Graph and Interpret Coordinate Points
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5th Grade Math › Graph and Interpret Coordinate Points
A landscaper is planning a garden with four flower beds positioned at coordinates (2, 3), (5, 3), (5, 7), and (2, 7). She wants to install a circular fountain at the center of this arrangement and then place benches at positions that are exactly 1.5 units away from the fountain in both the north and east directions. If she installs 3 such benches, and each unit on the coordinate plane represents 2 meters, what is the total cost if each bench costs $150?
$375 because there is a bulk discount for multiple bench installation
$450 because there are 3 benches at $150 each, regardless of location
$600 because the installation cost doubles due to the remote location
$300 because the benches are placed at a discounted location rate
Explanation
This question tests coordinate plane understanding while incorporating real-world application. The center of the rectangle formed by the four flower beds is at ((2+5)/2, (3+7)/2) = (3.5, 5). The benches are placed 1.5 units north and east of this center at (5, 6.5). However, the key insight is that the cost calculation is straightforward: 3 benches × $150 each = $450, regardless of their coordinate positions. The coordinate information is relevant for placement but doesn't affect the total cost calculation.
In a video game, a character moves on a coordinate grid. The character starts at the origin (0, 0) and follows these moves in order: right 3 spaces, up 2 spaces, right 4 spaces, up 1 space, left 2 spaces. After completing all these moves, the character needs to return to the starting point by moving in straight lines only (no diagonal moves). How many total spaces must the character move to get back to the origin?
10 spaces because the character must move 6 spaces left and 4 spaces down
9 spaces because the character must move 5 spaces left and 4 spaces down
8 spaces because the character must move 5 spaces left and 3 spaces down
12 spaces because the character must move 7 spaces left and 5 spaces down
Explanation
Track the character's position: Start (0,0) → right 3 to (3,0) → up 2 to (3,2) → right 4 to (7,2) → up 1 to (7,3) → left 2 to (5,3). The final position is (5,3). To return to (0,0), the character must move 5 spaces left and 3 spaces down, for a total of 5 + 3 = 8 spaces.
A farmer plants crops in a rectangular field represented on a coordinate plane. The southwest corner is at (1, 2), the southeast corner is at (9, 2), and the northwest corner is at (1, 7). If the farmer wants to install a water sprinkler at the center of the field, and then build a fence post exactly 2 units north and 1 unit east of the sprinkler, what will be the coordinates of the fence post?
(7, 6) because the center is at (6, 4) plus the movement
(6, 7) because the center is at (5, 5) plus the movement
(5, 6.5) because the center is at (4, 4.5) plus the movement
(6, 6.5) because the center is at (5, 4.5) plus the movement
Explanation
First, find the fourth corner: northeast corner is at (9, 7). The center of the rectangle is at the midpoint of the diagonal from (1, 2) to (9, 7). Center = ((1+9)/2, (2+7)/2) = (5, 4.5). Moving 1 unit east and 2 units north from the center: (5+1, 4.5+2) = (6, 6.5).
Maya is tracking the temperature of her science experiment over several days. She records the data on a coordinate plane where the x-axis represents the day number and the y-axis represents the temperature in degrees Celsius. On day 3, the temperature was 25°C. On day 7, the temperature was 41°C. If the temperature increased at a steady rate, what was the temperature on day 5?
31°C because the temperature increases by 3°C each day
33°C because the temperature increases by 4°C each day
29°C because the temperature increases by 4°C each day
35°C because the temperature increases by 5°C each day
Explanation
First, find the rate of change: From day 3 to day 7 is 4 days, and the temperature went from 25°C to 41°C, an increase of 16°C. So the rate is 16°C ÷ 4 days = 4°C per day. From day 3 to day 5 is 2 days, so the temperature increased by 2 × 4°C = 8°C. Therefore, on day 5: 25°C + 8°C = 33°C.
Use the coordinate grid in the first quadrant. The x-axis shows the number of minutes a class spends cleaning up, and the y-axis shows the number of supplies put away. Two points are plotted: $M(4, 12)$ and $N(6, 9)$. Which statement is incorrect? (Points represent minutes and supplies.)
At point M, the class cleaned for 4 minutes and put away 12 supplies.
At point N, the class cleaned for 6 minutes and put away 9 supplies.
Point N shows more minutes than point M.
Point N shows more supplies put away than point M.
Explanation
Points on a coordinate grid represent pairs of values from two related categories. In this case, the x-axis represents the number of minutes a class spends cleaning up, while the y-axis represents the number of supplies put away. Ordered pairs like (4, 12) and (6, 9) are read as x-value first, then y-value, indicating minutes and supplies for each. Thus, points M and N connect to the situation by comparing 4 minutes with 12 supplies to 6 minutes with 9 supplies. A common misconception is assuming more time means more supplies without checking values, but compare coordinates directly. Graphs help represent information by enabling quick comparisons of efforts. They visualize productivity and highlight inaccuracies effectively.
