Download Chapter One: The Building Blocks Of Algebra

Chapter One:
The Building Blocks
Of Algebra
Index:
A: Rates, Patterns, and Problem Solving (U1L1)
B: Variables and Expressions (U1L2)
C: Commutative and Associative Properties (U1L3)
D: Distributive Property (U1L4)
E: Equivalent Expressions (U1L5)
F: Mindful Manipulations (U1L6)
G: Exponents and Multiplication (U1L7)
H: Complex Equivalency (U1L8)
I: Structure (U1L9)
J: Translating English to Algebra (U1L10)
1
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Rates, Patterns, and Problem Solving 1A
Lesson 1
Algebra at its core is all about using the properties of numbers (how they behave) to manipulate
unknowns, called variables. In particular, Algebra is used to recognize patterns, turn them into
mathematical relationships, and the use these relationships for useful purposes.
Today's lesson is to explore and have an understanding of rates or ratios.
Exercise 1: Answer the following rate/ratio questions using multiplication and division. Show your
calculations. Don't forget your units! :)
(a) If there are 12 eggs per carton, then how many
eggs do we have in 5 cartons?
(b) If a car is traveling at 65 miles per hour,
then how far does it travel in 2 hours?
(c) If a pizza contains 8 slices and there are 4 people
eating, how many slices are there per person?
(d) If a biker travels 20 miles in one hour, how
many minutes does it take per mile traveled?
Rates show up everywhere in the real world, whether it is your pay per hour of work or the texts you can
send per month. Rates are all about multiplication and division because they ultimately are a ratio of
two quantities, both of which are changing or varying.
Exercise 2: A runner is traveling at a constant rate of 8 meters per second. How long does it take for the
runner to travel 100 meters?
(a) Experiment solving this problem by setting up a
table to track how far the runner has moves after
each second.
(b) Create an equation that gives the distance D, that
the person has run if you know the amount of time, t,
they have been running.
(c) Now, set up and solve a simple algebraic equation
based on (b), that gives the exact amount of time it
takes for the runner to run 100 meters.
2
The previous exercise showed how we can take a pattern and extend it into the world of algebra, a world
that contains symbols and conventions that may seem strange, but hopefully somewhat familiar from
previous work. In this final exercise, we will take larger problems to see how rates, patterns, and algebra
can combine to solve a more challenging problem.
Exercise 3: A man is walking across a 300 foot long field at the same time his daughter is walking
towards him from the opposite end. The man is walking at 9 feet per second and the daughter is moving
at 6 feet per second. How many seconds will it take them to meet somewhere in the middle?
(a) Draw a diagram to help keep track of where the man and his daughter are after 1 second, 2 seconds, 3 seconds,
etc. Create a table as well that helps keep track of how far each of them has traveled as time goes on.
(b) What must be true about the distances the two have traveled when they meet somewhere in the middle?
(c) Create equations similar to Exercise #3 to predict the distance the father has traveled and the distance the
daughter has traveled.
(d) Create and solve an equation to predict the exact amount of time it takes for the father and daughter to meet in
the middle.
3
Name:_____________________________________________________
Algebra I
Homework
Date:_________________ Period:_________
Rates, Patterns, and Problem Solving 1A
1) Answer the following rate questions based on either multiplication or division. Think carefully about which is
requires (they will be mixed up). Show the calculation and units that you use.
2) If there are 4 quarts in a gallon, 2 pints in a quart, and 2 cups in a pint, then how many cups are in a gallon?
Show your calculations or explain how you arrived at your answer.
3) A person driving along the road moves at a rate of 56 miles per hour driven. How far does the person drive in
1.5 hours? Show the calculation you use in your answer and given your answer in proper units.
4
4) Mr. Weller has 32 students in his class. He wishes to place them into 8 groups of equal size. Which of the
following represents the number of students per group?
(1) 256
(2) 2
(3) 6
(4) 4
5) Seating in theaters or auditoriums is often arranged such that rows closer to the stage have less seats than rows
farther away. An example of a seating chart for a theater is shown below:
(a) Assuming this pattern continues, fill out the following table:
(b) Jonathan tries to mathematically model the number of seats in a given row. He tries to come up with an
equation for the number of seats and determines:
, where S is the number of seats in row, r
Does this equation work for
answers.
