math-ell

## Math ELL Content Modification Template

| Section | Details |
| :--- | :--- |
| **Role** | I am a high school mathematics teacher at Chicago Public Schools. |
| **Content Area** | Algebra 2 |
| **Grade Level(s)** | 11th Grade |
| **Learning Target** | Students will be able to solve quadratic equations by factoring. |
| **Success Criteria** | Students can accurately find the real roots of 4 out of 5 quadratic equations by using the zero product property. |
| **Input Problem (Original Word Problem)** | "The height of a ball thrown from a cliff is given by the equation $\text{h}=-16\text{t}^2+48\text{t}+160$, where $\text{t}$ is the time in seconds after it is thrown. When does the ball hit the ground?" |
| **Modification Task** | Generate modified versions of the problem and its solution to support English Language Learners (ELLs) at different proficiency levels. |
| **Modification Focus** | Adapt the problem's language, provide visual aids, and scaffold the steps. Use simplified vocabulary and sentence structure for lower levels. |

***

## Modified Content for ELL Proficiency Levels (REACH Scale)

### 1. Level 1 (Entering)

The goal is to use **simple, high-frequency words** and a **visual representation**. Focus on identifying the main question and the necessary condition.

**Modified Problem:**
Look at the picture:
The formula for the ball's **height** is: $\text{h}=-16\text{t}^2+48\text{t}+160$.
* **h** = height (up/down)
* **t** = time (seconds)
  **Question:** When is the **height (h) zero (0)**? (When does it hit the **ground**?)

**Problem Setup (Cloze style):**
$0 = -16\text{t}^2+48\text{t}+160$
**Solve for t.**

**Support/Scaffolding:**
* **Key Concept:** The ground is $\text{h}=0$.
* **Vocabulary:** **Ball** (round object), **Thrown** (Tossed), **Ground** (earth/floor), **Height** (how high), **Time** (seconds).

***

### 2. Level 2 (Beginning)

The goal is to use **short, simple sentences** and **break the problem into explicit questions**.

**Modified Problem:**
A ball is thrown from a high cliff. The **height** ($\text{h}$) of the ball at different **times** ($\text{t}$) is described by this **equation**:
$$\text{h}=-16\text{t}^2+48\text{t}+160$$

1.  What is the height ($\text{h}$) of the ball when it touches the **ground**? (Hint: The ground is $\text{h}=0$.)
2.  **Substitute** this value into the equation.
3.  **Solve** the new equation for the time ($\text{t}$).

**Word Bank:**
| Term | Simple Meaning |
| :--- | :--- |
| **Height (h)** | Distance up from the ground. |
| **Time (t)** | Measured in seconds. |
| **Equation** | A math rule with an equals sign. |
| **Ground** | Where the height is zero. |

***

### 3. Level 3 (Developing)

The goal is to introduce **academic vocabulary** with a glossary and **guide the students through the initial setup** of the problem.

**Modified Problem:**
The motion of a ball is modeled by the **quadratic equation**: $\text{h}=-16\text{t}^2+48\text{t}+160$. The variable $\text{h}$ **represents** the vertical height in feet, and $\text{t}$ **represents** the time in seconds.

**Task:** Use the **zero product property** to find the time ($\text{t}$) when the ball's height is **zero**.

**Guided Setup:**
1.  **Set the equation** equal to zero: $0 = -16\text{t}^2+48\text{t}+160$.
2.  **Simplify** the equation by dividing all terms by the greatest common factor (GCF), which is -16.
    $$\frac{0}{-16} = \frac{-16\text{t}^2}{-16} + \frac{48\text{t}}{-16} + \frac{160}{-16}$$
3.  **New Equation:** $\text{t}^2$ \_\_\_\_\_\_ $\text{t}$ \_\_\_\_\_\_ $0$.
4.  **Factor** the new equation and solve for $\text{t}$.

**Glossary:**
* **Quadratic Equation:** An equation with a variable raised to the power of two ($\text{t}^2$).
* **Zero Product Property:** If $\text{A} \times \text{B} = 0$, then $\text{A}=0$ or $\text{B}=0$.
* **Represent:** To stand for or model.

***

### 4. Level 4 (Expanding)

The goal is to use **grade-level academic vocabulary** and **modify the sentence structure** to be more direct. Provide a **hint** to start the factoring process.

**Modified Problem:**
A physics problem models the **vertical trajectory** of a projectile using the following **quadratic function**:
$$\text{h}(\text{t})=-16\text{t}^2+48\text{t}+160$$
where $\text{h}(\text{t})$ is the height (in feet) and $\text{t}$ is the elapsed time (in seconds). **Determine** the time at which the ball **impacts the ground**.

**Hint/Starting Point:**
1.  Set the function equal to zero (0) to represent the ground impact.
2.  **Simplify** the equation by dividing by the greatest common factor of $-16$: $\text{t}^2 - 3\text{t} - 10 = 0$.
3.  **Factor** the resulting quadratic expression.

**Scaffolded Task:** Find two numbers that multiply to -10 and add to -3.

***

### 5. Level 5 (Bridging)

The goal is to present **content at or near grade level**, focusing on clarifying **complex phrasing** and **multi-step reasoning**.

**Modified Problem:**
The **vertical displacement** of a ball thrown from a cliff is defined by the function $\text{h}(\text{t})=-16\text{t}^2+48\text{t}+160$. Using the **method of factoring** and the **zero product property**, **calculate** the **positive real root** of the equation to ascertain the exact time ($\text{t}$) when the ball **reaches zero height** (impacts the ground).

**Clarification Focus:**
* **Vertical Displacement:** Clarify that this is just another term for height.
* **Positive Real Root:** Emphasize that in this context, time cannot be negative, so only the positive solution is relevant.
* **Acertain:** Define this as 'to find out or determine with certainty.'

***

### 6. Level 6 (Reaching)

**Modified Prompt:**
**Original Input:** "The height of a ball thrown from a cliff is given by the equation $\text{h}=-16\text{t}^2+48\text{t}+160$, where $\text{t}$ is the time in seconds after it is thrown. When does the ball hit the ground?"

**Focus:** Minimal to no modification needed. The content is presented in its original form. Students are expected to independently set up, simplify, factor, and select the correct positive root, justifying why the negative root is not physically possible.