Lesson 7.6Comparing Distances on the Plane
Start hereWords, worked example, and sentence starters
Learning target I can find the distance between two points on the coordinate plane using absolute value.
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| DistanceSpanish: Distancia | How far apart two points are. It is never negative. | always a positive length |
| Absolute valueSpanish: Valor absoluto | How far a number is from zero. It is never negative. | |−7| = 7 |
| Horizontal distanceSpanish: Distancia horizontal | How far apart two points are going left or right. | From (-2, 3) to (5, 3): count from -2 to 5 = 7 units across |
| Vertical distanceSpanish: Distancia vertical | How far apart two points are going up or down. | uses the y-coordinates |
| Coordinate planeSpanish: Plano cartesiano | A grid with a line going across and a line going up to plot points. | Two number lines crossing at (0, 0), creating four quadrants |
| IntegerSpanish: Número entero | Whole numbers and their opposites, like -2, -1, 0, 1, 2. | ..., -3, -2, -1, 0, 1, 2, 3, ... |
How it worksWorked exampledeciding whether to add or subtract, then comparing
These numbers are not on your problems. The steps are. Follow them with your own numbers.
From the hub H(4.25, 5.75), Ana walks to A(4.25, −3.25) and Beto walks to B(−6.75, 5.75). Who walks farther?
- Ana's trip shares an x-coordinate, so compare the y-coordinates 5.75 and −3.25.
- Their signs differ, so the trip crosses the x-axis: 5.75 + 3.25 = 9
- Beto's trip shares a y-coordinate; the x-coordinates 4.25 and −6.75 differ in sign.
- Add those distances: 4.25 + 6.75 = 11 . Then compare 9 with 11.
Answer: Ana walks 9 units and Beto walks 11 units, so Beto walks farther.
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
From the hub G(1.75, 8.25), Cruz walks to C(1.75, 2.25) and Dara walks to D(−5.25, 8.25). Who walks farther?
- Cruz's trip shares an -coordinate, so compare the -coordinates.
- Those two coordinates have the sign, so I :
- Dara's trip shares a -coordinate; its x-coordinates have signs.
- Find Dara's distance: . Then compare the two trips:
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- The two points share the -coordinate, so I compare the -coordinates.
- Their signs are , so I the absolute values.
- One trip is units and the other is units, so is farther.
- A distance is always positive because .
Word bank distanceabsolute valuecomparequadrantaxisaddsubtractsame signopposite signsfartherunitscoordinate
Watch outA common mistake
A common mistake in Distance on the Coordinate Plane is subtracting the coordinates as if both points were on the same side of zero, instead of adding their absolute values when the points are on opposite sides. For example, to find the distance between (-6, 3) and (2, 3), a student might subtract 6 − 2 = 4, but since -6 and 2 are on opposite sides of zero, the absolute values must be added instead: |-6| + |2| = 6 + 2 = 8. The correct distance is 8 units, not 4. Before you submit, ask: 'Are these two points on opposite sides of zero — and if so, did I add the absolute values instead of subtracting?'
Lesson 7.6Comparing Distances on the Plane
Apply Day · Version ASupported practice
Learning target I can find the distance between two points on the coordinate plane using absolute value.
-
1Complete each step, then write the final answer.
Find the distance between (−1, 3) and (5, 3).
Show your workHint 0 is between −1 and 5. Add the distances from 0.
Final answer: -
2Complete each step, then write the final answer.
Find the distance between (2, 1) and (2, 9).
Show your workHint Both y-values are positive. Subtract them.
Final answer: -
3Complete each step, then write the final answer.
Find the distance between (−4, 2) and (−4, −5).
Show your workHint The y-values have different signs. Add the distances from 0.
Final answer:
Lesson 7.6Comparing Distances on the Plane
Apply Day · Version BCore practice
Learning target I can find the distance between two points on the coordinate plane using absolute value.
-
1Complete each step, then write the final answer.
Leo finds the distance from (2, −1) to (2, −7). He adds 1 + 7 = 8. Is he right? What is the distance?
Show your workFinal answer: -
2Complete each step, then write the final answer.
Find the distance between (−3.5, 1) and (2.25, 1).
Show your workFinal answer: -
3Complete each step, then write the final answer.
Find the distance between (−5, 3) and (4, 3).
Show your workFinal answer:
Lesson 7.6Comparing Distances on the Plane
Apply Day · ChallengeExtension
Learning target I can find the distance between two points on the coordinate plane using absolute value.
-
1Circle the letter of the best answer. Show how you know.
A chest sits at (−3, 2) and a cave at (4, 2). How far apart are they?
- A1 unit
- B2 units
- C5 units
- D7 units
Show your work -
2Complete each step, then write the final answer.
Kim says the distance from (5, −4) to (5, 9) is 9 − 4 = 5. She always subtracts the distances from 0. What is the correct distance?
Show your workFinal answer: -
3Complete each step, then write the final answer.
Find the distance between (−9, −5) and (−2, −5).
Show your workFinal answer: