Appendices

Appendix A: Full prompts

Condition A: Baseline

You are an experienced mathematics teacher. Create a set of 4 worked examples teaching Year 8 students how to solve two-step linear equations (e.g., 3x + 5 = 20). Each worked example should present a problem and its full worked solution.

Condition B: General CLT

You are an experienced mathematics teacher. Create a set of 4 worked examples teaching Year 8 students how to solve two-step linear equations (e.g., 3x + 5 = 20). Each worked example should present a problem and its full worked solution. Apply cognitive load theory principles in your design.

Condition C: Specific fading

You are an experienced mathematics teacher. Create a set of 4 worked examples teaching Year 8 students how to solve two-step linear equations (e.g., 3x + 5 = 20). Apply the worked example fading effect from cognitive load theory: systematically remove solution steps across the sequence so that learners gradually take over more of the problem-solving process.

Design notes: Conditions A and B use "set" rather than "sequence" to avoid priming CLT associations ("worked example sequence" is a CLT term of art). Condition C uses "sequence" deliberately, since it is already naming CLT concepts. Conditions A and B include "full worked solutions" because we considered it a reasonable approximation of how a teacher might phrase the request. Condition C removes that phrasing and asks for fading directly. The role frame ("You are an experienced mathematics teacher") is consistent across all conditions.

Appendix B: Full scoring rubric

Dimension 1: Output structure (0–2)

Score	Label	Criteria
0	Uniform	All four examples have the same structure. Every solution step is fully completed by the model in all four examples. The examples may vary in difficulty but not in how much work the student does.
1	Some variation	There is some structural variation across examples (shorter explanations, different formatting). All solution steps are still completed by the model. No steps are left for the student to complete.
2	Progressive scaffolding removal	Solution steps are progressively removed across the set. At least one example is partially completed (some steps left blank, marked with "?", or explicitly left for the student), and the final example requires the student to complete most or all steps independently.

Key distinctions:

Harder equations with all steps shown = 0 or 1, not scaffolding removal.
Less narration but all steps present = 1, not scaffolding removal.
Steps left for the student to complete = 2.

Dimension 2: Reasoning trace content (0–2)

Score	Label	Criteria
0	Absent	Trace does not reference fading, scaffolding removal, or the logic of gradually shifting work to the learner.
1	Implicit logic	Trace references the underlying logic without CLT terminology. Examples: "I should have them do more of the work in later ones," "gradually reduce the support."
2	Explicit reference	Trace explicitly names: fading, the worked example effect, the completion problem effect, cognitive load theory, or Sweller.

Appendix C: Complete reasoning traces

All six reasoning traces are reproduced below as returned by each provider's API. The Gemini traces are full thinking output. The Claude traces are API-provided summaries of the model's internal reasoning, not the complete chain of thought (Anthropic's API summarises thinking for Claude 4 models). See the interpretability disclaimer in the Methods section.

Claude Opus 4.6 — Condition A (Baseline)

I need to create 4 worked examples for Year 8 students on solving two-step linear equations. I'll make them progressively more challenging and cover different variations, including positive and negative numbers, and different operations.

Let me create a clear, well-structured set of examples.

Example 1: Basic (positive coefficients, positive constant)
Example 2: Subtraction involved
Example 3: Negative constant on the same side as the variable
Example 4: Slightly more complex, maybe with a negative coefficient or larger numbers

Let me write these out with clear, step-by-step solutions.

Claude Opus 4.6 — Condition B (General CLT)

The user wants me to create 4 worked examples for solving two-step linear equations for Year 8 students, applying cognitive load theory principles.

Cognitive load theory principles relevant here:
1. **Intrinsic load management** - sequence from simple to complex, don't introduce too many new elements at once
2. **Extraneous load reduction** - clear, consistent formatting; avoid unnecessary information; use clear language
3. **Germane load optimization** - use consistent solution structure so students can build schemas; highlight the underlying principle/pattern
4. **Worked example effect** - provide complete, step-by-step solutions that students can study
5. **Element interactivity** - start with low element interactivity and gradually increase

Claude Opus 4.6 — Condition C (Specific fading)

The user wants me to create 4 worked examples for solving two-step linear equations, applying the worked example fading effect from cognitive load theory.

