Point-Slope Form of a Line (Lesson 1.6)
Unit 1: Sequences and Linear Functions
Day 1: Recursive Sequences
Day 2: Applications of Arithmetic Sequences
Day 3: Sum of an Arithmetic Sequence
Day 4: Applications of Geometric Sequences
Day 5: Sequences Review
Day 6: Quiz 1.1 to 1.4
Day 7: Linear Relationships
Day 8: Point-Slope Form of a Line
Day 9: Standard Form of a Linear Equation
Day 10: Quiz 1.5 to 1.7
Day 11: Unit 1 Review
Day 12: Unit 1 Test
Unit 2: Linear Systems
Day 1: Linear Systems
Day 2: Number of Solutions
Day 3: Elimination
Day 4: Larger Systems of Equations
Day 5: Quiz 2.1 to 2.4
Day 6: Systems of Inequalities
Day 7: Optimization Using Systems of Inequalities
Day 8: Quiz 2.5 to 2.6
Day 9: Unit 2 Review
Day 10: Unit 2 Test
Unit 3: Function Families and Transformations
Day 1: Interpreting Graphs
Day 2: What is a function?
Day 3: Translating Functions
Day 4: Quiz 3.1 to 3.3
Day 5: Quadratic Functions and Translations
Day 6: Square Root Functions and Reflections
Day 7: Absolute Value Functions and Dilations
Day 8: Equations of Circles
Day 9: Quiz 3.4 to 3.7
Day 10: Unit 3 Review
Day 11: Unit 3 Test
Unit 4: Working with Functions
Day 1: Using Multiple Strategies to Solve Equations
Day 2: Solving Equations
Day 3: Solving Nonlinear Systems
Day 4: Quiz 4.1 to 4.3
Day 5: Combining Functions
Day 6: Composition of Functions
Day 7: Inverse Relationships
Day 8: Graphs of Inverses
Day 9: Quiz 4.4 to 4.7
Day 10: Unit 4 Review
Day 11: Unit 4 Test
Unit 5: Exponential Functions and Logarithms
Day 1: Writing Exponential Functions
Day 2: Graphs of Exponential Functions
Day 3: Applications of Exponential Functions
Day 4: Quiz 5.1 to 5.3
Day 5: Building Exponential Models
Day 6: Logarithms
Day 7: Graphs of Logarithmic Functions
Day 8: Quiz 5.4 to 5.6
Day 9: Unit 5 Review
Day 10: Unit 5 Test
Unit 6: Quadratics
Day 1: Forms of Quadratic Equations
Day 2: Writing Equations for Quadratic Functions
Day 3: Factoring Quadratics
Day 4: Factoring Quadratics. Part 2.
Day 5: Solving Using the Zero Product Property
Day 6: Quiz 6.1 to 6.4
Day 7: Completing the Square
Day 8: Completing the Square for Circles
Day 9: Quadratic Formula
Day 10: Complex Numbers
Day 11: The Discriminant and Types of Solutions
Day 12: Quiz 6.5 to 6.9
Day 13: Unit 6 Review
Day 14: Unit 6 Test
Unit 7: Higher Degree Functions
Day 1: What is a Polynomial?
Day 2: Forms of Polynomial Equations
Day 3: Polynomial Function Behavior
Day 4: Repeating Zeros
Day 5: Quiz 7.1 to 7.4
Day 6: Multiplying and Dividing Polynomials
Day 7: Factoring Polynomials
Day 8: Solving Polynomials
Day 9: Quiz 7.5 to 7.7
Day 10: Unit 7 Review
Day 11: Unit 7 Test
Unit 8: Rational Functions
Day 1: Intro to Rational Functions
Day 2: Graphs of Rational Functions
Day 3: Key Features of Graphs of Rational Functions
Day 4: Quiz 8.1 to 8.3
Day 5: Adding and Subtracting Rational Functions
Day 6: Multiplying and Dividing Rational Functions
Day 7: Solving Rational Functions
Day 8: Quiz 8.4 to 8.6
Day 9: Unit 8 Review
Day 10: Unit 8 Test
Unit 9: Trigonometry
Day 1: Right Triangle Trigonometry
Day 2: Solving for Missing Sides Using Trig Ratios
Day 3: Inverse Trig Functions for Missing Angles
Day 4: Quiz 9.1 to 9.3
Day 5: Special Right Triangles
Day 6: Angles on the Coordinate Plane
Day 7: The Unit Circle
Day 8: Quiz 9.4 to 9.6
Day 9: Radians
Day 10: Radians and the Unit Circle
Day 11: Arc Length and Area of a Sector
Day 12: Quiz 9.7 to 9.9
Day 13: Unit 9 Review
Day 14: Unit 9 Test
Learning Targets
Write and graph linear equations in point slope form.
