6.DS.3 Lesson 2-5-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

2.5 Small Group · Group 2

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

Range and Interquartile Range · Range · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How spread out is the data?
1Range = greatest − least, and it covers every value. Interquartile range = Q3 − Q1, and it covers only the middle half, so one faraway value cannot stretch 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 measure the spread of the test scores
  1. Here are 20 history test scores in order: 68, 70, 73, 74, 75, 76, 77, 80, 82, 83, 84, 85, 86, 86, 88, 88, 90, 90, 93, 94. Ordering them first is what makes every step after this possible.
  2. Range: I subtract the least value from the greatest. 94 − 68 = 26. The scores span 26 points from end to end.
  3. Now the quartiles. The median splits the 20 scores into two halves of 10. The median is halfway between the 10th and 11th values: (83 + 84) ÷ 2 = 83.5.
  4. Q1 is the median of the lower 10 values: halfway between 75 and 76, so Q1 = 75.5. Q3 is the median of the upper 10: halfway between 88 and 88, so Q3 = 88.
  5. Interquartile range: IQR = Q3 − Q1 = 88 − 75.5 = 12.5. The middle half of the students' scores differed by only 12.5 points, even though the full range was 26.
3Second Model — Try it together — then prove it
  1. Class A's box plot reads: least 62, Q1 74, median 81, Q3 87, greatest 90. Find the range first.
  2. Range = 90 − 62 = 28 points from the lowest score to the highest.
  3. Now the IQR: subtract Q1 from Q3. IQR = 87 − 74 = 13 points across the middle half.
  4. Say what each one means: the whole class's scores span 28 points, but the middle half of the class is packed into 13 points.
  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
  • Range and Interquartile Range (Rango y rango intercuartílico) — Two numbers that describe spread: the range covers all the data, and the interquartile range covers only the middle half.
  • Range (Rango) — The distance from the least value to the greatest value, found by subtracting: greatest − least.
  • Quartile (Cuartil) — One of the three values that cut an ordered data set into four equal-sized groups.
  • Interquartile range (Rango intercuartílico) — The distance across the middle half of the data, found by subtracting the first quartile from the third: Q3 − Q1.
  • Variability (Variabilidad) — How much the values in a data set differ from one another — whether they sit close together or far apart.
  • Measure of variation (Medida de variación) — A single number that describes the spread of a whole data set. The range and the IQR are both measures of variation.
5Watch out
  • A common mistake with Range and Interquartile Range is adding the two quartiles instead of subtracting them — writing IQR = Q3 + Q1. With Q1 = 42 and Q3 = 58 that gives 100, which is larger than the data set's entire range of 20. Every measure of variation is a DIFFERENCE, and the IQR can never exceed the range, because the middle half is part of the whole span. If your IQR comes out bigger than your range, you added when you should have subtracted.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 DRAG SORT

    A value far above the rest is added to a data set. Sort what happens.

    Target Categories: The range grows a lot The IQR barely changes
    • Because it uses the greatest value
    • Because it uses only Q1 and Q3
    • It measures the whole span of the data
    • It measures only the middle half
  2. 2 DRAG SORT

    A coach wants ONE number for each question. Sort each question under the measure that answers it.

    Target Categories: Range Interquartile range
    • How far apart are our best and worst games?
    • How consistent is a typical night for this team?
    • How much did the middle half of our scores vary?
    • What is the full span from lowest to highest?
    • Which measure survives one record-breaking game?
  3. 3 MULTIPLE CHOICE

    Class A and Class B both have a range of 40 on a test. Class A's IQR is 8; Class B's is 30. What does that tell you?

    1. AClass A's scores cluster tightly with a few far-out scores; Class B's are spread through the whole range
    2. BClass A did better than Class B
    3. CNothing — equal ranges mean the classes performed the same
    4. DClass B has more students than Class A
    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 4 MULTIPLE CHOICE

    Claim: “The interquartile range can never be greater than the range.” Always, sometimes, or never true?

    1. AAlways — the middle half sits inside the full spread, so it cannot be wider than it
    2. BSometimes — it depends whether the data set has an outlier
    3. CNever — the IQR uses more values, so it comes out larger
    4. DSometimes — it is true only for data sets with an even count
    ✏️ Workspace & Solution Steps
  2. 5 MULTIPLE CHOICE

    Which data set has the greater range: A (12, 14, 15, 18) or B (5, 9, 14, 25)?

    1. AB, because 25 − 5 = 20
    2. BA, because it has larger values
    3. CThey are equal
    4. DYou cannot tell without the median
    ✏️ Workspace & Solution Steps
  3. 6 MULTIPLE CHOICE

    A coach records practice times, then one runner is injured mid-run and posts a time far above the rest. Which measure of spread should the coach report, and why?

    1. AThe IQR — it describes the middle half and is not moved by one extreme time
    2. BThe range — it uses every value, so it is the most complete
    3. CEither — one unusual value cannot change a measure of spread much
    4. DNeither — spread cannot be reported once there is an outlier
    ✏️ Workspace & Solution Steps
🗣️ 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