The Little Book of Algebra

THE LITTLE BOOK OF ALGEBRA
SUCCESSFUL TRANSITIONS

COMPILED BY ANNE RYAN

A Note on Successful Transitions

Successful Transitions is a ten week programme discovering the fundamentals
of Algebra. The programme is based on initial discussions with the students who
identified the areas of learning they find difficult. Students indicated that
mathematics, particularly Algebra, was the trickiest area in their learning
throughout Secondary School.
What the students have accomplished is commendable. They have kept up with
a class that mirrors that of a(n) University/ College course and produced a group
project through research and critical thinking culminating in a final presentation
to a group of people. This achievement is to be remembered going forward in
their learning journey.
Anne Ryan
Programme Manager, Successful Transitions

The Little Book of Algebra by Anne Ryan is licensed under
a Creative Commons Attribution-NonCommercial 4.0 International
License.

To share or adapt this content please reference:
Ryan, A. (2014) Little Book of Algebra. CC BY-NC. Available:
https://www.linkedin.com/in/annekathleenryan

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The Little Book of Algebra
Table of Contents

A Note on Successful Transitions ............................................................................................................ 1
Section 1 – Algebra (History & Purpose)................................................................................................. 4

Algebra – History & Purpose............................................................................................................... 4
Why Do I Need Algebra?..................................................................................................................... 5

Practice – Section 1......................................................................................................................... 6
Section 2 – Solving for X.......................................................................................................................... 9

Why do We Solve for ‘X’ ..................................................................................................................... 9
Equations ............................................................................................................................................ 9
Solving for the Unknown (X) .............................................................................................................10
Steps for Solving the Unknown (X): ..................................................................................................10

Practice – Section 2.......................................................................................................................11
Section 3 – Order of Operations ...........................................................................................................14

What is the Order of Operations? ....................................................................................................14
Practice – Section 3.......................................................................................................................16

Section 4 – Squares & Cubes.................................................................................................................19
Practice – Section 4.......................................................................................................................20

Section 5 – Quadratic Equations...........................................................................................................23
Quadratic Equations .........................................................................................................................23
Solving Quadratic Equations .............................................................................................................23
Practice – Section 5.......................................................................................................................25

Section 6 – Graphing (Part 1) ................................................................................................................28
Why is graphing an equation helpful? ..............................................................................................28
Steps for Graphing a Linear Equation: ..............................................................................................28
Steps for Graphing a Quadratic Equation: ........................................................................................29
Practice – Section 6.......................................................................................................................30

Section 7 – Graphing (Part 2) ................................................................................................................32
Real-World Mathematics ..................................................................................................................32
Practice – Section 7.......................................................................................................................33

Section 8 – Maths in Nature .................................................................................................................35
Field Trip to Lough Gur .....................................................................................................................35

Useful Videos (Links) Used During Successful Transitions ....................................................................41
Section 1 – Algebra (History & Purpose)...........................................................................................41

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Section 2 – Solving for X....................................................................................................................41
Section 3 – Order of Operations (BODMAS Song!!!) ........................................................................41
Section 6 – Graphing (Part 1) ............................................................................................................41
Section 7 – Graphing (Part 2) ............................................................................................................41
Section 8 – Maths in Nature .............................................................................................................41
Answer Key ...........................................................................................................................................42

Answers – Section 1 ......................................................................................................................42
Answers – Section 2 ......................................................................................................................46
Answers – Section 3 ......................................................................................................................47
Answers – Section 4 ......................................................................................................................48
Answers – Section 5 ......................................................................................................................49
Answers – Section 6 ......................................................................................................................50
Answers – Section 7 ......................................................................................................................51
References ............................................................................................................................................52
Appendix – Lough Gur Stone Circle & Heritage Centre ........................................................................54
Part 1 – Theory & Technology Used to Build the Stone Circle..........................................................55
Part 2 – Activity .................................................................................................................................59

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Section 1 – Algebra (History & Purpose)

Algebra – History & Purpose
o As far as we know, algebra came from Egypt. It dates from when the pyramids

were built, around 2,630 BC. The ancient Egyptians used the word 'aha',
meaning 'heap', to mean an unknown number. In the same way, we might
use the letter X today. Maths exercises have even been discovered dating
back to 1,650 BC which were clearly set as exercises for young
mathematicians. These exercises were written in word problems like the one
below:

Method of calculating a quantity,

Multiplied by 1 1/2 added 4 it has come to 10.

