Transcript Let`s Do Algebra Tiles

Let’s Do
Algebra Tiles
Algebra Tiles
Manipulatives used to enhance
student understanding of subject
traditionally taught at symbolic level.
 Provide access to symbol
manipulation for students with weak
number sense.
 Provide geometric interpretation of
symbol manipulation.

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.

Algebra Tiles
Algebra tiles can be used to model
operations involving integers.
 Let the small indigo square represent
+1 and the small red square (the flipside) represent -1.
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
The indigo and red squares are
additive inverses of each other.
Zero Pairs
Called zero pairs because they are
additive inverses of each other.
 When put together, they cancel each
other out to model zero.

Addition of Integers
Addition can be viewed as
“combining”.
 Combining involves the forming and
removing of all zero pairs.
 For each of the given examples, use
algebra tiles to model the addition.
 Draw pictorial diagrams which show
the modeling.

Addition of Integers
(+3) + (1) =
3
+ 1
=
(6) + (+6) =
6
+6
12
(-2) + (-1) =
-2 + (-1) = -3
4
Addition of Integers
(-3) + (-1) =
-3
+ (-1) = -4
(-4) + (-5) =
-4
+(-5)
-9
(-6) + (-6) =
-6+ (-6) = -12
Addition of Integers
(+3) + (-1) =
3
+ (-1) =
2
(+4) + (-4) =
4
+
0
+
5=
(-2) + 5 =
(-2)
3
After students have seen many examples of addition,
have them formulate rules.
(-4) =
Addition of Integers
Addition can be viewed as
“combining”.
 Combining involves the forming and
removing of all zero pairs.
 For each of the given examples, use
algebra tiles to model the addition.
 Draw pictorial diagrams which show
the modeling.

Subtraction of Integers
Subtraction can be interpreted as
“take-away.”
 Subtraction can also be thought of as
“adding the opposite.”
 For each of the given examples, use
algebra tiles to model the subtraction.
 Draw pictorial diagrams which show
the modeling process.

Subtraction of Integers
(+5) – (+2) =
Remove or take away +2 of the tiles:
So,….What’s left?
(+1) – (+1) =
(-4) – (-3) =
(-7) – (-2) =
Subtracting Integers
(+3) – (-5) =
+3 “remove” -5, but you do not have any
negatives to remove.
Add zero pairs!
Subtracting Integers
Zero pair
(+3) – (-5) =
Now you can remove -5
And, your left with……
Now, try: (-4) – (+1) =
Subtracting Integers
Zero pair
(-4) – (+1) =
Add a zero pair.
Now you can remove +1
And, your left with……
Now, try these:
(+3) – (-3) =
(-8) – (+3) =
4 – (-7) =
-6 – (+9) =
After students have seen many examples, have them formulate rules for
integer subtraction.
So, what are the rules for
subtracting integers?
Look at the second number to see if you can
remove that many from what you have.
 If not, add zero pairs.
 Remove the tiles you need to remove.
 Count tiles left.

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Your end result will be to simply add the
opposite of whatever the sign is.
Multiplication of Integers
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Integer multiplication builds on whole
number multiplication.
Use concept that the multiplier serves as
the “counter” of sets needed.
For the given examples, use the algebra
tiles to model the multiplication. Identify the
multiplier or counter.
Draw pictorial diagrams which model the
multiplication process.
Multiplication of Integers

The counter indicates how many rows
to make. It has this meaning if it is
positive.
(+2)(+3) =
(+3)(-4) =
Multiplication of Integers

If the counter is negative it will mean
“remove” __ sets of __
(-2)(+3)
Remove 2 sets of +3. Nothing to
remove so start with zeros.
Multiplication of Integers
With many examples, students may develop a rule

If the counter is negative it will mean
“take the opposite of.” (flip-over)
(-2)(+3)
Then flip
Two sets of 3
(-3)(-1)
Then flip
Multiplication of Integers
After students have seen many
examples, have them formulate rules
for integer multiplication.
 Have students practice applying rules
abstractly with larger integers.

Division of Integers
Like multiplication, division relies on
the concept of a counter.
 Divisor serves as counter since it
indicates the number of rows to
create.
 For the given examples, use algebra
tiles to model the division. Identify the
divisor or counter. Draw pictorial
diagrams which model the process.

Division of Integers
(+6)/(+2) =
(-8)/(+2) =
Division of Integers

A negative divisor will mean “take the
opposite of.” (flip-over)
(+10)/(-2) =
Division of Integers
(-12)/(-3) =

After students have seen many
examples, have them formulate rules.
Distributive Property
NYS Common Core Learning Standards

3.OA.5 5. Apply properties of operations as strategies to multiply and
divide. Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8
x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)

6.NS.4 Use the distributive property to express a sum of two whole
numbers 1–100 with a common factor as a multiple of a sum of two
whole numbers with no common factor. For example, express 36 + 8
as 4 (9 + 2).

