5th Grade - Plug In Values And Plot Points

 
     
 
     
 
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5th
Algebra
Plug in Values and Plot Points
Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.
Be able to figure out equations from word problems, choose values for x and figure out y using a t-chart, and graph the resulting ordered pairs on a coordinate plane.
 

Sample Problems

(1)

Xavier is 2 years younger than his brother. If x stands for Xavier’s age, which equation represents his brother’s age?


a) y = x + 2

b) y = x – 2

c) y = x * 2

d) y = x / 2


(a)


(2)

Which number in the ordered pair (2,3) is telling you how many steps to go to the right of the origin?

(the x-coordinate, 2, is telling you to go 2 steps to the right)


(3)

If the equation is y = x + 3, fill in the values in the t-chart:

x

y



0 3

1

-1



(y should be 4 and then 2)

(4)

Given this t-chart, which ordered pair is NOT a solution?


x y


0 1

2 3

-1 0


(1, 0), (0, 1), (2, 3), (-1, 0)


((1,0) is not a solution)


(5)


Which graph does NOT represent a linear equation?












(the green curve is not linear)


Learning Tips

(1)

Vocabulary


Linear equation– an equation that uses only numbers, y, x, and the operations of addition, subtraction, multiplication, and division; also called a linear function. The graph of a linear equation is a line.


Integer values- whole numbers, both positive and negative and zero. Examples are infinite and include 1, 2, 3, -1, -2, -3, 945, and so on.


Equation- a mathematical sentence that uses an equal sign to show two quantities (shown using numbers and operators) are equal. For example, y = x + 8 is an equation.



Expression – a mathematical phrase that has numbers and operators but no equal sign or inequality sign.


The graph of an equation- the picture formed by plotting all the points that solve the equation.


Graphing an equation- the process of figuring out and plotting the points that solve an equation.


Ordered pair- two numbers in the form of (x,y) that give the location of a specific point; also called coordinates. For example, (2, 3) is an ordered pair.


Origin- the intersection of the horizontal and vertical axes, which has the ordered pair (0,0).


Point- one specific location on a plane; usually represented by a dot.


T-chart- an chart with two columns: x values and y values.


x-coordinate-the first number in an ordered pair; it tells the number of steps to take left or right along the horizontal axis.


y-coordinate- the second number in an ordered pair; it tells the number of steps to take up or down along the vertical axis.


Cartesian plane- a grid formed by a horizontal axis (called the x-axis) and a vertical axis (called the y-axis); also called a coordinate plane.


Solve an equation- figuring out which values for x and y make the equation true.



(2)

Two Parts: Linear + Equation


Let’s break down “linear equation” into two parts: an “equation” is a mathematical sentence that sets one group of numbers and operators (plus sign, minus sign, multiplication sign, or division sign) equal to another group of numbers and operators. These are examples of equations:


1 + 2 = 3


y = x + 5


The “linear” part tells us that the graph, or picture, of that equation makes a line. Do you see the word “line” in “linear”?


So here is a graph of a linear equation:













(The linear equation is the slanted line.)


The colored lines are also graphs of linear equations:























Did you see how a vertical line is the graph of a linear equation? Do you think a horizontal line represents a linear equation? It sure does!


(3)

Writing Equations


When you are trying to figure out the equation that fits a situation, your usual choices will involve the operations of addition, subtraction, multiplying, and dividing. To figure out which is correct, pick a sample number and see which equation works in your sample.


For example:


Jane is 5 years older than her brother. If x stands for Jane’s age, which equation represents her brother’s age?


First, let’s be clear what is x and what is y:

x = Jane’s age

y = her younger brother’s age


Let’s pick a sample by saying Jane is 12. If Jane is 5 years older than her brother, then her brother must be 7. So, now we need to find an equation where Jane’s age (x) is 12 and her brother’s age (y) is 7.

Here are some obvious ones to try using the operators we mentioned above:

y = x + 5 7 = 12 + 5 ?? Nope!


y = 5 – x 7 = 5 – 12 ?? Nope!


y = x – 5 7 = 12 – 5 Yes! This one works!


Then try a few other real life examples to make sure this equation works. If three different ones check out, then you can be pretty confident that you have the right equation.