(c) The correct equation is:
? What about
and
? Show calculations that support your yes/no
. Verify that this equation matches your table for
and
.
5
(d) According to the formula from part (c), how many seats are in the 15th row? Show your calculation.
(e) Finally, let’s say we know that a certain row has 91 seats in it. Which row is it? Try to set up and solve a simple
equation that gives you this answer.
6
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Variables and Expressions 1B
Lesson 2
Algebra is the process of using the properties of numbers to manipulate unknown or changing quantities.
These quantities are known as variables and are often represented using letters to distinguish them from
numbers we do know (which we just use the numbers for). When we group (combine) numbers together, we get
what is known as an expression.
An expression is any combination of numbers that we know and ones that we don’t (variables).
Examples:
or
The Order of Operations are rules that govern which mathematical operations are done first.
In order to eliminate any confusion on which steps to follow, the Order of Operations was created. We use the
acronym PEMDAS (MD and AS are interchangeable).
Parenthesis (Start with the innermost and most out)
Exponents
Multiplication/Division (From left to right)
Addition/Subtraction (From left to right)
Exercise 1: Review order of operations by giving the value of each of the following purely numerical expressions.
Do these without a calculator in order to review basic middle school number concepts.
(a)
(b)
(c)
(d)
(e)
(f)
7
Knowing your order of operations is absolutely essential! Once we move past expressions that contain only
numbers to ones that contain variables, you need to be able to "read" an expression and understand what is being
done to the variable.
Exercise 2: If the letter x represents some unknown quantity, explain the calculation that each of the following
expressions involving x represents.
(a)
(b)
(c)
If you can read an algebraic expression (one that has variables), then you should also be able to evaluate the
expression.
Evaluating an Expression: Finding the result of the calculations of an expression when all variables are known.
Example: Evaluate
when
Exercise 3: For each given expression, explain in steps what the calculation is doing and then find its value for the
given variable values.
(a) Evaluate
value.
when
(b) Evaluate the expression
them:
Calculation:
. First explain what calculations are occurring in the expression and then find its
when
. Show the calculations you do and the order in which you do
Explanation:
8
(c) Evaluate the expression
for when
Exercise 4: What is the value of the expression
(1) -3
(2) -8
(3) 3
. Show the steps in your calculation.
when
(4) 7
?
9
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Variables and Expressions 1B
Homework
1) Using order of operations, evaluate the following numerical expressions. Do not use a calculator for this
section.
(a)
(b)
(c)
2) Evaluate the following expressions for the values of x given. Show the steps in your calculation.
(a)
when
(b)
when
(c)
when
3) Robert just got his first job and is saving 45 dollars a week. He also has 155 dollars saved from his birthday that
just passed. To see how much money he will have in his bank account, Robert came up with the following
expression:
, where is the number of weeks that he has been saving.
(a) Exactly how much will he have saved in 6 weeks?
(b) After his first month he had more than he expected to have due to interest the bank provided. This let
Robert come up with a better expression,
will he have in 1 year?
, where
is the number of weeks. How much
10
4) Using order of operations, solve the following. Use your calculator to verify your answer.
(a)
(b)
(c) Explain what changed from the expression in (a) to (b) and why that changed your answer.
5) Andrew received a 95 on his last test and the only question he got wrong was the following:
(a) Read through the question and Andrew’s
work. Find and circle his mistake.
(b) Explain what he did wrong and what he
should have done.
(c) Using your knowledge and abilities, show Andrew how to evaluate the expression correctly. State the
correct value.
11
Review Section:
(1) Robin has 9 packages of gum. There are 15 pieces of gum in each package. How many pieces of gum does
Robin have?
(2) Sixty three people are going to the zoo. There are 3 vans to take people to the zoo. How many people will go
in each van if the same number go in each van?
12
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Commutative and Associative Properties 1C
Lesson 3
Numbers combine through the operations of addition, subtraction, multiplication, and division to produce other
numbers. Sometimes, how they combine is dictated by convention, like with the order of operations. Other
times, though, properties about numbers exist simply due to how these operations work.
Exercise 1: Add the following numbers without using a calculator.