6.EE.3 Apply the properties of operations to generate equivalent
expressions. For example, apply the distributive property to the
expression 3 (2 + x) to produce the equivalent expression 6 + 3x;
apply the distributive property to the expression 24x + 18y to produce
the equivalent expression 6 (4x + 3y); apply properties of operations to
y + y + y to produce the equivalent expression 3y.
Distributive Property
NYS Common Core Learning Standards

7.NS.2a a. Understand that multiplication is extended from
fractions to rational numbers by requiring that operations
continue to satisfy the properties of operations, particularly the
distributive property, leading to products such as (–1)(–1) = 1
and the rules for multiplying signed numbers. Interpret products
of rational numbers by describing real-world contexts.

8.EE.7b Solve linear equations with rational number coefficients,
including equations whose solutions require expanding
expressions using the distributive property and collecting like
terms.
Distributive Property
Use the same concept that was
applied with multiplication of integers,
think of the first factor as the counter.
 The same rules apply.
3(12) You can split this up into
manageable parts: 3(10 + 2)
 Three is the counter, so we need
three rows of (10 + 2)

Distributive Property

3(10 + 2)
30

+ 6
36
Now try these

2(16)
4(14)
2(19)
2(16) = 2(10 + 6)
20
+
32
12
4(14) = 4(10 + 4)
= 40 +16
= 56
2(19) = 2(20 – 1)
=
– 2
40
= 38
Distributive Property
This also works with a
variable in the parentheses.
3(X + 2)
3x + 6

Now try these:
3(x + 4)
2(x – 2)
4(x – 3)
Distributive Property
3(x + 4)
2(x – 2)
2x – 4
3x + 12
4(x – 3)
4x – 12
Solving Equations
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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.
Variables can be isolated by using zero
pairs.
Equations are unchanged if equivalent
amounts are added to each side of the
equation.
Solving Equations

Use the green rectangle as X and the
red rectangle (flip-side) as –X (the
opposite of X).
X+2=3
Solving Addition Equations
X+2=3
Therefore:
X
= 1
Now try these: x + 3 = 8
5 + x = 12
Solving Addition Equations
X+3=8
Therefore:
X
= 5
Solving Addition Equations
5 + x = 12
Therefore:
X
= 7
Solving Addition Equations
9=x+4
Therefore:
5
= x
Now try these: 8 = x + 6
14 = 10 + x
Solving Addition Equations
8=x+6
Therefore:
2
= x
Solving Addition Equations
14 = 10 + x
Therefore:
4
= x
Solving Subtraction Equations
X–2=3
Therefore:
X
= 5
Now try these: x – 3 = 8
5 – x = 12
Solving Subtraction Equations
X–3=8
Therefore:
X
= 11
Solving Subtraction Equations
5 – x = 12
Then flip all to make the x positive
Therefore:
X
= -7
Solving Subtraction Equations
9=x–4
Therefore:
13
= x
Now try these: 8 = x – 6
14 = 10 – x
Solving Subtraction Equations
8=x–6
Therefore:
14
= x
Solving Subtraction Equations
14 = 10 – x
Then flip the sides to make the x positive
Therefore:
-4
= x
Multiplication Equations
2X = 6
Then split the two sides into 2 even groups.
Therefore:
x = 3
Now, try these: 4x = 8
3x = 15
Multiplication Equations
4X = 8
Then split the two sides into 4 even groups.
Therefore:
x = 2
Multiplication Equations
3x = 15
Then split the two sides into 3 even groups.
Therefore:
x = 5
Multiplication Equations
4x = -16
Then split the two sides into 4 even groups.
Therefore:
x = -4
Now, try these: -10 = 2x
-2x = -10
Multiplication Equations
-10 = 2x
Then split the two sides into 2 even groups.
Therefore:
x = -5
Multiplication Equations
-10 = -2x
Then split the two sides into 2 even groups.
But to make the x’s positive, you have to flip
both sides.
Therefore:
x = 5
Division Equations
=6
We can’t split or cut the x into 2 parts here.
So let’s manipulate the two sides to fit the whole
x. It takes 2 groups of the 6 to make the whole
x. So make enough of the number side to
make the whole x—make another group of 6.
Therefore:
x = 12
Division Equations
=9
We can’t split or cut the x into 3 parts here.
So let’s manipulate the two sides to fit the whole
x. It takes 3 groups of the 9 to make the whole
x. So make enough of the number side to make
the whole x—make two other groups of 9.
Therefore:
x = 27
Division Equations
=3
We can’t split or cut the x into 5 parts here.
So let’s manipulate the two sides to fit the whole
x. It takes 5 groups of the 3 to make the whole
x. So make enough of the number side to make
the whole x—make 4 other groups of 3.
Therefore:
x = 15
Division Equations
2=
We can’t split or cut the x into 6 parts here.
So let’s manipulate the two sides to fit the whole
x. It takes 6 groups of the 2 to make the whole
x. So make enough of the number side to make
the whole x—make 5 other groups of 2.
Therefore:
x = 12
Division Equations
-2 =
We can’t split or cut the x into 6 parts here.
So let’s manipulate the two sides to fit the whole
x. It takes 6 groups of the -2 to make the whole
x. So make enough of the number side to make
the whole x—make 5 other groups of -2.
Therefore:
x = -12
Division Equations
= -3
We can’t split or cut the x into 5 parts here.
So let’s manipulate the two sides to fit the whole
x. It takes 5 groups of the -3 to make the whole
x. So make enough of the number side to make
the whole x—make 4 other groups of -3.
Therefore:
x = -15
Division Equations
= -9
We can’t split or cut the x into -3 parts here.
So let’s manipulate the two sides to fit the whole
x. It takes -3 groups of the -9 to make the whole
x. So make enough of the number side to make
the whole x—make two other groups of -9.
Only this time, the x is negative, so we have to flip
everything in order to make it positive.
Two Step Equations
2X + 4 = 6
The first thing is to get the x’s by themselves. Do this by
subtracting 4 from both sides of the equation.
Then, get rid of zero pairs.
Now, split the two sides into equal parts.
Therefore:
x = 1
Two Step Equations
Now, try this:
4x + 2 = 10
First, make your
equation with
algebra tiles
Next, add/subtract
#’s to get the x’s by
themselves.
And get rid of the
zero pairs.
Therefore: x = 2
After that, divide
the sides into even
groups of x’s & #’s
Two Step Equations
How about this one:
3x – 2 = 13
First, make your
equation with
algebra tiles
Next, add/subtract
#’s to get the x’s by
themselves.
And get rid of the
zero pairs.
Therefore: x = 5
After that, divide
the sides into even
groups of x’s & #’s
Solving EquationsVariables on both sides
2X + 3 = X – 5
2X + 3 = X - 5
-X -X
X+3=-5
Solving EquationsVariables on both sides
2X + 3 = X – 5
X+3=-5
-3 -3
X
= -8
Solving Equations
Check by substituting X with - 8
2X + 3 = X – 5
Modeling Polynomials
Algebra tiles can be used to model
expressions.
 Aid in the simplification of
expressions.
 Add, subtract, multiply, divide, or
factor polynomials.

