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2.6: Coordinates and Line Graphs

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Introduction

Garden Data and Graphs

Tania and Alex have had a terrific summer. They have harvested many, many vegetables and are now ready to put up a small farm stand in the front of their house.

Alex has decided to draw a map of the area and figure out where to put the stand. He likes the idea of using a grid, where 1 box or unit of the grid is equal to 4 feet. That way he can figure out exactly where everything goes. Alex enjoys being organized like that.

There are three things that he wishes to put on his grid:

  • The garden plot which is in the back yard-12 feet directly behind the house.
  • The house-which is 16 feet from Smith St. and 16 feet from Walker St.
  • The farm stand

The house is bordered by Smith and Walker streets, so Alex would like to put the farm stand near the corner so that people on both streets will see it.

Alex begins drawing his map, but is soon stuck. Here is how far he gets.

In the meantime, Tania is already planning for next year’s garden. She takes a trip to the nearby organic farm to gather some data. When she meets with Mr. Jonas the farmer, he shows her a line graph that shows vegetable growth for the past four years. Tania is fascinated. Here is what she sees.

Mr. Jonas tells Tania that according to his calculations, the farm will produce twice as much in 2009 as it did in 2008.

Tania leaves the farm with the data and a lot of excitement.

She decides to redraw the line graph at home with the new calculations for 2009.

The minute she gets home, she realizes that she is confused and can’t remember how to draw a line graph.

This is where you come in. There is a lot to learn in this lesson, pay attention so that you can help Alex draw his map and Tania draw her line graph at the end of the lesson.

What You Will Learn

In this lesson, you will learn the following skills:

  1. Identify elements of a coordinate grid (origin, vertical and horizontal axes, ordered pairs.)
  2. Graph given points on a coordinate grid (1^{st} Quadrant)
  3. Make a line graph to display given data over time.
  4. Collect, organize, display and analyze real-world data using line graphs.

Teaching Time

I. Identify the Elements of a Coordinate Grid

What is a coordinate grid?

A coordinate grid is a graph that allows us to locate points in space. You have probably seen a coordinate grid when you have looked at a map. A map often has letters on one side and numbers on the other side so you can use a letter and a number to locate a city or a specific place. We use a coordinate grid to locate points in two-dimensional space. A pair of numbers, called coordinates, tells us where the point is. We can graph any point in space on the coordinate grid.

What does a coordinate grid look like?

Here is what a coordinate grid looks like.

You can see that this coordinate grid has two lines, one that is vertical and one that is horizontal.

It also has one point where the two lines meet

Each of these parts has a special name. Let’s look at naming the parts of a coordinate grid.

What are the names of the parts of a coordinate grid?

To understand this better, let’s look at the diagram.

The horizontal axis or the line that goes across is called the x axis.

The vertical axis or the line that goes up and down is called the y axis.

The point where the two axes meet is called the origin. The origin has the value of (0,0).

You can understand the origin a little more if you know about the x and y axis.

Every line on the x axis has a different value. The values start at 0 with the origin and go to 17 on the horizontal axis. Each line has a value of 1.

Every line on the y axis has a different value. The values start at 0 with the origin and go to 9 on the vertical axis. Each line has a value of 1.

Now that we know the parts of the coordinate grid, we can look at graphing points on the grid.

II. Graph Given Points on a Coordinate Grid (1^{st} Quadrant)

How do we graph points on a coordinate grid?

To graph a point on the coordinate grid, we use numbers organized as coordinates.

A coordinate is written in the form of an ordered pair. In an ordered pair, there are two numbers put inside a set of parentheses. The first number is an x value and the second number is a y value (x, y).

Let’s look at an ordered pair.

Example

(3, 4)

This ordered pair has two values. It has an x value of 3 because the x value comes first. It has a y value of 4.

Each ordered pair represents one point on a coordinate grid.

Next, we can graph this ordered pair on the coordinate grid.

We are going to work in one part of the coordinate grid. You will learn about the other sections later.

If we graph (3,4) as one point on the coordinate grid, we start at the origin and count three units on the x axis first. Then working from the 3, we count up four since the y coordinate is four. That is where we put our point.

What about if we have an ordered pair with a 0 in it?

Sometimes, we will have a zero in the ordered pair.

Example

(0, 4)

This means that the x value is zero, so we don’t move along the x axis for our first point. It is zero so we start counting up at zero. The y value is four, so we count up four units from zero.

Notice that this point is actually on the y axis.

You try a few. Identify the coordinates for each lettered point on the coordinate grid.

  1. A = _______
  2. B = _______
  3. C = _______

Take a minute and check your work with a peer.

III. Make a Line Graph to Display Given Data Over Time

We have already learned about a few different ways to visually display data.

A line graph is a graph that helps us to show how data changes over time.

