multiplication of Integers
• After students have seen
many examples, students
will begin to see the
pattern for integer
multiplication.
• Have students practice
applying the concept with
larger integers. Fade away
the manipulatives.
23
Division of Integers
• Division is putting things
into sets of equal size.
• We start with a total
number of tiles – either
positive or negative.
• The divisor tells us how
many sets to make.
• The answer is how many
tiles are in each set.
• If the divisor is negative, we
interpret that as ―the
opposite of.‖
24
Division of Integers
(+6)/(+2) = 3
• Put the tiles into two equal groups of 3 positives
(-8)/(+2) = -4
• Put the tiles into two equal groups of 4
negatives
25
Division of Integers
A negative divisor means
―take the opposite of‖ (flip-over)
(+10)/(-2) = -5
• Divide ten tiles into two equal
groups of +5
• Find the opposite of +5
26
Division of Integers
(-12)/(-3) =
+4
• Divide 12 negatives into 3
equal groups of -4.
• Find the opposite of -4.
27
Division of Integers
It’s Your Turn!
• (+8)/(+4)
• (+12)/(-3)
• (-12)/(+6)
• (-10)/(-5)
28
Polynomials
and
Algebra Tiles
29
Polynomials
―Polynomials are unlike the
other ‗numbers‘ students
learn how to add, subtract,
multiply, and divide. They
are not ‗counting‘
numbers. Giving
polynomials a concrete
reference (tiles) makes
them real.‖
David A. Reid, Acadia
University
30
Modeling Polynomials
• Algebra tiles can be used
to model expressions.
• Model the simplification of
expressions.
• Add, subtract, multiply,
divide, or factor
polynomials.
31
Modeling Polynomials
2x2
4x
3 or +3
32
More Polynomials
• Represent each of the
given expressions with
algebra tiles.
• Draw a pictorial diagram
of the process.
• Write the symbolic
expression.
x+4
33
More Polynomials
2x + 3
4x – 2
34
More Polynomials
• Examples from page 7
35
Zero Pairs
• Called zero pairs because they are
additive inverses of each other.
• When put together, they model zero.
• Don‘t say ―cancel out‖ for zeroes!
Instead say ―they form a zero pair‖ or
―they add up to zero.‖
+
-
+
36
Find the Additive Inverse
Model and find the additive inverse for:
• 5x + 1
• 3x2 + 1
• -2 + 4x
• -4x2 - 7
37
Simplifying Polynomials:
Combining Like Terms
• 4x – 4x + 1
• 3x – 4x – 1
• 5x2 + 2x – x
• -5 + 6x2 -4x
• 3x + 2x + 5x
38
Simplifying Polynomials:
Distributive Property
• Use the same concept that
was applied with
multiplication of integers:
think of the first factor as
the number of sets.
• Consider the following:
3(x + 2)
• We need three sets of x + 2
39
Distributive Property
3(x + 2)=
3·x + 3·2 = 3x + 6
We have three x‘s
and three groups of +2
3x
+6
40
Distributive Property
Model each of the following:
1. 2(x + 2)
2. 4(x + 3)
3. 3(2x – 3)
4. -3(3x -2)
41
Adding and Subtracting
Polynomials
• Use algebra tiles to simplify
each of the given
expressions. Combine like
terms. Look for zero pairs.
Draw a diagram to
represent the process.
• Write the symbolic
expression that represents
each step.
42
More Polynomials
2x + 4 + x + 1
= 3x + 5
Combine like terms to get
three x’s and five positives
43
More Polynomials
3x – 1 – 2x + 4
• This process can be used
with problems containing
x2.
(2x2 + 5x – 3) + (-x2 + 2x + 5)
(2x2 – 2x + 3) – (3x2 + 3x – 2)
44
Polynomials
Background Knowledge
Activity
Background Knowledge (SIOP)
“For maximum learning to occur,
planning must produce lessons
that enable students to make
connections between their own
knowledge and experience and
new information being taught.”
Echevarria, Jana, Vogt, MaryEllen, Short,
Deborah. (2008) Making Content Comprehensible
for English Learners. SIOP Model. Pearson. 3rd
Edition. P. 23
Ask Yourself
What connections can I
make with students to
introduce
Polynomials?
Let‘s Build Background!
Objective: Today you will
write a definition for the
following vocabulary
words:
Monomial
Trinomial
Binomial
Polynomial
Warm Up
Warm up: With a shoulder buddy determine how
many terms are in each expression? How do you
count the terms?
7x + 2
4x3
9x4 + 11 + 3
2
1
3
A term is _________________________
Categorize Activity
1) Look at your expression
and determine if it has 1
term, 2 terms, or 3 terms.
2) Go to the poster that
represents your
expression.
3) With your group
determine what picture
best represents your
expressions and be ready
to explain why.