What is the quantity that says it?

Then you calculate the difference of this 10 to this 4. Then 6 results.

Then you divide 1 by 1 1/2. Then 2/3 result.

Then you calculate 2/3 of this 6. Then 4 results.

Behold, it is 4, the quantity that said it.

What has been found by you is correct.

o “Al-Jabr” in Arabic meaning reunion of broken parts was documented in the
year 830 AD as the ‘Science of Restoration and Balance’. Persian
Mathematician Al-Khwarizmi in Israel was the first to write linear and
quadratic equations, but not how we know them today. They were written
in paragraph or word form only.

o 1637 AD – Algebraic Symbols, Powers & Operations were developed by
French Mathematician René Descartes. These are the symbols that help us
write equations in the form we use today (x = y²). Think of it like a new
language.

o Simply put, Algebra is about finding the unknown or it is about putting real
life problems into equations and then solving them. It is a branch of
mathematics that substitutes letters for numbers. An algebraic equation

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represents a weighing scale balance, what is done on one side of the scale
with a number is also done to the other side of the scale. The numbers are
the constants.

Why Do I Need Algebra?
o Algebra develops your thinking, specifically logic, patterns, problem solving,

deductive and inductive reasoning.
o The more math you have, the greater the opportunity for jobs in engineering,

actuary, physics, programming etc.
o Algebra is the foundation for Higher-maths. Higher-maths can help you in

college or universities, both while you are there and when you are sitting
your Leaving Cert.
o Maths is a doable subject and just takes Practice!! Practice will enable you to
feel more comfortable to go onto Higher-maths.

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Practice – Section 1
Each of these word-problems can be answered using Algebraic Equations.
There is a range of problem levels from easy to hard. Answers can be found on
page 42. Remember all levels of mathematics’ can be understood through
practice. Through practice you can progress onto higher levels and use maths
in other areas of learning such as engineering, business studies and in science-
related subjects like chemistry, biology and physics.

Question 1:
A football team lost 5 yards and then gained 9.

What is the team's progress?
Workings:

Question 2:
Maria bought 10 notebooks and 5 pens costing €2 each.

How much did Maria pay?
Workings:

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Question 3:
A customer pays €50 for a coffee maker after a discount of €20.

What is the original price of the coffee maker?
Workings:

Question 4:
Half a number plus 5 is 11.

What is the number?
Workings:

Question 5:
The sum of two consecutive even integers is 26.

What are the two (2) numbers?
Workings:

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Question 6:
The ratio of two numbers is 5 to 1. The sum is 18.

What are the two (2) numbers?
Workings:

Question 7:
The area of a rectangle is 24 cm2. The width is two less than the length.

What are the two (2) numbers?
Workings:

Question 8:
The sum of two numbers is 16. The difference is 4.

What are the two (2) numbers?
Workings:

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Section 2 – Solving for X
Why do We Solve for ‘X’
Take a look at this very short yet e(x)tremely interesting video about why we
commonly use the variable X as the unknown in Algebra:
https://www.youtube.com/watch?v=YX_OxBfsvbk
Equations

o Let’s think about the word EQUATION
o Dictionary Defines the word Equation –
 The act of equating or making equal.
 Equally balanced state; equilibrium.

o An Equation is a Mathematical sentence stating that two things are
equal =

o The = sign is telling you…the left-side of the equation is equal to, or the
same as, the right-side of the equation.

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Solving for the Unknown (X)
Steps for Solving the Unknown (X):

Step One – Combined Like Terms
Step Two – Isolate Like-Terms that Contain the Variable (e.g. 2x – 4x = -2x)
Step Three – Isolate the Variable You Wish To Solve For

Balancing the Scale 8 + 4x – 8 = 8 + 8

When Solving Equations and Isolating
the Unknown (x), Whatever you do to
one side you must do to the other side.

