8.5 Small Group · Group 2

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

Visual Models · Graph Inequalities · Number line · Procedural Fluency

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — How do I graph an inequality on a number line?

1Closed circle for ≤ or ≥ (included), open circle for < or > (not included); shade right for greater, left for less — 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

Graph t ≥ 2. The boundary number is 2.
The symbol ≥ includes 2, so I draw a closed (filled) circle at 2.
'Greater than or equal to' means larger values work, so I shade to the right.
Now the graph shows 2, 3, 4, and every value above 2.

3Second Model — Try it together — then prove it

Graph x < 5. What number is the boundary?
The symbol < does NOT include 5, so we use an open (empty) circle at 5.
Which way do we shade for 'less than'? To the left, toward smaller numbers.
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

Graph Inequalities (Representar desigualdades en una recta numérica) — Show every solution to an inequality on a number line with a boundary point and an arrow.
Number line (Recta numérica) — A straight line where numbers are placed in order; numbers get smaller to the left and larger to the right.
Open circle (Círculo abierto) — An empty circle showing a number is NOT included.
Closed circle (Círculo cerrado) — A filled-in circle showing a number IS included.
Solution set (Conjunto solución) — All the numbers that make the inequality true.
Inequality (Desigualdad) — A math sentence that compares two sides with <, >, ≤, or ≥.

5Watch out

A common mistake in Graph Inequalities is treating ≥ and ≤ the same as > and < and drawing the wrong circle. For example, students often graph x ≥ 6 with an OPEN circle (as if it were x > 6), which wrongly excludes 6 even though 6 itself makes x ≥ 6 true. A second common mistake is choosing the right circle but shading the wrong direction — for x < 9 students sometimes shade right instead of left. Always check the symbol twice: once for the circle (does it have an 'or equal to' line?) and once for the shading direction (does the symbol point toward larger or smaller numbers?).

SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 NUMBER LINE

Read the graph description and write the inequality it represents. Then graph it.
✏️ Workspace & Solution Steps
2 FILL TABLE

Complete the table comparing how different symbols affect the graph.

Inequality Circle Type Shade Direction Is x = boundary a solution?
x > 6
x ≥ 6
x ≤ 6

✏️ Scratchpad / Reasoning

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

3 MULTIPLE CHOICE
Clue: the witness ran more than 15 feet, so d > 15. On the number line, what do you draw?
1. AOpen circle at 15, shade right
2. BClosed circle at 15, shade right
3. COpen circle at 15, shade left
4. DClosed circle at 15, shade left
✏️ Workspace & Solution Steps
4 MULTIPLE CHOICE
Clue: the getaway car held at most 4 suspects, so s ≤ 4. How is this graphed?
1. AClosed circle at 4, shade left
2. BOpen circle at 4, shade left
3. CClosed circle at 4, shade right
4. DOpen circle at 4, shade right
✏️ Workspace & Solution Steps
5 MULTIPLE CHOICE
Which graph represents x < 5?
1. AOpen circle at 5, shaded left
2. BClosed circle at 5, shaded left
3. COpen circle at 5, shaded right
4. DClosed circle at 5, shaded right
✏️ Workspace & Solution Steps

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

6 ERROR ANALYSIS
Level 2 Extension — Crack the Graphing Mistake
1. 1Clue inequality:k > 19 (the loot weighed more than 19 pounds)
2. 2Circle type:Closed circle at 19
3. 3Shade direction:Shade right
4. 4Final graph:Closed circle at 19, shaded right
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation

🗣️ 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 ratio remains equivalent because both quantities are scaled by the exact same multiplicative factor.

BUT Two ratios may look similar, but inverting the order of terms fundamentally changes the comparison.

SO The recipe requires 3 parts flour to 2 parts water, so the unit rate is 1.5 cups of flour per cup of water.

📐 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

Inequality	Circle Type	Shade Direction	Is x = boundary a solution?
x > 6
x ≥ 6
x ≤ 6