nicholekurtz.weebly.com€¦ · web viewfor example, page 11 describes squares, rectangles and...

Nichole Kurtz March 27, 2009 Unit Part 2

1. The Traditional Textbook:

Geometry Part 1 has many of the strengths and weaknesses that are common to

traditional textbooks. Chapter 6 is the primary chapter that concerns polygons and triangles.

The best feature in this chapter is the Quadrilateral Hierarchy Theorem and its related diagram.

On page 319, the theorem states that, “Every property true of all figures of one type in the

hierarchy is also true of all figures [that are connected beneath it].” The diagram shows the

interrelatedness of squares, rectangles, parallelograms, etc. According to Monaghan, our

textbooks and our need to have a “one-to-one object-word match” (p. 9) have distorted the way

students perceive certain polygons. However, this textbook is working to reverse this one-to-one

object-word match by showing how different polygons are related. The Geometry Part 1

textbook also gives accurate definitions that will help students “define geometric vocabulary”

including “isosceles, equilateral, acute, obtuse … perpendicular/parallel sides [and] congruent

angles/sides” (Ohio Content Standards). Even though the students are not discovering the

properties that classify polygons, they are still seeing multiple examples of each figure.

Some weaknesses include an entirely worksheet-based curriculum and the order of the

information presented. The worksheet-based curriculum gives students many opportunities to

practice what they have learned. However, worksheets are often not problem-based, and they do

not help students develop their problem solving skills (NCTM Standards). Secondly, the order

of the information is not the way I would teach it. The textbook presents reflection symmetries

in section 1 of chapter 6, but then there are four more sections before rotational symmetry is

discussed. In addition, regular polygons are not discussed until the very end of the chapter, but I

think regular polygons should be discussed before the symmetries because the students will be

aware that all of the sides are congruent when they are looking for symmetries.


References:

Monaghan, F. (2000, July). What difference does it make? Children's views of the differences between some quadrilaterals. Educational Studies in Mathematics, 42(2), 179-196. Retrieved February 13, 2009.

The University of Chicago School Mathematics Project (1998). Geometry part 1. Glenview, Illinois: Scott Foresman and Company.

1. The Connected Mathematics 2 Workbook:

The Connected Mathematics 2 workbook entitled Shapes and Designs also has many

strengths and weaknesses. My favorite lesson is the sorting shapes activity on page 10. Rather

than telling the students twenty different properties that classify shapes, the lesson allows the

students to place shapes into groups that make sense to them. In addition, I like the order of the

lessons in the workbook. It places regular polygons after shape classification and before the

symmetries. Finally, I like the real-world examples and problems that the workbook offers. The

skateboarder in the park problem on page 61 is an excellent example of a real-world situation

that involves requires problem-solving skills (NCTM Standards).

However, there are a few weaknesses as well. The workbook still uses a one-to-one

object-word match. For example, page 11 describes squares, rectangles and parallelograms as

singular rather than interconnected quadrilaterals (Monaghan, 2000). In addition, I do not think

that the workbook spends enough time discussing types of triangles and their properties. In fact,

one of the only times triangle properties are explicitly mentioned is on page 11. In general, both

books have strengths and weaknesses, but I would prefer to use the Connected Mathematics

workbook when designing my lesson plans.

Reference:

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2006). Shapes and designs: Two-dimensional geometry. New Jersey: Pearson Education, Inc.


2. The Unit’s Sequence:

Day 1: Triangles and their Angles

Individual Project and Class Discussion

Indicator 2: Use standard language to define geometric vocabulary: vertex, face, altitude, diagonal, isosceles, equilateral, acute, obtuse and other vocabulary as appropriate.

Goal: The students will show that the sum of the angles of any triangle must equal 180 degrees. The students will classify triangles as right, acute and obtuse.

Beginning (5-7 minutes):The students will watch a Discovery Channel video that discusses the real-world use of triangles in construction and other fields.

Middle (40 minutes):(25 minutes) The students will use Geometer’s Sketchpad to discover that the sum of every triangle’s angles must equal 180 degrees.(15 minutes) I will give a short PowerPoint about right, acute and obtuse angles. Then we will have a class discussion about these types of triangles. For example, is it possible to have two obtuse angles in one triangle? Why? Is it possible to have two right angles in one triangle? Etc.

