## Image- Types of Solutions to Linear Equations - Section 3: Three Types of Solutions

# Selling Cake Pops Day 4 of 5

Lesson 9 of 15

## Objective: Students will be able to write and solve equations graphically from a profit model.

## Big Idea: Extend writing expressions into solving equations through modeling a student run business. Linear modeling as entry point to build students' algebra confidence.

*49 minutes*

#### Warm up and Homework Review

*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 resource Selling Cake Pops Day 4, Video Narrative, Warm Up specifically explains this lesson’s Warm Up- Selling Cake Pops Day 4, which asks students to find the missing number in an equation.

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

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As with the previous lessons using the Cake Pop scenario, this lesson is a transition to the Common Core Algebra 2 standards.

I start by having the students make a list of the profit expressions from the first day of this lesson. We are going to look at graphing (using a calculator) as a method of solving equations. We will use this technique over and over again as we solve new types of equations over the school year.

This is the first time the students will be using the graphing portion of the calculators. In the PowerPoint- Selling Cake Pops Day 3 I provide steps I have students do with (the TI-84s) so that we all start "on the same page". An overhead calculator or Smartview is helpful to this lesson for modeling purposes.

This lesson begins by asking students to identify the number of cake pops sold in order to reach certain profit targets and are solved algebraically and graphically. The first problem is viewable in the standard (10x10) viewing screen, but subsequent problems will require window manipulation. This provides an opportunity for students to use their knowledge of the problem at hand to reason out a suitable window (**Math Practice 5**). Specific scaffolding techniques and lesson procedures are located in the PowerPoint- Selling Cake Pops Day 3 notes.

For an additional TI-84 tutorial please see http://www.atomiclearning.com/k12/en/ti_84.

#### Resources

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#### Three Types of Solutions

*7 min*

This lesson is meant to be spread in two days so each class will reach a certain point in this section.

We have already discussed equations with one solution. Now we are going to talk about equations with no solutions or infinite solutions in context, algebraically and graphically (**Math Practice 2**).This discussion will set us up for future types of equations with unusual or no solutions.

Infinite Solutions

*When is the profit the same if we take the market this week in Caldwell vs. the market last week in Caldwell.* This is a bit of a trick question. I start by doing a think-pair-share on what we expect the answer to be followed by the students writing and solving the equation. Students will get something like 0=0 and we talk about what this means in terms of our solution. Finally, we graph it and the students describe how the graph shows infinite solution.

No Solutions

*Last year, Caldwell didn't charge a weekly set fee. They only charged $45 per season and the 5%. Write an expression representing Caldwell last year.* The students write a new expression for this problem. For those that are struggling, it is useful to note that the percent didn’t change, just what is being charged each week.

Now, I ask the students to find out how many cake pops need to be sold so that the profit is the same between this year’s Caldwell market vs. last year’s Caldwell market. The students write an equation, solve it, and end up with no solution. I make sure that they note that the variables cancel and the remaining equation is not an equality. They then graph these on the calculator and describe the shape of linear graphs with no solution.

Finally, I ask the students to write an example of the Types of Solutions to Linear Equations and draw a picture of what the graph solution may look like. You can do this as a class if your students struggle a bit or have them do it themselves. This summary is the BIG IDEA for today.

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#### Exit Ticket

*2 min*

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

Today's Exit Ticket, located in the PowerPoint, asks the students to look at the graph of the two sides of an equation and identify the solution from the graph.

#### 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: Turn Your Words Into Money
- LESSON 2: Crossing the Snake River
- LESSON 3: Interpreting Algebraic Expressions Day 1
- LESSON 4: Interpreting Algebraic Expressions Day 2
- LESSON 5: Mind Reading with Algebra
- LESSON 6: Selling Cake Pops Day 1 of 5
- LESSON 7: Selling Cake Pops Day 2 of 5
- LESSON 8: Selling Cake Pops Day 3 of 5
- LESSON 9: Selling Cake Pops Day 4 of 5
- LESSON 10: Selling Cake Pops Day 5 of 5
- LESSON 11: Representing Inequalities
- LESSON 12: Modeling Compound Inequalities
- LESSON 13: Modeling Expressions and Equations Review Day 1
- LESSON 14: Modeling Expressions and Equations Review Day 2
- LESSON 15: Modeling Expressions and Equations Test