2.5 Small Group · Group 2

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

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

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

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.
Range: I subtract the least value from the greatest. 94 − 68 = 26. The scores span 26 points from end to end.
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.
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.
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

Class A's box plot reads: least 62, Q1 74, median 81, Q3 87, greatest 90. Find the range first.
Range = 90 − 62 = 28 points from the lowest score to the highest.
Now the IQR: subtract Q1 from Q3. IQR = 87 − 74 = 13 points across the middle half.
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.
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

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 MULTIPLE CHOICE
Team A's salaries have a range of $14M and an IQR of $7M. Team B's have a range of $7M and an IQR of $2.5M. Both teams have the same median salary. What does this tell a player?
1. ATeam B's salaries cluster much more tightly around the median
2. BTeam B pays more overall
3. CTeam A has more players
4. DTeam B's highest salary is greater
✏️ Workspace & Solution Steps

SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]

2 MULTIPLE CHOICE
On a box plot, Q1 = 20 and Q3 = 36. What is the interquartile range?
1. A16
2. B56
3. C28
4. D36
Put the data in order

Work
3 MULTIPLE CHOICE
Which measure of spread uses ONLY the middle half of the data?
1. AThe interquartile range
2. BThe range
3. CThe median
4. DThe mode
✏️ Workspace & Solution Steps
4 MULTIPLE CHOICE
A box plot shows: least 68, Q1 75.5, median 83.5, Q3 88, greatest 94. What are the range and the IQR?
1. ARange 26, IQR 12.5
2. BRange 12.5, IQR 26
3. CRange 26, IQR 83.5
4. DRange 20, IQR 12.5
✏️ Workspace & Solution Steps

SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]

5 ERROR ANALYSIS
Fix our table's thinking
1. 1The problem:Which measure of spread uses ONLY the middle half of the data?
2. 2A classmate at our table answered:The range
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation
6 ERROR ANALYSIS
Find the error
1. 1Setup:Data: 15, 18, 20, 22, 25, 60.
2. 2Step 1:The range is 60 − 15 = 45.
3. 3Step 2:The range of 45 shows that a typical value differs from the others by about 45.
4. 4Step 3:So this data set has high variability everywhere.
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