2.10 Small Group · Group 2

Second Challenge Form · Non-Routine Extension · Same Standard, New Problems

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

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

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.
Mean: 22 + 24 + 20 + 25 + 23 + 21 + 58 = 193, then 193 ÷ 7 ≈ 27.6.
Median: in order they are 20, 21, 22, 23, 24, 25, 58. The middle (4th) value is 23.
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

Data set A: 4, 5, 6, 7, 8. Are there any outliers? If not, which measure should we use, mean or median?
Data set B: 4, 5, 6, 7, 40. Which value is an outlier here?
For data set B, would the outlier pull the mean up or down? So which measure best shows a typical value?
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

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 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]

1 MULTIPLE CHOICE
A gymnast's scores are: 8.5, 8.8, 8.7, 8.6, 8.9. There are no outliers. Which measure best represents a typical score?
1. AMean
2. BMedian
3. CNeither — the data is too small
4. DBoth are equally bad
✏️ Workspace & Solution Steps
2 MULTIPLE CHOICE
A runner's mile times are: 7:10, 7:15, 7:12, 7:20, 12:00. The 12:00 was due to a cramp. Which measure best represents a typical mile?
1. AMedian — the outlier 12:00 pulls the mean too high
2. BMean — it uses all the data
3. CMode — it shows the most common time
4. DNeither measure works
✏️ Workspace & Solution Steps
3 MULTIPLE CHOICE
A baseball player's batting averages over 6 seasons are: .280, .295, .290, .285, .300, .110. The .110 was an injury-shortened season. Which measure better represents the player's typical batting average?
1. AMedian, because the .110 outlier pulls the mean down
2. BMean, because it uses all the data
3. CMean, because .110 is a real season
4. DNeither measure works
✏️ Workspace & Solution Steps

SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]

4 ERROR ANALYSIS
Fix our table's thinking
1. 1The problem:A gymnast's scores are: 8.5, 8.8, 8.7, 8.6, 8.9. There are no outliers. Which measure best represents a typical score?
2. 2A classmate at our table answered:Both are equally bad
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation
5 ERROR ANALYSIS
Spot the Mistake — Choosing the Mean Despite an Outlier
1. 1Data:A receiver's catches per game: 12, 14, 13, 15, 11, 60 (the 60 was a record-setting playoff game).
2. 2Find the mean:Sum = 125, Count = 6, Mean = 125 ÷ 6 ≈ 20.8.
3. 3Find the median:Ordered: 11, 12, 13, 14, 15, 60 → Median = (13 + 14) ÷ 2 = 13.5.
4. 4Choose a measure:The student reports the mean (≈ 20.8) as the typical number of catches because it uses every game.
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation
6 ERROR ANALYSIS
Find the Reasoning Error
1. 1Data:Player minutes per game: 32, 34, 30, 35, 33, 5
2. 2Find mean:Sum = 169, Count = 6, Mean = 28.2
3. 3Find median:Ordered: 5, 30, 32, 33, 34, 35 → Median = (32+33) ÷ 2 = 32.5
4. 4Conclusion:The mean (28.2) is the best measure 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

📐 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.

Student Mastery Self-Assessment:

1 · Need More Support

2 · Getting Closer

3 · Got It / Solid

4 · Master / Can Teach It