6.DS.4 Lesson 2-8-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

2.8 Small Group · Group 2

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

Visual Models · Mean · Average · Procedural Fluency

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — What is the mean?
1Mean = total of all the values ÷ how many values there are — 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. Miguel's 4 days of earnings are shown with counters. When I combine all of the counters, there are 60 of them, which means Miguel earned $60 in all.
  2. Now I redistribute the 60 counters equally among the 4 days. Each of the 4 groups ends up with 15 counters.
  3. The arithmetic does exactly the same thing: I find the total, 60, and divide it by the number of days, 4. So 60 ÷ 4 = 15.
  4. We can use $15 to describe Miguel's typical daily earnings. That leveled value is the mean.
3Second Model — Try it together — then prove it
  1. New data set: a dog walker earns $9, $11, $16, and $12 over 4 days. First, what is the total?
  2. Next, how many values are in the data set? That number is what we divide by.
  3. Last, divide the total by 4. Is your answer bigger than the smallest day and smaller than the biggest day? It should be.
  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 (Media) — The average of a data set. Add all the values, then divide by how many values there are.
  • Average (Promedio) — Another word for the mean — the amount each value would be if the total were shared equally.
  • Measure of center (Medida de centro) — A single number used to describe the middle, or typical value, of a whole data set.
  • Balance point (Punto de equilibrio) — The value where the total distance of the data above it equals the total distance below it. The mean is the balance point.
  • Data set (Conjunto de datos) — The whole collection of values that were gathered or measured.
5Watch out
  • The most common mistake with the mean is adding correctly and then dividing by the wrong number — or not dividing at all. Students report the total (for 8, 10, 6, 12 they say the mean is 36) or they divide by a count that does not match the data, such as dividing a 4-value total by 5, or skipping a zero when counting the values. Before you finish, do two things: count the values in the data set out loud, and check that your mean lands between the smallest and largest value. A mean of 36 for data that never goes above 12 is impossible.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

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

    Data SetTotalHow Many ValuesMean
    6, 9, 12, 9
    20, 14, 16, 18, 22
    5, 8, 8, 11, 13, 15
    ✏️ Scratchpad / Reasoning
  2. 2 NUMBER LINE

    Data: 10, 12, 14, 16, 18. Place a marker at the mean of this data set.

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

    A data set has a mean of 20. Every value in the data set is then increased by 4. What is the new mean?

    1. A24
    2. B20
    3. C80
    4. D5
    Put the data in order
    Work
  2. 4 MULTIPLE CHOICE

    A data set of 5 numbers has a mean of 12. A sixth value of 12 is added. What happens to the mean?

    1. AIt stays 12 — adding a value equal to the mean cannot pull it either way
    2. BIt rises, because the total went up
    3. CIt falls, because there are more values to divide by
    4. DThere is not enough information without the original five numbers
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 5 OPEN RESPONSE

    Sofia realized she made an error: she actually scored 12 points in Game 8, not 10. Her other games are unchanged (12, 20, 16, 11, 17, 14, 15, and 18). How does this change the number of points she needs in Game 10 to average 15 points per game? Show how you know.

    ✏️ Mathematical Justification & Response
  2. 6 ERROR ANALYSIS

    The Missing Zero

    1. 1Data set:Goals allowed in 4 games: 0, 3, 2, 7
    2. 2Goal:Find the MEAN number of goals allowed per game.
    3. 3Add the values:0 + 3 + 2 + 7 = 12
    4. 4Divide:The student says the shutout does not count as a game, so 12 ÷ 3 = 4 goals per game.

    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