8.7 Small Group · Group 2

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

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

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

Clue: 'A total divided equally among 4 people gave each $85.'
This has an exact total, so I use an equation. Let t be the total.
Divided among 4 to get 85 means t ÷ 4 = 85, so t = 85 × 4 = 340.
Check reasonableness: $340 split 4 ways is $85 each. That makes sense!

3Second Model — Try it together — then prove it

Clue: 'The lookout earned less than $50.' Equation or inequality?
'Less than' is a limit, so it is an inequality. Let e be the earnings.
'Less than 50' means e < 50. Is $30 a reasonable value? Yes.
Now prove it: say why that move had to work at all — not just that it did.
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 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 FILL TABLE

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

Word Problem	Model	Solution	Reasonable?
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

2 BALANCE SCALE

Algebraic Balance Model

Solve this two-step problem: 3x + 5 = 26
✏️ Workspace & Solution Steps

SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]

3 MULTIPLE CHOICE
Room 2: The vault keypad clue reads, 'A number of stolen coins plus 6 equals 14.' Solve x + 6 = 14 to find the code x.
1. Ax = 8
2. Bx = 20
3. Cx = 6
4. Dx = 2
✏️ Workspace & Solution Steps
4 MULTIPLE CHOICE
Room 3: A sign warns, 'The safe holds at most 20 gold bars.' Which inequality models the number of bars x?
1. Ax ≤ 20
2. Bx ≥ 20
3. Cx > 20
4. Dx = 20
✏️ Workspace & Solution Steps
5 MULTIPLE CHOICE
Room 4: A locked drawer clue reads, 'Five identical evidence boxes weigh 60 pounds in all.' Solve 5w = 60 for each box's weight w.
1. Aw = 12
2. Bw = 55
3. Cw = 300
4. Dw = 65
✏️ Workspace & Solution Steps

SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]

6 ERROR ANALYSIS
Level 2 Extension — The Final Door: Catch the Detective's Mistake
1. 1Clue:Solve the keypad inequality 3x ≤ 18 to find how many bars x fit.
2. 2Step 1:3x ≤ 18
3. 3Step 2:Divide only the left side by 3: 3x ÷ 3 ≤ 18
4. 4Step 3:x ≤ 18
5. 5Answer:At most 18 bars fit.
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