##
* *Reflection: Modeling
Absolute Value Equations - Section 2: Modeling Absolute Value

The highlight of this class was the Modeling Temperature problem. The students had just had a basic introduction to absolute value and worked with a simple modeling problem. I introduced the Problem and asked them to write a model that found the maximum and minimum temperatures. I then walked around the classroom writing down their models on a personal white board which I then transferred to the large whiteboard.

Here are the Models from one class. While close, none of the students hit a precise model. I padded the list with two accurate models (the fifth and eighth ones).

Once they were all of the board, I asked the students to talk to their partners about which of the eight models they thought were accurate, which were not, and why. This is the key to the entire process. I don't tell them which is correct or incorrect, I follow their lead. I do insist that they raise their hand to offer an opinion as this allows me to manage the flow and not allow specific students to dominate. An example of the conversation, one student who wrote │x│= 80, and claimed it as his to the class, eliminated it as a possibility since it doesn't model either the min or the max. It was interesting that many students initially hit on │x - 5│= 80. We discussed the necessity of checking our models for reliability and the students quickly saw that this only modeled the max but not the min. Many of the other models had the same issue.

Finally, we narrowed it down to │x – 80│= 5 which proved to find 75 and 85. One student asked whether │80 – x│= 5 would produce the same results or not so I had each partner discuss this (I like to turn question like this back on the students). After checking, we found that it did.

One of the best things occurred at the end of this activity when I asked the students to discuss with each other why this model works. A student in the discussion stated that it worked because the 80 is the average and the five is the difference between the average and each extreme. This really struck the other students and when asked to model the next Problem, most of the students immediate wrote the correct equation.

*Modeling: Multiple Models*

# Absolute Value Equations

Lesson 18 of 24

## Objective: Students will be able to model and solve absolute value equations. How cold is it?

#### Warm Up and Homework Check

*10 min*

I include **Warm ups **with a **Rubric **as part of my daily routine. My goal is to allow students to work on **Math Practice 3 **each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson’s Warm Up- Absolute Value Equations which asks students to model an absolute value situation.

I also use this time to correct and record the previous day's Homework.

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#### Modeling Absolute Value

*20 min*

This lesson is a precursor to the Common Core Algebra 2 standards. Absolute value equations are part of the Algebra 1 standards but this is something my students need to see before studying transformation of absolute value functions which are in the Algebra 2 standards.

We are going to begin our discussion of absolute value by looking at this situation: *The average monthly temperature in a northern Canadian city is 0 degrees Fahrenheit. The actual January temperature for that city may be about 5 degrees Fahrenheit warmer or colder. *I will ask my students to determine the maximum and minimum temperature, an easy task, and then write an equation with algebra to model this situation, which is not so easy (**Math Practice 1**)*.* I will be doing the **Note Card Activity** with this problem.

There is a good chance that you will get someone who comes up with an absolute value equation. If not, ask some guiding questions to the class like “What math concept do you think this is modeling?”

Once the students have gone through this introduction to absolute value equations, there are three additional modeling situations temperature, assessment grades and biology(**Math Practice 4**). The first two problems are relatively simple. My goal with these problems is to give the students an opportunity to experiment with building the absolute value models. Students will struggle with building the equation properly (**Math Practice 2**). I recommend to students that they test their equations to ensure that their model works. Depending on the speed that the students learn the model, I will continue to do the note card activity or not.

The final scenario will require some additional reasoning. A portion of the model is given and the students must use it to build the complete absolute value model. This problem should provide some great discussion around appropriate models. My lesson Absolute Value Equations presentation includes detailed presenter notes.

*expand content*

The remaining portion of this lesson has students review solving Absolute Value Equations. I have build this portion very carefully using increasing levels of difficulty. The final problems deal with absolute value equations that may have extraneous solutions.

Please see the PowerPoint for detailed presentation plans.

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#### Exit Ticket

*3 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

This Exit Ticket asks students to solve a multi-step absolute value equation.

#### Resources

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The goal of this Assignment for this section is to reinforce the day’s lesson. The first six problems are straight absolute value problems of varying difficulty. The next two are modeling problems similar to those seen in the lesson. The final problem gives the students an absolute value equation and asks them to write a scenario to model it.

#### Resources

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Mathematical Modeling: Properties of Functions Day 1 of 2
- LESSON 2: Mathematical Modeling: Properties of Functions Day 2 of 2
- LESSON 3: Mathematical Modeling: Linear Functions
- LESSON 4: Mathematical Modeling: Linear Functions Design Project
- LESSON 5: Finding Equations of Parallel and Perpendicular Lines
- LESSON 6: Inverse Functions
- LESSON 7: System of Equations Day 1 of 2
- LESSON 8: Systems of Equations Day 2 of 2
- LESSON 9: Modeling Systems of Equations
- LESSON 10: Function Review
- LESSON 11: Function Mid Test
- LESSON 12: The Tortoise and The Hare Project Day 1
- LESSON 13: The Tortoise and The Hare Project Day 2
- LESSON 14: The Tortoise and The Hare Project Day 3
- LESSON 15: The Tortoise and the Hare Project Day 4
- LESSON 16: The Tortoise and the Hare Piece-wise Functions
- LESSON 17: Step Functions
- LESSON 18: Absolute Value Equations
- LESSON 19: Absolute Value Inequalities
- LESSON 20: Absolute Value Functions: Transformations Day 1
- LESSON 21: Absolute Value Functions: Transformations Day 2
- LESSON 22: Functions Test Review Day 1
- LESSON 23: Functions Test Review Day 2
- LESSON 24: Functions Test