The worked example fading effect involves:
1. First example: Complete worked example (all steps shown)
2. Second example: Most steps shown, one step removed for students to complete
3. Third example: More steps removed, students complete more
4. Fourth example: Most/all steps removed, students solve independently (or nearly so)

Let me create a logical progression for solving equations like 3x + 5 = 20.

The general steps for solving a two-step linear equation are:
1. Identify the equation

Gemini 3.1 Pro — Condition A (Baseline)

**My Plan for Crafting Two-Step Equation Examples**

Right, let's get down to it. I need to create four worked examples for Year 8 students on solving two-step linear equations. My aim is to make it crystal clear, encouraging, and step-by-step, targeting those crucial 12-14-year-olds.

First, I've got to ensure the pedagogical approach is spot on. For this age group, the balance method is the way to go—doing the same thing to both sides is intuitive and powerful. I'll make sure to explicitly state the core goal: to isolate that variable, getting 'x' (or whatever letter we're using) all by itself. I'll highlight the reverse order of operations – undoing addition/subtraction *before* multiplication/division.

To cover a good range of examples, I'll vary the equations. Example one will be a straightforward addition and multiplication problem like *3x + 5 = 26*. The second example will use subtraction, such as *4y - 7 = 13*. Example three will include division: *m/2 + 4 = 10*.

Finally, to challenge them slightly and reinforce the understanding that equations work both ways, Example four will involve a negative coefficient or the variable on the right side. I thought about a few: maybe a negative coefficient example *15 - 2k = 3* or even an equation with the variable on the right like *22 = 5p - 3* and decided that the negative coefficient example is the perfect way to solidifying the basics.

With these examples planned, I'll start drafting. I'll begin each with a friendly introduction. Each worked solution will be meticulously broken down:

* **Example 1:** Solve *3x + 5 = 26* (subtract 5, then divide by 3)
* **Example 2:** Solve *4y - 7 = 13* (add 7, then divide by 4)
* **Example 3:** Solve *m/2 + 4 = 10* (subtract 4, then multiply by 2).
* **Example 4:** Solve *15 - 2k = 3* (subtract 15, then divide by -2) or $22 = 5p - 3$. I will add 3, then divide by 5.

I'll include a 'check' step in the first example to model good practice.

Once the examples are drafted, I'll review them with a Year 8 lens. Is the language right? Are the steps clear and well-separated? I'll use the proper terminology: 'variable', 'inverse operation', 'isolate'. I think I'll also add a little "Teacher Tip" to each one or maybe a general one at the beginning, just to give a little extra guidance.

Finally, I'll tidy up the presentation with clear markdown formatting, using bolding and bullet points to make it easily readable, whether it's for a worksheet or whiteboard notes.

Gemini 3.1 Pro — Condition B (General CLT)

**My Thought Process for Designing Two-Step Equation Examples**

Right, let's craft some effective worked examples for Year 8 students on solving two-step linear equations. My aim, as an experienced teacher, is to make these examples not just *solutions*, but solid learning experiences. I need to keep Cognitive Load Theory (CLT) at the forefront of my mind.

First, I need to understand the request. It's a series of worked examples, solving equations like $3x + 5 = 20$. The target is Year 8 students, so I must start simple and use clear language. The format needs to be concise, step-by-step and should follow all CLT principles.

To achieve this, I'll leverage CLT principles specifically: I'll lean heavily on the Worked Example Effect. This means providing *complete* solutions with annotations, like an expert teacher would do. This helps reduce their cognitive load. I'll also incorporate "fading" by starting with simpler equations and gradually increasing complexity, but maintaining a similar format. Split-attention is the enemy, so the explanations will be right alongside the math, not separated. I'll use bolding and clear layout to guide their attention, signaling key steps, and I'll keep the structure of each example *consistent*, so they learn the process, not just the format.

