HS Geometry

Transformation in the Plane HSG.CO.A

stephensmithcwu CCSS-Math Learning Progressions, HS Geometry

This learning progression will be taught in a sophomore level Geometry course at Ellensburg High School. The Common Core State Standard (CCSS) domain and cluster for this learning progression is: CCSS.MATH.CONTENT.HSG.CO.A. There are two standards that the students will be learning: HSG.CO.A.1 and HSG.CO.A.2, and HSG.CO.A.4. The math practices (MP) that will be used by students during this progression will be MP1, MP3, and MP5.

This learning progression will be broken into three separate lessons. The first lesson will cover HSG.CO.A.1 and HSG.CO.A.2. The second lesson will cover HSG.CO.A.4, but it will specifically address rotations and translations. The third lesson will also cover HSG.CO.A.4, but it will focus on reflections.

CCSS.MATH.CONTENT.HSG.CO.A

Experiment with transformations in the plane

HSG.CO.A.1

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

HSG.CO.A.2

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

HSG.CO.A.4

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Link to the full learning progression: Transformation Learning Progression

Similarity Transformations using Dilation HSG.SRT.A

stephensmithcwu CCSS-Math Learning Progressions, HS Geometry

This learning progression is for a High School Geometry class. The Common Core State Standard (CCSS) domain and cluster for this learning progression is: CCSS.MATH.CONTENT.HSG.SRT.A. There are two standards that the students will be learning: HSG.SRT.A.1 and HSG.SRT.A.2. The math practices (MP) that will be used by students during this progression will be MP1, MP3, and MP5.

The textbook used in the class is McDougall Littell’s Geometry 10th edition. In teaching this learning progressions, we assume that students have a strong grasp of previous concepts required for learning similarity transformations. These concepts are HSG.CO.A.1, HSG.CO.A.2, HSG.CO.A.5, HSG.CO.B.6, and HSG.CO.C.9.

CCSS.MATH.CONTENT.HSG.SRT.A

Understand similarity in terms of similarity transformations

HSG.SRT.A.1

Verify experimentally the properties of dilations given by a center and a scale factor:

HSG.SRT.A.1.A

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

HSG.SRT.A.1.B

The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

HSG.SRT.A.2

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Read the whole learning progression here: Dilation Learning Progression

Experiment with transformations in the plane G.CO.A

camachoo Assessment of CCSS-Math, CCSS-Math Learning Progressions, HS Geometry

This learning progression is designed for a 10th grade Geometry Class. In this unit students learn about transformations in the plane such as translation, reflection, rotation and glade reflections. The CCSS are

CCSS.MATH.CONTENT.HSG.CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). CCSS.MATH.CONTENT.HSG.CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. CCSS.MATH.CONTENT.HSG.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 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

To teach this material I use different teaching methods as well as different activities to elicit students learning. For example, I use modeling to work detail examples and give students a visual aid in the board. At the beginning of the lesson, I use a short warm up that helps students review the material learned in the previous lesson. During and after instruction, students also have the opportunity to work with others and ask questions.  To assess students learning, I use their responses to class discussions, questions, and answers to problems in different activities For summative assessment, in this learning progression students will show their knowledge by preforming the different transformations to a shape they select and writing explanations of the changes applied to the figure.

Complete learning progression here High School Geometry

HSG.CO.A.5 Transformations

sargentca CCSS-Math Learning Progressions, Common Core State Standards Mathematics, Geometry Curriculum Aligned with CCSS-Math, HS Geometry

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

Creating A Tangent To A Circle: Dare To Tangent G.C.A

schloemerj HS Geometry, Technology Teaching Issues

Lesson Title: Dare to Tangent

Unit Title: Circles and Their Properties

Teacher Candidate: Jenell Sellers

Subject, Grade Level, and Date: Geometry, 10th Grade, December 2nd, 2016

 

Placement of Lesson in Sequence and Lesson Rationale

This lesson is the fourth lesson of a four part unit where students will be learning about the properties and of circles. The lessons will progress from 1) Learning about the similarities between all circles, 2) Identifying the relationship between angles, radii and chords, 3) Constructing inscribed and circumscribed about a triangle, and 4) Creating tangents to a circle from a point outside of the circle.