Hint: Although order of operations tells us we should add from left to right, think about an easier way to sum
these numbers. Show how you summed them.
Addition and multiplication have two very important properties with very technical names. The next exercise will
review these properties.
Exercise 2: Fill in the missing blanks for each property.
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The Commutative Property and Associative Property essentially give us permission to rewrite addition and
multiplication problems in different orders than what are normally given.
Exercise 3: Give an example that shows that subtraction is not commutative.
Even though subtraction is not commutative, we should remember a very important fact about subtraction: it can
always be made into the addition of opposites. (5-3 is the same as 5+-3)
Exercise 4: Change the following expression involving addition and subtraction into one only involving addition
and then use the commutative and associative properties to quickly determine the value of this expression.
Through this, please review some properties of negative numbers.
We should be able to now extend the commutative and associative properties for numbers we know to numbers
we don’t know (variables). One of the very nice ways to illustrate the usefulness of these properties is in
combining two or more expressions.
Exercise 5: Please recall the following quickly:
(a)
(b)
(c)
Does anyone know what this is called? ________________________________________________________
Exercise 6: In the following exercise we showed how two linear expressions are combined using various
properties. List what the properties are:
14
Exercise 7: Combine the expressions below. Replace subtraction by addition of opposites, if needed.
(a)
(b)
(c)
(d)
(e)
(f)
15
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Commutative and Associative Properties 1C Homework
Homework
1) Combing the expressions below. Replace subtraction by addition of opposites, if needed.
2) Use the associative property to rewrite the following. You do NOT need to simplify these.
(a)
(b)
(c)
3) Use the commutative property to rewrite the following. You do NOT need to simplify these.
4) Sophia and Emily are twin sisters and best friends. They’re saving up for concert tickets and agreed to pay for
the tickets together when they have enough money. They both created equations to see how fast they were making
money and came up with the following expressions:
Sophia:
Emily:
where is the number of weeks they have been saving.
(a) Combine their expressions to see how much they are making together.
(b) Using the expressions, see if they will have about $350 in four weeks. If not, how much will they be
short?
(c) If their friend Becky also wants to join and is making money according to the expression
create a new expression for the total and see if they will have about $525 for the three of them after four
weeks.
16
5) List which of the associative and commutative properties are being used in each step.
6) An example of an algebraic expression is:
(1)
(2)
(3)
(4)
Review Section:
8) If
then
7) An example of an algebraic equation is:
(1)
(2)
(3)
(4)
equals ____________.
10) What is the value of the expression
and
?
(1) -192
(2) -108
(3) -48
(4) 48
when
9) What is the first step in simplifying the expression
?
(1) square 5
(2) add 4 and 5
(3) subtract 3 from 2
(4) multiply 3 by 4
11) Brett was given the problem: “Evaluate
when
.” Brett wrote that the answer was 41.
Was Brett correct? Explain your answer.
17
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
The Distributive Property 1D
Lesson 4
Working with the Distributive Property
Example:
Since in the order of operations, multiplication comes before addition and subtraction, we must get rid of the
multiplication before you can combine like terms. We do this by using the distributive property:
Now you can combine the like terms:
Final answer:
Exercise 1: Simplify the following using the Distributive Property.
(a)
(b)
(c)
(d)
18
Exercise 2: Consider the product of
(a) Evaluate using the standard algorithm.
(b) Represent the equivalent product
(Hint: Use the Distributive Property.)
as repeated addition of 10 and 5. Find the product.
As you can see, we don’t need a calculator to multiply! We can use the Distributive Property to multiply big
numbers!
HINT: When breaking up the problems, try to break down the numbers into 10’s.
Example:
make into 4(10+2) or 6(28) make into 6(30-2)
Exercise 3: Evaluate each product by using the Distributive Property to make it easier. For (b), express 18 as a
subtraction problem. Do not use a calculator!
(a) 7(23)
(b) 9(18)
Evaluate 4: The Distributive Property can be used twice in order to multiply two digit numbers. For example, find
the product (12)(28) by evaluating (10+2)(20+8). Show each step in your calculation. Do not use a calculator
unless it is to check.
