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Reveal Math · Unit 7 · Supplemental Resources

Integers & Coordinate Plane

Standards: 6.NS.C.5 (Positive/Negative) · 6.NS.C.6 (Number Line/Coordinate Plane) · 6.NS.C.7 (Absolute Value) · 6.NS.C.8 (Graphing)

🖨️ Differentiated Practice Worksheets

Ready-to-print practice at three levels — pick the right fit for each student.

Level 0 · Extra Support Scaffolded steps, worked examples, extra space. Level 1 · Support Guided practice on the core skills. Level 2 · Enrichment Stretch problems and deeper reasoning.

Visual Vocabulary

0 −3 +3
Integer
Número entero
A whole number that can be positive, negative, or zero. Examples: −5, 0, 3
0 −4 +4 opposites
Opposite
Opuesto
Two numbers the same distance from 0 but on different sides. +4 and −4 are opposites.
0 −4 |−4| = 4
Absolute Value
Valor absoluto
The distance from 0 on a number line. Always positive or zero. |−4| = 4
I (+,+) II (−,+) III (−,−) IV (+,−) y x origin
Coordinate Plane
Plano de coordenadas
A grid with an x-axis (horizontal) and y-axis (vertical) that cross at the origin (0, 0).
(3, −2) x = 3 (right 3) y = −2 (down 2) Quadrant IV
Ordered Pair
Par ordenado
Two numbers (x, y) that tell the exact location of a point. x goes left/right, y goes up/down.
Quadrant The 4 sections of the coordinate plane I, II, III, IV
Quadrant
Cuadrante
One of the four sections of the coordinate plane. Numbered I, II, III, IV counter-clockwise from top-right.

Sentence Frames

The number   is a   (positive/negative) integer because it is   (above/below) zero.
The opposite of   is   because they are the same distance from zero but on   sides.
The absolute value of   is   because it is   units from zero.
The point ( ,  ) is in Quadrant   because x is   and y is  .
To plot (3, −4), I start at the origin, move   units right, then   units down.
The distance between   and   on the number line is   because | | + | | =  .

Step-by-Step: Plot a Point on the Coordinate Plane

Example: Plot the point (−3, 2)

x y 0 1 2 3 4 −1 −2 −3 −4 1 2 3 4 −1 −2 −3 −4 (−3, 2) left 3 up 2
1

Start at the origin (0, 0) — where the two axes cross.

2

Look at x = −3. Negative means left. Move 3 units left.

3

Look at y = 2. Positive means up. Move 2 units up.

4

Draw a dot. This is (−3, 2). It is in Quadrant II (x negative, y positive).

Step-by-Step: Find Distance Between Two Points

Example: Distance between (−3, 2) and (4, 2)

1

Check: Are the points on the same horizontal line? Yes! Both have y = 2.

2

Find the difference in x-values: |−3| + |4| = 3 + 4 = 7 (they are on opposite sides of 0).

3

The distance is 7 units.

Practice Problems

Problem 1

What is the opposite of 7?

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The opposite has the same number but a different sign.
−7
Problem 2

What is |−9|?

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Absolute value is the distance from 0. It is always positive.
9 — −9 is 9 units from zero.
Problem 3

Which is greater: −3 or −8?

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On a number line, the number farther to the right is greater. −3 is to the right of −8.
−3 > −8 — −3 is closer to zero, so it is greater.
Problem 4

In which quadrant is the point (4, −5)?

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x is positive (right), y is negative (down). That is the bottom-right section.
Quadrant IV
Problem 5

What is the opposite of −12?

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Change the sign. Negative becomes positive.
12
Problem 6

Put these in order from least to greatest: 3, −5, 0, −1, 4

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Negative numbers are always less than positive numbers. More negative = smaller.
−5, −1, 0, 3, 4
Problem 7

What is the distance between −3 and 5 on a number line?

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They are on opposite sides of 0. Add |−3| + |5|.
8 units — |−3| + |5| = 3 + 5 = 8
Problem 8

In which quadrant is (−2, −6)?

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Both x and y are negative. That is the bottom-left section.
Quadrant III
Problem 9

What are the coordinates of a point that is 3 units left and 4 units up from the origin?

