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

8.6 Small Group · Group 2

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

Solve and Graph Inequalities · Solve · Procedural Fluency · Reasoning & Critique

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do I solve an inequality and then graph it?
1Solve with inverse operations, then graph the solution set: open circle for < or >, closed for ≤ or ≥ — 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. Solve x + 3 > 53. The inverse of adding 3 is subtracting 3.
  2. I subtract 3 from both sides: x + 3 − 3 > 53 − 3, so x > 50.
  3. The symbol > does not include 50, so I use an open circle at 50.
  4. 'Greater than' shades right. Check: x = 60 gives 60 + 3 = 63 > 53. True!
3Second Model — Try it together — then prove it
  1. Solve x − 5 ≤ 8. What is the inverse of subtracting 5?
  2. We add 5 to both sides: x ≤ 8 + 5, so x ≤ 13.
  3. ≤ includes 13, so we use a closed circle and shade left.
  4. Substitute x = 10: 10 − 5 = 5 ≤ 8. True?
  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
  • Solve and Graph Inequalities (Resolver y representar desigualdades) — Isolate the variable, then show the complete solution set on a number line.
  • Solve (Resolver) — To find the number that makes it true.
  • Graph (Graficar) — To show answers on a number line with circles and shading.
  • Solution (Solución) — A number that makes the equation or inequality true.
  • Substitute (Sustituir) — To put a number in place of a letter.
  • Solution set (Conjunto solución) — All the numbers that make the inequality true.
5Watch out
  • A common mistake in Solve and Graph Inequalities is shading the number line in the wrong direction after solving. For example, a student solves x + 4 > 10 correctly to get x > 6, but then shades LEFT toward smaller numbers instead of RIGHT toward greater numbers. The shading direction always matches the solved symbol, not the original problem: '>' and '≥' shade right (greater numbers), '<' and '≤' shade left (smaller numbers) — check this AFTER you solve, not before. To catch this mistake, substitute one shaded value and one unshaded value back into the original inequality: the shaded value should make it true, and the unshaded value should make it false.
SECTION 1 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 1 MULTIPLE CHOICE

    Solve: x + 6 > 14

    1. Ax > 8
    2. Bx > 20
    3. Cx < 8
    4. Dx = 8
    ✏️ Workspace & Solution Steps
  2. 2 MULTIPLE CHOICE

    Solve: x − 9 ≤ 3

    1. Ax ≤ 12
    2. Bx ≤ 6
    3. Cx ≥ 12
    4. Dx = 12
    ✏️ Workspace & Solution Steps
  3. 3 MULTIPLE CHOICE

    Is x = 5 a solution to x + 4 > 12?

    1. ANo, because 5 + 4 = 9, and 9 is not greater than 12
    2. BYes, because 5 + 4 = 9
    3. CYes, because 5 > 4
    4. DNo, because 5 < 12
    ✏️ Workspace & Solution Steps
SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 4 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:Solve: x + 6 > 14
    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

    Catch the Open vs. Closed Circle Mistake

    1. 1Inequality:x − 4 ≥ 9
    2. 2Inverse operation:Add 4 to both sides
    3. 3Simplify:x ≥ 13
    4. 4Graph:Open circle at 13, shade 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 Solving Error

    1. 1Inequality:x − 8 ≥ 15
    2. 2Inverse operation:Subtract 8 from both sides
    3. 3Simplify:x ≥ 7
    4. 4Graph:Closed circle at 7, shade 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