6.AT.11 Lesson 9-1-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

9.1 Small Group · Group 2

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

independent variable · dependent variable · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Which quantity is doing the changing — and which is responding?
1The independent variable is subject to choice. The dependent variable changes in response to it — 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 sort Akela's experiment
  1. The two quantities are hours of sunlight per day and average plant height after 3 weeks.
  2. Akela CHOSE where to put each tray, so she chose the sunlight. Sunlight is subject to choice — that makes it the independent variable.
  3. She did not choose how tall the plants got. Height responded. That makes height the dependent variable.
  4. Say it as a sentence: plant height depends on sunlight.
3Second Model — Try it together — then prove it
  1. Claire burned 123 Calories on a 30-minute jog and 246 Calories on a 60-minute jog.
  2. Which did Claire choose — the minutes, or the Calories?
  3. She chose how long to jog. So time is independent, and Calories burned is dependent.
  4. Sentence frame: ___ depends on ___ .
  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
  • independent variable (variable independiente) — The quantity that is subject to choice. It does not depend on the other quantity.
  • dependent variable (variable dependiente) — The quantity that changes in response to the independent variable.
  • depends on (depende de) — One quantity depends on another when changing the first causes the second to change.
  • quantity (cantidad) — Something that can be counted or measured.
  • relationship (relación) — How two quantities change together.
5Watch out
  • Naming the variables backward — calling the responding quantity 'independent' because it is the one the question asks about.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 MULTIPLE CHOICE

    Two quantities are related. Which statement is ALWAYS true?

    1. AThe independent variable is subject to choice
    2. BThe independent variable is always the larger number
    3. CThe dependent variable is always measured in dollars
    4. DThe independent variable is always listed first in a table
    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 2 MULTIPLE CHOICE

    Which question would you ask to find the DEPENDENT variable?

    1. AWhich quantity responds when the other one changes?
    2. BWhich quantity is larger?
    3. CWhich quantity has units?
    4. DWhich quantity comes first alphabetically?
    ✏️ Workspace & Solution Steps
  2. 3 MULTIPLE CHOICE

    Claire decides to walk instead of jog for the same 30 minutes. What happens to the two variables?

    1. ATime stays the independent variable; Calories burned still depends on it
    2. BCalories burned becomes the independent variable
    3. CNeither is a variable any more
    4. DTime becomes the dependent variable
    ✏️ Workspace & Solution Steps
  3. 4 MULTIPLE CHOICE

    If Claire runs exactly 3 miles and records how long it takes, which is the independent variable now?

    1. ADistance (3 miles)
    2. BTime it takes
    3. CHer speed
    4. DCalories burned
    ✏️ Workspace & Solution Steps
  4. 5 MULTIPLE CHOICE

    Claim: “The dependent variable always has the larger values.” Is this claim always, sometimes, or never true?

    1. ASometimes — the size of the numbers has nothing to do with which variable is which
    2. BAlways — the dependent variable is the result, so it is bigger
    3. CNever — the independent variable is always the larger one
    4. DOnly when both quantities are measured in the same unit
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 OPEN RESPONSE

    A pizza shop decides each morning how many pizzas to make, then weighs out the cheese it needs. Devon says: “Cheese is the independent variable, because the shop buys the cheese before it makes any pizzas.” Devon is wrong. What rule did Devon use, and why does that rule fail here?

    ✏️ Mathematical Justification & Response
✍️ 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