# HSG.CO.A.5 Transformations

This learning progression focuses on the first half of a unit on Transformations in a high school Geometry class, consisting of mostly 9th and 10th graders. The first lesson will cover 7.1 Rigid Motion in a Plane, which will just briefly introduce the concepts of transformations and what each transformation means. The second lesson will cover 7.2 Reflections, and will give students a more in depth understanding of reflections, and how to use them to find coordinates in a plane. The third lesson covers 7.3 Rotations more in depth, and then for the final lesson in the learning progression, students will have a review to make sure they understand these concepts before moving on to the second half of the chapter.

CCSS Content Standards:

CCSS.MATH.CONTENT.HSG.CO.A.5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g. graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

CCSS Mathematical Practice Standards:

CCSS.MATH.PRACTICE.MP1

Make sense of problems and perservere in solving them.

CCSS.MATH.PRACTICE.MP5

Use appropriate tools strategically.

CCSS.MATH.PRACTICE.MP6

Attend to precision.

Throughout this learning progression, students will be getting new notes in the first three lessons, while also working on practice problems and homework assignments in every lesson. I will be implementing a lot of group work during this learning progression, because it is helpful for students to compare their answers with peers so that they can work together to figure out the correct answers. For the last lesson, they will do an activity for the review, where they are put in groups, and rotating through different stations that will be focusing on the main ideas from each concept. I will be giving them entry tasks daily as their formative assessments in this learning progression to check their understanding, along with checking their homework assignments, and going over the most missed problems so that they can see common errors.

The full learning progression is attached here: edTPA Learning Progression

This learning progression focuses on Solving Quadratic Equations using multiple methods including by inspection, taking square roots, completing the square, the quadratic formula, and factoring. This learning progression would be taught in a high school Algebra I class, consisting of mostly freshmen. In the first lesson students will be learning these methods and when to use each one of the methods, and thought the next two lessons they will be reviewing all of these methods and getting more practice on them during the activities planned for each of the following two lessons.

CCSS Content Standards:

CCSS.MATH.CONTENT.HSA.REI.B.4.A

Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.

CCSS.MATH.CONTENT.HSA.REI.B.4.B

Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

CCSS Mathematical Practice Standards:

CCSS.MATH.PRACTICE.MP1

Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP5

Use appropriate tools strategically.

CCSS.MATH.PRACTICE.MP8

Look for and express regularity in repeated reasoning.

The learning progression is attached here:

sargent_learningprogression

# A-REI Solving Systems of Equations

A-REI Solving Systems of Equations

Alignment 1:

A-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

A-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.

Solve the following systems of equations. First, find an approximate answer (using a graph) and then algebraically by the given method.

a.        x-2y=10

3x+y=9

Graphically:                                                                Algebraically using Substitution:

b.       6x+3y=18

3x-3y=9

Graphically:                                                               Algebraically using Elimination:

c.         y=x2+3x-4

y=x-1

Graphically:                                                               Algebraically (student’s choice):

d.         y=2x-2

x2+y2=4

Graphically:                                                               Algebraically (student’s choice):

Assessment Commentary and Solutions are attached here: im-assessment-solving-systems-of-equations

# Lets Roll The Dice HSS.MD.B.5.A

A great way to get students engaged in the mathematical concepts they are learning is to give them problems that involve real world things that they can actually relate to. Everyone at some point in time has rolled dice during some sort of game, so this picture can relate to students and get them interested in learning more about probability.

Using this picture, ask the students which numbers they think the dice will land on. This lesson can involve theoretical probability and experimental probability. They can start by calculating the theoretical probabilities of different combinations of numbers that the dice could land on. Then they can either use two actual dice, or even the dice rolling app on a graphing calculator to find the experimental probability. This activity gives students the opportunity to have a hands on experience with the concept of probability.

This problem is aligned with:

CCSS.MATH.CONTENT.HSS.MD.A.3
(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

CCSS.MATH.CONTENT.HSS.MD.B.5.A
Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

# Chill Out: Modeling with Exponential Functions HSF.IF.C.8.B

Have you ever burned your tongue when taking a big gulp of that hot drink you just got? This is something that almost everyone has experienced. This would be a relatable activity for students to participate in to determine how long it takes for a hot drink to cool down using the Vernier equipment and software.

Newton’s Law of cooling gives a model, which states that the temperature difference (Tdiff) between a hot object and its surroundings decreases exponentially with time. In the model T0 is the initial temperature and k is a positive constant.

Tdiff = T0 e-kt

In this activity, we will use a Vernier temperature probe to collect data as the hot water that the probe is placed in cools. This activity is applicable because after collecting the data, you can find the line of best fit for the data. By completing this activity, students will be able to see that modeling using regression lines of data is applicable to everyday events.Objectives of this lesson:

• Record temperature versus time cooling data
• Model cooling data with an exponential function.

Equipment needed:

• EasyTemp or Go!Temp or Temperature Probe and data-collection software
• TI-Nspire handheld or computers and TI-Nspire software
• Hot water

Content standards:

CCSS.MATH.CONTENT.HSF.IF.C.8.B

Use the properties of exponents to interpret expressions for exponential functions.

CCSS.MATH.CONTENT.HSF.LE.A.2

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs.

CCSS.MATH.CONTENT.HSA.CED.A.1

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

The Activity is from Vernier’s website: Chill Out Activity

# Exploration of the Euler Line using Geogebra- HSG.CO.D.12

This lesson focuses on the use of modeling in Geogebra to help students explore the concept of the Classical Triangle Centers (centroid, orthocenter, and circumcenter) and how they relate to the Euler Line. In the activity for this lesson, students will be asked to make constructions of triangles using Geogebra, and manipulate the vertices of the triangles to answer a series of questions relating to the centroid, orthocenter, and circumcenter. They will also be asked questions that will help them to further explore the relationship between these three centers of a triangle.