6.AT.9 Lesson 8-5-group2 🟣 Group 2 · Set B
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

8.5 Small Group · Group 2

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

Visual Models · Graph Inequalities · Number line · Procedural Fluency

Interactive Workbench & Models →
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
  1. Graph t ≥ 2. The boundary number is 2.
  2. The symbol ≥ includes 2, so I draw a closed (filled) circle at 2.
  3. 'Greater than or equal to' means larger values work, so I shade to the right.
  4. Now the graph shows 2, 3, 4, and every value above 2.
3Second Model — Try it together — then prove it
  1. Graph x < 5. What number is the boundary?
  2. The symbol < does NOT include 5, so we use an open (empty) circle at 5.
  3. Which way do we shade for 'less than'? To the left, toward smaller numbers.
  4. Now prove it: say why that move had to work at all — not just that it did.
  5. 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 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 1 MULTIPLE CHOICE

    Which graph represents x ≥ 3?

    1. AClosed circle at 3, shaded right
    2. BOpen circle at 3, shaded right
    3. CClosed circle at 3, shaded left
    4. DOpen circle at 3, shaded left
    ✏️ Workspace & Solution Steps
  2. 2 MULTIPLE CHOICE

    Which graph represents x > 8?

    1. AOpen circle at 8, shaded right
    2. BClosed circle at 8, shaded right
    3. COpen circle at 8, shaded left
    4. DClosed circle at 8, shaded left
    ✏️ Workspace & Solution Steps
  3. 3 MULTIPLE CHOICE

    A number line shows a closed circle at 8 and is shaded to the left. Which inequality does this represent?

    1. Ax ≤ 8
    2. Bx < 8
    3. Cx ≥ 8
    4. Dx > 8
    ✏️ Workspace & Solution Steps
SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 4 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:A number line shows a closed circle at 8 and is shaded to the left. Which inequality does this represent?
    2. 2A classmate at our table answered:x ≥ 8

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

    ✏️ Corrected Mathematical Work & Explanation
  2. 5 ERROR ANALYSIS

    Spot the Common Mistake

    1. 1Read:The case file needs fewer than 9 clues, so c < 9. Graph it.
    2. 2Circle choice:Open circle at 9
    3. 3Shade direction:Shade right
    4. 4Final graph:Open circle at 9, shaded right

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

    ✏️ Corrected Mathematical Work & Explanation
  3. 6 ERROR ANALYSIS

    Find the Graphing Error

    1. 1Inequality:x > 6
    2. 2Circle type:Closed circle at 6
    3. 3Shade direction:Shade right
    4. 4Graph:Closed circle at 6, shaded right

    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