(a)
(b) 24(16)
19
One common mistake that students make is not realizing that the distributive property works for division as well
as multiplication. For division, the property would look in symbolic form like:
Exercise 5: Express each of the following quotients as binomials in simplest form. Show your calculations. Some
of your answers will contain fractional coefficients.
(a)
(b)
(c)
(d)
20
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
The Distributive Property 1D Homework
Homework
1) Simplify the following using the Distributive Property.
(a)
(c)
–
–
(e)
(b)
–
(d)
(f)
–
–
– )
2) Simplify the following expressions using the distributive property. Show your calculations.
(a)
(b)
(c)
(d)
(e)
(f)
21
3) Using the equivalent expressions provided, find the value of the product on the left by evaluating the expression
on the right.
(a)
(b)
(c)
4) Using your knowledge of the distributive property, rewrite the following and evaluate without using your
calculator. See #3 if you need a help on how to do these.
(a) 6(38)
(b) 7(35)
(c) 5(58)
5) In the lesson we saw that we can multiply two digit numbers by using the distributive property twice. Use this
knowledge to multiply the following terms. Show the calculations that lead to your answers.
(a) (22)(31)
(b) (52)(11)
6) Which property is illustrated by the equation
(1) associative
(2) commutative
?
(3) distributive
(4) identity
22
Review Section:
7) If M and A represent integers,
(1) commutative
(2) associative
8) What is the value of
(1) -2
, if
and
(2) 2
is an example of which property?
(3) distributive
(4) closure
?
(3) 10
(4) 14
9) Mr. Stanton asked his students to write an algebraic expression on a piece of paper. He chose four
students to go to the board and write their expression.
Robert wrote:
Meredith wrote:
Steven wrote:
Cynthia wrote:
Which student wrote an algebraic expression?
(1) Robert
(2) Meredith
(3) Steven
(4) Cynthia
23
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Equivalent Expressions 1 E
Lesson 5
The idea of equivalent expressions, or equivalency, is extremely important. It is the basis of how many if not
most of our algebraic manipulations.
Exercise 1: Consider the three expressions below. By substituting in the values of x given, determine which two
expressions are equivalent. Show your calculations of the expressions' values and circle your final answers.
Exercise 2: Which property, the commutative, associative, or distributive, justifies the equivalency of the two
expressions you determined to be equivalent above?
Exercise 3: Which of the following expressions is equivalent to
response. Test at least one value of x to check your answer.
(1)
(2)
(3)
? Show your work to justify your
(4)
24
Exercise 4: Which of the following expressions is equivalent to
? Again, show your work by
thinking carefully about order of operations and the properties we have learned about. Finally, check your answer
by substituting a value of x. Show this check.
(1)
(2)
(3)
(4)
Exercise 5: Which of the following expressions is equivalent to
the space provided.
(1)
(2)
(3)
(4)
? Explain how you made your choice in
Factoring will be one of the most important skills that we want to reach fluency with, but for now we will do some
fairly easy factoring by simply applying the distributive property in "reverse" if you will.
This is called the Greatest Common Factor: Take the greatest common factor (GCF) for the numerical coefficient.
When choosing the GCF for the variables, if all the terms have a common variable, take the one with the lowest
exponent (we will do this later in the year).
Exercise 6: Factor each of the following expressions by writing an equivalent expression that is in the form of a
product. Check your work by using the Distributive Property.
(a)
(b)
(c)
25
Name:_____________________________________________________
Algebra I
Homework
Date:_________________ Period:_________
Equivalent Expressions 1 E Homework
1) Use the Associative, Commutative, and Distributive Properties to write the expression given as an equivalent
expression in simplest form.
26
2) Factor each of the following by using the distributive property (look at Exercise 6 from your notes if you need
help).
3) Taylor is factoring the following expression but notices she got the wrong answer when checking her work.
Identify what she did wrong and show her the appropriate way to factor:
4) Four friends have an assortment of Snack bars that cost dollars each, Munch bars that cost dollars each and
Chewies that cost dollars each that they sell to raise money for a trip they are taking. They decide to split the
money from the sales evenly between the four friends. They create an expression to make sure that everyone gets
the same amount of money. The amount each friend receives is given by the expression below:
(a) Write an equivalent expression that simplifies the
amount that each friend will earn in terms of the unit
costs,
and .
