6.1 Small Group · Group 2

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

Visual Models · Division of Fractions · Dividend · Procedural Fluency

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — What does it mean to divide by a fraction?

1Dividing by a fraction smaller than 1 gives a BIGGER answer, because many small pieces fit inside. Dividing a fraction BY a whole number gives a smaller answer, because one piece is being shared out — 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

I want to cut a 3-foot strip of tape into 1/4-foot pieces. I ask: how many 1/4-foot pieces fit into 3 feet?
First I look at just 1 foot. How many 1/4-foot pieces fit in 1 foot? Four, because 4 fourths make one whole.
Now I have 3 feet. So I do 3 groups of 4: 3 × 4 = 12.
So 3 ÷ 1/4 = 12. There are 12 pieces. The answer is bigger than 3 because each piece is very small.

3Second Model — Try it together — then prove it

Let's try 2 ÷ 1/3. The question it asks is: how many 1/3-size pieces fit into 2 wholes?
How many thirds are in 1 whole? Three.
We have 2 wholes, so 2 × 3 = 6. That means 2 ÷ 1/3 = 6.
The answer is bigger than the 2 we started with, because each piece is smaller than one whole.
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

Division of Fractions (División de fracciones) — Finding how many groups of one fraction fit inside another amount.
Dividend (Dividendo) — The total number being divided into equal groups (the number inside the division bracket).
Divisor (Divisor) — The number of equal groups you are dividing into (the number outside the division bracket).
Quotient (Cociente) — The answer when you divide.
Reciprocal (Recíproco) — A fraction turned upside down.
Unit fraction (Fracción unitaria) — A fraction with 1 on top, like 1/4.

5Watch out

A common mistake in Interpret Division of Fractions is assuming that dividing always makes the answer smaller. For 4 ÷ 1/3, students reason 'dividing shrinks the number' and compute 4 ÷ 3 ≈ 1.3 instead of asking how many 1/3-foot pieces fit into 4 feet (each whole foot holds 3 thirds, so 4 × 3 = 12). Before you submit, check: if the divisor is a fraction smaller than 1, the quotient must be BIGGER than the dividend, not smaller.

SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 MATCHING GAME
Match each division expression to its quotient.
1. 5 ÷ 1/2
2. 6 ÷ 1/3
3. 2 ÷ 1/6
4. 4 ÷ 1/4
5. 3 ÷ 1/5
6. 7 ÷ 1/2
- A10
- B18
- C12
- D16
- E15
- F14
2 BAR MODEL

How many 1/6-size pieces fit into 3 wholes?
✏️ Workspace & Solution Steps
3 MULTIPLE CHOICE
Agent Cole has 8 feet of evidence rope and cuts it into 1/3-foot pieces for tagging. What does 8 ÷ 1/3 mean?
1. AHow many 1/3-foot pieces fit into 8 feet
2. B1/3 of the 8 feet of rope
3. C8 feet shortened by 1/3 of a foot
4. D8 pieces that are each 1/3 foot long
✏️ Workspace & Solution Steps
4 MULTIPLE CHOICE
What does 3 ÷ 1/4 mean?
1. AHow many 1/4-size pieces fit into 3
2. B3 groups of 1/4
3. C3 minus 1/4
4. D1/4 of 3
✏️ Workspace & Solution Steps

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

5 MULTIPLE CHOICE
Picture clue: a 5-inch fingerprint strip is split into 1/6-inch slivers under the microscope. Which division does the picture show, and what is the count?
1. A5 ÷ 1/6 = 30 slivers
2. B5 ÷ 6 = 5/6 of a sliver
3. C1/6 ÷ 5 = 1/30 sliver
4. D5 × 1/6 = 5/6 of an inch
✏️ Workspace & Solution Steps

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

6 ERROR ANALYSIS
Level 2 Extension — Which Question Is Being Asked?
1. 1Case file:A 9-foot strip of evidence tape is cut into 1/4-foot pieces. How many pieces?
2. 2Detective writes:This is 1/4 ÷ 9 — split the 1/4-foot piece among 9 sections.
3. 3Detective's answer:1/36 of a foot per piece
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 The mathematical model proves the solution because each step preserves quantitative equivalence.

BUT An estimate provides a quick benchmark, but an exact proof is required for precision.

SO The quantities follow the standard rule, so the final result is verified with certainty.

📐 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