Central Focus and Purpose

The central focus of this lesson is to assist students in understanding the function of creating a tangent to a circle. The students will be given directions on how to create a tangent by hand and also in Geogebra. Each student will be provided with a compass and a straight-edge and also a laptop. Students will use the class period to complete both activities

CCSS.MATH Content and Practice Standards

The Common Core State Standards that are covered throughout this four lesson unit are as follows;;

CCSS.MATH.CONTENT.HSG.C.A.1

Prove that all circles are similar.

CCSS.MATH.CONTENT.HSG.C.A.2

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

CCSS.MATH.CONTENT.HSG.C.A.3

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

CCSS.MATH.CONTENT.HSG.C.A.4

Construct a tangent line from a point outside a given circle to the circle.

Prior Content Knowledge and Pre-Assessment

The prior content knowledge that the students will be building off of will be that of understanding of right triangles and the basic trigonometry.

Learning Target(s) and Plan for Clarifying Intending Learning

The learning target for this lesson will be “I can create a tangent to a circle using technology and by hand”.

 

Strategy for using assessments to guide student learning
The goal for this lesson is to allow students to make the connection between completing a task by hand and also doing the same task via computer/using technology. This will help them also make more of a connection to how they might have to complete such a task in a job.

 
Success Criteria (criteria for interpreting student success of the learning target) Plan for providing feedback and students’ monitoring of their own learning
The success criteria for this lesson will be an exit slip that they will have to complete in the last 5-7 minutes of the class period. The exit slip will consist of questions that will allow the students to express how well they understand the concept of creating a tangent line to a circle from a point outside of the circle and what help they might need in doing so.

 

 The plan for feedback for the students during this lesson is to attend to each group during the lesson to see how they are doing on their graphs and to ask them open ended questions in order to get them to think a little bit more critically about the task at hand. I will also address all the misconceptions that are written down on the exit slips in the beginning of class the next day so that all of the students can anonymously get their questions answered.

 

Academic Language Demands
Language Function Vocabulary & Symbols Secondary Language Demand
·         In this lesson, students will use the vocabulary to:

-Correctly address different pieces of the circle

-Construct a tangent to a circle both by hand and using Geogebra.

 

 

 

 

 ·         Circle

·         Tangent

·         Point

·         Center

·         Radius

·         Diameter

·         Bisect

 

 Mathematical Precision:

Students will have to pay close attention to the directions that they are given. Students will also have to understand the notation and the vocabulary in order to correctly follow the directions.

Syntax:

Students will have to

Discourse:

Students will be working on their own, however since the desks in the classroom are arranged in groups of four the students can easily collaborate with each other. They may discuss the activities during the class period but they also must each turn in their own work.

 

Language Support (instructional and assessment strategies)
Language Instruction Guided Practice Independent Practice
The teacher will begin the class period by explaining what a tangent line is; a drawing on the whiteboard in front of the class maybe necessary. To create a better visual for the students, the teacher may explain a tangent in a real-world scenario; the tires of a car on the road, for example.

The teacher will also show the students how to bisect a line and how to create a circle using their compass and straight-edge. These will be the two things that they will have to know how to do while they are creating the tangent by hand.

 

 The teacher will guide the students by providing hints and tips on how to create the tangent to a circle when they are stuck either creating it by hand or when they are using Geogebra. Their handout will have instructions but it still might be possible that they get confused or lost.  If the students need answers to questions the teacher will be monitoring the class throughout the period so that they are easily accessible to the students. After the teacher has introduced the lesson and handed out the worksheets and the laptops and the students are all signed in with Geogebra open and the materials needed are on their desk students will be able to get started. They may collaborate with those who are sitting in their group as long as each student turns in their individual work.

 

 

Differentiation, Cultural Responsiveness, and Accommodation for Individual Differences

Accommodations have been made for several students. One student with autism gets up and walks around a lot during class so in order to help him focus he sits on an exercise ball so that he can bounce in his seat so that he’s not disrupting the class. This actually works extremely well; he can still tell you exactly what his thought process is even though he’s still bouncing up and down.

Another student with autism needs to sit closer to the front of the room in order to help him visually because he has fairly poor eye sight. Every now and again he will have some outbursts out of frustration but he only needs reassurance that he’s doing okay. There is a para-educator for the classroom that helps out with his frustrations and calms him down as well.

The ELL students in my class have a hard time with the language barrier, so in order to help them out with the language barrier and to also, hopefully, improve their English reading skills I have worked with the Spanish teacher and he helped me translate the worksheet done in English into Spanish. The ELL students will receive both sheets so that they are able to see the Spanish and English translation and make a connection between the Spanish words and the English words.

