6.AT.9 Lesson 8-4-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
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8.4 Small Group · Group 2

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

Write Inequalities · Inequality · Procedural Fluency · Reasoning & Critique

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do I write an inequality from words?
1'At least' means ≥ and 'at most' means ≤; 'more than' means > and 'fewer/less than' means < — 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. Clue: 'The suspect is at least 18 years old.'
  2. I let a stand for the age. 'At least 18' means 18 or more.
  3. 'At least' includes 18, so I use the symbol ≥.
  4. My inequality is a ≥ 18. So a could be 18, 19, 20, and up.
3Second Model — Try it together — then prove it
  1. Clue: 'A bag holds fewer than 10 items.' What does 'fewer than' mean?
  2. It means less than, and it does NOT include 10, so we use <.
  3. We let n be the number of items: n < 10.
  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
  • Write Inequalities (Escribir desigualdades) — Represent a range of possible values with <, >, ≤, or ≥.
  • Inequality (Desigualdad) — A math sentence that compares two sides with <, >, ≤, or ≥.
  • Greater than (Mayor que) — The > sign, showing one number is bigger.
  • Less than (Menor que) — The < sign, showing one number is smaller.
  • At least / At most (Al menos / A lo más) — 'At least' means ≥. 'At most' means ≤.
  • Variable (Variable) — A letter that stands for a number that is unknown or can change.
5Watch out
  • A common mistake in Write Inequalities is using a strict symbol (> or <) for a phrase that actually includes the boundary value. For 'the suspect is at least 18 years old,' students often write a > 18 instead of the correct a ≥ 18 — this wrongly excludes 18 itself, even though 'at least 18' means 18 counts as a match. The same error happens in reverse with 'at most': for 'the getaway bag holds at most 12 items,' students write n < 12 instead of n ≤ 12. Always ask: does the key phrase include the number ('at least,' 'at most,' 'no more than,' 'no fewer than' → use ≥ or ≤) or exclude it ('more than,' 'fewer than' → use > or <)?
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Identify the key phrase, symbol, and write the inequality for each situation.

    SituationKey PhraseSymbolInequality
    A package weighs less than 50 pounds.
    You need at least $25 to buy the game.
    No more than 30 students can join the club.
    ✏️ Scratchpad / Reasoning
  2. 2 DRAG SORT

    Test x = 10 in each inequality. Sort by whether it works.

    Target Categories: x = 10 IS a solution x = 10 is NOT a solution
    • x ≥ 10
    • x ≤ 15
    • x > 8
    • x > 10
    • x < 10
    • x ≤ 5
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    A clue says the suspect carried at least 12 stolen gems. Which inequality shows the number g of gems?

    1. Ag ≥ 12
    2. Bg > 12
    3. Cg ≤ 12
    4. Dg < 12
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    A witness reports that fewer than 5 people entered the building. Which inequality shows the number n of people?

    1. An < 5
    2. Bn ≤ 5
    3. Cn > 5
    4. Dn ≥ 5
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Which inequality represents: 'The temperature is less than 40 degrees'?

    1. At < 40
    2. Bt > 40
    3. Ct ≤ 40
    4. Dt = 40
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 ERROR ANALYSIS

    Level 2 Extension — Catch the Symbol Mix-Up

    1. 1Clue:The suspect is at least 12 years on the force, where y is the years.
    2. 2Find the key phrase:'At least 12' means 12 is allowed and anything larger.
    3. 3Write the inequality:y > 12

    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