This lesson teaches students how to solve real-world and mathematical problems with the coordinate plane. Students will: 
- solve real-world problems by graphing points on a coordinate plane. 
- solve mathematical problems, graph points on the coordinate plane. 
- interpret the coordinate values of points within the context of a circumstance. 

- How do spatial relationships, such as shape and dimension, help to create, construct, model, and portray real-world situations or solve problems? 
- How may geometric properties and theorems be utilized to describe, model, and analyze problems? 
- How may using geometric shape features help with mathematical reasoning and problem solving? 

- Coordinate Plane: Formed by the intersection of two number lines (called axes) that meet at right angles at their zero points. Used to locate points in the plane or in space by means of two numbers that represent the distance the point is from the horizontal axis and the vertical axis.
- Origin: The point at which the number lines of a coordinate plane intersect. As an ordered pair, the point (0,0).
- x-Axis: The horizontal number line of a coordinate plane. Used to show horizontal distance.
- x-Coordinate: The first number in an ordered pair, it designates the distance a point is along the horizontal axis.
- y-Axis: The vertical number line of a coordinate plane. Used to show vertical distance.
- y-Coordinate: The second number in an ordered pair, it designates the distance a point is along the vertical axis.

- at least two copies of the First Quadrant worksheet (M-5-3-2_First Quadrant and KEY) for each student
- a copy of Activity 2 Key (M-5-3-2_Activity 2 KEY) for each student. Once cut apart, each copy of the key is sufficient for three students.

- At the end of Activity 1, assess the students' constructed maps to determine their level of understanding.

Scaffolding, Active Engagement, and Metacognition 
W: After learning how to plot points on the coordinate plane and identify the parts of the coordinate plane, students will have an opportunity to apply their knowledge. 
H: Students will be asked to make a map. This will help students see the relationship between their mathematical knowledge and their real-world knowledge, as well as provide them the opportunity to apply what they have learned. 
E: Students will design their own maps using various information sources (e.g., supplied coordinates, coordinate information, and point relationships on the same plane). All of these different types of information or "clues" enable students to encounter and explore the coordinate plane in a variety of ways. 
R: Students will practice coordinate plane skills through a task with a single right response based on detailed instructions. Students' thinking improves when they have more freedom to practice and rehearse their skills. Finally, students are provided a large open space in which to practice delivering and receiving instructions. 
E: Students will evaluate their work by comparing their maps with neighbors. They will also evaluate their work in Activity 3 by comparing their drawings to the original drawings and determining where errors occurred (in the drawing or the directions). Observation during this period will decide whether more practice or instruction is required. 
T: Use the Extension section to customize the lesson to match the needs of the students. The Routine section is intended for usage throughout the year to assist students review lesson concepts. The Small Group section is designed to give additional education and practice opportunities for students who would benefit from these resources. The Expansion section includes challenge exercises and suggestions for students who want to go above and beyond the standard requirements. 
O: The lesson is structured in an experimental way, progressing from easy to more difficult activities. Students will be immediately intrigued by the description of an imaginary town and their task to create a map of the town. Students are still engaged in Activity 2 because of its flexibility. They'll be curious to see how different the other maps are from their own, and if anyone else made the same map they did. Finally, students remain engaged during Activity 3 because they have greater freedom and can choose a letter from their own name, which instills ownership in their painting. They get to play the role of the teacher when they write instructions and give them to another student. 

Activity 1

Depending on how long it has been since students worked with the coordinate plane, a brief reminder of the order in which ordered pairs are "followed" may be required. 

Make sure each student has at least one copy of the First Quadrant worksheet (M-5-3-2_First Quadrant and KEY), and have them work with a partner. 

"Take a copy of your worksheet. On the worksheet's topmost coordinate plane, each group will create a map using the same instructions. During the exercise, I can repeat any instructions you need, but I cannot answer any additional questions. You can discuss what to do with your partner, but not with any other groups."

Read the directions below, leaving time after each for students to complete them. 

"You're creating a map of a town you've never visited. When you arrive in town, your first visit will be a welcome center, which provides tourist information. This welcome center is somewhat unusual. They don't have any maps of the town, but they will help you make your own. You are given a coordinate plane and a pencil, and the person at the welcome center—that's me—begins to give you directions so you can find your way around town." 

"Plot a point at (4, 4) and label it Library." 

"Plot a point at (8, 4) and label it School."

Because this instruction is near the start of the activity, it may be important to walk around and ensure that students have successfully followed the two directions thus far. 

