The Art of Geometry SRT

 

 

CCSS.Math.Content.HSG.SRT.B.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

This picture will be covered in my geometry class. The purpose of the picture is to start talking about how mathematics is a form of art. You can find mathematics in every type of art. The lesson that will be covered to relates to this picture is finding corresponding and supplementary angles. The assignment will be for the students to create their own drawings and label the types of triangles and angles in their picture.

Finding height using trig ratios SRT.C.8

Title (primary CCSS Math with Title)

CCSS.MATH.CONTENT.HSG-SRT.C.8.

  • Similarity, Right Triangles, & Trigonometry

Alignment to Content Standards
Define trigonometric ratios and solve problems involving right triangles
Use trigonometric ratios and Pythagorean Theorem to solve right triangles in applied problems.

Task
Word Problem:
Isabel just planted a new tree and attaches a guy wire to help support the tree while its roots take hold. A 6-foot wire is attached to the tree and to a pole in the ground. From the pole in the ground the angle of elevation of the connection with the tree is 52º. Find to the nearest tenth of a foot, the height of the connection point on the tree. Use the diagram below to help model the problem. Label diagram if desired.

Figure 1
Commentary
The purpose of this task is so students can apply their combination of skills of similar triangles, ratios, right triangle trigonometry and their knowledge on the Pythagorean theorem to solve problems in mathematics related to real life scenarios.

Solution

The “angle of elevation” is from the ground up.
It is supposed that the tree is vertical which makes it perpendicular with the ground.
This problem is a sine problem since it involves opposite and hypotenuse.
sin52°=h/6; 6sin⁡52°=h; h≈4.7 feet

Finding Properties of Scale Factor G-SRT.1

Assessment Task:

Solve the following problems based on the pictures given. You may use a calculator. Show work and answers on separate piece of paper. Use units when appropriate.

  1. What scale factor makes the sides of B equal the sides of A?
  2. What scale factor makes the area of B equal the area of A?
  3. The squares are now cubes. What scale factor makes the volume of B equal the volume of A?
  4. What is the relationship do you see between the scale factors of side length, area, and volume? Describe that relationship as a ratio and in words. Do you think this will always work? Why?

Finding Properties of Scale Factor Task, Commentary, and Solutions

CCSS.Math.HS.G-SRT.1 “Verify experimentally the properties of dilation given by a center and a scale factor”

Pizza Problem GMD.A

Image result for pizza
Problem:
If I ordered a pizza with an area of 200 inches and a height of 3 inches then what is the volume of this pizza?
CCSS-Math:
HSG-GMD.A.3 Geometric Measurement & Dimension
Explain volume formulas and use them to solve problems.

I choose this activity because who does not love pizza. We are in football season again and everyone is always enjoying a slice of pizza. This gives the students a chance to work out a math problem that involves something they actually might like and get them interested in how to find volumes of cylinders or any other figure. The teacher could  present this problem in many different ways. The teacher could actually order a pizza for the class and have them get into groups of 4 and try and solve it. Once each group gets the solution to the problem then they could share their answers to the whole class and then enjoy a delicious slice of pizza.

Pumpkin Pie Problem GMD.A

The Problem:

If I had a whole pumpkin pie with a diameter of 8 inches, and a height of 2 inches, what is the maximum amount of pie I could eat? Find the volume of the pie.

CCSS-Math:

HSG-GMD.A.3 Geometric Measurement and Dimension: explain volume formulas and use them to solve problems.

The relevance of this activity is that it can be used around Thanksgiving time, so the students have a real life example of how and when they could solve for the volume of a shallow cylinder. The teacher could have a couple different approaches for student engagement. He/she could have his/her students discuss in their groups what they think the solution is, the students could have this as an individual entry task, or there could simply be a class discussion of the process to solve this problem. All of those options could engage students in their current understanding of finding the volume of a cylinder, as well as allow the teacher to lead the lesson based off of the students’ findings. If the teacher wanted to take that extra step, he/she could potentially bring in an actual pumpkin pie for the students to reference (and possibly eat).

Bouncing Into Math CCSS.MATH.5.G.A.2

Some math teachers might ask the question “how can I adapt this curriculum to relate to real world scenarios?” One way to start is by taking an interest in what your students are interested in. There’s a time for lecture and there’s a time for, well, bouncing balls and having fun. This activity features measurement of a tennis ball bouncing and when it slows down, compared to a rubber ball bouncing and when it slows down, using the Ipad app called Video Physics. You might be asking yourself now “how in the world does this relate to the real world?” Well, once this activity is performed by questioning young minds, the students will want to know more. They’ll want to use the technology and math and relate it to people and if they were to keep jumping, at what point would those bounces become smaller and slower. This is an activity to get the students up out of their seats and “perform” mathematics in the classroom.

