The coordinate plane in four quadrants, and relating two variables
Plotting and finding distances in all four quadrants, and using tables and graphs to relate a dependent and an independent variable
About four lessons of 45 to 60 minutes
What is on the other side of zero?
In Grade 5, every coordinate point lived in the friendly first quadrant, where x and y were always positive. Real maps, temperature charts, and bank balances need the other side of zero too: negative coordinates. Extending the coordinate plane to all four quadrants lets you plot any point, in any direction from the origin.
This unit also builds a second, related skill: writing an equation that connects two changing quantities, such as total cost depending on how many items you buy, and showing that relationship in a table and a graph at the same time.
- The point (-3, 4)3 left and 4 up from the origin -- Quadrant II
- Distance between (-4, 3) and (5, 3)same y-value, so the distance is 5 - (-4) = 9 units
- Reflecting (3, -5) over the x-axislands at (3, 5): the x stays, the y flips sign
- y = 5x for concert tickets at $5 each3 tickets cost y = 5 x 3 = $15: a dependent variable (cost) tracking an independent one (tickets)
What students will be able to do
Students will graph points in all four quadrants of the coordinate plane and find distances between points that share an x-coordinate or a y-coordinate by counting along the axes, and will write and use an equation to show how a dependent variable changes in relation to an independent variable, representing the relationship in a table and a graph.
- I can identify which quadrant a point is in from the signs of its coordinates.
- I can find the distance between two points that share an x-value or a y-value.
- I can reflect a point over the x-axis or the y-axis.
- I can write an equation relating a dependent variable to an independent variable.
- I can complete a table of values and graph the relationship for an equation such as y = 3x.
Standards this unit teaches
- 6.NS.C.8Common Core (US)Coordinate plane problems
Solve problems by graphing points in all four quadrants and finding distances along the axes.
- 6.EE.C.9Common Core (US)Relate two variables
Use equations to show how a dependent variable changes with an independent variable, with tables and graphs.
- AC9M6SP02Australian Curriculum v9 (ACARA)Coordinate systems (Year 6)
Build a grid coordinate system to locate positions in a space and use coordinates and directions to describe position and movement.
- AC9M7A05Australian Curriculum v9 (ACARA)Patterns, tables and graphs (Year 7)
Make a table of values from a growing pattern or a function rule, then describe and plot the relationship on the Cartesian plane. Australia's matching descriptor for relating two variables with tables and graphs sits at Year 7, so that half of this unit runs about a year ahead of the ACARA placement.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 5 coordinate plane and number patterns teaching unitthe first-quadrant coordinate plane this unit extends to all four quadrants
- Grade 6 integers and negative numbers teaching unitreading and ordering negative numbers, needed for quadrants II, III and IV
- Grade 6 exponents and algebraic expressions teaching unitevaluating an expression by substitution, used to complete each table row
- How to teach coordinatesthe grid and axes this unit extends to negative values
- How relate two variables and function tables workthe dependent/independent variable relationship this unit builds
Words to teach and display
- Quadrant
- one of the four regions the coordinate plane is divided into by the x-axis and y-axis
- Reflection
- flipping a point or shape over a line, such as the x-axis or y-axis, to a mirror-image position
- Dependent variable
- the quantity whose value depends on, and changes because of, another quantity (often y)
- Independent variable
- the quantity that is chosen or changes freely, driving the dependent variable's value (often x)
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. All four quadrants
ConcreteExtending both axes past zero in the negative direction splits the coordinate plane into four quadrants. Quadrant I (top right) has both x and y positive, the only quadrant used in Grade 5. Quadrant II (top left) has x negative and y positive. Quadrant III (bottom left) has both negative. Quadrant IV (bottom right) has x positive and y negative.
To plot a point with a negative coordinate, move in the opposite direction: a negative x means move left instead of right, a negative y means move down instead of up.
Which quadrant is the point (-3, 4) in?
- The x-coordinate, -3, is negative, so the point is to the left of the origin.
- The y-coordinate, 4, is positive, so the point is above the origin.
- Negative x, positive y is Quadrant II.
Answer: (-3, 4) is in Quadrant II.
- What are the signs of x and y in Quadrant III?
- Which quadrant has a positive x and a negative y?
2. Distances along the axes, and reflections
PictorialWhen two points share the same y-value, they lie on a horizontal line, and the distance between them is the difference of their x-values (as a positive number). When two points share the same x-value, they lie on a vertical line, and the distance is the difference of their y-values.
Reflecting a point over the x-axis keeps the x-value the same and flips the sign of the y-value: (3, -5) reflects to (3, 5). Reflecting over the y-axis keeps the y-value the same and flips the sign of the x-value: (3, -5) reflects to (-3, -5).
Find the distance between (-4, 3) and (5, 3). Then reflect (2, 6) over the x-axis.
- (-4, 3) and (5, 3) share the same y-value, 3, so they lie on a horizontal line.
- The distance is the difference of the x-values: 5 - (-4) = 9 units.
- Reflecting (2, 6) over the x-axis keeps x = 2 and flips the sign of y: (2, -6).
Answer: The distance is 9 units. The reflection of (2, 6) over the x-axis is (2, -6).
- Why is the distance between (-4, 3) and (5, 3) not just 5 - 4?
- What changes and what stays the same when you reflect a point over the x-axis?
3. Relating two variables with tables and graphs
AbstractMany real situations have one quantity that depends on another: the total cost of concert tickets depends on how many tickets you buy. If tickets cost $5 each, the equation y = 5x relates the number of tickets (x, the independent variable, chosen freely) to the total cost (y, the dependent variable, which changes because of x).