Step Four – Substitute Answer in the Original Equation (check it balances)

Isolating the unknown (X) to find what it equals (=) is the answer!!

Positive (+) & Negative (-) Numbers:
2 + 2 = (+) 4

- 10 + - 10 = (-) 20
6 + - 4 = (+) 2
- 8 + 4 = (-) 4

3 – 10 = (-) 7
When Multiplying or Dividing:

(+, +) = +
(+, -) = -
(-, +) = -
(-, -) = +

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Practice – Section 2 2. x + 3 = 5
(Solving for X) 4. x + 1 = 9
6. 46 = 47 + -1x
Level: Easy 8. 2x = 10
1. x + -6 = 9 10. 4x = 16

3. -32 = 3 + x

5. 12 = -1x + 1

7. 29 - 1x = 13

9. 10x = 130

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Level: Medium 2. 2(3x - 4) = 3x + 1
1. 12 + 5x - 8 = 12x - 10

3. 3x = 5(x + 3) – 3 4. 4x + 7 - 6x = 5 - 4x + 4

5. 3(5x - 2) + 4x = 9x + 6 - 2x 6. 5(2x + 3) = 3(4x + 1) - 2(3x + 2)

7. 2.3x + 1.2 + 2.5x = 9.2 - 4.3x 8. 3.8(x + 0.4) = 1.14

9. 2.25(x - 4.2) = x + 3.28 10. 3.12 + 8.4x + 9.33 = 6.2x - 4.2

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Level: Hard

1. + 1 = + 1 2. 2 ( + 5) = 4
2 3 3 2 3 9

3. 2 +4 = 1 4. 1 � 2 + 34� = 1
3 −1 2 5 4

5. +3 + ( − 1) = 4 6. 4 = 2
2 5 −2 3

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Section 3 – Order of Operations

What is the Order of Operations?

o In the previous section we looked at solving equations by finding the X. But
sometimes expressions are complicated and need to be simplified first
before you can isolate the unknown.

o However, you may encounter misunderstandings or even incorrect answers
to solving a equations purely by the order in which you perform each
operations in an equation.

o Look at this quick example to see where confusion can happen:

Suppose two classes are going on a field trip to the Lough Gur. There are
28 people in one class and 22 people in the other class. The teachers want to
order a snack for all of the students, so they order 2 packages of crisps for each
student. They also want to order 4 extra packets for the teachers. How many
packages of crisps should the teachers order?
Well, here is where order of operations comes in:

2 + 2 x (28+22) =
Without using the Order of Operations 200
When using the Order of Operations 102 and Correct Answer

o Another way to look at the Order of Operations is that an equation is a
mathematical sentence. In language we have grammatical rules to help us
write and communicate with one another. The Order of Operations are the
grammatical rules used in Maths, Science, Engineering, Computer Science
and a number of other disciplines for answers to be the same anywhere in
the world.

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Use a Mnemonic memory devise to Help you Remember the Order of
Operations

B Brackets First
O Orders (i.e. Powers and

Square Roots, etc.)

D Division and Multiplication
M (left-to-right)

A S Addition and Subtraction
(left-to-right)

The BODMAS Song is a fun way to remember the Order of Operations which is
used throughout all of Mathematics and in other disciplines such as Physics,
Computer Science, Engineering and Teaching!!

Available from: https://www.youtube.com/watch?v=Z-FKjqL6NyQ

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Practice – Section 3 2. 7 × 6 – 4
(Order of Operations) 4. 17 × 3 + 2

Level: Easy
1. 8 + 96 ÷ 2

3. 38 × 3 – 65

5. 9 × 3 + 8 6. 51 – 21 × 2

7. 90 ÷ 2 + 4 8. 58 × 2 – 85

9. 24 ÷ 12 – 1 10. 42 – 15 ÷ 5

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Level: Medium 2. 5 × (7 + 11) – 56 ÷ (2 × 4)
1. (7 + 41) ÷ 2 – 15

3. (30 × 3) – 24 4. 7 + 48 ÷ (18 – 16)