In this activity the students will need to get into groups of at least three and have an Ipad, a worksheet, a tennis ball, and a rubber ball. Each student will have his/her own job: recorder, video-taper, and ball bouncer. Each student will get to perform each task within the groups of three. The students will collect and record data pertaining to the tennis ball and the rubber ball and how each ball’s acceleration and height of the bounces differ slightly. Prior to the students gathering data, they will draw their prediction of what each ball’s graph will look like. This will give the students more insight to how the math aligns with reality. After they complete the worksheet given during this lesson, the class can discuss their findings as well as how else they could use this technology in their day to day lives.

This assignment relates to the Common Core State Standard

CCSS.MATH.5.G.A.2 Graph points on the coordinate plane to solve real-world and mathematical problems.

Bouncing Into Math Worksheet-1oj7bir

Geogebra’s a Ka-HOOT

In this lesson, Geogebra and Kahoot will be used as our technology to better support the students in the class and hopefully to help them learn and remember the information well. Students can use the technology of Geogebra as they can move different points around and show that they understand how changing parts of a triangle changes the circumscribed circle without making multiple drawings. Kahoot is a good technology because it allows both the students and teacher to have instant feedback on how well the information was understood. It also allows the students to be having fun and working both with each other in groups and against other groups to win the most points.

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.

Geometry Using Techonology

Special Quadrilaterals HSG.CO.A.3

This learning progression is to be taught to a 10th grade geometry class. This learning segment is about special quadrilaterals; specifically the rectangle, rhombus, trapezoid, and kite and the properties that each figure possesses. The mathematical practices that will be used are MP1, MP2, MP3, and MP5. The standards that will be taught are

HSG.CO.A.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

HSG.CO.C.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Full learning progression: edtpa learning progression

Polygons and Parallelograms G.CO

High School Geometry: Polygons and Parallelograms

This learning progression will be taught at Ellensburg High School in a Geometry course. The students in the class this learning progression is being taught in are all good students. By that I mean each student is a diligent and hard worker. The course text is Geometry: Reasoning, Measuring, Applying by McDougal Littell. The learning progression covers sections 6.1-6.3, after which will be a summative assessment. Section 6.1 covers the basics of Polygons; what they are, definitions related to them, procedures for determining a figure is a polygon, and the different types of polygons. Section 6.2 covers the properties of parallelograms; related theorems and proofs. Section 6.3 covers the theorems and proof of parallelograms being quadrilaterals. I have planned to spend a single lesson on each section with another lesson afterward designated to a quiz. Below are the prior CCSS which have been covered that are related this learning progression:

Experiment with transformations in the plane

CCSS.MATH.CONTENT.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.

Prove geometric theorems

CCSS.MATH.CONTENT.HSG.CO.C.9

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Make geometric constructions

CCSS.MATH.CONTENT.HSG.CO.D.12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

The students have mastered these concepts and CCSS through activities and tasks assigned in earlier learning progressions. Thus, the students are prepared and have the appropriate academic knowledge to begin learning CCSS.MATH.CONTENT.HSG.CO.C.11 which covers proving theorems about parallelograms. As I explain the tasks within the learning progression I will explain how the students’ prior academic knowledge was necessary; as well as, how the new concepts build off of the students’ prior knowledge will help them gain a deeper understanding of the new concepts. However, before beginning the explanation of the tasks and assessments I would first like to discuss my instructional strategies.

When teaching the learning progression I will use direct instruction and the use multi-media to communicate the learning targets, the concepts, and the directions for the tasks. The initial discussion will not provide examples. I want the students to struggle and make their own inquiries about how to solve the posed problems. I will then make use of grouping to have students work together on the tasks and make inquiries as a group or as an entire class. I will also allow students who have figured something out to travel to another group and share what they know after they first share with their own group (not a requirement though). This will work effectively because I know that my students are diligent workers and that they have proven they work well in groups. I also know that it will promote them to work together and share procedures and reasoning if I give them the opportunity to do so. Lastly, this learning progression will only include formative assessment except for the last quiz since all work will be done in groups and I want to assess discussion and inquiry not mathematical correctness. For the most part answers will be gone over as a whole class so students that I ca make sure all students have the correct solutions, procedures and reasoning to be studying.

