7.1 Small Group · Group 2

Tier 1 Extension · Non-Routine Synthesis · Misconception Traps · Rigorous CER Proofs

Visual Models · Integer · Opposite · Procedural Fluency

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

Badwater Basin is 86 meters below sea level, and sea level is defined as 0.
Below 0 means a negative number, so I write the elevation as −86 meters.
On a number line, I count 86 units to the LEFT of 0 and mark that point.
The opposite of −86 is 86. Both are 86 units from 0, just on different sides.

3Second Model — Try it together — then prove it

One day in January the temperature in Toronto was 9°C and in Reykjavik it was −9°C.
Which one is to the left of 0 on the number line? The negative sign tells us: −9.
How many units from 0 is 9? How many units from 0 is −9? Both are 9 units.
So 9 and −9 are opposites — same distance from 0, opposite sides.
Now prove it: say why that move had to work at all — not just that it did.
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 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 FILL TABLE

Write the opposite of each integer.

Integer Opposite
-15
8
0
-100

✏️ Scratchpad / Reasoning
2 NUMBER LINE

Graph these integers: -9, 9, 0, -4, 6
✏️ Workspace & Solution Steps

Integer	Opposite
-15
8
0
-100

SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]

3 MULTIPLE CHOICE
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.?
1. A12°C and -12°C
2. B-12°C and 12°C
3. C12°C and 24°C
4. D0°C and 12°C
✏️ Workspace & Solution Steps
4 MULTIPLE CHOICE
A football team starts at the line of scrimmage, which is 0. During one series they had a 10-yard loss, a 4-yard gain, a 3-yard loss, and a 21-yard gain. Where does the series leave them?
1. A12 yards past the line of scrimmage
2. B8 yards past the line of scrimmage
3. C18 yards past the line of scrimmage
4. D38 yards past the line of scrimmage
✏️ Workspace & Solution Steps
5 MULTIPLE CHOICE
Claim: “The opposite of a number is always negative.” Always, sometimes, or never true?
1. ASometimes — it is negative only when the original number is positive
2. BAlways — that is what taking the opposite does
3. CNever — opposites are always positive
4. DSometimes — it depends whether the number is a whole number
✏️ Workspace & Solution Steps

SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]

6 OPEN RESPONSE

Describe two different real-world situations that use negative integers. For each one, tell exactly what 0 means in that situation, write one negative integer it could produce, and explain what the opposite of that integer would mean in the same situation.
✏️ Mathematical Justification & Response

🗣️ MATHEMATICAL DISCOURSE & TALK MOVES SMP.3 / Construct Viable Arguments

Group Discussion Prompt: How does your visual model justify your mathematical solution?

🗣️ Partner A:
Partner A: "I modeled this by identifying the relationship and..."

👂 Partner B:
Partner B: "I agree with your step because the standard mathematical rule states..."

✍️ The Writing Revolution (TWR) · Sentence Expansion Because · But · So

Complete each sentence stem to demonstrate precise mathematical reasoning:

BECAUSE The mathematical model proves the solution because each step preserves quantitative equivalence.

BUT An estimate provides a quick benchmark, but an exact proof is required for precision.

SO The quantities follow the standard rule, so the final result is verified with certainty.

📐 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.

✍️ AUTHOR YOUR OWN EXTENSION CHALLENGE

Create an original multi-step word problem aligned to this standard. Include constraints, and write the complete step-by-step mathematical proof below.

✏️ Author Workspace & Complete Solution Key

Student Mastery Self-Assessment:

1 · Need More Support

2 · Getting Closer

3 · Got It / Solid

4 · Master / Can Teach It