5. (21 ÷ 3) + 27 – (8 × 5) + 75 6. (78 – 65) × 5

7. (34 + 15) ÷ 7 8. 3 × (16 ÷ 2) + 16 × 5

9. (10 + 5) × 3 – 25 10. (21 × 3) + 17

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Level: Hard 2. 9 × (72 ÷ 8) + 2 × 32
1. (24 ÷ 8) + 72 – 84

3. (14 × 5) + 25 4. 122 ÷ (4 + 8) × 2

5. 12 × (3 + 4) – (34 ÷ 3) × 2 6. 7 + 66 ÷ (24 – 5)

7. (20 × 2) + 92 8. (62 + 23) ÷ (2 × 5) – 8 + 32

9. 14 + 75 – (31 × 32) 10. (14 – 16) × 34

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Section 4 – Squares & Cubes
o Squaring Numbers allows us to consider the area of an object or space and

Cubing Numbers allows us to investigate the volume of an object or space.
o When squaring your answer is a number multiplied by itself once (squared or

) providing a square or parallelogram.

Squaring
o When cubing your answer is a number multiplied by itself twice (cubed or

) resulting in a three-dimensional cube.

Cubing

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Practice – Section 4

(Squaring & Cubing Numbers)

Fill Out the Table – Squaring Numbers

Question? Numerical Answer Draw its Square

1

4

9

16

25

36

49

64

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81

100

Fill Out the Table – Cubing Numbers

Question? Numerical Answer Draw its Cube













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Section 5 – Quadratic Equations
Quadratic Equations
Quadratic Equations are written in the form + + = . Later on in
Section 8 – Maths in Nature there is an explanation for practical uses of
Quadratic Equations and their symmetry (page xxx).
Solving Quadratic Equations
There are a number of ways to solving Quadratic Equations. The main methods
that came up during the course were:

1. Factoring – Finding what you need to multiply to make the Quadratic. The
table below is quite useful for when you are given Quadratic Equations to
solve using factorisation.

(Signs When Factoring Quadratics)

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2. Using the Quadratic Formula:

= − ± √ −


3. Graphing Quadratic Equations – finding the answer where the quadratic
parabola crosses the x-axis (see the next Section 6 for Steps on Graphing
Quadratic Equations).

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Practice – Section 5
(Solving and Forming Quadratic Equations)
Table 1

Solve the following Quadratic Equations by Factoring

1. + + = 2. + − =

3. − − = 4. − + =

5. − + = 6. + − =

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Table 2

Solve the following Quadratic Equations using the Quadratic Formula

= − ± √ −


1. 2 2 − 7 + 6 = 0 2. 9 2 − 8 − 1 = 0

3. 5 2 − 17 + 6 = 0 4. 8 2 + 10 − 7 = 0

5. 3 2 + 2 − 8 = 0 6. 4 2 + 12 + 5 = 0

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Table 3

Form the Quadratic Equation using the variable ‘x’ for given pair of roots:

1. 2 and 3 2. 1 and 7

Quadratic Equation: Quadratic Equation:

3. 3 and -4 4. 5 and 0
Quadratic Equation: Quadratic Equation:

5. -1 and -2 6. 3 and -8
Quadratic Equation: Quadratic Equation:

7. 6 and 3 8. -7 and 8
Quadratic Equation: Quadratic Equation:

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Section 6 – Graphing (Part 1)

Why is graphing an equation helpful?
It provides us with a visual representation of our mathematical formulas and
their answers (i.e. where the line passes through the x-axis and the y-axis is also
the numerical answer for x and y respectively).
There are two types of equations we looked at and their graphs: Linear Equation
Graphs and Quadratic Equation Graphs. Each of the steps for Graphing both of
these types of equations are set out below along with practice questions. In the
next section (Section 7) graphing will be used to solve actual questions, this is
called modelling.

Steps for Graphing a Linear Equation:

Step One – Make sure the linear equation is in y-intercept form:
y = mx + b

Step Two – Plot the b number on the Y-axis.
Step Three – Convert m into a fraction to show you the rise over run

(slope) of the line.
Step Four – Start extending the line from b using slope, or rise over run.
Step Five – Using a ruler, continue extending the line being sure to use the

slope, m, as a guide.