A point is plotted at $,(9, 4),$ on the coordinate grid. The x-axis is the number of pencils in a box, and the y-axis is the number of boxes. Points represent how many pencils and boxes there are. Which claim about $,(9, 4),$ is incorrect?
There are 9 pencils in each box.
The point uses x for pencils and y for boxes.
The point shows 9 boxes and 4 pencils.
There are 4 boxes.
Explanation
Points on a coordinate grid represent quantities in organizational contexts, such as supplies. Here, the horizontal x-axis connects to pencils in a box, and the vertical y-axis connects to the number of boxes. The ordered pair (9, 4) is read as 9 on x first, then 4 on y, meaning 9 pencils per box and 4 boxes. This point connects to the situation by describing a specific setup of pencils and boxes. A misconception is swapping values to claim 9 boxes and 4 pencils. Graphs help represent information by structuring inventory data. They allow quick verification of quantities and claims.
A science class measures plant growth. The x-axis is the number of weeks, and the y-axis is the plant’s height in centimeters. Points on the grid represent values in this situation.
Which point matches the description: “After 6 weeks, the plant is 18 cm tall”?
$(6, 6)$
$(18, 18)$
$(18, 6)$
$(6, 18)$
Explanation
Points on a coordinate grid represent pairs of values that illustrate connections between two measurements in a given context. In this plant growth scenario, the x-axis connects to the number of weeks passed, while the y-axis connects to the plant's height in centimeters. The ordered pair (6, 18) is read by noting the x-value of 6 first for the horizontal position and then the y-value of 18 for the vertical position. This point connects to the situation by indicating that after 6 weeks, the plant has reached a height of 18 centimeters. A common misconception is confusing the order, thinking it means 18 weeks and 6 centimeters, but the first number always corresponds to the x-axis. Graphs help represent information by making it easier to see changes and patterns in data over time. They provide a visual way to interpret and compare measurements in real-world situations.
Two points are plotted on the coordinate grid: $R(2, 5)$ and $S(2, 9)$. The x-axis is the number of laps run, and the y-axis is the number of minutes it took. Points represent running times. Which statement is true?
R and S show the same number of laps but different numbers of minutes.
S took fewer minutes than R.
R and S show different numbers of laps but the same number of minutes.
R took 2 minutes to run 5 laps.
Explanation
Points on a coordinate grid represent values for comparing multiple instances, like running performances. In this case, the horizontal x-axis connects to laps run, and the vertical y-axis connects to minutes taken. Ordered pairs (2, 5) and (2, 9) are read with x first for laps, then y for minutes. These points connect to the situation by showing the same laps but different times for R and S. A common misconception is assuming different laps when x-values are identical. Graphs help represent information by highlighting similarities and differences. They make it easy to compare data like speeds or times.
Use the coordinate grid in the first quadrant. The x-axis shows the number of stickers a student buys, and the y-axis shows the total cost in dollars. Which point matches the description: “A student buys 4 stickers for a total cost of $6$ dollars”? (Each point represents stickers and total cost.)
$(4, 0)$
$(6, 0)$
$(6, 4)$
$(4, 6)$
Explanation
Points on a coordinate grid represent pairs of values from two related categories. In this case, the x-axis represents the number of stickers a student buys, while the y-axis represents the total cost in dollars. An ordered pair like (4, 6) is read as the x-value first, followed by the y-value, indicating 4 stickers and $6. Thus, the point (4, 6) connects to the situation by showing a purchase of 4 stickers costing $6 total. A common misconception is confusing the axes, such as plotting cost first instead of stickers, but remember x is always the horizontal axis. Graphs help represent information by displaying relationships between quantities like purchases and costs. They make it easier to understand and analyze real-world data patterns.
On the coordinate grid, the x-axis is the number of cups of water poured into a plant, and the y-axis is the plant’s height in inches. Points represent measurements. Which point matches the description: “The plant is 6 inches tall after 2 cups of water”?
$(6, 6)$
$(6, 2)$
$(2, 6)$
$(2, 2)$
Explanation
Points on a coordinate grid represent pairs of values that describe a specific measurement or outcome. Here, the horizontal x-axis connects to the number of cups of water given to the plant, and the vertical y-axis connects to the plant's height in inches. The ordered pair is read as (x, y), so for a description matching 2 cups and 6 inches, it would be (2, 6). This point connects to the situation by indicating the plant reached 6 inches after receiving 2 cups of water. One misconception is confusing the axes, such as plotting height first, but x always precedes y. Graphs help represent information by showing how one factor, like water, affects another, like growth. They allow us to track changes and make predictions based on patterns.