3.6 Small Group · Group 2

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

Visual Models · Convert Measurements within the Same System · Conversion factor · Procedural Fluency

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — Converting inside one measurement system

1Write the conversion factor as a unit ratio (12 inches : 1 foot), then multiply to go to the smaller unit and divide to go to the larger one. The amount never changes — only the unit does — 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 check the ride height rule

A ride requires riders to be at or above 44 inches tall. A child is 4 feet tall. The two measurements use different units, so I cannot compare them yet.
Both units belong to the customary system, and there are 12 inches in one foot. That relationship is the conversion factor. As a ratio it is 12 : 1 — a unit ratio, because it compares an amount to exactly 1 foot.
Inches are smaller than feet, so there will be MORE of them: I multiply. 4 × 12 = 48.
The child is 48 inches tall. Now the comparison is fair: 48 inches is greater than 44 inches, so the child can go on the ride.

3Second Model — Try it together — then prove it

Diamond wants to drink 4 liters of water. Her bottle is marked in milliliters.
The conversion factor is 1,000 mL : 1 L. On a double number line, every 1 liter above matches 1,000 milliliters below.
Milliliters are smaller, so multiply: 4 × 1,000 = 4,000. So 4 liters = 4,000 milliliters.
Her bottle holds 500 milliliters. How many fills? Solve ? × 500 = 4,000. Since 8 × 500 = 4,000, she fills the bottle 8 times.
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

Convert Measurements within the Same System (Convertir medidas dentro del mismo sistema) — Rewriting a measurement in a different unit of the SAME system, such as feet into inches or liters into milliliters.
Conversion factor (Factor de conversión) — The ratio that links two units in the same measurement system, such as 12 inches to 1 foot. Within one system it is exact.
Unit ratio (Razón unitaria) — A ratio that compares an amount to exactly 1 unit of another quantity, like 12 inches to 1 foot.
Convert (Convertir) — To change a measurement from one unit to another without changing how much it is.
Measurement system (Sistema de medidas) — A family of units that go together — customary units like inches and feet, or metric units like milliliters and liters.
Double number line (Recta numérica doble) — Two number lines lined up so matching tick marks show two units for the same amount.

5Watch out

A common mistake when converting within a measurement system is choosing the operation by habit instead of by unit size — multiplying every time. Converting 36 inches to feet by multiplying gives 36 × 12 = 432 feet, taller than a 40-story building. Ask first which unit is smaller: it takes MORE small units to make the same amount, so going to a smaller unit multiplies, and going to a larger unit divides.

SECTION 1 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]

1 MULTIPLE CHOICE
Rosa converts 48 ounces to pounds by dividing: 48 ÷ 16 = 3. Theo builds up instead: 16, 32, 48 — that is 3 pounds. Whose reasoning is sound?
1. ABoth — dividing and building up the ratio answer the same question
2. BOnly Rosa — building up is guessing, not converting
3. COnly Theo — you cannot divide when converting units
4. DNeither — 48 ounces is not a whole number of pounds
2 MULTIPLE CHOICE
A child is 4 feet tall. A ride requires 44 inches. Can the child ride?
1. AYes, because 4 feet = 48 inches, which is more than 44
2. BNo, because 4 is less than 44
3. CYes, because 4 feet = 44 inches exactly
4. DYou cannot compare feet and inches
✏️ Workspace & Solution Steps
3 MULTIPLE CHOICE
Diamond needs 4 liters of water. Her bottle holds 500 milliliters. How many times must she fill it?
1. A8 times
2. B4 times
3. C2 times
4. D40 times
✏️ Workspace & Solution Steps

SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]

4 ERROR ANALYSIS
Fix our table's thinking
1. 1The problem:A child is 4 feet tall. A ride requires 44 inches. Can the child ride?
2. 2A classmate at our table answered:No, because 4 is less than 44
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation
5 ERROR ANALYSIS
Find the error
1. 1Setup:A recipe needs 2 pounds of flour. The scale reads in ounces (16 oz = 1 lb).
2. 2Step 1:The unit ratio is 16 ounces : 1 pound.
3. 3Step 2:2 × 16 = 32, so 2 pounds = 32 ounces.
4. 4Step 3:The scale reads 40 ounces, so there is not enough flour.
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation
6 OPEN RESPONSE

Michael's mother is setting four fence posts. Each post needs 250 ounces of concrete mix, one bag holds 80 ounces, and each bag needs 1 quart of water. How many bags and how many cups of water does she need? Show your conversions.
✏️ Mathematical Justification & Response

📐 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