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Steps for Graphing a Quadratic Equation:
Step One – Make sure the Quadratic Equation is in the form:

= + +

Now label a, b, c

(example)
= + +

a=1

b=2

c=1

Step Two – Find the Axis of Symmetry using the Formula = −


= − = − = −
( )

The Axis of Symmetry is a Vertical Line -1 on the X-Axis

Step Three – Make a Table and Find the Vertex* and a few (X,Y) Coordinates

(The Vertex* is the lowest/highest point of the parabola and is found by
plugging in the answer to Step Two above back into the Quadratic Equation to
find the coordinate for Y – this is your Vertex Coordinate)

XY
-1 0
01
14

Step Four – Use the Axis of Symmetry to Finish the Parabola
(All Points on the Right of the Graph are exactly the same on the Left)

A Helpful and Fun video on How to Graph a Quadratic Equation (Step-by-Step)
(https://www.youtube.com/watch?v=_vkHw7caCN4).

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Practice – Section 6
(Graphing Linear & Quadratic Equations)

Table 1

Graph the Following Linear Equations using the Steps Above
1. = − +
2. = +

xy xy

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Table 2

Graph the Following Quadratic Equations using the Steps Above
3. = − − 4. = − −

xy xy

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Section 7 – Graphing (Part 2)
Graphs are Models and can be used to represent real-life problems. It is the
same techniques described in Section 6 above only now the numbers and
variables are coming from story problems (i.e. word problems). You can
determine profits, how much time a problem may take to resolve itself, etc. The
problems are endless, the way you choose to solve the problem is up to you!!!
Before you head to the practice questions in this section take a look at the video
link below. It describes how one Maths teacher is using new technologies and
real-life scenarios to teach Algebra to students just like you.
Real-World Mathematics
Dan Myer – https://www.youtube.com/watch?v=jRMVjHjYB6w

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Practice – Section 7

(Linear & Quadratic Modelling)

Using the same steps above for Linear Graphs and Quadratic Graphs solve
these two questions through modelling.

Question 1 y= Equation
x= Define your variables:
Paul opens a savings account with €350.
He saves €150 per month. Assume that Write your equation:
he does not withdraw money or make
any additional deposits.

a. After how many months will Paul
have more than €2,000?

y=

Table of Values Graph
XY

Points to Graph:
(,)
( ,)
(,)

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Question 2 Equation
Define your variables:
A bottlenose dolphin jumps out y=
of water. The path the dolphin x=
travels can be modelled by Write your equation:
= − . + . y=

a. What is the maximum height
the dolphin reaches?

b. How far did the dolphin jump?

Table of Values Graph
XY

Points to Graph:
(,)
( ,)
(,)

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Section 8 – Maths in Nature
Field Trip to Lough Gur
As part of Successful Transitions, we took our Algebra out to Lough Gur. See
Appendices at the end of this workbook for the Stone Circle work done with
Dr. Vincent Casey from University of Limerick.

Here are the Pictures you took:

1. An Object in Nature Falling
2. Sunlight Reflected on the Water
3. Symmetry in Nature
4. Fibonacci Number(s) in Nature (1, 1, 2, 3, 5, 8, 13, 21, 34, 55…)
5. Ripple Circle on the Water’s Surface

Question to Think About & Discuss about these Pictures:
What do all five pictures you are taking have in common?

Answers & Conclusions:

o All five of these pictures can be represented by an equation. Therefore they
can all be measured and explored in greater depth. Where you may have
questions (X) you can now find the answers.

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o Mathematics takes what we cannot see and makes it visible and understood.
Solving problems/questions through our mathematical sentences
(equations).

o Once again, Mathematics takes what is invisible in our would and makes it
visible, discovering patterns and regularities of all kinds in our everyday
natural surroundings.

o A lot of what we have looked at is explored in greater depth in physics and
biology studies, but as you can see maths is the foundation which helps to
answer many questions within these disciplines.

o Below are the descriptions of each of the pictures and their respective
mathematical equations.

