Grade 6 Mathematics · Unit 7: Integers, Rational Numbers, and the Coordinate PlaneStandard 6.NOS.9

Lesson 7.67.6 Small Group · Group 2

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can find the distance between two points on the coordinate plane using absolute value, explain why the method works, and use it on a problem I have not seen before.

Why This Matters This is your challenge small group for Lesson 7.6. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
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. uses the x-coordinates
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. x-axis across, y-axis up
IntegerSpanish: Número entero Whole numbers and their opposites, like -2, -1, 0, 1, 2. ..., -3, -2, -1, 0, 1, 2, 3, ...

How it worksWorked example

These numbers are not on your problems. The steps are. Follow them with your own numbers.

Each unit is 1 meter. Find the distance from P(−9, 4) to Q(3, 4), and from P to R(−9, 11).

  1. P and Q share y = 4, so use x. −9 and 3 have opposite signs: 9 + 3 = 12 m.
  2. P and R share x = −9, so use y. 4 and 11 are both positive: 11 − 4 = 7 m.

Answer: PQ = 12 m and PR = 7 m

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

Find the distance between (2, 3) and (2, 8).

Answer:

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 the segment runs .
  • I used absolute value because .
  • The points sit on sides of the axis, so I the two distances.
  • The distance is units, which means on this map.

Word bank distanceabsolute valuehorizontalverticalcoordinate planeaxisunitsaddsubtractx-coordinatey-coordinatesegment

Watch out!A 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

Grade 6 Mathematics · Unit 7: Integers, Rational Numbers, and the Coordinate PlaneStandard 6.NOS.9

Lesson 7.67.6 Small Group · Group 2

Group 2Challenge practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can find the distance between two points on the coordinate plane using absolute value, explain why the method works, and use it on a problem I have not seen before.

Why This Matters This is your challenge small group for Lesson 7.6. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.

  1. 1Complete each step, then write the final answer.
    0−8−7−6−5−4−3−2−11−4−3−2−11(−7, −2)(−1, −2)
    Both points are left of the y-axis.

    Find the distance between (−7, −2) and (−1, −2).

    1. 1Which coordinate is the same? Which way does the line go?
    2. 2Are −7 and −1 on the same side of 0?
    3. 3Find the distance.
    Final answer:
  2. 2Complete each step, then write the final answer.
    0−11234−4−3−2−112345(1.5, −2.5)(1.5, 4)

    Find the distance between (1.5, −2.5) and (1.5, 4).

    1. 1Which coordinate is the same? Which way does the line go?
    2. 2Find the distance.
    3. 3Why do we add 2.5 and 4 here, and not subtract?
    Final answer:
  3. 3Complete each step, then write the final answer.

    Find the distance between (−3.5, 1) and (2.25, 1).

    Show your work
    Final answer:
  4. 4Complete each step, then write the final answer.

    Point P is at (−2, 5). Point Q has the same x-coordinate and is 8 units from P. Q is in Quadrant III. What is the y-coordinate of Q?

    Show your work
    Final answer:
  5. 5Complete each step, then write the final answer.

    A hiking map uses miles. The camp is at (−4.5, −2) and the lake is at (3, −2). How far is the camp from the lake?

    Show your work
    Final answer:
  6. 6Complete 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 work
    Final answer:

Explain your thinkingKim subtracted when she should have added. How can the signs tell you which to do? Is there one rule that always works?

ChallengeWrite your own problem

Write a new word problem that uses this lesson's idea. Then solve it and show every step.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help