Polynomial
(many terms)
(A polynomial is a monomial or the sum or
difference of two or more monomials )
1 term
4x
9y2
-8cd2
9a
-5x2y
xy
2 terms
6c – 5
7x + 4
-a + 2xy
3 terms
4x2 + 2x +5
7x + 4
x-2
5x2 – 2x3
6c-5
5x2 – 2x + 3
t + 14 – xy
4x2 + 2x + 5
5x2 – 2x + 3
X2 + 5 + t
monomial
binomial
A monomial is an
expression with
1 term. (product, variables)
A binomial is an
expression with
2 terms.
trinomial
A trinomial
is an expression with
3 terms.
Write Your Definition
Notes for Polynomials
Directions: Write a definition and illustrate the word.
A monomial is:
A polynomial is:
A binomial is:
A trinomial is:
Reflect
• Did we introduce the concept of
polynomials?
• What was connection we made for
our students?
• Could you use this type of activity in your
classroom?
Polynomial Partner Practice!
54
Substitution
• Algebra tiles can be used
to model substitution.
Represent original
expression with tiles. Then
replace each rectangle
with the appropriate tile
value. Combine like terms.
3 + 2x
let x = 4
55
Substitution
3 + 2x =
3 + 2(4) =
3+8=
11
let x = 4
56
Solving Equations
• Algebra tiles can be used to explain and justify the
equation solving process. The development of the
equation solving model is based on two ideas.
• Equivalent Equations are created if equivalent operations
are performed on each side of the equation. (Which
means to use the additon, subtraction, mulitplication, or
division properties of equality.) What you do to one side of
the equation you must do to the other side of the equation.
• Variables can be isolated by using the Additive Inverse
Property ( & zero pairs) and the Multiplicative Inverse
Proerty ( & dividing out common factors). The goal is to
isolate the variable.
57
Solving Equations
x+2= 3
-2 -2
x =1
•x and two positives are the same as
three positives
•add two negatives to both sides of the
equation; makes zeroes
•one x is the same as one positive
58
Solving Equations
-5 = 2x
÷2 ÷2
2½ = x
•Two x’s are the same as five negatives
•Divide both sides into two equal partitions
•Two and a half negatives is the same as one x
59
Solving Equations
1
 x
2
· -1 · -1
1
 x
2
•One half is the same as one negative x
•Take the opposite of both sides of the equation
•One half of a negative is the same as one x
60
Solving Equations
x
 2
3
•
3 •3
x = -6
•One third of an x is the same as two
negatives
•Multiply both sides by three (or make
both sides three times larger)
•One x is the same as six negatives
61
Solving Equations
2x+3=x–5
-x
-x
x + 3 = -5
+ -3 + - 3
x = -8
•Two x’s and three positives are the same as one x
and five negatives
•Take one x from both sides of the equation; simplify
to get one x and three the same as five negatives
•Add three negatives to both sides; simplify to get x
the same as eight negatives
62
Solving Equations
3(x – 1) + 5 = 2x – 2
3x – 3 + 5 = 2x – 2
3x + 2 = 2x – 2
– 2 or + -2
3x = 2x – 4
-2x -2x
x = -4
“x is the same as four negatives”
63
Multiplying Monomials
(2x)(3)
Make 3 sets of two x’s
64
Multiplying Polynomials by Monomials
x(x – 1)
Fill in each section of
the area model
x2 – 1x
65
Multiplying Polynomials by Monomials
x(2x + 3) = 2x2 + 3x
Fill in each section of
the area model
66
Dividing Monomials
•4x/2x = 2x
Find Zero Pairs and simplify
67
Round Robin Practice
68
Multiplication
• Multiplication using ―area
model‖
(12)(13)
• Think of it as (10+2)(10+3)
• Multiplication using the
array method allows
students to see all four subproducts.
69
Multiplication using “Area
Model”
(12)(13) = (10+2)(10+3) =
100 + 30 + 20 + 6 = 156
10 x 10 =
102 = 100
10
10xx22==20
20
10 x 3 = 30
2x3=6
70
Multiplying Polynomials
(x + 2)(x + 3)
Fill in each section of the
area model
Combine like terms
x2 + 2x + 3x + 6 = x2 + 5x + 6
71
Multiplying Polynomials
(x – 1)(x +4)
Fill in each section of the
area model
Make Zeroes or
combine like terms
and simplify
x2 + 4x – 1x – 4 = x2 + 3x – 4
72
Multiplying Polynomials
(x + 2)(x – 3)
(x – 2)(x – 3)
(x + 4)(x – 2)
73
Factoring Polynomials
• Algebra tiles can be used
to factor polynomials. Use
tiles and the frame to
represent the problem.
• Use the tiles to fill in the
array so as to form a
rectangle inside the frame.