1. An Object in Nature Falling: discovered by Galileo in 1589 by dropping
two objects from the Leaning Tower of Pisa = � where t is the time
and d is the distance, so if you know the distance you can work out the
time (this assumes the object is in a vacuum).

It was Newton’s Law of Universal Gravitation equation in 1687 that
factored in the gravitational pull between two objects and relationship
between their masses (weight).

=


Where:
F is the force between the masses,
G is the gravitational constant (6.673×10−11 N·(m/kg)2),
m1 is the first mass,
m2 is the second mass, and
r is the distance between the centres of the masses.
(From Article in Nova by Peter Tyson – “Describing Math with Nature”)

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2. Sunlight Reflected in the Water
Speed of Light is c = 2.99792458 x 108 m/s

(or 299,792,458 meters per second) in a vacuum.

Fun Fact: Sunlight takes about 8 minutes 17 seconds to travel the average
distance from the surface of the Sun to the Earth

Here we can Here some of the
see the sun’s sun’s light is
refracted
reflection. through the
water.

o As light travels through different materials, it scatters off of the molecules in
the material and is slowed down. The amount by which light slows in a given
material is described by the index of refraction, n.

o The Index of Refraction of a material is defined by the speed of light in
vacuum c divided by the speed of light through the material v:
n = c/v

The Index of Refraction of some common materials are:

Material N
Vacuum 1
Water 1.33 m/s
1.003 m/s
Air 1.54
Salt 1.635
Asphalt 2.42
Diamond 2.6
Lead

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3. Symmetry in Nature (& Everyday life): + + = Quadratic

Equation graphically is a parabola with the axis of symmetry found using

this equation = − .
2

Calculating Areas

People frequently need to calculate the area of things like rooms, boxes or plots of land. An example might
involve building a rectangular box where one side must be twice the length of the other side. So long as this
ratio is satisfied, the box can be any size, as a person wants it. Say, however, that a person has only 4 square
feet of wood to use for the bottom of the box. Knowing this, the person needs to create an equation for the area
of the box using the ratio of the two sides. This means the area -- the length times the width -- in terms of x
would equal x(2x), or 2x^2. This equation must be less than or equal to 4 for the person to successfully make a
box using the constraint of having only 4 square feet of wood. Once solved, he now knows that one side of the
box must be sqrt(2) feet long, and the other must be 2[sqrt(2)] feet long (adopted from Reference 1).

Figuring Out a Profit

Sometimes calculating a business' profit requires using a quadratic function. If you want to sell something -- even
something as simple as lemonade -- you need to decide how many things to produce so that you'll make a profit.
Let's say, for example, that you're selling glasses of lemonade, and you want to make 12 glasses. You know,
however, that you'll sell a different number of glasses depending on how you set your price. At $100/glass,
you're not likely to sell any, but at $0.01/glass, you'll probably sell 12 glasses in less than a minute. So, to decide
where to set your price, use P as a variable. Let's say you estimate the demand for glasses of lemonade to be at
12 - P. Your revenue, therefore, will be the price times the number of glasses sold: P(12 - P), or 12P - P^2. Using
however much your lemonade costs to produce, you can set this equation equal to that amount and choose a
price from there (adopted from Reference 2).

Quadratics in Athletics

In athletic events that involve throwing things, quadratic equations are highly useful. Say, for example, you want
to throw a ball into the air and have your friend catch it, but you want to give her the precise time it will take
the ball to arrive. To do this, you would use the velocity equation, which calculates the height of the ball based
on a parabolic (quadratic) equation. So, say you begin by throwing the ball at 3 meters, where your hands are.
Also assume that you can throw the ball upward at 14 meters per second, and that the earth's gravity is reducing
the ball's speed at a rate of 5 meters per second squared. This means that we can calculate the height, using the
variable t for time, in the form of h = 3 + 14t - 5t^2. If your friend's hands are also at 3 meters in height, how
many seconds will it take the ball to reach her? To answer this, set the equation equal to 3 = h, and solve for t.
The answer is approximately 2.8 seconds (adopted from Reference 2).