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Left means negative x. Up means positive y.
(−3, 4)
Problem 10

Find the distance between (2, 5) and (2, −3).

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Same x-value, so it is a vertical distance. Use the y-values: |5| + |−3|.
8 units — |5| + |−3| = 5 + 3 = 8

Real-World Connections

Temperature

When the temperature is −5°F, it is 5 degrees below zero. If it goes up 12 degrees, the new temperature is −5 + 12 = 7°F. Absolute value tells us how cold: |−5| = 5 degrees below zero.

Bank Account

Positive numbers = money you have. Negative numbers = money you owe. If you have $20 and spend $35, your balance is 20 − 35 = −$15 (you owe $15).

City Map

A city grid works like a coordinate plane. Your school is at the origin. The library is 3 blocks east and 2 blocks north = (3, 2). The park is 4 blocks west and 1 block south = (−4, −1).

Elevator

Ground floor = 0. Going up 5 floors = +5. Going to the basement 2 floors down = −2. The distance between floor −2 and floor 5 is |−2| + |5| = 7 floors.

Challenge Problems

Challenge 1

Point A is at (−4, 3) and Point B is at (5, 3). What is the distance between them? What is the midpoint?

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Same y-coordinate, so horizontal distance. For midpoint, average the x-values.
Distance = 9 units, Midpoint = (0.5, 3) — Distance: |−4| + |5| = 9. Midpoint x: (−4 + 5) ÷ 2 = 0.5.
Challenge 2

A rectangle on the coordinate plane has vertices at (−3, 4), (5, 4), (5, −2), and (−3, −2). Find its perimeter and area.

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Find the length (horizontal distance) and width (vertical distance).
Perimeter = 28, Area = 48 — Length: |−3| + |5| = 8. Width: |4| + |−2| = 6. P = 2(8+6) = 28. A = 8 × 6 = 48.
Challenge 3

Arrange from least to greatest: |−7|, −4, |3|, −|−5|, 0

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First simplify: |−7| = 7, |3| = 3, −|−5| = −5.
−5, −4, 0, 3, 7 — That is: −|−5|, −4, 0, |3|, |−7|
Challenge 4

A triangle has vertices at (0, 0), (6, 0), and (6, 8). Find the perimeter. (Hint: you will need the distance formula or Pythagorean theorem for the slanted side.)

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Two sides are along the axes: 6 and 8. The hypotenuse: √(6² + 8²) = √(36 + 64) = √100.
24 units — Bottom = 6, Right side = 8, Hypotenuse = √100 = 10. Perimeter = 6 + 8 + 10 = 24.
Challenge 5

A point is reflected across the x-axis. Its new coordinates are (3, −7). What were the original coordinates?

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Reflecting across the x-axis changes the sign of the y-coordinate.
(3, 7) — The y-coordinate flips: −7 becomes 7. The x stays the same.
Challenge 6

Point P is at (−2, 5). It is reflected across the y-axis to get P', then P' is reflected across the x-axis to get P''. What are the coordinates of P''?

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Reflecting across y-axis: change sign of x. Then reflecting across x-axis: change sign of y.
(2, −5) — P' = (2, 5), then P'' = (2, −5). Double reflection = both signs change.
Challenge 7

The temperature at midnight was −8°C. By noon it rose 15°C. Then it dropped 7°C by evening. What was the evening temperature? What was the total change from midnight to evening?

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−8 + 15 = noon temp. Then subtract 7 for evening.
Evening: 0°C. Total change: +8°C — −8 + 15 = 7. 7 − 7 = 0. Change: 0 − (−8) = 8.
Challenge 8

A square has one vertex at (−1, −1) and the opposite vertex at (3, 3). Find the coordinates of the other two vertices and the area.

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The other vertices pair opposite x and y values: (−1, 3) and (3, −1). Side length is the distance between adjacent vertices.
Vertices: (−1, 3) and (3, −1). Area = 16 — Side = |3 − (−1)| = 4. Area = 4² = 16.
Challenge 9

Find all integers x such that |x| < 4.