(b) If Snack bars cost $3 each, Much bars cost $5
each, and Chewies cost $4.50 each, then how much
does each friend earn?
27
Review Section:
5) State which property (Associative, Commutative, or Distributive) was used to get from one equivalent
expression to the next.
6) If
and
, what is the value of
?
7) Simplify the following:
8) Simplify the following:
28
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Mindful Manipulations 1 F
Lesson 6
Many times the techniques of algebra can seem like mindless moving of symbols from here to there without any
obvious purpose. In the Common Core, we seek to challenge students to do mindful manipulations. In other
words, always have a reason for the manipulation you are doing.
Exercise 1: Consider the expressions
and
(a) Find the value of both expressions when
.
.
(b) What is the ratio of the larger outcome to the smaller? Why did the ratio turn out this way? What
property can you use to justify this?
Now for some challenging problems! You are now going to be asked to find the value of an expression without
knowing the value of x. Let's try some!
Exercise 2: The expression
is equal to 7 for some value of x (don't solve for it!). Determine the values of
each of the following expressions for the same value of x. Show your reasoning.
(a)
(b)
29
The last exercise forces us to do mindful manipulations because we have to think about how expressions relate to
each other and how to write equivalent expressions. Let's do some more of these types of puzzles!
Exercise 3: The expression
has a value of 10 for some value of x (don't solve for it!). Do mindful
manipulations on each of the following to find the values of these expressions for this same value of x.
(a)
(d)
(b)
(c)
(e)
Exercise 4: If the expression
has a value of -3 for some value of x, then what is the value of the expression
for the same value of x? Show the calculations that lead to your choice below.
(1) 11
(2) -15
(3) 3
(4) -6
30
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Mindful Manipulations 1 F Homework
Homework
1) Evaluate each expression for the given value of x. Do these without the aid of a calculator to practice your
mental arithmetic.
(a)
for
(b)
for
(c)
for
2) If the expression
has a value of -5, then which of the following represents the value of
you arrived at your answer.
(1) -2
(2) -9
(3) -15
(4) -42
? Show how
3) The expression
is equal to 9 for some value of x. Without finding the value of x, determine the values for
each of the following expressions. Show how you arrived at each answer.
(a)
(b)
(c)
4) The expression
expression
(1) 0
(2) 15
(3) 14
(4) -10
(d)
has a value of -5 for some value of x. For the same value of x, what is the value of the
Show your reasoning for this problem in the space provided.
31
5) The number of feet that Jennifer can run in a given time period is given by the expression
. Her friend
Erika can run a distance given by the expression
Erika claims that she can only run half of what Jennifer can
run plus an additional 2 feet. Is she correct?
(a) Let’s build up some evidence by playing around with various values of . Fill out the following chart for
both Jennifer and Erika’s distances given the value of .
(b) The table provides good numerical evidence that what Erika says is true. Show by using mindful
manipulations of the expression
that Erika’s distance is always 2 feet more than half of Jennifer’s.
Review Section:
6) Evaluate:
when
and
7) When solving the equation
for Tom’s first line he wrote
Which property justifies Tom’s first step?
(1) Addition Property of Equality
(3) Commutative Property of Addition
(2) Distributive Property
(4) Multiplication Property of Equality
8) Match each of the following expressions with its equivalent.
_____1)
A)
_____2)
B)
_____3)
C)
_____ 4)
_____ 5)
–
D)
E)
32
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Exponents and Multiplication 1 G
Lesson 7
We've used exponents a little so far, but they will become much more important to us as our studies in Algebra
progress. So, in the next few lessons we are going to work with some basic exponents. Recall that an exponent is a
way to indicate repeated multiplication by the same number.
Exercise 1: Write out what each of the following exponents means as an extended product and find its value.
(a) 24
(b) 32
(c) 53
Of course, just as with numbers, variables can also be raised to exponents (other than 1).
Exercise 2: Write out what each of the following terms involving exponents means as an extended product (do not
solve!). Consider carefully your order of operations and remember that exponents come before multiplication.
(a) x3
(b) x2y4
(c) (2x)2
(d) 4x4y3
(e) (9x2)3
(f) (-4x3)2
Exercise 3: Write out each of the following products and then express them in the form xn.