End (2-3 minutes): The students will write in their journals about what they learned and what they still need help understanding.

Assessment: I will collect the Geometer’s Sketchpad printouts to see who still needs help.____________________________________________________________________________

Day 2: Side Lengths and Triangles

Individual Project and Class Discussion


Goal: The students will show that the sum of the two shorter sides must be greater than the longest side in order to create a triangle. The students will be able to classify triangles as isosceles, scalene or equilateral.

Beginning (3 minutes):The students will answer the questions on the board in their journals, which will be review from the previous day. For example, I would draw three triangles and have them classify the triangles as right, acute or obtuse. Then, I would have them draw an example of each in their journals.


Middle (40 minutes): (20 minutes) The students will use Geometer’s Sketchpad to discover that the sum of the two shorter sides must be greater than the longest side in order to create a triangle.(20 minutes): I will give a brief PowerPoint about scalene, equilateral and isosceles triangles. The students will then work in pairs to practice measuring and classifying five pre-cut triangles. They will record their measurements, triangle names and rationale for naming the triangles on a worksheet.

End (5-7 minutes):I would have five different groups volunteer to share their answers on the SMART BOARD where I would have a copy of the worksheet on the screen.

Assessment: I would collect the Geometer’s Sketchpad worksheets and the triangle naming worksheets to see if anyone needs help.____________________________________________________________________________

Day 3: Triangle Classification

Individual Project

Indicator 3: Use multiple classification criteria to classify triangles; e.g., right, scalene triangle.NCTM Indicator: Create and use representations to organize, record, and communicate mathematical ideas.

Goal: The students will be able to classify triangles by both their angle name and their side name (e.g., right scalene).

Beginning (3 minutes): The class will answer the questions on the board in their journals, which will be review from the previous day. For example, I would draw three triangles and have them classify the triangles as scalene, isosceles or equilateral. Then, I would have them draw an example of each in their journals.

Middle (40 minutes): I will create a table on the SMART BOARD like the following:

Equilateral Isosceles Scalene

Acute

Obtuse

Right


Then the students would each draw the table on a large piece of cardstock using rulers. I would provide a bag of pre-cut straws of varying lengths at each table. The students need to make a triangle for each box out of the pre-cut straws and tape it to their cardstock. If the triangle is impossible to make, they need to describe why it is impossible in the box where the triangle should be taped. Their projects are due by the end of the block.

End (5-7 minutes): We will have a class discussion about the triangles that were impossible to make and why.

Assessment: I would collect their projects. I would also assign some homework problems that gave scenarios with triangles. The students would then need to sketch the triangle in the scenario and name it by both names._____________________________________________________________________________

Day 4: Sorting Shapes and Polygons

Whole Group Discovery

Indicator 1: Classify and describe two-dimensional and three-dimensional geometric figures and objects by using their properties; e.g., interior angle measures, perpendicular/parallel sides, congruent angles/sides.

Goal: The students will sort the shapes into groups that make sense to them and discuss properties of those shapes.

Beginning (10 minutes):As the students walk in the door, I would hand out an anticipation guide. This guide has four different columns. The first column has different statements about polygons. The second column says “Beginning of Class,” and the students write agree or disagree in the column after reading each statement. The third column says “End of Class,” and the fourth column leaves room for “Comments.” They need to complete the “Beginning of Class” column. Then, we would go over any homework problems that the students did not understand last night.

Middle (35-38 minutes): (25-28 minutes) The students would work in pairs to complete the Sorting Shapes activity in the Connected Mathematics 2 workbook. Each student would answer questions A, B, C, and D on a worksheet that provides the questions and answers.(10 minutes) We would have a class discussion about the properties they found. I would write down the main findings on tag board and post them around the room.

End (3 minutes):The students would complete the “End of Class” column and add any comments about why their opinions changed or did not change in the “Comments” column. I would remind the students to study for a quiz on triangles tomorrow.


Assessment: I would collect the anticipation guides and the homework problems._____________________________________________________________________________

Day 5: Quadrilateral Hierarchy Theorem

Teacher-Led Discussion


Goal: The students will continue yesterday’s lesson by classifying shapes into more than one category.

