6.NOS.6 Lesson 7-1-group2 🟣 Group 2 · Set B
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

7.1 Small Group · Group 2

Second Challenge Form · Non-Routine Extension · Same Standard, New Problems

Visual Models · Integer · Opposite · Procedural Fluency

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do you use integers to describe quantities above and below zero?
1Pick what 0 means in the situation first. Then a quantity above 0 is a positive integer and a quantity below 0 is a negative integer, and its opposite is the same distance from 0 on the other side — and you can say why it is true, and where it would stop being true.
  • You already can do this. So we go up a level, not up a number: find what is always true, show it a second way, and be ready to defend it with a reason instead of an answer.
2Worked Example — Watch me
  1. Badwater Basin is 86 meters below sea level, and sea level is defined as 0.
  2. Below 0 means a negative number, so I write the elevation as −86 meters.
  3. On a number line, I count 86 units to the LEFT of 0 and mark that point.
  4. The opposite of −86 is 86. Both are 86 units from 0, just on different sides.
3Second Model — Try it together — then prove it
  1. One day in January the temperature in Toronto was 9°C and in Reykjavik it was −9°C.
  2. Which one is to the left of 0 on the number line? The negative sign tells us: −9.
  3. How many units from 0 is 9? How many units from 0 is −9? Both are 9 units.
  4. So 9 and −9 are opposites — same distance from 0, opposite sides.
  5. Now prove it: say why that move had to work at all — not just that it did.
  6. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • Integer (Número entero) — The counting numbers, their opposites, and 0.
  • Opposite (Opuesto) — A number that is the same distance from 0 as another number, but on the other side of 0.
  • Negative number (Número negativo) — A number less than 0, written with a − sign. On a number line it sits to the left of 0.
  • Positive number (Número positivo) — A number greater than 0. On a number line it sits to the right of 0.
  • Number line (Recta numérica) — A straight line where numbers are placed in order. It extends to the left of 0 to show values less than 0.
5Watch out
  • The most common mistake in this lesson is thinking that the opposite of a negative integer is still negative — for example, saying the opposite of -7 is -7, or that the opposite of -12 is -24. Opposites are the same DISTANCE from 0 but on OPPOSITE sides of 0, so taking an opposite always changes the direction and never changes the distance: the opposite of -7 is 7 and the opposite of -12 is 12. The one exception is 0, which is 0 units from itself and is therefore its own opposite. Before you submit, count the units from 0 and check that your answer is on the other side.
SECTION 1 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 1 MULTIPLE CHOICE

    A diver is at −18 m and a drone is at +18 m. Priya says they are the same distance from sea level. Wes says the drone is higher, so it is farther away. Who is right?

    1. APriya — both are 18 m from 0, and distance from 0 ignores direction
    2. BWes — +18 is greater than −18, so it must be farther
    3. CBoth, because distance and height mean the same thing
    4. DNeither — you cannot compare a depth with a height
    ✏️ Workspace & Solution Steps
  2. 2 MULTIPLE CHOICE

    A classmate says “0 has no opposite, because you cannot make it negative.” What is the best response?

    1. A0 is its own opposite — it sits at distance 0 from itself, so flipping sides changes nothing
    2. BThey are right; 0 is the one number with no opposite
    3. CThe opposite of 0 is −0, which is a different number
    4. D0 has two opposites, +0 and −0
    ✏️ Workspace & Solution Steps
  3. 3 MULTIPLE CHOICE

    Which list of integers is ordered from LEAST to GREATEST?

    1. A-8, -3, 0, 5
    2. B-3, -8, 0, 5
    3. C0, -3, -8, 5
    4. D5, 0, -3, -8
    ✏️ Workspace & Solution Steps
  4. 4 MULTIPLE CHOICE

    Badwater Basin is at -86 meters. Which elevation is FARTHER below sea level than Badwater Basin?

    1. A-120 meters
    2. B-40 meters
    3. C0 meters
    4. D86 meters
    ✏️ Workspace & Solution Steps
SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 5 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:At 6 a.m. in Reykjavik the temperature was -12°C. At 2 p.m. it was the opposite of the 6 a.m. temperature, and at 11 p.m. it was the opposite of the 2 p.m. temperature. What were the temperatures at 2 p.m. and 11 p.m.?
    2. 2A classmate at our table answered:12°C and 24°C

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
  2. 6 ERROR ANALYSIS

    Spot the Mistake: Reykjavik's Temperature Swing

    1. 1Problem:At 6 a.m. in Reykjavik the temperature was -12°C. At 2 p.m. it was the opposite of the 6 a.m. temperature. What was the temperature at 2 p.m.?
    2. 2Student thinking:The opposite of a temperature should still be a cold temperature, so I keep the negative sign and just make the number bigger. That gives a temperature farther from 0.
    3. 3Student answer:The temperature at 2 p.m. was -24°C.

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
📐 CER MATHEMATICAL PROOF MATRIX (CER 2.0) SMP.3 / Proof & Justification

Writing Task: Justify why your mathematical solution is accurate and complete.

C Claim
State your direct mathematical answer/claim with units.
E Evidence
Cite exact numbers, calculations, dimensions, or graph data.
R Reasoning
Explain the mathematical theorem, property, or definition connecting evidence to claim.
Student Mastery Self-Assessment:
1 · Need More Support
2 · Getting Closer
3 · Got It / Solid
4 · Master / Can Teach It