Download Using Algebra Tiles to Represent Expressions

Using Algebra Tiles to Represent Expressions
Algebra tiles are shaped tiles that can be used to represent
expressions and to help solve equations (equations have an
equal sign, expressions don’t).
There are 3 different tiles that are used:
Unit tile – used to represent the value ‘1’
X tile – used to represent the
variable ‘X’
X2 tile – used to represent the
variable ‘X2’
These tiles either come in two different colours, or have
different colours on each side. One colour represents
positive values, and the other represents negative values.
Example: Show x2 + 4x – 5 using algebra tiles.
Solution:
Working with Exponents
‘Powers’ are a shorter way to write repeated multiplication.
They are written with a base and an exponent. The base
tells us the factor to multiply by. The exponent tells us
how many times to complete the multiplication.
For example:
Exponent
45 = 4 x 4 x 4 x 4 x 4
Base
Tips:
 When solving equations, remember to follow
BEDMAS rules.
 Some calculators have x2 buttons
 Some calculators also have yx buttons for other
exponents.
Exponent Rules
To help us solve complicated equations with exponents,
there are a set of rules we can follow:
Product Rule: When multiplying powers with the same
base, add the exponents and write the product as a
single power.
Xa x Xb = Xa+b
Example:
22 x 25 = 22+5
= 27
But how do we know it works? Try it the long way:
22 x 25 = 2 x 2 x 2 x 2 x 2 x 2 x 2
= 27
Quotient Rule: When dividing powers with the same base,
subtract the exponents and write the quotient as a
single power.
Xa  Xb = Xa-b
Example:
46  44
= 46-4
= 42
Try it the long way: (cancel out the pairs)
46  44 =
4x4x4x4x4x4
4x4x4x4
= 42
Power of a Power Rule: A power of a power can be
written as a single power by multiplying the
exponents.
(Xa)b = Xa x b
Example: (32)4
= 32x4
= 38
Try it the long way:
(32)4
= (3 x 3) x (3 x 3) x (3 x 3) x (3 x 3)
= 38
Communicating With Algebra – Polynomials
A polynomial is an expression made up of one or more
terms, connected by addition or subtraction. They are
classified based on the number of terms they have:
One term
Two terms
Three terms
monomial
binomial
trinomial
x, 3y, 4a, etc.
2x – 3, 5b + 2, etc.
4a + 2b – 7, 2x – 3y + 4, etc.
The terms in a polynomial are made up of 2 different parts:
the variable and the coefficient.
 The variable is represented with a letter, and includes
any exponents that are attached to it.
 The coefficient is a number that is part of a term.
 When there is a coefficient without a variable, it is
called a constant.
Example: 2b2 the variable is b2 and the coefficient is 2
3x
the variable is x and the coefficient is 3
6
the constant is 6
When we are describing polynomials we can also describe
them based on the degree of a specific term, or of the entire
polynomial.
Degree of a term: the sum of the exponents in a single
term.
Example:
2b2c3
 2+3=5
 The degree of the term is 5.
Degree of a polynomial: the degree of the term with the
highest degree within the polynomial
Example:
3x2 + 6x – 2
 3x2 is the term with the highest degree.
 The degree of the term is 2
 The polynomial is a second degree polynomial.
Polynomials – Collecting Like Terms
When we work with complicated polynomials it is good to
simplify the expression to make it easier to use. When we
simplify an expression we gather the like terms together.
We decide if terms are alike or not based on the variable.
For example:
6x, x, 7x, -2x, and –3x are all like terms.
-x2, 4x2, and 6x2 are all like terms
4, 5, 8, and 2 are all like terms
6x and –4x2 are not like terms, because the variables
are different.
Once the like terms are grouped together you can add or
subtract them to make simpler expression.
Example: 300 + 4x + 2 x + 150
=
=
300 + 150 + 4x + 2x
450 + 6x
Add and Subtract Polynomials
To add two or more polynomials together, we remove the
brackets, and then collect like terms.
Example:
(4x + 3) + (7x + 2)
= 4x + 3 + 7x + 2
= 4x + 7x + 3 + 2
= 11x + 5
Remember: the sign (+ or -) must go with the term
Example:
(0.5v2 + 2v) + (-2.4v2 – 3v)
= 0.5v2 + 2v - 2.4v2 – 3v
= 0.5v2 - 2.4v2 + 2v – 3v
= - 1.9 v2 – v
the “-“ stays with the term
if the coefficient is 1, we
leave it off
To subtract two polynomials, we add its opposite.