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Which integers are less than 4 units from zero?
−3, −2, −1, 0, 1, 2, 3 — Seven integers total.
Challenge 10

A parallelogram on the coordinate plane has vertices at (−4, 0), (0, 3), (6, 3), and (2, 0). Find its area.

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Base = horizontal distance of bottom side. Height = vertical distance between parallel sides.
18 square units — Base = |2 − (−4)| = 6. Height = |3 − 0| = 3. Area = 6 × 3 = 18.

Real-World Investigations

Investigation 1: Map Your Neighborhood

Choose a location near your home as the origin. Draw a coordinate grid and plot at least 8 locations (your house, school, store, park, etc.) as ordered pairs. Label each point with coordinates. Find the distance between at least 3 pairs of locations. Which two places are farthest apart? Create a "walking route" that visits all locations and calculate the total distance.

Investigation 2: Stock Market Tracker

Track a stock or the daily high temperature for 10 days. Record each day's change as a positive or negative number. Create a coordinate plane graph where the x-axis is the day and y-axis is the cumulative change. On which days was the cumulative change positive? Negative? What was the total change over all 10 days? Which day had the biggest absolute change?

Investigation 3: Game Board Designer

Design a board game on a coordinate grid. Create at least 3 different polygon regions (a triangular "danger zone," a rectangular "safe zone," a parallelogram "bonus area"). Specify each polygon using coordinates. Calculate the area and perimeter of each region. Write the rules explaining what happens when a player lands in each zone.

Brain Teasers

Teaser 1: The Absolute Truth

Is it possible for |a| + |b| to equal |a + b|? When does this happen? When does |a| + |b| > |a + b|? (Hint: try positive and negative combinations.)

Show Answer
|a| + |b| = |a + b| when a and b have the same sign (or one is zero). |a| + |b| > |a + b| when a and b have different signs. Example: |3| + |−5| = 8, but |3 + (−5)| = |−2| = 2. This is the triangle inequality.

Teaser 2: Quadrant Puzzle

A point (a, b) is in Quadrant II. In which quadrant is (−a, −b)? What about (b, a)? Prove your answers.

Show Answer
If (a, b) is in Quadrant II, then a < 0 and b > 0. So −a > 0 and −b < 0, meaning (−a, −b) is in Quadrant IV. For (b, a): b > 0 and a < 0, so (b, a) is in Quadrant IV as well.

Teaser 3: The Elevator Riddle

An elevator starts at floor −3 (basement level 3). It goes up 7 floors, down 4 floors, up 2 floors, down 8 floors, and up 5 floors. What floor is it on? Without calculating each step, can you find a shortcut?

Show Answer
Shortcut: add all the changes. +7 − 4 + 2 − 8 + 5 = +2. Start at −3, move +2: Floor −1 (one level below ground).

Teaser 4: Coordinate Symmetry

What shape is formed by connecting these points in order: (2, 0), (0, 2), (−2, 0), (0, −2)? What is special about its position on the coordinate plane? Calculate its area.

Show Answer
It forms a square rotated 45 degrees (a diamond). It is centered at the origin with perfect symmetry across both axes. Side length = √(2² + 2²) = √8. Area = side² = 8 square units. (Or: it is a square with diagonal 4, so area = ½ × 4 × 4 = 8.)

Where This Math Leads Next

7th Grade: Integer Operations

You will add, subtract, multiply, and divide positive and negative numbers fluently. Rules like "negative times negative equals positive" build on the number line understanding you are developing now.

7th Grade: Proportional Relationships on Graphs

The coordinate plane becomes the place where you graph proportional relationships (y = kx) and identify the constant of proportionality from a graph.

8th Grade: Linear Equations & Slope

Plotting points leads to graphing lines. You will learn slope (rise over run) and y-intercept to write equations like y = mx + b. Every point on the line is an ordered pair that satisfies the equation.

High School: Transformations & Distance Formula

Reflections across axes become full geometric transformations (translations, rotations, dilations). The distance formula d = √((x₂−x₁)² + (y₂−y₁)²) extends what you learn about horizontal and vertical distance.

Self-Assessment Checklist