Beginning (20 minutes):(5 minutes) The students will watch the Math’s Mansion video about polygons.(10-15 minutes) The students will take a short quiz about triangles.

Middle (25 minutes): Using the worksheets from yesterday, and the properties we have learned about quadrilaterals, the students and I will construct the Quadrilateral Hierarchy Diagram on page 319 of the Geometry Part One textbook.

End (3 minutes):The students will write in their journals about what they learned and what is still unclear.

Assessment: I would grade the quizzes, and I would collect their journals every Friday._____________________________________________________________________________

Day 6: Parallelograms with Geometer’s Sketchpad

Individual Project


Goal: The students will use Geometer’s Sketchpad to show how interior angle measures are related to quadrilaterals.

Beginning (5-10 minutes):I would return the quizzes and their journals. We would talk about any problems with which the class struggled.

Middle (30 minutes): Have the students use Geometer’s Sketchpad and the following worksheet I created:


Instructions:1. Open Geometer’s Sketchpad. Click on Start, then click Programs, then Math, then GSP.

2. Go to my website. Copy and paste the parallelogram into your Geometer’s Sketchpad screen.

3. Measuring the Angles: Using the arrow tool, select (highlight) points A, then B, then C (order is very important when measuring angles).

4. Now highlight Measure – Angle from the menu bar. A measurement should appear in the top left corner.

5. Repeat these steps for the other three angles. *Remember that the letter of the angle you are measuring should be in the middle. We already measured angle B by measuring ABC. This is the same angle as CBA because B is still in the middle.*

Measure ABC or CBA = _________________Measure DAB or BAD = _________________Measure DCB or BCD = _________________Measure ADC or CDA = _________________

What do you notice about the relationships of these angles?______________________________________________________________________________

______________________________________________________________________________

6. Using the pointer arrow, highlight all four angle measures.

7. Select Graph – Tabulate from the menu bar. A table with all four angle measures should appear in the top left corner.

8. Changing the Parallelogram: Using the arrow, click on a blank spot on the screen to make sure nothing is selected.

9. Highlight point A and drag it to a new spot. Drop it there.

What happens to the shape of your parallelogram?

______________________________________________________________________________

______________________________________________________________________________

What happens to the angles of your parallelogram?

______________________________________________________________________________

______________________________________________________________________________


What happens to the relationship of the angles in my parallelogram?

______________________________________________________________________________

______________________________________________________________________________

*Print your drawing to turn in with this worksheet.*

Extension:10. Select Graph – Grid Form – Square Grid. Predict what will happen if you rotate your

parallelogram by 90 degrees:

________________________________________________________________________

11. Highlight all four edges and all four points on your parallelogram. Select Transform — Rotate. Click Enter. What happened? Was your prediction correct?

________________________________________________________________________

________________________________________________________________________

End (10 minutes):The class will discuss what they learned about interior angles of parallelograms.

Assessment: I would collect and grade the Geometer’s Sketchpad printouts.

Research: Kerchner, J. Area of a parallelogram. http://faculty.kutztown.edu/schaeffe/GSP/Kerchner_GSP.pdf._____________________________________________________________________________

Day 7: Marshmallow Quadrilaterals

Individual Project

Indicator 1: Classify and describe two-dimensional and three-dimensional geometric figures and objects by using their properties; e.g., interior angle measures, perpendicular/parallel sides, congruent angles/sides.NCTM Indicator: Create and use representations to organize, record, and communicate mathematical ideas.

Goal: The students will classify the quadrilaterals by their sides and angles.

Beginning (5 minutes):We will review the Quadrilateral Hierarchy Diagram that we made as a class on Day 5. I will pass out the directions for the assignment, the card stock, the toothpicks and the marshmallows.

Middle (40 minutes): I will give the students two large pieces of cardstock taped together. First, the students should write the heading “Quadrilateral” at the top of the page. They will work individually


constructing their six quadrilaterals out of toothpicks and marshmallows. These will include a square, a rectangle, a rhombus, a parallelogram, a kite and a trapezoid. By the end of the day, all six shapes should be constructed and taped down in an order similar to the Quadrilateral Hierarchy Diagram.