Finding a Speed

Quadratic equations are also useful in calculating speeds. Avid kayakers, for example, use quadratic equations
to estimate their speed when going up and down a river. Assume a kayaker is going up a river, and the river
moves at 2 km/hr. Say he goes upstream -- against the current -- at 15 km, and the trip takes him 3 hours to go
there and return. Remember that time = distance / speed. Let v = the kayak's speed relative to land, and let x =
the kayak's speed in the water. So, we know that, while traveling upstream, the kayak's speed is v = x - 2 (subtract
2 for the resistance from the river current), and while going downstream, the kayak's speed is v = x + 2. The total
time is equal to 3 hours, which is equal to the time going upstream plus the time going downstream, and both
distances are 15km. Using our equations, we know that 3 hours = 15 / (x - 2) + 15 / (x + 2). Once this is expanded
algebraically, we get 3x^2 - 30x -12 = 0. Solving for x, we know that the kayaker moved his kayak at a speed of
10.39 km/hr (adopted from Reference 2).

(Reference: “Everyday Examples of Situations to Apply Quadratic Equations” by Robert Wandrei)

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4. Fibonacci Number(s) in Nature (1, 1, 2, 3, 5, 8, 13, 21, 34, 55…)
1, 2, 3, 4, 5, 6, 7, 8 9 10

Binet's Fibonacci Number Formula
−(1− )
= √5 Where = Golden Ratio 1.618034

= the Fibonacci number you are seeking

e.g. if you are searching for the 10th Fibonacci Number in the sequence
(where = 10)

=a. 10 10−(1− )10
√5

b. 55 = 1.61803410−(1−1.618034)10
√5

c. 55 = 122.9918779 −(8.130620236 10−3)
√5

d. 55 = 122.9837473 = 55.00000382
√5

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5. Ripple Circle on the Water’s Surface

The one-dimensional wave equation (pictured)
describes how much any material is displaced, over time, as the wave proceeds.
The curly "d" symbols scattered through the equation are mathematical
functions known as partial differentials, a way to measure the rate of change of
a specific property of the system with respect to another.
On the left is the expression for how fast the material is deforming (y) in space
(x) at any given instant; on the right is a description for how fast the material is
changing in time (t) at that same instant. Also on the right is the velocity of the
wave (v). For a wave moving across the surface of a sea, the equation relates
how fast a tiny piece of water is physically deforming, at any particular instant,
in space (on the left) and time (on the right).

(Discussed by Alok Jha in The Guardian – “A Short History of Equations”)

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Useful Videos (Links) Used During Successful Transitions

Section 1 – Algebra (History & Purpose)
Algebra: An Upgrade for Your Mind
Nigel Nisbet at TedXYouth@BommerCannyon
Available at: https://www.youtube.com/watch?v=LLsCcyY3kBU

Section 2 – Solving for X
Why is ‘X’ the Unknown
Terry Moore – Ted Talks
Available at: https://www.youtube.com/watch?v=YX_OxBfsvbk

Section 3 – Order of Operations (BODMAS Song!!!)
The BODMAS Song!!
Films of a Mathematical Nature
Available at: https://www.youtube.com/watch?v=Z-FKjqL6NyQ

Section 6 – Graphing (Part 1)
How to Graph a Quadratic Equation (Step-by-Step)
Mr. Hensley’s Class
Available at: https://www.youtube.com/watch?v=_vkHw7caCN4

Section 7 – Graphing (Part 2)
Real-World Math
Mr. Dan Meyer
Available at: https://www.youtube.com/watch?v=jRMVjHjYB6w

Section 8 – Maths in Nature
The Magic of Fibonacci Numbers
Mr. Arthur Benjamin – Mathematician (TedTalk)
Available at: https://www.youtube.com/watch?v=SjSHVDfXHQ4

T h e L i t t l e B o o k o f A l g e b r a Page 41

Answer Key
Answers – Section 1

1. For lost, use negative (-). For gain, use positive (+).
-5 + 9 = 4

Answer: Team’s Progress is 4 Yards

2. Use distributive property to solve the problem.
2 × (10 + 5) = (2 × 10) + (2 × 5)
= 20 + 10 = 30
Answer: Maria Paid €30