 

Materials – Instructional and Technological Needs (attach worksheets used)

The materials needed for this activity are:

-Handouts with the instructions

-Compass

-Straight-edge

-Laptop with the Geogebra Application

-Pencil and Eraser

Instructional Plan (detailed explanation for thing the lesson)
Pacing Teacher Activities Student Activities
Before Before the class period starts the teacher will need to ensure that each student will be able to have their own laptop and that each laptop also will have Geogebra downloaded already; this is to minimize wasted time and distractions by the students being online. The teacher will also need to ensure that each student will have a compass and a straightedge. Extra paper will also be necessary in case the students will want to practice or make a mistake. The student activity will be to create a tangent line to a circle using the Geogebra application on a provided laptop and also creating the same construction by hand.
During The teacher will introduce the topic of tangents and explain what a tangent is. In order to make a real-world connection for the students to relate to, the teacher might be able to explain that a tangent can also be thought of a tire of a car and the road that it’s driving on; a circle with a line touch the edge but not crossing through. The teacher will then explain that the students will be constructing a tangent to a circle using the Geogebra application on the laptops that are provided and also by hand with the provided tools. The teacher will monitor each table group and help students who have any questions and those who need assistance. If there seems to be a common misconception then the teacher may address the whole class and correct the common error that is occurring throughout the class. The students will have to turn in their worksheets at the end of class. They will also be able to print out their work that was done in Geogebra so that the teacher will be able to assess whether they understand the Geogebra application enough to operate it for another lesson. After students receive all of the necessary materials they will begin working on their handouts. Students may collaborate with their tablemates if need be. The students will be given instructions on how to create the construction but then they must be able to also do it on their own.
After Ten minutes before class is to end the teacher will instruct the students to put away all of the materials and to clear their desks so that they can fill out the exit slip. The students will be able to complete the exit slip within the last 5-7 minutes of class. Afterwards the teacher will make sure that all of the laptops have been returned and properly stored in the laptop cart and that all of the compasses and straight-edges are back in their appropriate boxes. The teacher will collect all of the exit slips and all of the handouts and prints from the students and review what the students have been able to accomplish throughout the lesson. The students will need to swiftly put all of the materials away and turn in their handouts with the print out of their work in Geogebra. The last 5-7 minutes of class they will be working on their exit slip until the bell rings and they are dismissed.

 

 

Dare to Tangent

Learning Target: I can create a tangent to a circle using technology and by hand.

Ready for a challenge?! Your mission today is to construct a tangent to a circle. This will be done by hand and also using Geogebra, the master of all things Geometry!

You may begin with either activity. Manage your time wisely, students of mine, for everything must be turned in in its entirety at the end of class!

 

Geogebra Instructions:

Construct, in Geogebra, a tangent line to a circle through a point not on the circle.

Directions:

  1. Using the ‘Graphics’ tab, get rid of the axes that are visible when you open the application.
  2. Using the Circle tool, create a circle of a size that is convenient to work with.
  • Create a point that is outside of the circle, call it P.
  1. Now, using the line tool to create the line AP
  2. Using the other line tool, bisect AP.
  3. Using the point of intersection, create a circle whose center is the intersection and whose radius diameter is AP.
  • Notice how this creates two intersections between the two circles you have created? Plot a point on each of these intersections using the Point tool.
  • Now, create a line that connects the original point that is not on the circle and the two points that you just created. Voila!
  1. Clap your hands once, raise a fist in the air and proclaim “Boomjam!” You have just created tangents to the circle using a point not on the circle! Print out your masterpiece as it will have to be turned in with the rest of the handout.

 

 

Manual Instructions:

Construct, using a compass and a straight-edge, on a separate piece of paper the following using the scheme that is provided below:

Given a circle C and a point P not on C, construct a line through P and tangent to C.

Given:

by-hand-construction

Scheme:

  1. Draw AP
  2. Bisect AP; giving point D
  • Create circle with center D and radius AD; giving points T and S
  1. Draw PT and PS
  2. Clap your hands twice, raise your fist in the area and yell “Boomjam Again!” You just created a tangent to a circle by hand!

 

 

 

Exit Slip

 

What was the most difficult/challenging aspect to the activity today?

 

 

 

Did you receive any help during this challenge? From whom (peers, friends, teacher, etc.)?

 

 

 

On a scale from Basic, Proficient to Mastery, how comfortable are you in performing this task in a stressful setting (quiz, exam)?