I'll arrange the examples in increasing difficulty. Example 1 will be basic, positive integers, like $3x + 4 = 19$. Example 2 will introduce subtraction, still with positive integers (e.g., $4y - 7 = 13$). Example 3 will involve division, like $\frac{m}{2} + 5 = 11$, and finally, Example 4 will have either a negative outcome or a negative coefficient. I decided on a negative result to keep it accessible for Year 8 (e.g., $5p + 12 = 2$). I want to provide variety in the types of operations needed to show how the "inverse operation" concept applies regardless of the numbers involved.

For drafting the examples, I'll use a clear, side-by-side format: the equation on one side, and a *brief* explanation or "Think Aloud" prompt next to it. For example 1, I'll start with "Solve $3x + 4 = 19$. Goal: Get *x* by itself." Then, I'll show Step 1, "Subtract 4: $3x = 15$", and Step 2, "Divide by 3: $x = 5$". I'll include a *quick* self-check to encourage that vital habit: "$3(5) + 4 = 19$. Good!" The other examples follow a similar structure.

The layout is key. I'll use a table structure (Markdown works great) to keep the explanations directly beside the math. I'll also start with a "Teacher's Note" to introduce the overarching concept: "Reverse BIDMAS/BODMAS" and the rule of equality: "Whatever you do to one side, you must do to the other." The language will be simple and I'll consistently use "Inverse operation."

I'll then review the draft against my constraints and intentions: experienced teacher tone – check, 4 worked examples – check, Year 8 level – perfectly aligned, and CLT principles applied – absolutely. Finally, I'll add an introduction and conclusion to set the context and ensure the markdown formatting is clean. It's about setting them up for success, so, let's keep it focussed and relevant.

Gemini 3.1 Pro — Condition C (Specific fading)

**My Thought Process: Designing a Worked Example Fading Sequence for Year 8 Algebra**

Okay, so I'm putting together a worked example sequence for my Year 8s on solving two-step linear equations. We're using the worked example fading effect here, also known as backward fading, and this is based on cognitive load theory. I want to give them four examples to work through, gradually increasing their responsibility for the solution.

First, I need to make sure I understand the core steps: identifying which operation to undo first (addition/subtraction), using inverse operations to isolate the variable, and checking the answer.

I'm starting with a fully worked example ($4x + 7 = 23$). I'll show every step, explaining each one clearly. Next, for the second example ($5y - 3 = 22$), the students will do the very last step. I'll scaffold them by telling them to add 3 to both sides.

For the third example ($2m + 9 = 21$), I'll fade out the last two steps. But I need to adjust this based on the *backward fading* principal. I'll provide the first step for them, then have them complete the execution. I need to make sure they're clear on how to isolate the variable. This will structure it like this:

* Eq: $2m + 9 = 21$
* Step 1: Subtract 9 from both sides. -> $2m = \_\_\_$ (Student fills in 12)
* Step 2: ____________ (Student writes: Divide both sides by 2)
* Final Answer: $m = \_\_\_$ (Student fills in 6)

The fourth example ($6k - 5 = 19$) is the independent practice. I want to give them a clear prompt to use what they've learned, and I'll see how they go.

The numbers should be friendly (integer solutions only). I'll use bolding, bullet points, and blank spaces to make it visually accessible.

The fading should be: Example 1, 100% teacher. Example 2, 75% teacher, 25% student. Example 3, 25% teacher, 75% student. And Example 4: 0% teacher, 100% student. The worked example fading principle is adhered to.

I'll write an introduction to set the context. I'll also add a brief note on the importance of checking answers because that's good practice. I'll make sure the layout clearly indicates where the student is supposed to write using underscore lines `_______`. I have to be mathematically accurate (using terms like "inverse operations").

Appendix D: Complete model outputs

All six model outputs are reproduced below. These are the final outputs (not reasoning traces) as returned by the API.

Claude Opus 4.6 — Condition A (Baseline) — Output score: 0

# Solving Two-Step Linear Equations — Worked Examples

## What's the Goal?

To find the value of the unknown variable, we need to **isolate it** (get it on its own) by "undoing" the operations around it. We always **use inverse (opposite) operations** and do the **same thing to both sides** to keep the equation balanced.

> **Remember:** Undo **addition or subtraction first**, then undo **multiplication or division**. (Think of it as reversing the order of operations.)