3. Let x be the original price.
x - 20 = 50

x - 20 + 20 = 50 + 20
x + 0 = 70
x = 70

Answer: The Original Price of the Coffee Maker was €70

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4. Let x be the number. Always replace "is" with an equal (=) sign.
(1/2)x + 5 = 11

(1/2)x + 5 - 5 = 11 - 5
(1/2)x = 6

2 × (1/2)x = 6 × 2
x = 12

Answer: The Number is 12

5. Let 2n be the first even integer & let 2n + 2 be the second integer.
2n + 2n + 2 = 26
4n + 2 = 26
4n + 2 - 2 = 26 - 2
4n = 24
n=6

So the first even integer is 2n = 2 × 6 = 12 and the second is 12 + 2 = 14
Answer: The two Consecutive Even Integers are 12 & 14

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6. Let x be the first number. Let y be the second number
x/y=5/1
x + y = 18

Using x / y = 5 / 1, we get x = 5y after doing cross multiplication
Replacing x = 5y into x + y = 18, we get 5y + y = 18
6y = 18
y=3
x = 5y = 5 × 3 = 15

As you can see, 15/3 = 5, so ratio is correct and 3 + 15 = 18, so is the sum.
Answer: The two numbers are 3 & 15

7. Let x be the length and let x - 2 be the width
Area = lenth × width = x × ( x - 2) = 24
x × ( x - 2) = 24
x2 + -2x = 24
x2 + -2x - 24 = 0

Since -24 = 4 × -6 and 4 + -6 = -2, we get:
(x + 4) × ( x + -6) = 0

This leads to two equations to solve:

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x + 4 = 0 and x + -6 = 0
x + 4 = 0 gives x = -4. Reject this value since a dimension cannot be negative

x + -6 = 0 gives x = 6
Therefore, length = 6 and width = x - 2 = 6 - 2 = 4
Answer: The Rectangle’s Length is 6cm and Width is 4cm

8. Let x be the first number. Let y be the second number
x + y = 16
x-y=4

Solve the system of equations by elimination by adding the left sides and the
right sides gives:

x + x + y + -y = 16 + 4
2x = 20
x = 10

Since x + y = 16, now replace 10 for x to find y
10 + y = 16

10 - 10 + y = 16 - 10
y=6

Answer: The Numbers are 10 & 6

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Answers – Section 2
(Solving for X)

Easy 2. 2
4. 8
1. 15 6. 1
3. -35 8. 5
5. -11 10. 4
7. 16
9. 13

Medium 2. 3
4. 1
1. 2 6. -4
3. -6 8. -0.1
5. 1 10. -7.57
7. 0.88
9. 10.18

Hard

1. 1 2. − 13 − 4 1
3 3

3. -9 4. 1
1
5. 5 6. 8

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Answers – Section 3
(Order of Operations)

Easy 12. 38
14. 53
11. 56 16. 9
13. 49 18. 31
15. 35 20. 39
17. 49
19. 1 8. 83
10. 31
Medium 12. 65
14.104
7. 9 16. 80
9. 66
11. 69 2. 99
13. 7 4. 6
15. 20 6. 13
8. 31
Hard 10. –162

1. –32
3. 102
5. 30
7. 121
9. –190

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Answers – Section 4

(Squaring & Cubing Numbers)

Squaring Numbers

Question? Numerical Answer
1
4
9
16
25
36
49
64
81

100

Squaring Numbers

Question? Numerical Answer
1
8
27
64

125
216
343
512
729
1000

T h e L i t t l e B o o k o f A l g e b r a Page 48

Answers – Section 5

Table 1

Solve by Factoring

1. (x+3)(x+1) 2. (x+4)(x-2)
3. (x+2)(x-3) 4. (x-3)(x-4)
5. (x-5)(x-1) 6. (x+5)(x-2)

Table 2

Solve Using Quadratic Formula

1. 3 or 2 2. 1 or − 1
2 9

3. 3 or 2 4. 1 or − 7
5 2 4
4 1 3
5. -2 or 3 6. 2 or − 2

Table 3 . − + =
Forming Quadratic Equations (Answers) . − =

. − + = . + − =
. + − = . − − =
. + + =
. − + =

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