8 Grade
Common Core
Math
th
Booklet 2
Expressions and Equations
Main Idea of Expressions and Equations:
Understand the connections between proportional relationships,
lines, and linear equations.
What are Lines, Linear Equations, and Proportional Relationships?
Lines are straight, one-dimensional figures that extend infinitely in both
directions.
Examples of lines:
Linear equations are equations that have two variables. A variable is a
letter that is used to express an unknown number such as the letter x and
y.
These are examples of linear equations: y = 7x + 20 y = 2x + 1
y=x
Any time you graph a linear equation it will be in a straight line. For
example, this is the graph of y= 2x+1
HOW TO GRAPH y = 2x + 1:
In order to plot a linear equation on a graph,
you will need to find the coordinates of the
points on the graph. So by plugging in a
value (number) for x, you can solve the
equation to find the value of y. This will
become one of your coordinate pairs (x,y).
For example, if we plug in the number 1 for
the letter x, we get: y = 2 * 1 + 1 . So, y = 3.
That means the coordinate point would be
(1,3). You can do this to find more
coordinate points and connect them with a
ruler to make the line graph of y= 2x + 1.
Proportional Relationships are relationships between two quantities in
which the two quantities vary directly with one another and stay in
proportion.
Examples:
x
2x
In this example, we have two
squares. The second square is
twice as large as the first square.
Its side lengths are 2 times as long
as the smaller square. We see that
the small square has a side length
of x which means the larger square
has a side length of 2x.
Small
Square
side
(x)
Large
Square
side
(2x)
1
2
2
4
3
6
4
8
5
10
This table shows potential
values for x (small square
side) and it shows that the
large square side (2x) is twice
as much. It increased
proportionally.
8th Grade Common Core Math Standard:
Standard 8.EE.B.5: Graph proportional relationships, interpreting the unit
rate as the slope of the graph. Compare two different proportional
relationships represented in different ways. For example, compare a
distance-time graph to a distance-time equation to determine which of two
moving objects has greater speed.
What the student learns: Students learn how to graph a proportional
relationship (as explained on pages 1-2). They understand that a unit rate
in a math problem is the slope of the line graphed. They also learn to
compare different proportional relationships, such as equations and
graphs to find to find what they are looking for.
Standard examples:
David's Mile Time (In Minutes) y axis
2.5 2 Miles
(12 , 2) (9 , 1.5) 1.5 1 (6 , 1) (3 , 0.5) 0.5 x axis
0 0 1 2 3 4 5 6 7 8 9 10 11 12 Time
Question: What is the slope of the line in the above graph?
Answer: The way we find the slope of the line is by finding the
between two coordinate points on the line.
!!!"#$ !" !
!!!"#$ !" !
For example, if we pick the coordinate points (6,1) and (12,2) we see that
the line connecting those two coordinate points on the graph rises 1 unit
(or mile) on the y axis (from 1 to 2) and runs 6 units (or minutes) on the x
axis (from 6 to 12). When we plug those unit changes into the above
!
formula we see that the slope of the line is ! with 1 representing the
change in y and 6 representing the change in x. This slope represents
David running 1 mile for every 6 minutes at a constant rate.
𝟏
So, who runs faster, David or Alex if Alex runs at this speed: y = 𝟓x.
!
Answer: Alex runs faster because the slope of his line is !. That means
!
Alex runs 1 mile in 5 minutes, whereas the slope of David’s line is , so
!
David runs 1 mile in 6 minutes. Alex is faster.
Standard 8.EE.B.6: Use similar triangles to explain why the slope m is the
same between any two distinct points on a non-vertical line in the
coordinate plane; derive the equation y = mx for a line through the origin
and the equation y = mx + b for a line intercepting the vertical axis at b.
What the student learns: Students learn how to use triangles that are
similar (proportional) to explain how the slope between any two points on a
line (not a vertical line, they don’t have a slope) is the same. They learn
why the equation of a line is y = mx + b where m is the slope and b is the
y-intercept.