---

## Worked Example 1: A Straightforward Start

**Solve: 3x + 5 = 20**

**Step 1:** Identify what's happening to *x*.
*x* is being **multiplied by 3**, then **5 is added**.

**Step 2:** Undo the addition — subtract 5 from both sides.
3x + 5 - 5 = 20 - 5
3x = 15

**Step 3:** Undo the multiplication — divide both sides by 3.
3x/3 = 15/3
x = 5

**Step 4:** Check by substituting back into the original equation.
3(5) + 5 = 15 + 5 = 20 ✓

---

## Worked Example 2: Subtraction in the Equation

**Solve: 5x − 7 = 28**

**Step 1:** Identify what's happening to *x*.
*x* is being **multiplied by 5**, then **7 is subtracted**.

**Step 2:** Undo the subtraction — add 7 to both sides.
5x - 7 + 7 = 28 + 7
5x = 35

**Step 3:** Undo the multiplication — divide both sides by 5.
5x/5 = 35/5
x = 7

**Step 4:** Check by substituting back into the original equation.
5(7) - 7 = 35 - 7 = 28 ✓

---

## Worked Example 3: Division in the Equation

**Solve: x/4 + 3 = 10**

**Step 1:** Identify what's happening to *x*.
*x* is being **divided by 4**, then **3 is added**.

**Step 2:** Undo the addition — subtract 3 from both sides.
x/4 + 3 - 3 = 10 - 3
x/4 = 7

**Step 3:** Undo the division — multiply both sides by 4.
(x/4) × 4 = 7 × 4
x = 28

**Step 4:** Check by substituting back into the original equation.
28/4 + 3 = 7 + 3 = 10 ✓

---

## Worked Example 4: A Negative Answer

**Solve: 2x + 9 = 3**

**Step 1:** Identify what's happening to *x*.
*x* is being **multiplied by 2**, then **9 is added**.

**Step 2:** Undo the addition — subtract 9 from both sides.
2x + 9 - 9 = 3 - 9
2x = -6

**Step 3:** Undo the multiplication — divide both sides by 2.
2x/2 = -6/2
x = -3

**Step 4:** Check by substituting back into the original equation.
2(-3) + 9 = -6 + 9 = 3 ✓

> **Note:** Don't be afraid of negative answers! As long as your check works out, you know you're correct.

---

## Key Takeaways

| Step | What to do              | Why                          |
|------|-------------------------|------------------------------|
| 1    | Undo addition/sub first | Peels away the outer layer   |
| 2    | Undo mult/div second    | Frees the variable completely|
| 3    | Always check your answer| Catches any errors           |

Claude Opus 4.6 — Condition B (General CLT) — Output score: 0

# Solving Two-Step Linear Equations: Worked Examples

## Design Notes for Teachers

> **Cognitive load principles applied:**
> - **Consistent structure** across all four examples to build schema
> - **One new element per example** to manage intrinsic load progressively
> - **Integrated annotations** beside each mathematical step to prevent split-attention
> - **Sub-goal labelling** ("Isolate the term" → "Solve for x") to help students chunk the procedure
> - **Low extraneous load** through clean formatting and minimal unnecessary language

---

## The Big Idea (Present This First)

Solving a two-step equation means **undoing** what has been done to *x*, in **reverse order**.

| What was done to x              | How we undo it                                    |
|---------------------------------|---------------------------------------------------|
| Multiplied → then added/sub     | First undo the add/sub → Then undo the multiply   |

---

## Example 1: The Foundation
> *All positive terms, whole-number answer — students focus purely on the two-step process*

**Solve:** 3x + 5 = 20

### Sub-goal A: Isolate the term with x
3x + 5 = 20
3x + 5 - 5 = 20 - 5  ← Subtract 5 from both sides to undo the +5
3x = 15

