Activities

1. Santana lives 2 blocks east and five blocks north of the school as indicated on the graph below.  What point represents the school?

2. What point would be plotted on the coordinate grid if we started at the origin and moved 3 units east and 2 units north?
   
   
3. The equipment at a school playground is plotted on the coordinate grid.  The school is plotted at the origin.  What are the coordinates of the school?
   
   
4. Trees are to be planted at the local park.  The location for each tree is plotted on the coordinate grid. The parking lot is at the origin on the grid.  What are the coordinates of the tree the farthest distance from the parking lot?

5. How many points can be plotted on the x-axis?

6. The endpoints of one of the longest sides of a rectangle are plotted on the coordinate grid at (1, 4) and (6, 4).  How many possible pairs of points can represent the other longest side if the shorter side is 4 units long?
   
   
   1. 1
   2. 2
   3. 4
   4. None, it is not possible given these conditions
      
      
7. Downtown Philadelphia is a laid out on a coordinate grid. The grocery store is at (1, 6), the courthouse is at (6, 4) and the library is at (0, 0).  Karen’s house is located at (3, 3).  What is located the farthest distance from Karen’s house?
   
   
8. Which of the following points is the shortest distance to the origin?
   
   
   1. (1, 9)
   2. (2, 3)
   3. (4, 2)
   4. (3, 4)
      
      
9. The three vertices of a square are plotted on the coordinate grid.  What is the fourth vertex?
   (5, 2); (3, 7); (10, 4)      
   
   
10. The endpoints for the hypotenuse of a right triangle are (8, 11) and (3, 4).  What is one possible point for the third vertex?
    
    
11. Lily runs the local bakery and is looking for a pattern from her collected data to try to determine how sweet her desserts should be.  She plotted the data on the coordinate grid, letting the x-coordinate represent the cups of sugar used in her cakes and the y-coordinate represent the number of cakes sold.

Based upon the graph, which statement is true?

1. Lily used the same amount of sugar when she sold 1 and 4 cakes
2. When Lily used 3 cups of sugar, she sold the most cakes
3. Zero cups of sugar was the least amount Lily used in her cakes
4. Five cakes were sold when she used 1 cup of sugar

12. How many points are exactly 3 units from the point (3, 4)?
    
    
    1. 1
    2. 2
    3. 3
    4. 4
       
       
13. Which point represents the fourth vertex in a parallelogram if (7, 1), (3, 4) and (1, 1) are the other three vertices?
    
    
    1. (9, 4)
    2. (9, 3)
    3. (8, 4)
    4. (10, 3)
       
       
14. Grady and his friends are going on a scavenger hunt through their little town.  The treasure map they are given has the following points plotted:  (2,8); (7,3); (11,6); (5,2)  

The following are the directions to reach the treasure:
1 - Start at the point with the smallest x-coordinate
2 - Proceed to the point farthest away from the starting point
3 - Take the y-coordinate of the second point and divide it in half
4 - Proceed to the point with the answer to step 3 as the y-coordinate
5 - Next move to the only point left
6 - Switch the x and y-coordinates around for this point
7 - The answer is the location of the treasure
What is the location of the treasure?

15. Plot the point (7, 3) on the coordinate grid.  Move 2 right, 6 up, 3 right, 2 down, and 1 left.  What is the location after the final move?

16. The endpoints of one of the longest sides of a rectangle are plotted on the coordinate grid at (1, 4) and (6, 4).  How many possible pairs of points can represent the other longest side if the shorter side is 2 - 4 units long?  Explain
    
    
17. Marcus sells candy bars at the local basketball games.  He is keeping track of the amount of money he spends to purchase the candy and the amount of money he makes per game by plotting the information on a coordinate grid. The x-coordinate represents what he spent per game to buy the candy and the y-coordinate represents the total sales of candy per game. Which statement is true?
    
    
    1. The point (35, 5) indicates that Marcus made $35 for one game
    2. Marcus plotted the point (56, 102), which means he made a profit of $56
    3. The point (12, 8) indicates that Marcus had 4 candy bars that were not sold
    4. When the y-coordinate is greater than the x-coordinate, Marcus made a profit
       
       
18. Explain how to calculate the shortest distance to the origin, following the grid lines, from the points listed below without plotting the points.  Which point is the shortest distance from the origin?  (6, 3); (3, 5); (1, 6); (2, 8)
    
    
19. Devon must choose between two locations in NYC for her family to live.  In order to make the choice easier she lists the following conditions that must be met: 

• Less than 5 blocks from the grocery store
• 10 blocks or less from school
• 18 or less blocks from work. 

She plots everything on the coordinate grid to make her calculations.  Determine if either location meets her criteria and explain your reasoning.