(a) x2x3
(b) x5x2
(c) x4x4
33
Exercise 4: So what's the pattern? What did you notice? Can you give a general rule for what happens when you
multiply two terms that have the same base?
When multiplying Monomials:
Coefficients:
__________________ the coefficients.
Variables:
When multiplying the variables of monomials you keep the base and ________ the exponents
(Remember if there is no exponent written, the exponent is 1.)
Exercise 5: Quickly write each of the following products as a variable raised to a single power.
(a) x4x9
(b) x2x3x4
(c) y2y6
Often you will need to be able to multiply more complicated terms and write them in as convenient (simplest)
form as possible. The next exercise walks us how to do this and the laws and properties needed.
Exercise 6: The steps to simplifying the product:
justifies each step.
to simplest terms are shown below. Write in what
Exercise 7: Rewrite each of the following as equivalent expressions in simplest exponential form.
(a)
(b)
(c)
(d)
(e)
(f)
34
Name:_____________________________________________________
Algebra I
Homework
Date:_________________ Period:_________
Exponents and Multiplication 1 G Homework
1) Rewrite each of the following terms as extended products. Consider carefully your order of operations and
remember that exponents come before multiplication. You do not need to simplify the products.
(a)
(b)
(c)
(d)
(e)
(f)
2) Write out each of the following products and then express them in simplest exponential form.
(a)
(b)
(c)
3) Rewrite each of the following as equivalent expressions in simplest exponential form. There is one that cannot
be simplified. Circle the letter that can’t be simplified.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
35
4) One of the most common uses of exponents is when dealing with scientific notation. Recall that
is
written in scientific notation where 10 is being raised to the 4th power. If
is the length of a park in meters
and
is the width in meters, what is the area of the park, if it is in the shape of a rectangle? It may help to
write the terms out as an extended product and then regroup them.
5) So far we have come up with an exponent rule for multiplying two monomials with like bases. We saw this to
be
. We can also find a rule for simplifying the expression
Try the following questions and
see if you can find the pattern that helps simplify this type of expression.
(a) Rewrite the following terms as extended products and then express them in the form
(i)
or
.
(ii)
(b) Looking back at part (a), see if you can see a connection between your answer and the question Make a
general rule for all terms in the form of
6) To get you thinking about topics in the future: For multiplying with exponents, we multiply the coefficients and
add the exponents.
What do you think happens when we divide with exponents? ______________________ the coefficient and
_______________________ the exponents.
Try one:
36
Review Section:
7) Use the order of operations to explain why the equation
is false.
8) The expression
has a value of 9 for some value of x (don't solve for it!). Do mindful manipulations on the
following to find the values of these expressions for this same value of x.
(a)
(b)
37
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Complex Equivalency 1 H
Lesson 8
At this point, we should now have a grasp on how to work with exponents. In this lesson, we will continue to
explore expressions that are equivalent but look different. We will primarily be sticking with linear expressions
(those where x is only raised to the first power) and quadratic expressions (where x is raised to the second power).
Recall that two expressions are equivalent if they return equal values when values are substituted into them.
Exercise 1: Consider the product
which by substituting in two values of x to check.
It is equivalent to one of the expressions below. Determine
Now, let's find out how to multiply two simple binomials using a variety of properties.
Exercise 2: The steps in finding the product of
each step.
are shown below. Write down the justification for
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Multiplying Binomials:
-Binomials contain two different terms.
-These terms are grouped together by parenthesis and separated by either an addition or subtraction
symbol.
-When we multiply binomials, we will use Double Distribution. Another Method is the "Box Method",
but we will see this later in the year.
Let's look at this example:
Exercise 3: Write out each of the following as equivalent trinomials (an expression involving three terms).
(a)
(b)
(c)
(d)
(e)
(f)
Exercise 4: Jeremy has noticed a pattern that he thinks is always true. If he picks any number and finds the
product of one number larger and one number smaller than it, the result will always be one less than the square of
his number.
(a) Test two numbers and see if Jeremy's pattern holds true.
(b) Give an algebraic explanation that shows that Jeremy's pattern will work for any number. Use let
statements to clearly define your variables.
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Exercise 5: Which of the following expressions is equivalent to the product
calculation that you use to find your choice and then test using a value of x.
