6.AT.8 Lesson 8-7-group2 🟣 Group 2 · Set B
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

8.7 Small Group · Group 2

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

Equations and Inequalities Problem Solving · Model · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Should I model this problem with an equation or an inequality?
1'Equals' or 'exactly' points to an equation (=); 'at least, at most, more than, fewer than' points to an inequality — 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: 'A total divided equally among 4 people gave each $85.'
  2. This has an exact total, so I use an equation. Let t be the total.
  3. Divided among 4 to get 85 means t ÷ 4 = 85, so t = 85 × 4 = 340.
  4. Check reasonableness: $340 split 4 ways is $85 each. That makes sense!
3Second Model — Try it together — then prove it
  1. Clue: 'The lookout earned less than $50.' Equation or inequality?
  2. 'Less than' is a limit, so it is an inequality. Let e be the earnings.
  3. 'Less than 50' means e < 50. Is $30 a reasonable value? Yes.
  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
  • Equations and Inequalities Problem Solving (Resolución de problemas con ecuaciones y desigualdades) — Use an equation for one exact value or an inequality for a range of values in a real situation.
  • Model (Modelar) — To show a real-life situation with an equation or inequality.
  • Equation (Ecuación) — A math sentence with an equal sign showing both sides are the same.
  • Inequality (Desigualdad) — A math sentence that compares two sides with <, >, ≤, or ≥.
  • Reasonableness (Razonabilidad) — Checking if your answer makes sense.
  • Variable (Variable) — A letter that stands for a number that is unknown or can change.
5Watch out
  • A common mistake in Equations and Inequalities Problem Solving is matching the wrong model to a key phrase. For 'the getaway car held no more than 6 bags,' students often write the equation b = 6 instead of the inequality b ≤ 6, because they see one number and assume it must be exact. The same mistake happens in reverse: for 'the reward split equally among 5 people gave each $40,' students sometimes write the inequality 5p ≥ 40 instead of the equation 5p = 40, because 'split equally' sounds like it could mean a range. Before you submit, ask: does this phrase give one exact value (equation, =) or a limit/range (inequality, <, >, ≤, ≥)?
SECTION 1 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 1 MULTIPLE CHOICE

    A museum has some paintings. After adding 12, they have 45. Which equation models this?

    1. Ap + 12 = 45
    2. Bp − 12 = 45
    3. C12p = 45
    4. Dp / 12 = 45
    ✏️ Workspace & Solution Steps
  2. 2 MULTIPLE CHOICE

    A detective needs more than 8 hours to finish the investigation. She has already worked 3 hours. Which inequality represents the additional hours h she needs?

    1. A3 + h > 8
    2. B3 + h = 8
    3. Ch > 3
    4. D8 + h > 3
    ✏️ Workspace & Solution Steps
  3. 3 MULTIPLE CHOICE

    A student solved 4n = 52 and got n = 13. Is the answer reasonable if n represents the number of notebooks in a box?

    1. AYes — 13 notebooks per box is reasonable
    2. BNo — 13 is too many notebooks
    3. CNo — the answer should be 48
    4. DYes — but only if n is a fraction
    ✏️ Workspace & Solution Steps
SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 4 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:A student solved 4n = 52 and got n = 13. Is the answer reasonable if n represents the number of notebooks in a box?
    2. 2A classmate at our table answered:No — 13 is too many notebooks

    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. 1Problem:Seven evidence lockers hold 84 case files in all, split equally. How many files per locker?
    2. 2Model:7f = 84
    3. 3Solve:f = 84 − 7 = 77
    4. 4Answer:77 files per locker.

    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 Problem-Solving Error

    1. 1Problem:A parking lot holds at most 200 cars. There are 134 already. How many more can park?
    2. 2Model:134 + c = 200
    3. 3Solve:c = 66
    4. 4Answer:Exactly 66 more cars can park.

    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