 

 

 

 

 

 

 

 

 

 

 

Exit Slip

 

What was the most difficult/challenging aspect to the activity today?

 

 

 

Did you receive any help during this challenge? From whom (peers, friends, teacher, etc.)?

 

 

 

On a scale from Basic, Proficient to Mastery, how comfortable are you in performing this task in a stressful setting (quiz, exam)?

 

 

 

tires-on-the-road

Interpreting the cross-sections of 3-dimensional objects: HSG-GMD.B.4

bennett77 HS Geometry, Picture Problems, Technology Teaching Issues , ,

Image result for object cut in half

 

 

HSG-GMD.B.4 has to do with a student’s ability to visualize relationships between 2 and 3 dimensional objects. Given the example above you see that a 3-dimensional sphere is related to a 2-dimensional circle. This relation is this; a circle is the cross-section of a sphere. No matter where or at what angle you take the cross-section it will be a circle and not matter the size of the sphere is will be a circle as seen by the rings in the cross-section. This could be better represented with 3 spheres cut in different places showing that the resulting cross-section is always a circle.

Calculating Cost of Wheels HSG.GMD.A.3

lonesc Assessment of CCSS-Math, HS Geometry, Uncategorized

roadroller

A construction company is designing new road rollers. The wheels of each road roller are made out of steel. The wheels are designed in a cylindrical shape in addition to having a smaller cylinder cut out of the center for the axel to fit through as shown in the picture below. The radius of the larger cylinder measures 2 ft., the radius of the cut-out cylinder measures 0.5 ft., and the length of the wheel measures 6 ft. From the diagram, this would mean R = 2 ft., r = 0.5 ft., and h= 6 ft. If the cost of steel is $2.40 per cubic foot, what is the cost of the steel to construct one wheel? Show your calculations and round your answer to two decimals.

cylinder

To view an assessment commentary and solution, follow the link below.

math-325-assessment-item-blog

Frisbee and Distance Problem HSG.SRT.C.6

berserc Common Core State Standards Mathematics, HS Geometry, Picture Problems

frisbee-picture

It can be difficult to create an interesting math problem that engages students. One way to grab students’ attention right from the start is to incorporate a relatable picture into the math problem. For instance, the picture above. Many math problems can be inspired by this image, but one problem in particular is a trigonometry problem involving a right triangle created from throwing the Frisbee.

An example of this problem could be:

Imagine you are throwing a Frisbee to a friend. If you throw the Frisbee linearly, at a 12-degree angle, and the Frisbee is 10ft. directly above your friend’s head, how far away are you from your friend?

For this problem, students would be using their problem solving skills and the concept of trigonometry, to relate the given side lengths and degrees. The students will have to go through the process of creating a diagram or visual of the triangle being discussed, decide how the picture will look, and where the information should be placed. Once the diagram is complete, students must have the right angle placed at the friend, with the 10ft. marked as the leg above that friend, and the 12-degree angle placed at the student. After this, the student must use algebra to complete the problem and find the distance between the student and their friend. Therefore, this problem is comparable to any simple trigonometry problem. However, if the students begin by looking at this picture and imagining doing the activity of throwing a Frisbee to their friend, then the problem will become much more engaging and relatable.

The Common Core State Standard that this problem teaches is:

CCSS.MATH.CONTENT.HSG.SRT.C.6

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

 

 

GMD.B.4-Play Time with Play-Doh

soton Assessment of CCSS-Math, Geometry Curriculum Aligned with CCSS-Math, HS Geometry, Picture Problems

play-doh

There is something very therapeutic about playing with play-dough. All children of all ages whether they admit it or not enjoy playing with play-dough and what better than to incorporate it into learning math. Using playdough in a subject that stresses many students can be very beneficial and making visualizing math concepts making the problems easier to approach.

For example, given the high school geometry standard:

CCSS.MATH.CONTENT.HSG.GMD.A.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
We can work on solving problems like:                           play-dough-diagram
Given that the radius of a sphere of play-dough is 8 cm.
If the cylinder it needs to fit into is only 6 cm, what is the minimum height the cylinder must be in order to fit all the play-dough inside of it?

 

Allowing students first to try the problem hands on will work on engaging the students and with the visualization solving the problems will become less of a stressful situation. And from this, we can create other similar problems in which the shapes the problem works with changes. Such as what could change if we had a pyramid shape of play-dough that needed to fit into a cone shape container.