• Be prepared to use zero
pairs to fill in the array.
• Draw a picture.
74
Factoring Polynomials
3x + 3 = 3 · (x + 1)
2x - 6
= 2 * (x – 3)
Note the two are positive, this
needs to be developed
75
Factoring Polynomials
x2 + 6x + 8 = (x + 2)(x +4)
76
Factoring Polynomials
x2 – 5x + 6 = (x – 2)(x – 3)
77
Factoring Polynomials
x2 – x – 6 = (x + 2)(x – 3)
78
Factoring Polynomials
x2 + x – 6
x2 – 1
x2 – 4
2x2 – 3x – 2
2x2 + 3x – 3
-2x2 + x + 6
79
X-box
Factoring
Filling in the X
Trinomial
ax2 + bx + c
1) Place the product of a and c at
the top of the x.
2) Place the coefficient b at the
bottom of the x.
3) Fill the 2 empty sides with 2
numbers that are factors of the
product on the top and add to
give you the value of b on the
bottom.
Product of
a&c
b
Filling in the X
x2 + 9x + 20
1∙20=
20
5
Fill the 2 sides with
numbers that are factors
of 20 that add to give you
9.
4
9
Filling in the X
2x2 -x - 21
2(-21)=
-42
6
Fill the sides with two
numbers that are factors
of -42 that add to give you
-1.
-7
-1
X-box Factoring
• This is a guaranteed method for factoring
quadratic equations—no guessing
necessary!
• We will learn how to factor the
trinomial using the x-box method.
LET‘S TRY IT!
Objective: Students will be able to use the
x-box factoring method to factor nonprime trinomials.
Factor the X-box way
Example: Factor x2 -3x -10
(1)(-10)=
x
-10
-5
x
x2
-5x
GCF
+2
2x
-10
GCF
2
-3
GCF
x2 -3x -10 = (x-5)(x+2)
-5
GCF
Factor the X-box way
y = ax2 + bx + c
First and
Last
Coefficients
Product
GCF 1st
column
GCF 2nd
column
1st
Term
Factor
Factor
Last
term
ac
GCF 1st
Factor
row
Factor
b
Sum
Middle
coefficient
GCF 2nd
row
Factor the X-box way
Example: Factor 3x2 -13x -10
x
-5
3x
3x2
-15x
+2
2x
-10
-30
2
-15
-13
3x2 -13x -10 = (x-5)(3x+2)
Examples
Factor using the x-box method.
1. x2 + 4x – 12
a)
b)
-12
6
-2
4
x
x
x2
+6
6x
-2 -2x -12
Solution: x2 + 4x – 12 = (x + 6)(x - 2)
Examples continued
2. x2 - 9x + 20
a)
20
-4
-5
-9
b)
x
x
-4
x2 -4x
-5 -5x 20
Solution: x2 - 9x + 20 = (x - 4)(x - 5)
Examples continued
3. 2x2 - 5x - 7
a)
-14
-7
2
-5
2x
b)
-7
x
2x2 -7x
+1
2x -7
Solution: 2x2 - 5x – 7 = (2x - 7)(x + 1)
Examples continued
3. 15x2 + 7x - 2
a)
b)
-30
10
-3
7
3x
5x
+2
15x2 10x
-1 -3x
-2
Solution: 15x2 + 7x – 2 = (3x + 2)(5x - 1)
Extra Practice
1. x2 +4x -32
2. 4x2 +4x -3
3. 3x2 + 11x – 20
Reminder!!
Don‘t forget to check your
answer by multiplying!
Dividing Polynomials
• Algebra tiles can be used
to divide polynomials.
• Use tiles and frame to
represent problem.
Dividend should form array
inside frame. Divisor will
form one of the dimensions
(one side) of the frame.
• Be prepared to use zero
pairs in the dividend.
95
Dividing Polynomials
x2 + 7x +6
= (x + 6)
x+1
96
Dividing Polynomials
x2 + 7x +6
x+1
2x2 + 5x – 3
x+3
x2 – x – 2
x–2
x2 + x – 6
x+3
97
Conclusion
• Algebra tiles can be made using the
Ellison (die-cut) machine.
• On-line reproducible can be found
by doing a search for algebra tiles.
• Virtual Algebra Tiles at HRW
http://my.hrw.com/math06_07/nsm
edia/tools/Algebra_Tiles/Algebra_
Tiles.html
National Library of Virtual
Manipulatives
http://nlvm.usu.edu/en/nav/topic_t
_2.html
98
Resources
• David McReynolds
AIMS PreK-16 Project
• Noel Villarreal
South Texas Rural Systemic Initiative
• Jo Ann Mosier & Roland O‘Daniel
Collaborative for Teaching and
Learning
• Partnership for Reform Initiatives
in Science and Mathematics
99