Modeling Polynomials
Let the blue square represent x2, the
green rectangle xy, and the purple
square y2. The red square (flip-side of
blue) represents –x2, the red rectangle
(flip-side of green) –xy, and the large
red square (flip-side purple –y2).
 As with integers, the red shapes and
their corresponding flip-sides form a
zero pair.

Modeling Polynomials
X
x2
y2
y
1
Modeling Polynomials
xy

1
Modeling Polynomials

Represent each of the following with
algebra tiles, draw a pictorial diagram
of the process, then write the symbolic
expression.
2x2
4xy
3
Modeling Polynomials
2x2
4y
3
Modeling Polynomials
3x2 + 2y2
-2xy
-3x2 –xy
More Polynomials
Represent each of the given
expressions with algebra tiles.
 Draw a pictorial diagram of the
process.
 Write the symbolic expression.
x+4

More Polynomials
2x + 3
4x – 2
More 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.
2x + 4 + x + 2
-3x + 1 + x + 3

More Polynomials
2x + 4 + x + 2
-3x + 1 + x - 3
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)
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
Substitution
3 + 2x
let x = 4
3 + 2x
let x = -4
3 – 2x
let x = 4
Multiplying Polynomials
(x + 2)(x + 3)
x+2
x+3
Multiplying Polynomials
(x + 2)(x + 3)
x+2
x+3
=x2 + 5x + 6
Multiplying Polynomials
(x – 1)(x +4)
x-1
x+4
Multiplying Polynomials
(x – 1)(x +4)
x-1
x+4
Multiplying Polynomials
(x + 2)(x – 3)
(x – 2)(x – 3)
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.

Factoring Polynomials
3x + 3
Factoring Polynomials
x+1
3
3x+3= 3(x+1)
Factoring Polynomials
x2 + 6x + 8
Need to make a rectangle
Factoring Polynomials
x2 + 6x + 8
Factoring Polynomials
x2 + 6x + 8
x+2
x+4
Factoring Polynomials
x2 + 6x + 8
Factoring Polynomials
x2 + 6x + 8
Factoring Polynomials
x2 + 6x + 8
Factoring Polynomials
x2 – 5x + 6
Factoring Polynomials
x2 – 5x + 6
+
- -
-
+
-
+
+ + +
+
Factoring Polynomials
x2 – 5x + 6
X-2
x -3
+
-
-
+
+
- -
-
-
+
-
-
+
+ + +
+
Factoring Polynomials
x2 – x – 6
Factoring Polynomials
x2 – x – 6
Can add zeros to “fill in”
Factoring Polynomials
x2 – x – 6
Think about signs
Factoring Polynomials
x2 – x – 6
Factoring Polynomials
x2 + x – 6
x2 – 1
x2 – 4
2x2 – 3x – 2
2x2 + 3x – 3
-2x2 + x + 6
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.

Dividing Polynomials
X+1
x2 + 7x +6
x+1
Dividing Polynomials
x2 + 7x +6
x+1
Dividing Polynomials
2x2 + 5x – 3
x+3
x2 – x – 2
x–2
x2 + x – 6
x+3
Conclusion
“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