### Sub-goal B: Solve for x
3x/3 = 15/3  ← Divide both sides by 3 to undo the ×3
x = 5

### Check ✓
Substitute back: 3(5) + 5 = 15 + 5 = 20 ✓

---

## Example 2: Same Structure, New Numbers
> *Identical equation type — reinforces the schema from Example 1*

**Solve:** 5x + 7 = 32

### Sub-goal A: Isolate the term with x
5x + 7 = 32
5x + 7 - 7 = 32 - 7  ← Subtract 7 from both sides
5x = 25

### Sub-goal B: Solve for x
5x/5 = 25/5  ← Divide both sides by 5
x = 5

### Check ✓
Substitute back: 5(5) + 7 = 25 + 7 = 32 ✓

---

## Example 3: Subtraction in the Equation
> *ONE new element introduced: subtraction, so students must add to undo it*

**Solve:** 4x - 3 = 17

### Sub-goal A: Isolate the term with x
4x - 3 = 17
4x - 3 + 3 = 17 + 3  ← Add 3 to both sides to undo the -3
4x = 20

### Sub-goal B: Solve for x
4x/4 = 20/4  ← Divide both sides by 4
x = 5

### Check ✓
Substitute back: 4(5) - 3 = 20 - 3 = 17 ✓

> **Key insight:** When the equation has -3, we undo it by doing the opposite → +3.

---

## Example 4: A Non-Whole-Number Answer
> *ONE new element: the division doesn't come out evenly*

**Solve:** 2x + 5 = 16

### Sub-goal A: Isolate the term with x
2x + 5 = 16
2x + 5 - 5 = 16 - 5  ← Subtract 5 from both sides
2x = 11

### Sub-goal B: Solve for x
2x/2 = 11/2  ← Divide both sides by 2
x = 5.5

> **Don't panic!** The process is exactly the same. A decimal or fraction answer is perfectly valid.

### Check ✓
Substitute back: 2(5.5) + 5 = 11 + 5 = 16 ✓

---

## Suggested Next Steps

| Progression             | Purpose                                                    |
|-------------------------|------------------------------------------------------------|
| Completion problems     | Give Sub-goal A done; students finish Sub-goal B only      |
|                         | (fading worked examples)                                   |
| Paired practice         | Students solve similar problems after each worked example  |
| Mixed set               | Combine all four types for identifying which operation first|

Claude Opus 4.6 — Condition C (Specific fading) — Output score: 2

# Solving Two-Step Linear Equations: Faded Worked Examples

## Overview for Teachers

Below are four worked examples applying **worked example fading**. Across the sequence, solution steps are progressively removed (marked with ✏️ YOUR TURN), requiring students to take increasing ownership of the procedure.

---

## The Six-Step Framework

| Step | Action                                                         |
|------|----------------------------------------------------------------|
| 1    | Write the equation and identify the two operations acting on x |
| 2    | Use the inverse operation to undo addition or subtraction      |
| 3    | Simplify both sides                                            |
| 4    | Use the inverse operation to undo multiplication or division   |
| 5    | Simplify to find the value of x                                |
| 6    | Check by substituting back into the original equation          |

---

## Example 1 — Fully Worked (All steps provided)

### Solve: 3x + 5 = 20

Step 1 – Identify: x is multiplied by 3, then 5 is added. Undo in reverse order.
Step 2 – Subtract 5 from both sides: 3x + 5 - 5 = 20 - 5
Step 3 – Simplify: 3x = 15
Step 4 – Divide both sides by 3: 3x/3 = 15/3
Step 5 – Simplify: x = 5
Step 6 – Check: 3(5) + 5 = 15 + 5 = 20 ✓

---

## Example 2 — Lightly Faded (Final two steps removed)

### Solve: 2x + 7 = 15

Step 1 – Identify: x is multiplied by 2, then 7 is added.
Step 2 – Subtract 7 from both sides: 2x + 7 - 7 = 15 - 7
Step 3 – Simplify: 2x = 8
Step 4 – Divide both sides by 2: 2x/2 = 8/2

✏️ YOUR TURN — Step 5: Simplify to find x.
x = _____

✏️ YOUR TURN — Step 6: Check by substituting your value back into 2x + 7 = 15.
2(___) + 7 = ___ + 7 = ___
Does it equal 15? ______

---

## Example 3 — Moderately Faded (Final four steps removed)

### Solve: 4x − 3 = 25

Step 1 – Identify: x is multiplied by 4, then 3 is subtracted.
Step 2 – Add 3 to both sides: 4x - 3 + 3 = 25 + 3

✏️ YOUR TURN — Step 3: Simplify both sides.
4x = _____

✏️ YOUR TURN — Step 4: What operation do you perform on both sides now? Do it.

