2.2 Small Group · Group 2

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

Display Data with Histograms · Histogram · Procedural Fluency · Real-World Applications

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — What is a histogram?

1Each bar's height shows how many data values land inside that interval — 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: 3, 7, 12, 15, 18, 24. I will use intervals of 10: 0–9, 10–19, 20–29.
Interval 0–9: the values 3 and 7 fit, so the frequency is 2.
Interval 10–19: the values 12, 15, and 18 fit, so the frequency is 3.
Interval 20–29: only the value 24 fits, so the frequency is 1.
Now I draw three bars with heights 2, 3, and 1, touching each other. The tallest bar (10–19) is where most values are.

3Second Model — Try it together — then prove it

New data: 2, 4, 8, 11, 13, 16, 21. We will use intervals of 10 again: 0–9, 10–19, 20–29.
How many values fall in 0–9? Count them. How many fall in 10–19?
How many fall in 20–29? Which interval would have the tallest bar?
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

Display Data with Histograms (Representar datos con histogramas) — Group numerical data into intervals and show each interval's frequency with a bar.
Histogram (Histograma) — A bar graph that groups data into equal ranges. The bars touch.
Frequency (Frecuencia) — How many times a value shows up.
Interval (Intervalo) — A range of numbers used to group data.
Distribution (Distribución) — How the data is spread out.
Variability (Variabilidad) — How spread out the numbers are.

5Watch out

A common mistake in Display Data: Histograms is counting a boundary value in two intervals at once — for example, putting the value 10 in both the '0–9' bin and the '10–19' bin. Each data value belongs to exactly ONE interval. Before you submit, add up all your frequencies and check that the total matches the number of data values you started with.

SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 FILL TABLE

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

Interval (Yards) Frequency
Player rushing yards per game: 0, 5, 12, 15, 18, 22, 25, 28, 31, 35, 40, 45
0–9
10–19
20–29
30–39
40–49

✏️ Scratchpad / Reasoning
2 MATCHING GAME
Match each term to its description.
1. Histogram
2. Bar graph
3. Frequency
4. Interval
- ABars touch, shows frequency by interval
- BBars separated, shows categories
- CCount of values in an interval
- DRange of values grouped together
3 MULTIPLE CHOICE
Two histograms show the SAME 40 test scores — one uses intervals of 5, the other intervals of 20. Which statement is true?
1. AThe interval-of-5 histogram leaves some scores out
2. BThe interval-of-5 histogram has more bars and shows more detail about the shape
3. CThe interval-of-20 histogram has more bars
4. DBoth histograms must look exactly the same
✏️ Workspace & Solution Steps

Interval (Yards)	Frequency
Player rushing yards per game: 0, 5, 12, 15, 18, 22, 25, 28, 31, 35, 40, 45
0–9
10–19
20–29
30–39
40–49

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

4 MULTIPLE CHOICE
A histogram of points-per-game has its tallest bars on the LEFT (low scores) and a long tail stretching RIGHT toward a few high scorers. How is this distribution described?
1. ASkewed left
2. BSymmetric
3. CBimodal (two equal peaks)
4. DSkewed right
✏️ Workspace & Solution Steps
5 MULTIPLE CHOICE
Data: 4, 8, 9, 12, 14, 15, 17, 18, 19, 22, 24, 27, 31, 33. How many values fall in the 10–19 interval?
1. A9
2. B5
3. C6
4. D3
✏️ Workspace & Solution Steps

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

6 OPEN RESPONSE

A teacher collected test scores and made two different histograms — one with intervals of 5 (50–54, 55–59, etc.) and one with intervals of 20 (50–69, 70–89, 90–109). Both use the same data. How might the two histograms look different? Which interval size gives you more detail about the distribution? When might the larger interval be better?
✏️ Mathematical Justification & Response

🗣️ 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