How can we make a line graph?

To make a line graph, we need to have a collection of data that has changed over time.

Data that shows growth over years is a good example of appropriate data for a line graph.

Here is an example of some data.

Example

When Jamal was born, his parents planted a tree in the back yard. Here is how tall the tree was in each of the next five years.

2003 2 ft.

2004 3 ft.

2005 5 ft.

2006 9 ft.

2007 14 ft.

Now let’s make a line graph.

The first thing that we need is two axes, one vertical and one horizontal.

The vertical one represents the range of tree growth. The tree grew from 2 feet to 14 feet. That is our scale.

The horizontal axis represents the years when tree growth was calculated.

Next, we plot the points on the graph and connect them with a line.

Now that we understand how to graph ordered pairs and how to create a line graph, we are ready to help Alex and Tania with their work.

Real Life Example Completed

Garden Data and Graphs

Now that we have finished the lesson, we can work on helping Tania and Alex. Here is the problem once again:

Tania and Alex have had a terrific summer. They have harvested many, many vegetables and are now ready to put up a small farm stand in the front of their house.

Alex has decided to draw a map of the area and figure out where to put the stand. He likes the idea of using a grid, where 1 box or unit of the grid is equal to 4 feet. That way he can figure out exactly where everything goes. Alex enjoys being organized like that.

There are three things that he wishes to put on his grid:

  • The garden plot, which is in the backyard, 12 feet directly behind the house.
  • The house, which is 16 feet from Smith St. and 16 feet from Walker St.
  • The farm stand

The house is bordered by Smith and Walker streets, so Alex would like to put the farm stand near the corner so that people on both streets will see it.

Alex begins drawing his map, but is soon stuck. Here is how far he gets.

In the meantime, Tania is already planning for next year’s garden. She takes a trip to the nearby organic farm to gather some data. When she meets with Mr. Jonas the farmer, he shows her a line graph that shows vegetable growth for the past four years. Tania is fascinated. Here is what she sees.

Mr. Jonas tells Tania that according to his calculations that the farm will produce twice what it did in 2008.

Tania leaves the farm with the data and a lot of excitement.

She decides to redraw the line graph at home with the new calculations for 2009.

The minute she gets home, she is immediately confused and can’t remember how to draw a line graph.

Let’s start with Alex and underline all of the important information for Alex’s map.

Now that we have all of Alex’s data underlined, we can draw a coordinate grid to graph each object in the space.

Alex’s yard is bordered by two streets Walker and Smith. Walker is our horizontal axis and Smith is our vertical axis.

The first point to plot is the house. It is 16 feet from Smith and 16 feet from Walker. If each box on the grid is equal to 4 feet, then we have a house coordinate of (4, 4).

The next thing to plot is the garden. It is 12 feet behind the house. That gives us a garden coordinate of (4, 7)

Looking at the map, Alex decides to place his farm stand at (3, 3). Then it will be 12 feet from Smith St. and 12 feet from Walker St.

Now it is time to help Tania. She wants to create a line graph to show the 2009 data with the other data she gathered from the farm.

The first thing that she needs to do is to draw in 2 axes.

The horizontal axis shows the years: 2005, 2006, 2007, 2008, 2009

The vertical axis shows the number of vegetables harvested.

The highest number she has is in 2008 with 400 vegetables. However, the Mr. Jonas told her he expects to double this amount. This would give 2009 a total of 800 vegetables.

Our range for the vertical axis is from 0 to 800 in increments of 100 units.

Here is Tania’s line graph.

Vocabulary

Here are the vocabulary words that you will find in this lesson.

Coordinate grid
a visual way of locating points or objects in space.
Coordinates
the x and y values that tell us where an object is located.
Origin
where the x and y axis meet, has a value of (0, 0)
X axis
the horizontal line of a coordinate grid
Y axis
the vertical line of a coordinate grid
Ordered pair
(x, y) the values where a point is located on a grid
Line Graph
a visual way to show how data changes over time

Technology Integration

Khan Academy, The Coordinate Plane

James Sousa, Plotting Points on the Coordinate Plane

Other videos:

http://www.youtube.com/watch?v=vCeAj4cLPIA – This is a video that shows you how to locate and identify ordered pairs on a coordinate grid.

Time to Practice

Directions: Write the coordinates of each point.

1. A

2. B

3. C

4. D

5. E

6. F

7. G

8. H

9. I

10. J

11. K

12. L

Directions: Graph and label each point on the coordinate grid.

13. M(1, 3)

14. N(2, 4)

15. O(0, 6)

16. P(8, 6)

17. Q(1, 3)

18. R(4, 7)

19. S(7, 7)

20. T(9,0)

21. U(4, 6)

22. V(0, 5)

23. W(6, 8)

24. Y(1, 7)

25. Z(3, 4)

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