✏️ YOUR TURN — Step 5: Simplify to find x.
x = _____

✏️ YOUR TURN — Step 6: Check by substituting back into 4x − 3 = 25.

---

## Example 4 — Heavily Faded (Only Step 1 provided)

### Solve: 5x + 8 = 33

Step 1 – Identify: x is multiplied by 5, then 8 is added. Undo in reverse order.

✏️ YOUR TURN — Step 2: Perform the correct inverse operation on both sides.

✏️ YOUR TURN — Step 3: Simplify both sides.

✏️ YOUR TURN — Step 4: Perform the next inverse operation on both sides.

✏️ YOUR TURN — Step 5: Simplify to find x.
x = _____

✏️ YOUR TURN — Step 6: Check your answer.

---

## Summary of the Fading Sequence

| Step                     | Ex 1    | Ex 2    | Ex 3      | Ex 4      |
|--------------------------|---------|---------|-----------|-----------|
| 1. Identify operations   | Given   | Given   | Given     | Given     |
| 2. First inverse op      | Given   | Given   | Given     | Student   |
| 3. Simplify              | Given   | Given   | Student   | Student   |
| 4. Second inverse op     | Given   | Given   | Student   | Student   |
| 5. Find x                | Given   | Student | Student   | Student   |
| 6. Check solution        | Given   | Student | Student   | Student   |

## Answer Key

| Example   | Answer |
|-----------|--------|
| Example 1 | x = 5  |
| Example 2 | x = 4  |
| Example 3 | x = 7  |
| Example 4 | x = 5  |

## Teacher Notes

- **Why fading works:** Full worked examples reduce cognitive load for novices
  (Sweller & Cooper, 1985). Fading gradually shifts the cognitive work from
  the example to the learner (Renkl, Atkinson & Maier, 2002).
- **Step 1 is always provided** because identifying the structure is the
  conceptual anchor.
- **After Example 4**, students should be ready for fully independent practice.
- **Differentiation tip:** If a student struggles at Example 3, return them to
  Example 2 rather than jumping to Example 4.

Gemini 3.1 Pro — Condition A (Baseline) — Output score: 0

Hello there! As a mathematics teacher, I always tell my Year 8 students that solving algebra equations is exactly like untying a knot or working backwards through a puzzle.

Our golden rule is the **Balance Method**: whatever you do to one side of the equals sign, you *must* do to the exact same thing to the other side. Our ultimate goal is always to **isolate the variable** (get the letter completely by itself).

To do this, we use **inverse (opposite) operations**. We also work in reverse order of operations (reverse BIDMAS/BODMAS) — meaning we generally undo addition and subtraction *before* we undo multiplication and division.

Here are 4 worked examples to show you exactly how it's done.

---

### Example 1: The standard addition equation
**Problem:** Solve 3x + 5 = 26

**Teacher's thought process:**
I want 'x' by itself. Right now, x is being multiplied by 3, and then 5 is being added to it. To untie this knot, I need to get rid of the +5 first, and then the ×3.

**Worked Solution:**
* Step 1: Undo the addition. Subtract 5 from both sides.
  3x + 5 - 5 = 26 - 5
  3x = 21
* Step 2: Undo the multiplication. Divide both sides by 3.
  3x/3 = 21/3
  x = 7

* Check your answer: 3(7) + 5 = 21 + 5 = 26. It works!

---

### Example 2: Dealing with subtraction
**Problem:** Solve 4y - 7 = 13

**Worked Solution:**
* Step 1: Undo the subtraction. Add 7 to both sides.
  4y - 7 + 7 = 13 + 7
  4y = 20
* Step 2: Undo the multiplication. Divide both sides by 4.
  4y/4 = 20/4
  y = 5

* Check your answer: 4(5) - 7 = 20 - 7 = 13. Perfect!

---

### Example 3: Equations with fractions (division)
**Problem:** Solve m/2 + 4 = 10

**Teacher's thought process:**
Don't let the fraction panic you! A fraction bar just means "divide". So, 'm' is being divided by 2, and then 4 is added.

