6.DS.6d Lesson 2-10-group2 🟣 Group 2 · Challenge & Extension
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2.10 Small Group · Group 2

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

Visual Models · Appropriate Measures of Center · Mean · Procedural Fluency

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Should I use the mean or the median?
1Use the mean for data with no outliers; use the median when an outlier or a skewed shape would pull the mean away from typical — 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, the top scorer's points: 22, 24, 20, 25, 23, 21, 58. I notice 58 is much bigger than the rest, so it is an outlier.
  2. Mean: 22 + 24 + 20 + 25 + 23 + 21 + 58 = 193, then 193 ÷ 7 ≈ 27.6.
  3. Median: in order they are 20, 21, 22, 23, 24, 25, 58. The middle (4th) value is 23.
  4. The mean (27.6) is higher than 6 of the 7 games because 58 pulled it up. So the median (23) better shows a typical game.
3Second Model — Try it together — then prove it
  1. Data set A: 4, 5, 6, 7, 8. Are there any outliers? If not, which measure should we use, mean or median?
  2. Data set B: 4, 5, 6, 7, 40. Which value is an outlier here?
  3. For data set B, would the outlier pull the mean up or down? So which measure best shows a typical value?
  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
  • Appropriate Measures of Center (Medidas de centro apropiadas) — A mean, median, or mode chosen because it describes a data set fairly.
  • Mean (Media) — The average. Add all the numbers, then divide by how many there are.
  • Median (Mediana) — The middle number when you put them in order.
  • Outlier (Valor atípico) — A number that is much bigger or smaller than the rest.
  • Skewed (Sesgado) — When most data sits on one side with a tail on the other.
  • Data distribution (Distribución de datos) — How the data looks: where it sits and how spread out it is.
5Watch out
  • A common mistake in Appropriate Measures is defaulting to the mean just because "it uses all the data," without first checking whether an outlier or skewed shape is present. For example, given a player's points per game 14, 16, 15, 17, 16, 58, a student calculates the mean (136 ÷ 6 ≈ 22.7) and reports that as the typical game — but 58 is far from the cluster of 14–17, so it pulls the mean above 5 of the 6 games. The median (16) actually represents a typical game here. Before choosing a measure, always scan the data for an outlier or a long tail first: only use the mean once you've confirmed the data has neither.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Solve the mathematical problem. Show all of your work and reasoning.

    Data SetMeanMedianOutlier?Best Measure
    15, 18, 16, 17, 15
    15, 18, 16, 17, 72
    ✏️ Scratchpad / Reasoning
  2. 2 NUMBER LINE

    Data: 10, 12, 11, 13, 50. Place the mean (19.2) and median (12) on the number line. Notice how far apart they are because of the outlier.

    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    A scout records points per game: 14, 16, 15, 17, 16, 58. To report a TYPICAL game, which measure of center is best?

    1. AMedian, because 58 is an outlier that pulls the mean up
    2. BMean, because it uses every game
    3. CRange, because it shows the highest game
    4. DMode, because 16 appears twice
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    The equipment manager needs to order the MOST COMMON jersey size from this list: M, L, M, S, M, L, M. Which measure answers the question?

    1. AMode, because it names the value that appears most
    2. BMean, because it averages the sizes
    3. CMedian, because it finds the middle size
    4. DRange, because it shows the size gap
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    A coach wants to know how SPREAD OUT the team's free-throw makes are: 6, 7, 6, 8, 18. Which measure of spread is most affected by the lone 18?

    1. ARange, because it is the highest minus the lowest value
    2. BMedian, because it is in the middle
    3. CMode, because it is the most common
    4. DMean absolute deviation only, never the range
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 ERROR ANALYSIS

    Level 2 Extension — The Analyst's Wrong Measure

    1. 1Data:A player's points per game: 19, 21, 20, 22, 23, 2 (the 2 was a game cut short by an early ejection).
    2. 2Find mean:Sum = 107, Count = 6, Mean = 107 ÷ 6 ≈ 17.8
    3. 3Find median:Ordered: 2, 19, 20, 21, 22, 23 → Median = (20 + 21) ÷ 2 = 20.5
    4. 4Analyst's choice:The analyst reports the mean (17.8) as the player's typical scoring because it uses all the data.

    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