(4)

Using t-charts

When you have a linear equation and want to come up with some values to graph, a t-chart is a handy way to get yourself organized.

For example, let’s say your equation is y = 3 + x

Draw a t-chart like this:

x

Do you see why it’s called a t-chart?

Notice that x always goes first. y







Now, to fill in the t-chart, pick a few values for x. It’s a good idea to pick zero, some low integers that will be easy to work with, and a negative value. If you are not familiar with negative values, it is okay to leave them off. But when you get to higher grades, you will want to remember to pick a negative value.

So we might pick 0, 1, 2, and –1.

x y

0

1

2

-1

Then we plug in the values for x and see what y must be.

For example, when x = 0,

y = 3 + x

y = 3 + 0

y = 3

We do that for each x value, and here is our filled-in t-chart:

x y

0 3

1 4

2 5

-1 2

Once we have these values, we know the ordered pairs that we need to graph are:

(0, 3)

(1, 4)

(2, 5)

(-1, 2)

Do you see how these ordered pairs come straight off our t-chart?

Once you have graphed those points, you can connect the dots with a straight edge (like a ruler or the side of a piece of paper) to make your line.

And that is how you graph a linear equation using a t-chart.

(5)

Graphing Ordered Pairs


Once you have filled in your t-chart, you need to graph the ordered pairs. The x value (also called the x-coordinate) comes first, and the y value (called the y-coordinate) is always second.


The x-coordinate tells you how far to walk left or right along the horizontal x-axis. The y-coordinate tells you how far to walk up or down along the vertical y-axis. If the x value is positive, you will walk to the right. If the y value is positive, you will go up.


For example, graph (3, 4). This means, go 3 steps right and 4 steps up.



When you graph ordered pairs, remember to always walk before you jump (go right before you go up).


Extra Help Problems

(1)

The graph of a linear equation looks like what? (a straight line)

(2)

Which is NOT an integer: 0, 1, 2, -1, 2, 1/2 (1/2)

(3)

Is this an equation or an expression: y + x (expression)

(4)

Is an ordered pair in the form (x,y) or (y,x)?

(5)

Does the x-coordinate tell you to go up or to the right? (to the right)

(6)

What is the name of the point (0,0)? (the origin)

(7)

What is the first column of the t-chart labeled? (x)

(8)

Which do you do first, put points on the grid or fill in the t-chart? (fill in the t-chart)

(9)

What do you put on the grid to represent an ordered pair, a point or a line? (a point)

(10)

What is another name for a coordinate plane? (Cartesian plane)

(11)

When you figure out the equation for a situation, what should you do to check that it’s the right one? (pick a value for x and see if y turns out right)

(12)

What is the name of the first number in an ordered pair? (the x-coordinate)

(13)

What is the name of the second number in an ordered pair? (the y-coordinate)

(14)

What is the equation for the line that goes through these points: (0,0), (1,1), (2,2), (3,3), and so on? (y = x)

(15)

What is the equation for the line that goes through these points: (0,0), (1, 0), (2, 0), (3,0), (4,0), and so on? (y = 0)

(16)

What is the equation for the line that goes through these points: (0,0), (0, 1), (0, 2), (0,3), (0,4), and so on? (x = 0)

(17)

Which of these is not an integer value: 0, 100, 1000, 0.1 (0.1)

(18)

What is the horizontal line in a coordinate plane called? (the x-axis)

(19)

What is the vertical line in a coordinate plane called? (the y-axis)

(20)

If you are trying to graph the ordered pair (5, 12), how many steps do you take to the right, and how many steps do you take up? (5 steps to the right and 12 steps up)

(21)

If you are trying to graph the ordered pair (5, 1), how many steps do you take to the right, and how many steps do you take up? (5 steps to the right and 1 step up)

(22)

If you are trying to graphing ordered pairs for your equation and one point is off the line, is that point right or wrong? (wrong)

(23)

What is this organizer called?


x y




(a t-chart)


(24)

Which do you do first, figure out your equation or fill in your t-chart? (figure out your equation)

(25)

What does an equation always have that an expression does not? (an equals sign)

 

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