**Worked Solution:**
* Step 1: Undo the addition. Subtract 4 from both sides.
  m/2 + 4 - 4 = 10 - 4
  m/2 = 6
* Step 2: Undo the division. Multiply both sides by 2.
  (m/2) × 2 = 6 × 2
  m = 12

* Check your answer: 12/2 + 4 = 6 + 4 = 10. Spot on!

---

### Example 4: The variable is on the right side
**Problem:** Solve 22 = 5p - 3

**Teacher's thought process:**
Students often freeze when the letter is on the right side. Don't worry! An equals sign is just a balance scale; it doesn't matter which side the letter is on.

**Worked Solution:**
* Step 1: Undo the subtraction. Add 3 to both sides.
  22 + 3 = 5p - 3 + 3
  25 = 5p
* Step 2: Undo the multiplication. Divide both sides by 5.
  25/5 = 5p/5
  5 = p
* Step 3: Rewrite for neatness.
  p = 5

* Check your answer: 5(5) - 3 = 25 - 3 = 22. Great job!

Gemini 3.1 Pro — Condition B (General CLT) — Output score: 1

As an experienced mathematics teacher, I have designed these four worked examples specifically for Year 8 students.

To apply **Cognitive Load Theory (CLT)**, I have used the following principles:
* **The Worked Example Effect:** Providing step-by-step expert models to free up working memory.
* **The Split-Attention Effect:** Placing the explanatory text directly next to the mathematical steps (using a table format) so students don't have to look back and forth.
* **Signaling/Cueing:** Using bold text and consistent phrasing to draw attention to the most important information (the inverse operations).
* **Gradual Fading (Isolating Elements):** The examples gradually increase in complexity. We start with addition/multiplication, move to subtraction, introduce fractions (division), and finally include negative numbers.

---

### Teacher's Golden Rules for Solving Equations:
1. **Goal:** Get the letter (the variable) completely by itself.
2. **Balance:** Whatever you do to one side, you must do to the other.
3. **Reverse BIDMAS:** Undo addition/subtraction first, then undo multiplication/division second.

---

### Example 1: The Basic Two-Step Equation
**Problem:** Solve 3x + 4 = 19

| Mathematical Steps              | Teacher's Explanation                              |
|----------------------------------|----------------------------------------------------|
| 3x + 4 = 19                     | Here is our starting equation. I want 'x' by itself.|
| 3x + 4 - 4 = 19 - 4             | Step 1: Undo the addition. Subtract 4 from both sides.|
| 3x = 15                         | New, simpler equation. '3x' means '3 multiplied by x'.|
| 3x/3 = 15/3                     | Step 2: Undo the multiplication. Divide both sides by 3.|
| x = 5                           | Final Answer.                                      |
| Check: 3(5) + 4 = 15 + 4 = 19   | Replace x with 5. It works!                        |

---

### Example 2: Dealing with Subtraction
**Problem:** Solve 5m - 7 = 23

| Mathematical Steps              | Teacher's Explanation                              |
|----------------------------------|----------------------------------------------------|
| 5m - 7 = 23                     | I want 'm' by itself.                              |
| 5m - 7 + 7 = 23 + 7             | Step 1: Undo the subtraction. Add 7 to both sides. |
| 5m = 30                         | Now I have 5 multiplied by m equals 30.             |
| 5m/5 = 30/5                     | Step 2: Undo the multiplication. Divide by 5.      |
| m = 6                           | Final Answer.                                      |
| Check: 5(6) - 7 = 30 - 7 = 23   | Replace m with 6. It works!                        |

---

### Example 3: Dealing with Fractions (Division)
*Note: A fraction bar just means "divide".*
**Problem:** Solve y/4 + 3 = 8

| Mathematical Steps              | Teacher's Explanation                              |
|----------------------------------|----------------------------------------------------|
| y/4 + 3 = 8                     | I want 'y' by itself. The fraction bar means y ÷ 4.|
| y/4 + 3 - 3 = 8 - 3             | Step 1: Undo the addition. Subtract 3.             |
| y/4 = 5                         | Now I have y divided by 4 equals 5.                 |
| (y/4) × 4 = 5 × 4               | Step 2: Undo the division. Multiply by 4.          |
| y = 20                          | Final Answer.                                      |
| Check: 20/4 + 3 = 5 + 3 = 8     | Replace y with 20. It works!                       |