Make connections between finding terms of an arithmetic sequence and outputs of a linear equation.
Tasks/Activity | Time |
---|---|
Activity | 20 minutes |
Debrief Activity with Margin Notes | 10 minutes |
QuickNotes | 5 minutes |
Check Your Understanding | 15 minutes |
Activity: How Much Does my Pizza Cost?
Lesson Handouts
Media Locked
Media Locked
Answer Key
Media Locked
Homework
Media Locked
Experience First
Today students use the context of pizza prices to discover point-slope form of a line. Students are given the cost of a 4-topping pizza and the cost of a 6-topping pizza and must first determine the cost of each topping. They then use this information to find the costs of other pizzas by adding on or subtracting toppings. The goal is for students to be able to think of each pizza in relation to the pizza they are already given. Instead of needing to know a “base price” of the pizza and adding on, students can add or subtract toppings from the 4-topping or 6-topping pizza. For every topping added or subtracted, 1.75 is added or subtracted from the price. It is very intuitive for students to calculate the extra cost of upgrading from a 4-topping to a 7-topping pizza simply by finding the price of three additional toppings, yet they often don’t make the connection to the equation y - y1 = m(x - x1) as representing the same thing. In this case, y-y1 represents the change in price between two pizzas, which we can find by multiplying the cost per topping by the number of additional toppings. Note that this interpretation does not require a y-intercept because we are thinking about toppings in addition to or removed from the 4-topping pizza. In question 4, students record the costs of various pizzas in the table. Note that only the 4-topping pizza is given, because we want to prime students to see this order as the "anchor point". The debrief will focus on how all other values in the table can be found from knowing the cost of a 4-topping pizza (see red notes on answer key). The rest of the activity is focused on formalizing the equation, or general form, of this relationship between number of toppings and cost. In question 5, students verbalize the relationship between the 4-topping pizza and any pizza. They may struggle to identify how this verbal relationship can be expressed as an equation (specifically the (x-4) part.) To help them with the (x-4) portion, go back to question 2 and ask why they did not multiply the 1.75 by 5 since it was a 5-topping pizza. I like to feign confusion here and really press them for a convincing explanation. Then ask them what they would do for 9 toppings or 11 toppings (i.e. how much would you have to add on to $18.99?) Students should soon be able to articulate that you only have to add on 5-4, or 9-4, or 11-4 toppings since we only care about the toppings that are additional to the ones already included in the $18.99 price. Note that many students may be using the 7-topping pizza to find the cost of a 9-topping pizza and then adding on the cost of one more topping to get to the 10-topping pizza. This is great reasoning and should be encouraged. However, during the debrief make sure to point out how we could have added on directly from the 4-topping pizza, since this will be essential to really understanding the point-slope equation. If possible, look for groups that did the problem in two different ways and have both of them present their methods (one adding on from the previous pizza, one adding on directly from the 4-topping pizza). We like to discuss with our students that there are multiple ways to find the cost of another pizza as long as we have one "reference" pizza and know the cost of a topping. This is a nice lead-in to the big idea of point-slope form of a line. You can use any point on the line and a slope to find the equation of the line.
Formalize Later
Point-slope form of a line is one of the most useful and illuminating concepts in math that transcends from Algebra 1 all the way through Calculus. In this lesson we take the “adding on approach”, not the translation approach (although this is a great connection to make in Unit 3 when we talk about translations and transformations of functions!). The goal is for students to think conceptually about both versions of the point slope form of a line, one that highlights the change in y-values and one that highlights the final y-value. We encourage you to use both versions throughout the unit and course. We hope students see the value in being able to start with any point on a line, instead of having to first solve for a y-intercept. Question 2 on the Check Your Understanding further drives home this point. A common confusion for students when writing point-slope form of a line is which parts in the general form need to be substituted and which ones stay variables. By explaining that this is the equation of a line, students should understand why an independent and dependent variable needs to be present in the equation. The equation needs to represent any and all points on the line. In other words, the way we find the cost of a pizza with any number (x) toppings is to take the price of a pizza we know with x1 toppings (whose price is given by y1) and add on (or subtract) the additional toppings, in other words, you have to account for “how far away” we are from the given pizza, either to the left or right. It should make sense to students why there is a minus in the equation and not a plus, since we are thinking about distance away from x1. It makes sense that if we want a point to the left of x1, we will end up subtracting copies of the slope since x-x1 would be a negative value. While the use of (x1,y1) for the given point is standard notation, some students find it easier to think about the anchor point as (a, c); we avoid using b, as students generally interpret it as a y-intercept. In later courses, we will use (a, f(a)).