Example:
(3y + 5) – (7y – 4)
= (3y + 5) + (-7y + 4)
= 3y + 5 – 7y + 4
= 3y – 7y + 5 + 4
= - 4y + 9
(-7y + 4) is the opposite of
(7y – 4)
Polynomials – The Distributive Property
When we multiply a polynomial by a number (like ‘5’) or a
variable (like ‘x’), we use a rule called the Distributive
Property.
a(x + y) = ax + ay
Examples:
3(x + 2)
= 3x + 6
3 x x = 3x, 3 x 2 = 6
-6(3b – 3)
= -18b + 18
-6 x 3b = -18b, -6 x –3 = + 18
x(x – 3)
= x2 – 3x
X x X = X2, X x 3 = 3X
Notes:
 Sometimes a polynomial will include multiplication.
We must do the multiplication before simplifying.
 If we have brackets within brackets, solve the inner
brackets first, including any multiplication.
Name: ________________________________ Date: _____________________ Class:
_____
MPM 1D Chapter 3 Test – Polynomials
Knowledge & Understanding: _____ / 24
Application: _____ / 9
Communication: _____ / 7
Thinking: _____ / 6
Multiple Choice (10 Marks K)
Write the letter of the correct answer on the line beside the question.
_____ 1. Which algebraic statement best represents this model?
a) 3x + 3
b) 3
c) 3x2
d) 3x
_____ 2. What is the simplified form of the expression w2 x w4 x w?
a) w5
b) w7
c) w8
d) w9
_____ 3. Which of the following includes three like terms?
a)
3x
2x
-x2
b)
4a
4b
4c
c)
3y
-y
8y
2
d)
3m n 2mn 4mn2
_____ 4. 4x2 + 3x – 14 is an example of a…
a) term
b) monomial
c) binomial
d) trinomial
_____ 5. What is the degree of the term 3v3m?
a) 1
b) 3
c) 6
d) 9
_____ 6. What is the degree of the polynomial x2 + 6x – 2?
a) 1
b) 2
c) 3
d) 4
_____ 7. 3x – 5 + 2x + 3 can be simplified to…
a) 3x
b) 5x – 2
c) x – 2
d) 5x + 2
_____ 8. -2m(3m – 1) simplifies to…
a) -6m2 + 2m
b) -6m2 – 1
c) 6m2 – 2m
d) -6m2 – 2m
_____ 9. In the expression 6x + 4, the 4 is referred to as the…
a) term
b) constant
c) coefficient
d) variable
____ 10. In the term 45, the 5 is referred to as the…
a) exponent
b) power
c) base
d) raised
Short Answer (14 Marks K)
Complete all work in the space provided. Show your steps.
11. Evaluate/Simplify as appropriate. (6 Marks K)
a) (-3)4
b) 23 x 22 x 24
c) 2k + 5m – k – 6m
d) 3(y – 7)
e) (2/3)4
f) -4k(2k + 6)
12. Simplify. (8 Marks K)
a) (5x – 3) + (2x + 7)
b) (y2 – 3y) – (2y2 – 5y)
c) 2(q – 5) + 4(3q + 2)
d) -3(2m – 6) – (8 – 6m)
Full Response/Problems (9 Marks A, 7 Marks C, 6 Marks T)
Complete all questions in the space provided. Show all work, and provide explanations. Use the
back of the page if you need more space.
13. $50 is put into a bank account that pays interest so that the amount in the account will grow
according to the expression 50(1.03)n, where n is the number of years the money is left
in the account.
How much money will be in the account in…
a) 10 years
(4 Marks A, 2 Marks C)
b) 20 years
14. Sarah and Martin are planning to paint the lines on the school soccer field. Sarah says, “The
formula for perimeter is P = 2l + 2w.” Martin replies, “That’s not right, the formula for
perimeter is P = 2(l + w).”
Who is right? Is it possible that they are both right? How do you know?
(2 Marks A, 1 Mark C)
Find at least one other way to write a valid formula for the perimeter of a rectangle, and
explain why it works.
(1 Mark T, 1 Mark C)
15. The sum of the perimeter of two shapes is represented by 13y + 4x. The perimeter of one
shape is represented by 4y – 2x. What is the perimeter of the second shape?
(3 Marks A, 1 Mark C)
16. Three teachers team up to edit a Math textbook. The publisher pays each according to their
own contract:
Teacher
Mr. Law
Ms. Vorobej
Mr. Scharf
Flat Fee ($)
4000
2000
1000
Royalty ($ for n books sold)
none
4n
6n
Create a simplified expression that represents the total the publisher must pay the
teaching team.
(4 Marks T, 1 Mark C)
Determine the total payout if the textbook sells 5000 copies.
(2 Marks T, 1 Mark C)