---

### Example 4: Crossing into Negative Numbers
**Problem:** Solve 2k + 15 = 3

| Mathematical Steps              | Teacher's Explanation                              |
|----------------------------------|----------------------------------------------------|
| 2k + 15 = 3                     | I want 'k' by itself.                              |
| 2k + 15 - 15 = 3 - 15           | Step 1: Undo the addition. Subtract 15.            |
| 2k = -12                        | Careful: 3 - 15 gives a negative number (-12).     |
| 2k/2 = -12/2                    | Step 2: Undo the multiplication. Divide by 2.      |
| k = -6                          | Final Answer. (A negative ÷ a positive = negative).|
| Check: 2(-6) + 15 = -12 + 15 = 3| Replace k with -6. It works!                       |

Gemini 3.1 Pro — Condition C (Specific fading) — Output score: 2

Hello! As an experienced mathematics teacher, I love using the **worked example fading effect**. It is a fantastic strategy from Cognitive Load Theory. By gradually removing steps (scaffolding), we prevent students from being overwhelmed by the whole process at once.

Here is a sequence of 4 examples designed for Year 8 students learning to solve two-step linear equations. We will use the **backward fading** technique, where the final steps are removed first.

---

### Topic: Solving Two-Step Linear Equations
**Goal:** Isolate the variable (get the letter by itself) by using inverse (opposite) operations.
**Rule:** Whatever you do to one side of the equals sign, you must do to the exact same to the other side!

---

#### Example 1: Fully Worked Example (Study this carefully)
**Solve for x: 4x + 7 = 23**

Teacher's thought process: I need to get x by itself. First, I will undo the addition. Then, I will undo the multiplication.

* Step 1: Subtract 7 from both sides to remove the constant.
  4x + 7 - 7 = 23 - 7
  4x = 16
* Step 2: Divide both sides by 4 to isolate x.
  4x/4 = 16/4
* Final Answer:
  x = 4

(Check: 4 × 4 + 7 = 16 + 7 = 23. It works!)

---

#### Example 2: Partially Faded (You do the last step)
**Solve for y: 5y - 3 = 22**

* Step 1: Add 3 to both sides.
  5y - 3 + 3 = 22 + 3
  5y = 25
* Step 2: Divide both sides by 5.
  5y/5 = 25/5
* Final Answer:
  y = _______

---

#### Example 3: Mostly Faded (You do the last two steps)
**Solve for m: 2m + 9 = 21**

* Step 1: Subtract 9 from both sides.
  2m = _______  (Write the new number on the right side)
* Step 2: __________________________ (What do you divide by?)
  2m/___  =  ___/___
* Final Answer:
  m = _______

---

#### Example 4: Fully Faded (Independent Practice)
**Solve for k: 6k - 5 = 19**

Teacher's thought process: You've got this! Apply the exact same pattern we used in the first three examples. Show all your working out.

* Step 1: __________________________

* Step 2: __________________________

* Final Answer:
  k = _______

(Don't forget to check your answer by putting it back into the original equation!)

---

### Teacher Notes on Implementation:
* **Spacing:** Ensure plenty of blank vertical space in Example 3 and Example 4.
* **Monitoring:** By the time they reach Example 3, you will quickly see who has grasped the concept.
* **Variability:** I alternated between addition (+7, +9) and subtraction (-3, -5) to ensure students don't blindly subtract in Step 1 every time.

Appendix E: Methodology document

The full methodological plan, including hypotheses, data analysis plan, and implementation details, is available in the project repository as REPORT_4_METHODOLOGY.md.

Repository: github.com/trgallagher-research/AlignED-research-report-4

Citation

Gallagher, T. (2026). Do LLMs fade worked examples? A pilot study of pedagogical reasoning in frontier AI models. AlignED Report 4. https://trgallagher-research.github.io/AlignED-research-report-4/

Contact

For questions about this report or the AlignED benchmark series, contact Tim Gallagher via GitHub.

This report is part of the AlignED research report series.

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