20. When calculating the distance a point is from the origin, the x- and y-coordinates are added together.  What minor adjustments can be made to this method so all other points on the coordinate plane will have a method that works for calculating distance from an original point?  Explain.
    
    
21. Which point represents the fourth vertex in a parallelogram if (7, 1), (1, 1) and (11, 6) are the other three vertices?  Explain your thinking.
    
    
22. The endpoints for the hypotenuse of a right triangle are (8, 11) and (5, 3).  List all possible points for the other vertex.  Explain.
    
    
23. A parallelogram, when plotted on the coordinate grid, has a height of 5, vertices (3, 8) and (8, 3), and one vertex on the y-axis.  Find the other two vertices.  Explain.
    
    
24. In some areas of Boston, the city blocks form squares or rectangles.  A “Walk in History” is planned and all of the historic places are plotted on a coordinate grid below.  The starting point for the walk is not plotted on the grid.  The starting point is located 4 units south and 2
    units west of the plotted point with the largest x-coordinate value.  What are the coordinates
    for the starting point?  Justify your solution.

25. Determine what is wrong with each statement below based on the graph.

1. It never snowed 4 inches in December
2. Each time it snowed 1 inch, different students wore mittens
3. The point (2, 3) indicates that two students wore mittens when it snowed 3 inches.

Answer Key/Rubric

1. (0, 0)
   
   
2. (3, 2)
   
   
3. (0, 0)
   
   
4. (6, 6)
   
   
5. infinite

6. b – two pairs; can either be the points (1, 8) and (6, 8) or (1,0) and (6,0)
   
   
7. library
   
   
8. b - (2, 3)
   
   
9. (8, 9)
   
   
10. (8, 4) or (3, 11)
    
    
11. b
    
    
12. d
    
    
13. a
    
    
14. (2, 5)
    
    
15. (11, 7)

16. 6 pairs of points.  Explanations will vary.  The key points are explaining the calculation of 2, 3, and 4 as the length of the shorter side is three pairs of points.  This calculation is doubled because you can draw the same set of distances above or below the longest side.
    
    
17. d
    
    
18. (1, 6)  Explanations will vary.  Student must somehow explain that the x- and y-coordinates are added together to get the distance following along the grid lines.
    
    
19.  Neither location meets her criteria.  Explanations will vary.  For Location A, the criteria that is not met is the grocery store, as it is 6 blocks from the apartment.  For Location B, the criteria that is not met is work, as it is 20 blocks from the apartment.
    
    
20. Explanations will vary.  Students must somehow explain that the x-and y-coordinates are added together when the original point is the origin because the distance is how far over and up the point is from the origin.  When the original point changes to any other point, the distance is the difference of the x-coordinates of the two points added to the difference of the y-coordinates of the two points.  This method works because it is the same theory as the one from the origin.
    
    
21. (5, 6)  Explanations will vary.  Students must include that since the lines are parallel, the y-coordinate must be a 6 because of the point (11, 6).  To determine the x-coordinate, it is the same distance or same number of blocks over between the points (7, 1) and (11, 6), which is 4.  So from (1, 1), go over (right) 4 blocks and you get 5.
    
    
22. (8, 3) or (5, 11).  Explanations will vary.  Students must explain what relevance the hypotenuse is to calculating the other vertices and how to find the other vertices.  Example:  Since the hypotenuse endpoints are given and the hypotenuse is opposite the right angle, on the coordinate grid plot both points and use the y-coordinate of (5, 3) and the x-coordinate of (8, 11).  This point is the location of the right angle to the right of the hypotenuse.  Reverse the process for the other point.
    
    
23. (0, 3) and (11, 8) or (0, 8) and (5, 3).  Explanations will vary.  An example explanation:  Plot both points.  The combination of a height of 5 and one vertex on the y-axis indicated that the point on the y-axis had to be at either (0, 3) or (0, 8).  The placement of the fourth point is determined by the distance of the point graphed on the y-axis.  If I chose the point (0, 3) the parallelogram has a length of 8 units, so my fourth point must be 8 units over from (3, 8).
    
    
24. (4, 2)  Explanations will vary. The largest x-coordinate is the point that is the farthest distance from the origin on the x-axis which is 6 units (6, 6).  From this point, 4 units down or south and then 2 units left or west is the point (4, 2).
    
    
25.  Statement a is not true or at least we cannot tell from the graph because the graph only contains information about school days.  Statement b is not true or at least we cannot tell from the graph because the points represent the number of students and does not differentiate between students.  Statement 3 is not correct.  The point (2, 3) represents when it snowed 2 inches, three students wore mittens.