2.11 Small Group · Group 2

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

Add and Subtract Decimals · Decimal · Procedural Fluency · Real-World Applications

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — How do you add and subtract decimals?

1Line up the decimal points, add zeros so each number has the same number of decimal places, then add or subtract one column at a time — 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

The tank has 128.75 liters. A shuttle adds 46.8 liters. I line up the decimal points and annex a zero: 46.80.
I add: 128.75 + 46.80 = 175.55 liters.
Now the thrusters use 19.35 liters, so I subtract: 175.55 − 19.35 = 156.20.
The final fuel level is 156.20 liters. I check with estimation: about 130 + 47 − 19 = 158, which is close, so my answer is reasonable.

3Second Model — Try it together — then prove it

Let's add 12.45 and 7.8. First, how do we write 7.8 with two decimal places? (7.80.)
Line up the points and add the hundredths, then tenths, then ones. What do we get? (12.45 + 7.80 = 20.25.)
So the sum is 20.25. The points stay lined up in the answer.
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

Add and Subtract Decimals (Sumar y restar decimales) — Combine or find the difference between decimals by lining up equal place values.
Decimal (Decimal) — A number with a dot, like 0.5, that shows a part less than one.
Place value (Valor posicional) — What a digit is worth based on where it sits in a number.
Tenths (Décimas) — The first spot after the decimal point. It shows parts out of 10.
Hundredths (Centésimas) — The second spot after the decimal point. It shows parts out of 100.
Annex zeros (Agregar ceros) — Adding zeros at the end of a decimal so numbers line up evenly.

5Watch out

A common mistake in Add and Subtract Decimals is lining up the last digits (the right edges) of each number instead of lining up the decimal points — for example, adding 8.3 and 2.45 as if the 3 and the 5 belong in the same column. Before you add or subtract, add zeros so every number has the same number of decimal places (write 8.3 as 8.30), then check that the decimal points sit directly above one another.

SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 FILL TABLE

Compute each decimal sum or difference.

Problem Answer
5.63 + 2.7
12.4 − 8.56
9.08 + 4.925
25.1 − 13.47
7.006 + 3.94

✏️ Scratchpad / Reasoning
2 DRAG SORT
Estimate each sum or difference by rounding to the nearest whole number, then sort: is the result greater than 10 or less than 10?

Target Categories: Result Greater Than 10 Result Less Than 10

Problem	Answer
5.63 + 2.7
12.4 − 8.56
9.08 + 4.925
25.1 − 13.47
7.006 + 3.94

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

3 MULTIPLE CHOICE
Case File: A crew member's snack receipt shows $27.60 for dried fruit and $15.45 for protein bars. What is the total on the receipt?
1. A$43.05
2. B$42.05
3. C$43.15
4. D$42.95
✏️ Workspace & Solution Steps
4 MULTIPLE CHOICE
Case File: The oxygen gauge dropped from 9.6 units to a new reading after the crew used 4.75 units. How much oxygen is left?
1. A4.85
2. B5.85
3. C4.95
4. D5.15
✏️ Workspace & Solution Steps
5 MULTIPLE CHOICE
Case File: A supply log started the shift with $48.00 in the station snack fund. By the end, $31.27 had been spent. How much is left in the fund?
1. A$16.73
2. B$17.73
3. C$16.83
4. D$17.27
✏️ Workspace & Solution Steps

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

6 ERROR ANALYSIS
Level 2 Extension — The Misaligned Receipt
1. 1Problem:A receipt adds a $12.40 part and a $3.75 part: 12.4 + 3.75
2. 2Set up:Lined up the right edges so the 4 sits under the 5
3. 3Add:12.4 + 3.75 = 12.79
4. 4Final answer:$12.79
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