Build a table by substituting several values of x into the equation to find matching y-values, then plot those (x, y) pairs on a graph. For y = 5x: x = 1 gives y = 5, x = 2 gives y = 10, x = 3 gives y = 15. Plotting these shows the cost climbing steadily as tickets are added.
A pool fills at 8 litres per minute, so y = 8x relates minutes (x) to litres (y). Complete a table for x = 1 to 4, and find y after 6 minutes.
- x = 1: y = 8 x 1 = 8. x = 2: y = 8 x 2 = 16. x = 3: y = 8 x 3 = 24. x = 4: y = 8 x 4 = 32.
- For x = 6: y = 8 x 6 = 48.
Answer: Table: (1, 8), (2, 16), (3, 24), (4, 32). After 6 minutes, the pool has 48 litres.
- In y = 8x, which variable is independent and which is dependent?
- How do you find a table value for a given x, using the equation?
Common misconceptions and how to address them
MisconceptionA negative x-coordinate means the point does not really exist, or is 'off the grid'.
Why it happens: Students who only met the first quadrant in Grade 5 treat the coordinate plane as starting and ending at zero on both axes.
How to address it: Negative coordinates are just as real as positive ones, describing a point in the opposite direction from the origin. Extend a physical or drawn grid past zero in both directions and plot several negative-coordinate points to normalise them.
MisconceptionThe distance between two points is always found by subtracting the x-values, regardless of whether the points share an x-value or a y-value.
Why it happens: Students default to a single 'subtract the x-values' rule without checking which coordinate the two points actually share.
How to address it: First check which coordinate matches: same y-value means measure along a horizontal line using the x-values; same x-value means measure along a vertical line using the y-values. If neither coordinate matches, the points are not on a straight horizontal or vertical line, and this simple axis-distance method does not apply.
MisconceptionReflecting a point over the x-axis flips the x-value instead of the y-value.
Why it happens: Students assume 'the x-axis' means 'change the x', mixing up which axis you reflect across with which coordinate changes.
How to address it: Reflecting OVER a line keeps points on that line fixed and flips everything on the other side. Reflecting over the x-axis keeps every point's x-value the same (points on the x-axis do not move) and flips the sign of the y-value. Reflecting over the y-axis is the reverse.
MisconceptionIn a relationship like y = 8x, x and y are interchangeable and it does not matter which is called dependent.
Why it happens: Students see two related numbers and do not track which one is chosen freely and which one is caused by that choice.
How to address it: The independent variable is the one you choose or control (minutes elapsed); the dependent variable is the one that results from that choice (litres filled). Ask 'which one causes the other to change?' to identify the direction of dependence.
Guided practice (with answers)
1. Which quadrant is the point (-6, -2) in?
Answer: Quadrant III, since both x and y are negative.
2. Find the distance between (7, -3) and (7, 4).
Answer: 7 units. They share x = 7, so the distance is the difference of the y-values: 4 - (-3) = 7.
3. Reflect (2, 6) over the x-axis.
Answer: (2, -6). The x-value stays, the y-value flips sign.
4. For the table y = 4x, find y when x = 5.
Answer: 20, since 4 x 5 = 20.
5. A pool fills at 8 litres per minute (y = 8x). How many litres after 6 minutes?
Answer: 48 litres, since 8 x 6 = 48.
Independent practice worksheets
Set the matching ChalkBee coordinates and algebra worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with quadrant identification and axis distances, then move to the table-and-graph relating task.
Differentiation
- Use a large floor or wall grid extended into all four quadrants so plotting negative coordinates is a physical routine before an abstract one.
- Highlight the shared coordinate (matching x or matching y) in colour before computing an axis distance.
- Provide a partially completed table (x column filled, y column blank) so students only compute one value at a time before graphing.
- Keep early reflection practice to points already on a grid, checking the image against a physically folded mirror line.
- Find distances between points using the full distance formula once both axis-distance skills are secure (a preview of later grades).
- Reflect a whole shape (not just one point) over both axes and compare the two resulting images.
- Write an original real-world scenario with a dependent and independent variable, build its table, and graph the relationship.
- Investigate what happens to a graph's steepness when the equation's multiplier changes, such as comparing y = 3x to y = 8x.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling quadrants, axis distance, and relating two variables.
1. Which quadrant is the point (4, -9) in?
Answer: Quadrant IV, since x is positive and y is negative.
2. Find the distance between (-5, 2) and (3, 2).
Answer: 8 units. They share y = 2, so the distance is 3 - (-5) = 8.
3. For y = 2x + 1, find y when x = 4.
Answer: 9, since 2 x 4 + 1 = 8 + 1 = 9.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 all four quadrants (section 1), Lesson 2 axis distances and reflections (section 2), Lesson 3 relating two variables with tables and graphs (section 3), Lesson 4 mixed practice and the exit ticket.
- Language to keep saying: check the sign of each coordinate for the quadrant, find the shared coordinate before measuring distance, which quantity is chosen and which one results. These target the four misconceptions directly.
- This unit deliberately pairs 6.NS.C.8 (coordinate geometry) with 6.EE.C.9 (relating variables) because both rely on the same skill, plotting and reading ordered pairs, and because a table of values IS a set of ordered pairs waiting to be graphed.
- Curriculum note: ACARA v9's coordinate-system descriptor sits at Year 6 (AC9M6SP02), matching the coordinate-plane half of this unit closely. The table-and-graph relating-variables descriptor sits at Year 7 in Australia (AC9M7A05), so that half of this unit runs about a year ahead of the ACARA placement.
- Present mode and print both work: use Present to build the four-quadrant grid and the y = 8x table live with the class, then print for independent plotting and graphing practice.