Lesson 1: Polygons

The first lesson does not touch a CCSS but teaches the students the necessary prior-academic knowledge needed to learn HSG.CO.C.11. As stated above my lessons do not begin with lectures or examples of how to solve the related problems within the lesson’s tasks. I will begin this first lesson with some light-hearted dialogue with the students to ease them into a learning-mindset. Afterwards, I will hand out a worksheet which I created. I devised the worksheet so that once finished it can be used as a resource tool for studying and review. The worksheet has this prompt, “[d]irections: For the figures 1-13 state which figures are polygons and why. If it is a polygon name it based upon its number of sides and state whether it is concave or convex. Lastly, state whether or not the polygon is regular or not.” Attached to the worksheet is a sheet with 13 figures; some are polygons and some are not, some are concave polygons and some are convex polygons, and some are regular polygons.

How this activity will work is once I have handed it out students will be directed to tackle the first part of the prompt in their groups without help from me. Students have been working with polygons for quite a while but without me explicitly telling them that have been. Thus, it is my hope that through inquiry some of the students will be able to determine which figures are polygons and which are not. Then those students can share their reasoning with the class. If after about 2 minutes not students have made any progress I will pick a figure and state whether or not it is a polygon. Students will then have an example to go off of. By 4 minutes, whether students are finished or not, I will ask the class to quiet down and for any volunteers to share which they chose to be polygons or not to be polygons. Once all of the students had shared I will go over the definition of a polygon and then go through which are and are not on the worksheet. Just like how we as a class went through this part of the prompt so too will we with the other parts. By the end students will have helped each other create a resource tool with visual representations connected to descriptive details. Students will have examples to study and use as reference on homework problems. Which leads me to the last part of this lesson; a homework assignment which has students answering 5-10 questions picked from the section 6.1 of the text.

Lesson 2: Properties of Parallel Lines

As stated above students have gone over parallel lines and have also constructed parallel lines. Thus, I have decided to begin this lesson by having student construct a parallelogram without actually telling them that. I will direct them to construct two parallel lines on a sheet of paper and extend the lines across the whole sheet. Then I will direct students to create two more parallel lines that are not parallel to the other lines and also extend the lines across the whole sheet. Lastly, I will ask the students if any of them know what they had just created. Hopefully a student might know and share but if not I will explain what they created and how it relates to what we are in the lesson.

The next step in the lesson would be to direct students to discover any properties about the parallelogram they just created; for instance any congruent angles or sides, any supplementary angles, or the fact that their diagonals bisect each other. I will designate 10 minutes to this part of the lesson and while students are discovery the properties I will be walking from group to group posing specific questions to promote the right inquiries. Once those 10 minutes are over I will present the 4 related theorems and ask the students how many of the 4 theorems they discovered and make a tally up on the board (a little competition never hurt). I will then go over two examples of how to use these properties to solve proofs relating parallelograms. The end of the lesson would be designated to their homework assignment which would incorporate basic problems from section 6.2 and 2 hand-made, two-column proofs. Students would only have in-class time for their homework if as a class we went through the example proofs in the time allotted for it.

Lesson 3: Proving Quadrilaterals are Parallelograms

This lesson will be set up exactly the same as the second but with a different introduction. Since the students have already constructed a parallelogram there is no sense in repeating that. Instead, I will open the lesson with this question, “[w]hat are the properties of a quadrilateral?” I will give students some time to discuss this. After a while, 2-3 minutes or so, I will ask students to volunteer what their group concluded the properties of a quadrilateral are. After all of the students have shared what they wanted to I would then go over the properties of a quadrilateral and reference to the properties of a parallelogram we went over the day before. Students should be starting to make the connections in their head that these properties are the same or share similarities. To finish the introductory I would ask the students if a parallelogram is a quadrilateral. Hopefully I would get a volunteer to share their reasoning thus creating a discussion. If not I would lead a discussion as to why it is a quadrilateral. Then as I said before the rest of the lesson would follow the same structure as the one before.

Quiz

            All of the assignments, tasks, and activities in the lesson were solely graded upon participation and completion. This is because I always gave the students the answers to the in-class activities at the end which meant I graded only the students’ participation in discussions and group work. The homework assignments were only graded on completion and not correctness because I wanted to promote students to at least provide work for a problem even if they thought they could not do it. A student who does not think they can solve a problem won’t even start it but if completion is the only grade then they will at the least attempt it. Therefore, I have decided to give a quiz half way through chapter 6 so that the students and I have an accurate measure of the conceptual and procedural understanding and reasoning skill that should have been learned in section 6.1-6.3. This assessment will help me determine whether or not the students are ready to move on and it will help the students self-reflect on what they have learned and plan out what they still need to.