? Show the
(1)
(2)
(3)
(4)
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Name:_____________________________________________________
Algebra I
Homework
Date:_________________ Period:_________
Complex Equivalency 1 H Homework
1) Rewrite each expression as a simpler, equivalent expression by first using the Distributive Property and then
combining like terms.
2) Which of the following expressions is equivalent to
(1)
(2)
3) Continuing with the expression
? Test with a value of x. Show your test.
(3)
(4)
, do the following:
(a) By using the Distributive Property twice,
show that this expression is equivalent to
.
(b) Test the equivalency by finding the value
of
and
when
.
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4) When reading some schematics of a rectangular garden you see the binomial
and the binomial
feet represents the width.
(a)Write an expression that represents the total area of the garden in the form
distributive property.
feet represents the length
by using the
Recall that
(b) Test to make sure that your expression from above is equivalent to
values of x. Show your test for equivalency.
using the following
5) Mariah thinks that the following rule should always hold true. Should it? Find evidence for or against the
following equivalency rule by substituting various values in for and .
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6) Using your understanding of the distributive property, write an equivalent expression of
and . Hint: if you’re having trouble, try referring to problems 2 and 3.
in terms of
7) Challenge: Write an expression that is equivalent to:
HINT: Double Distribute first
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Review Section:
8) Find the product of
and
9) Peter has 42 pieces of candy to divide evenly between his 2 children. If he puts the pieces into two boxes, how
many pieces of candy are there per box?
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Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Structure 1 I
Lesson 9
The more you are able to see structure in the various expressions that you deal with, the easier it will be to
manipulate complex expressions. We will work with structure throughout the course. We can now look at some
larger structural issues that include equivalency.
Exercise 1: Consider the somewhat complex expressions
(a) Write an equivalent trinomial expression. Test the equivalency with a value of
. Show the test.
(b) Write an equivalent expression that is in the form of the product of the two binomials. Also, test the
equivalency with
.
Which type of equivalent expression we might need would depend on the context of what we were trying to do
with the math. For now, we want to get practice with writing various expressions in an equivalent form, and being
able to test that equivalency.
Example: Consider the expression
of the product of two binomials.
Write an expression that is in the form
Step 1: Find what is in common between both terms.
Step 2: See what remains once you "remove"
So what we have is:
Let's try one together!
Exercise 2: Consider the expressions
Write an equivalent expression that is in
the form of the product of two binomials. Test the equivalency with a value of x. Show your test.
45
Exercise 3: Consider the expression
the form of the product of two binomials.
Be careful, this one is subtraction!
Write an equivalent expression that is in
Exercise 4: Rewrite each of the following expressions as an equivalent product of two binomials.
(a)
(d)
(b)
(c)
(e)
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CHALLENGE
Exercise 5: The binomial
is equal to 7 for some value of n. What is the value of the expression shown
below for the same value of ? Do not solve for in this problem! Use mindful manipulations and look for
structrue to help solve this problem.
47
Name:_____________________________________________________
Algebra I
Homework
Date:_________________ Period:_________
Structure 1 I Homework
1) Rewrite each of the following expressions as an equivalent product of two binomials.
2) Which of the following choices is equivalent to the expression
calculations that least to your choice and check using a value of .
(1)
(2)
(3)
(4)
3) If
has a value of 5, then which of the following is the value of
leads to your answer.
(1) 30
(2) 25
(3) 15
(4) 10
? Show the
? Show the work that
4) When figuring out the amount of mulch needed for Alex’s backyard, he created an equation that approximates
the number of bags, , he’ll use. If his equation is
and
is equal to 2, how many
bags will he need? Show your mindful manipulations.
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5) Alex’s friend Pablo comes up with an exact equation to find out how many bags he needs. Use his equation to
find out how many bags will actually be needed if
, where the
quantity
equals 4. Show how you arrived at your answer. (I know this example has “three terms”, it is the
same work as if you were to do it with “three terms”.)
6) In most of the previous examples, there were only two terms. Extend your work with using the Distributive
Law “backwards” and write the following as a product of binomials.
(a)
(b)
Review Section:
7) If A = x – 4 and B = x – 6, what is the product of A and B?
