6.DS.6c Lesson 2-9-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
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2.9 Small Group · Group 2

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

Mean Absolute Deviation · Deviation · Procedural Fluency · Reasoning & Critique

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — What is the mean absolute deviation (MAD)?
1MAD = the average of how far each number is from the mean (always a positive distance) — 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. My data set: 6, 8, 10, 12. First I find the mean: 6 + 8 + 10 + 12 = 36, then 36 ÷ 4 = 9. The mean is 9.
  2. Next I find each deviation (value − mean): 6−9 = −3, 8−9 = −1, 10−9 = 1, 12−9 = 3.
  3. Then I take the absolute value of each (the distance, always positive): 3, 1, 1, 3.
  4. Finally I average those distances: 3 + 1 + 1 + 3 = 8, then 8 ÷ 4 = 2. So the MAD = 2 points.
3Second Model — Try it together — then prove it
  1. New data set: 9, 10, 11. First, what is the mean? Add the three numbers and divide by 3.
  2. Now find how far each number is from the mean. What is the distance for 9? For 10? For 11?
  3. Add those distances and divide by 3. Is this MAD small or large? What does that tell us about the spread?
  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
  • Mean Absolute Deviation (Desviación media absoluta) — The average distance of data values from the mean; it describes spread.
  • Deviation (Desviación) — How far a number is from the mean.
  • Absolute Value (Valor absoluto) — How far a number is from zero. It is never negative.
  • Spread (Dispersión) — How far apart the numbers are.
  • Data distribution (Distribución de datos) — How the data looks: where it sits and how spread out it is.
  • Variability (Variabilidad) — How spread out the numbers are.
5Watch out
  • A common mistake in Mean Absolute Deviation is adding the signed deviations (value − mean) instead of their absolute values, so the positives and negatives cancel out to 0. For 6, 8, 10, 12 (mean = 9), the deviations are −3, −1, 1, 3 — they sum to 0, but that is NOT the MAD. Before averaging, take the absolute value of every deviation so each becomes a positive distance: |−3| = 3, |−1| = 1, |1| = 1, |3| = 3. Sum = 8, so MAD = 8 ÷ 4 = 2. A MAD of exactly 0 is a warning sign — it only happens if every value in the data set equals the mean.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Deviation Detective: the quiz scores are 12, 8, 10, 14, 6 and the mean is 10. Complete each deviation and absolute deviation, then compute the MAD.

    ScoreDeviation (Score − Mean)Absolute Deviation
    12
    8
    10
    14
    6
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each MAD value to what it tells you about the data.

    1. MAD = 1.5
    2. MAD = 8.0
    3. MAD = 0
    4. MAD = 4.2
    • AVery consistent data
    • BVery spread out data
    • CAll values are identical
    • DModerate spread
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    A guard scored 2, 4, 6, 8 points in four scrimmages, so the mean is 5 points. The distances from the mean are 3, 1, 1, 3. What is the mean absolute deviation (MAD)?

    1. A1
    2. B2
    3. C5
    4. D8
    Put the data in order
    Work
  2. 4 MULTIPLE CHOICE

    Two shooters average 11 points a game. Shooter A's MAD is 2 points and Shooter B's MAD is 6 points. Which shooter is MORE consistent?

    1. AShooter A, because a smaller MAD means scores stay closer to the average
    2. BShooter B, because a bigger MAD means a bigger average
    3. CThey are equally consistent because they have the same mean
    4. DYou cannot tell consistency from MAD
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Claim: “A data set's MAD can equal 0.” Always, sometimes, or never true — and when?

    1. ASometimes — exactly when every value in the set is identical
    2. BNever — absolute values are always positive, so their average is too
    3. CAlways — every data set has SOME value equal to its mean
    4. DSometimes — whenever the mean happens to be 0
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 ERROR ANALYSIS

    Level 2 Extension — The Deviations That Vanished

    1. 1Data set:A center's rebounds: 3, 7, 11, 15, 19
    2. 2Find the mean:Sum = 55, Count = 5, Mean = 55 ÷ 5 = 11
    3. 3Find each deviation (value − mean):−8, −4, 0, 4, 8
    4. 4Find the MAD:MAD = (−8 + −4 + 0 + 4 + 8) ÷ 5 = 0 ÷ 5 = 0

    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 A question is statistical because it expects varied responses across different subjects rather than a single fixed value.
BUT A question might gather numbers, but it is not statistical if there is only one exact unchanging answer.
SO The researcher needed to understand group variation, so she collected data using a statistical survey question.
📐 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