Unit 9 · Standard 6.NS.8
Distance on the Coordinate Plane
Key Vocabulary Level 1 support
Picture first, then the word, then a plain-language meaning. Say each word out loud.
The distance from -3 to 4 on a number line is |-3| + |4| = 3 + 4 = 7 units
Distance
How far apart two points are. It is never negative.
|-3| = 3 and |4| = 4; to find distance: |4 - (-3)| = 7
Absolute value
How far a number is from zero.
From (-2, 3) to (5, 3): count from -2 to 5 = 7 units across
Horizontal distance
How far apart two points are going left or right.
From (4, -1) to (4, 6): count from -1 to 6 = 7 units up
Vertical distance
How far apart two points are going up or down.
Two number lines crossing at (0, 0), creating four quadrants
Coordinate plane
A grid with a line going across and a line going up that cross.
..., -3, -2, -1, 0, 1, 2, 3, ...
Integer
Whole numbers and their opposites, like -2, -1, 0, 1, 2.
Key Ideas & Notes
- Captain Vega has two treasure locations on her map: a buried chest at (-3, 2) and a hidden cave at (4, 2).
- Both are at the same height (y = 2) on the map.
- She needs to figure out how far apart they are so she knows how much rope to bring.
- Another pair of treasures is at (1, -3) and (1, 5) — same column, different rows.
- Plot each pair of treasure locations and find the distance between them.
Think About It
- What do the two points (-3, 2) and (4, 2) have in common?
- How can you count the distance between points in the same row?
- How does absolute value help when one point has a negative coordinate?
My Notes
Guided Examples
Example 1
What is the distance between (-4, 3) and (2, 3)?
Solution: Both points have y = 3, so this is a horizontal distance. |2 - (-4)| = |2 + 4| = 6 units.
Answer: C. 6 units
Example 2
What is the distance between (5, -2) and (5, 4)?
Solution: Both points have x = 5, so this is a vertical distance. |4 - (-2)| = |4 + 2| = 6 units.
Answer: C. 6 units
Example 3
A rectangle has vertices at (1, 1), (5, 1), (5, 4), (1, 4). What is its perimeter?
Solution: Width = 5 - 1 = 4. Height = 4 - 1 = 3. Perimeter = 2(4 + 3) = 14.
Answer: A. 14 units
Write About the Math The Writing Revolution
I can explain my work using the words distance, absolute value, horizontal distance, and vertical distance.
1. Kernel Sentence subject + verb
Model: Distance is how far apart two points are. It is never negative.Distancia es qué tan separados están dos puntos. Nunca es negativo.
Write a kernel sentence about distance. Use a subject and a verb.Escribe una oración base sobre distancia. Usa un sujeto y un verbo.
2. Sentence Expansion because · but · so
Kernel: Distance matters in mathDistancia importa en matemáticas
Expand the kernel three ways. Add a reason, a contrast, and a result.
Distance matters in math because ___.Distancia importa en matemáticas porque ___.
Distance matters in math, but ___.Distancia importa en matemáticas, pero ___.
Distance matters in math, so ___.Distancia importa en matemáticas, entonces ___.
3. Sentence Types 4 ways to write a math idea
Tell one true fact about distance.Di un hecho verdadero sobre distance.
Distance ___.
Ask a question about distance.Haz una pregunta sobre distance.
How does ___ ?¿Cómo ___ ?
Show excitement about distance.Muestra entusiasmo sobre distance.
Wow, ___ !¡Guau, ___ !
Tell a partner what to do with distance.Dile a un compañero qué hacer con distance.
First, ___ .Primero, ___ .
4. Explain Your Reasoning use a sentence starter
I subtracted / counted ___.Resté / conté ___.
The distance is ___ units.La distancia es ___ unidades.
I would measure distance to ___.Mediría la distancia para ___.
Try It
Solve on your own. Check the answer key when you are done.
1. A rectangle has vertices at (1, 1), (5, 1), (5, 4), (1, 4). What is its perimeter?
- 14 units
- 12 units
- 20 units
- 8 units
2. Two points are at (-3, -7) and (-3, 5). What is the distance between them?
- 12 units
- 2 units
- 7 units
- 8 units
Stretch Your Thinking Level 2 enrichment
Challenge task — explain your reasoning in full sentences.
A rectangle has vertices at (-4, -2), (3, -2), (3, 5), and (-4, 5). Find the length, width, perimeter, and area of the rectangle. Show your work using absolute value.
Sentence starter: The width is |___ - (___)| = ___ units. The height is |___ - (___)| = ___ units. The perimeter is 2(___ + ___) = ___ units. The area is ___ × ___ = ___ square units.
Reflect — Exit Ticket
What is the distance between the points (3, -2) and (3, 5)?
- 3 units
- 5 units
- 7 units
- 9 units
Answer Key & Teacher Guide
- Try It 1: A. 14 units — Width = 5 - 1 = 4. Height = 4 - 1 = 3. Perimeter = 2(4 + 3) = 14.
- Try It 2: A. 12 units — Same x = -3. Distance = |5 - (-7)| = |5 + 7| = 12 units.
- Exit Ticket: C. 7 units — Both points share x = 3, so this is a vertical distance. |5 - (-2)| = |5 + 2| = 7 units.
Writing (TWR) — what to look for
- Kernel sentence: A complete sentence needs a subject and a verb. Example: Distance is how far apart two points are. It is never negative.
- Expansion: because gives a reason, but shows a contrast or exception, so shows a result. Answers vary; each must keep the kernel idea and add the correct kind of detail.
- Sentence types: Statement ends with a period, question with "?", exclamation with "!", and a command starts with an action verb (a "bossy" verb).