Standard example:
Find how the slope of the line is the same between any two points by
using similar triangles.
A = Green
D = Black
B = Red
E = Orange
C = Purple
F = Dark Blue
y axis
14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 C A E
D F B
x axis 0 1 2 3 4 5 6 7 8 9 10 11 12 Answer: Slope =
!!!"#$ !" !
!!!"#$ !" !
A, C, and E are the changes in y (rise).
B, D, and F are the changes in x (run).
!
!
!
So, If this line were to have a constant slope, ! = ! = !
!
!
We see that A = 4 and B = 3, so ! = !
!
!
We also see that C = 4 and D = 3, so ! = !.
However, E = 8 and F = 6, so
!
!
=
!
!
!
!
But when we simplify the fraction !, we get !
!
!
!
The slope is constant because ! = ! = !
How to derive the equation of a line as y = mx + b where m is the
slope and b is the y intercept (the y intercept is where the line on a
graph crosses the y axis.)
If the line passes through the origin (0,0) we know the equation would be
y = mx rather than y = mx + b, because the y intercept is 0, so you don’t
need to add the + 0 to the equation as the value of b.
6 The slope of this line
!!!"#$ !" ! (!"#$)
is
!!!"#$ !" ! (!"#)
!
5 which is
!
or 2.
4 If we draw a right
triangle between the
two points, (0,0) and
(1,2), the triangle will
have side lengths of 2
and 1 (shown in the
red triangle on the
picture to the left.)
3 2 1 0 0 1 2 1
3 4 5 6 Now, if we were to use any point on
the blue line on the graph above, the
triangle formed by it and (0,0) would
have to be proportional to the red
triangle.
2
x
This green triangle is proportional to the red one,
!
!
!
which means ! = ! or ! = 2
y
In order to derive the equation y = mx, we multiply
!
both sides of the equation ! = 2 by x to get y = 2x.
This equation is in the form of y = mx where m,
which represents the slope of the line, is 2.
If the line does not pass through the origin, (0,0), it is in the form
y = mx + b where m is the slope and b is the y intercept.
Let’s find y = mx + b.
Remember, the slope
is m or as shown on
!
the purple triangle
7 6 1 5 !
!"#$
( !"# )
m
4 So the triangle drawn
between (1,3) and
(2,5) has a rise of m
and a run of 1.
3 2 1 0 0 1 2 3 4 5 6 7 1
As previously mentioned, we know that any other right triangle
on this graph will be proportional to the purple triangle.
m
x
The grey triangle has a height of y and a run of x. The
!
slope is then ! . Since the grey triangle is proportional to
the purple triangle we know that
y
!
!
!
!
= ! or m = !
If we multiply each side by x we see that mx = y .
We flip the equation around so it is in the familiar
form of y = mx.
To get the b in the equation, we check to see if the
line passes through the origin (0,0). In this case, it
passes through (0,1) which is the y- intercept so
we know the equation is y = mx + b. If it passed
through (0,0) it would simply be y = mx.
WHY THIS IS IMPORTANT
In real life, students will need to interpret data that comes in many different
forms. Often they may need to compare data sets to each other that are
not in the same form or same units.
For example, proportional relationships can be used when making
blueprints, drawings, or maps to make them to scale. The Miniland at
Legoland is all constructed to scale and proportional relationships were
most definitely used when it was being designed.
Slope is something that would be used if you wanted to track something
like an investment. For example, if you invest $10,000 for retirement.
You’d like to plot how much your money grew (calculate the unit rate of
change) from month to month (on average) after one year. If you graph
your initial investment (your first coordinate point on your graph) and the
ending value of investment after one year (your last coordinate point on
the graph), you can find the average amount of money your investment
earned per month by calculating the slope. In this example, you are using
different units: money, time and percentages. The information you get by
graphing your investment can help you compare your investment to others
so you can find the best deal possible.