6.AT.8 Lesson 8-7-group1 🟡 Group 1 · Support & Scaffolding
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

8.7 Small Group · Group 1

Tier 2 Intervention · Concrete-Representational-Abstract (CRA) · Dual-Language Anchors

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

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ CONCEPT SUMMARY & GUIDED NOTES: Let's build it together — 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.
  • To model means to write a real situation as math. I use an equation when there is one exact total, and an inequality when there is a limit or a range of allowed values. We'll walk through a worked example, try one together, then you'll try a few with hints right there when you need them.
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!
3Mathematical 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.
4Watch 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 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 MULTIPLE CHOICE

    Which model is correct for: 'A number divided by 6 equals 7'?

    1. An / 6 = 7
    2. B6n = 7
    3. Cn − 6 = 7
    4. Dn + 6 = 7
    ✏️ Workspace & Solution Steps
  2. 2 DRAG SORT

    Sort each problem: equation or inequality?

    Target Categories: Equation (=) Inequality (<, >, ≤, ≥)
    • Five boxes weigh exactly 60 pounds total. How much does each weigh?
    • A number plus 8 equals 20.
    • A shelf holds at most 30 books. There are already 17. How many more fit?
    • The temperature must stay above 32 degrees.
  3. 3 FILL TABLE

    Write a model, solve, and check reasonableness for each problem.

    Word ProblemModelSolutionReasonable?
    Five identical boxes weigh a total of 60 pounds. How much does each box weigh?
    A shelf can hold at most 30 books. There are already 17 books. How many more can be added?
    A baker split 84 cookies equally into bags of 12. How many bags did she use?
    ✏️ Scratchpad / Reasoning
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 4 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. 5 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. 6 MULTIPLE CHOICE

    A bus can carry at most 48 passengers. There are already 31 on board. Which inequality shows how many more can board?

    1. A31 + p ≤ 48
    2. B31 + p > 48
    3. C31 + p = 48
    4. Dp − 31 ≤ 48
    ✏️ Workspace & Solution Steps
🗣️ 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
Starter: My mathematical claim is that the solution is...
E Evidence
Starter: The evidence from the model/table demonstrates that...
R Reasoning
Starter: This proves my answer because the standard mathematical definition of...
Student Mastery Self-Assessment:
1 · Need More Support
2 · Getting Closer
3 · Got It / Solid
4 · Master / Can Teach It