"Mark a place halfway between the library and the school and label it Park. On the side of your worksheet, put the word Park, followed by the coordinates for the point that represents the park." (6, 4) 

"There is a farm 6 units above the park. Make a point of representing the farm's location. On the side of your page, write the term Farm, followed by the coordinates for the point that represents the farm." (6, 10)

"There is a straight road that connects the Farm to the School. Draw a line to represent the road." 

"Begin at the farm and move 3 units to the left and 4 units up. There is a lake at that location. Make a point of representing the lake's position. On the side of your worksheet, write the term Lake, followed by the coordinates for the point that represents the lake." (3, 14) 

"There is a straight road that connects the lake to the farm. The road then follows a straight line from the farm to the school. Draw the two parts of the road with straight lines."

"Finally, there is a straight road that connects the school to the park. Draw a line to represent the road." 

After students have done designing the final road, have each pair of student compare their drawings to those of another group. If their maps differ in any way, the foursome should have you repeat the instructions until the larger group decides which map is correct and fixes the incorrect map. 

Before proceeding, ensure that all students have the correct map and that any misunderstandings about directions and point plotting have been resolved.

Activity 2

"In the prior activity, everyone should have received the same map. The instructions were so specific that there was just one possibility at each point. In this task, you will have greater freedom to do whatever you want, but you must still follow the instructions. Begin on a new coordinate plane with a point at (8, 8). Label that point Library." 

"The Market is two units away from the Library, in any direction. Select a position that is two units (left, right, up, or down) from the Library and label the point Market." Possibilities include (6, 8), (10, 8), (8, 6), (8, 10).

"The Pool is three units distant from the Library and in the opposite direction you took to reach the Market. So, if you move the Market to the left of the Library, where should the Pool be? (To the right) "If you went up, where should the Pool go?" (Down) "Go ahead and locate the Pool, three units distant from the Library. Plot the point and label it Pool." Possible values: (11, 8), (5, 8), (8, 5), (8, 11). 

"The Arcade's coordinates are 10 and 2, however you can select the order. You can select whether the Arcade is at (10, 2) or (2, 10). Put the Arcade on your map and label it."

"City Hall shares the same y-coordinate with the Arcade and has an x-coordinate of 7. Put City Hall on your map and label it." (7, 2), (7, 10) 

"Put the police station last on your map. The coordinates of the police station sum up to 12, with one of the coordinate is 5. Put the Police Station on your map and label it." (5, 7), (7, 5) 

"Next to your coordinate plane, record the name and coordinates of each place you graphed. Remember to enter the x-coordinate first and the y-coordinate second."

Give each student a key to the activity (M-5-3-2_Activity 2 KEY), which you have divided into three parts. Students should share and examine each other's maps and coordinates to ensure that their map is correct.

Activity 3

"Now, each of you should choose a letter from your first or last name—it may be any letter you want—and we'll create instructions for someone to follow so s/he can graph the letter you choose. Has everyone selected a letter?"

After everyone has picked a letter, continue: "Determine where you want to begin drawing your letter. Remember to plan ahead and ensure that you have enough room on your coordinate plane to fit the entire letter. Plot the first point in your letter and record its coordinates on a separate sheet of paper. Now plot the second point and connect it with a line. Write down the coordinates for the second point beneath the first. Continue plotting and linking points, writing them down in order. If you get to a point in your letter where you don't want the new point to be connected to the old one, put the word STOP on your paper before you write the new point. Once you believe you have all of the coordinates required for someone else to graph your letter, double-check your work." 

After listing the coordinates and double-checking their work, students should give their instructions to someone else. 

"You now have the directions to create someone else's letter. Begin with the first point in the list and graph them in order. Unless the instructions specifically state otherwise, you should always connect each point to the one before it. When you're finished, compare your letter to your partner's and see if the pictures match." 

If the pictures do not match, the pair should collaborate to determine where the directions do not match the original picture or where the second picture does not match the instructions (or both). 

This activity can be repeated if students have difficulty writing accurate instructions.

Extension:

Use the strategies listed below to adjust the lesson to your students' needs throughout the year. 

Routine: These concepts can be repeated later in the year with new, more complicated maps, or students might investigate geometric principles by graphing them on the coordinate plane. For example, if students understand the many classifications of shapes (or parallel / perpendicular lines, etc.), they can write instructions for graphing these geometric items.

Small Group: Students can collaborate in a small group to develop a map of a town and "clues" to help another group understand the map. Their hints, like those in this lesson, can be simple (e.g., "The mall is located at (5, 6).").

Expansion: This lesson can be extended to include locations outside of Quadrant I as well as points with decimal coordinates. Maps and instructions can always become more difficult. Here's an example: "The Junior High is located 5 units from the Police Station; plot all the possible locations of the Junior High."