(1)
(3)
–
(2)
(4)
–
–
–
8) Simplify the following:
9) Evaluate
, when
and
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Name:_____________________________________________________
Algebra I
Lesson 10
Date:_________________ Period:_________
Translating English to Algebra 1 J
There will be many instances when we
have to translate phrases from English
into mathematical expressions. This is
a skill that take a lot of practice and
time to master. In this lesson, we will
begin to build this fluency.
Exercise 1: It is important to be able to recognize addition and subtraction in phrases. First, let's begin with some
numerical work and then transition to expressions that only contain variables.
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We also need to be able to translate multiplication and division. Multiplication is typically easier to spot and
translate. Let's try some!
Exercise 2: Translate each verbal statement into an expression and evaluate the expression if it is numerical.
Now we want to be able to put operations together to create more complex expressions. These can be tricky! It is
always important to read them carefully, think about your order of operations, and check them with a real number.
Exercise 3: Translate each of the following statements into an algebraic expression.
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Exercise 4: Neat patterns can occur repeatedly when you play around with numbers. A fairly easy one occurs
when you add a number to one less and one more than that number. Do this for a few numbers, x, and record the
results. Then, prove a general pattern by writing an expression for the sum of a number with a number one less
and a number one more than it.
52
Name:_____________________________________________________
Algebra I
Date:_________________ Period:_________
Translating English to Algebra 1 J Homework
Homework
1) Translate each of the following statements into an algebraic expression.
2) Translate each of the following statements into an algebra expression.
(a) If represents a number, then write an expression for the quotient of twice
(b) If
and 10 more than .
represents a number, then write an expression for one half the sum of and 7.
(c) If represents a number, then write an expression for 7 more than one half the number.
3) Which algebraic expression represents 15 less than
divided by 9?
(1)
(2)
(3)
(4)
4) Which expression represents “5 less than the
product of 7 and ”?
(1)
(2)
(3)
(4)
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5) Which verbal expression is represented by
?
(1) twice the sum of a number and four
(2) the sum of two times a number and four
(3) two times the difference of a number and four
(4) twice the product of a number and four
6) The Miller family made mathematical statements out of their ages as follows. Tom is four less than twice Gary’s
age. Rebecca is the youngest and she is two less than half of Gary’s age after it was increased by three. Sam’s age is
the ratio of seven more than Gary’s age to eight less than Gary’s age.
(a) Translate each of the Miller family members ages into algebraic expressions in terms of Gary’s age, .
Tom’s Age:
Rebecca’s Age:
Sam’s Age:
(b) If Gary is 11 years old, how old are each of the family members?
(c) Using Gary’s age, come up with an expression that represents your age in terms of . Be creative! For
example, if Mr. Weiler is 43 years old, then his age would be
.
7) Our future work in this course will necessitate that we work with what are known as consecutive integers.
Integers are the set of positive and negative whole numbers (as well as zero).
Integers: {…-4,-3,-2,-1,0,1,2,3,4…}
Consecutive integers are lists of integers that increase by one unit between them.
(a) Fill in the pattern with consecutive integers:
2, 3, _____,5, _____, _____, 8
(b) We cal also talk about consecutive even integers and consecutive odd integers. Fill in the patterns.
5, 7, 9, _____, 13, _____, _____
-10, -8, _____, _____, -2, _____
(c) Regardless of whether we have consecutive even integers or consecutive odd integers, to get from one
to another you add what number? If represents the first in a list of consecutive even (or odd) integers,
write out the next three terms.
What do we add to each term?
_____, _____, _____
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Review Section:
8) Expand the following:
9) Simplify the following:
10) Fill in the blanks: When multiplying with exponents, we must _________________ the coefficients and
____________________ the exponents. When dividing with exponents, we must ________________________ the coefficients and
____________________ the exponents.
Try these:
11) The length of a rectangle is two less than three times a number,
same number.
and the width is five morethan that
(a) Draw a diagtam that represents the rectangle. Be sure to label the sides in terms of the
unknown, .
(b) Using your diagram, find what the perimeter of the rectangle is in terms of . Write your
answer as a simplified binomial.
(c) What is the area of the rectangle in terms of ? Write your answer as a trinomial. Remember
the formula for area of a rectangle is
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