End (3 minutes):We will discuss the rest of the project, which will be finished tomorrow. The rest of the project requires that three sentences be written next to each shape. One sentence will describe the angles, one sentence will describe the sides and one sentence will describe why the shape is connected to other shapes.

Assessment: I would walk around and ensure that the students are on the right track._____________________________________________________________________________

Day 8: Marshmallow Quadrilaterals continued

Individual Project continued


Goal: The students will classify the quadrilaterals by their sides and angles.

Beginning (3-5 minutes):We will review the requirements for the sentences, and I will give an example. For the square: A square has four congruent, 90-degree angles. A square also has four congruent sides. The square is also a rhombus because the rhombus has four congruent sides. The square is also a rectangle because the rectangle has four congruent, 90-degree angles.

Middle (40 minutes): Any students who have not finished making their shapes need to finish quickly. Then they will write their sentences for each shape NEATLY.

End (3 minutes):The students will write in their journals about what they still do not understand about polygons and what they enjoyed about the unit so far.

Assessment: I would collect the projects.____________________________________________________________________________

Day 9: Coordinate Geometry

Teach-Led and Discovery Learning


Indicator 4: Identify and define relationships between planes; i.e., parallel, perpendicular and intersecting.

Goal: The students will draw and identify which figures are parallel, perpendicular and intersecting.

Beginning (3-5 minutes):The students will make a KWL chart in groups of two or three to show what they know, what they want to learn and (later) what they have learned.

Middle (40 minutes):

The students will create three-sided figures on the electronic geoboard and compare them those of another student. Have the students discuss similarities and differences among the figures.

Virtual Geoboard Example

As a class, discuss the properties common to each figure.

Students should work in pairs for this lesson. Have students create a four-sided figure and compare it with the one their partner made. Have students discuss the similarities and differences among the figures.

Tell students to use two rubber bands on the Virtual Geoboard E-Example to make a set of parallel line segments. Ask the students:

What is the important property of parallel lines? How are parallel lines used in the environment and in art?

Next, the students will use two rubber bands on the Virtual Geoboard to make intersecting lines. Ask the students:

What happens to the two intersecting lines? How do intersecting lines compare with parallel lines?

Have students use two rubber bands on the Virtual Geoboard to make perpendicular lines.

What is special about perpendicular lines?

http://standards.nctm.org/document/eexamples/chap4/4.2/index.htm%00%E5%A1%B9%EF%92%81%E1%B4%BB%E4%A1%BF%E2%B2%AF%E5%B6%82%E8%97%84%E6%8C%A7%00%00%EA%AE%A5#applet%00


How are perpendicular lines and parallel lines different? Why is it important to know the properties of perpendicular and parallel lines?

Tell students to make a quadrilateral with one set of parallel lines, two acute angles, and two obtuse angles.

What is name of your figure? How do trapezoids differ from other quadrilaterals?

Describe the properties of a trapezoid.

Ask students to make a quadrilateral that has two sets of parallel lines.

What is the name of your shape? Why is it important to know the properties of these figures compared with the figures in

the previous lesson? Describe the properties of a parallelogram and explain how it differs from other

quadrilaterals.

Have students make a shape that has five sides.

What is the name of this figure? How many vertices does it have? How does it compare with other figures we have made? Why is it important to know the properties of these figures compared with the figures

studied previously?

Repeat with figures of six and eight sides.

What is the name of this figure? How many vertices does it have? How does it compare with other figures we have made? Why is it important to know the properties of these figures compared with other figures?

End (5-7 minutes):Call on several students to make the shapes on the SMART BOARD. Have the students discuss similarities and differences. Have the students fill out the last column in their KWL charts.

Assessment: Observe students using geoboards and answering questions.

Research: NCTM Illuminations. http://illuminations.nctm.org/LessonDetail.aspx?ID=L555_____________________________________________________________________________

Day 10: Problem Solving

Group Work


NCTM Indicators: Build new mathematical knowledge through problem solving. Recognize reasoning and proof as fundamental aspects of mathematics. Develop and evaluate mathematical arguments and proofs

Goal: The students will solve real-world problems and prove their answers. Beginning (3-5 minutes):Students will answer a few questions on the board in their journals to review from yesterday.

Middle (40 minutes): Students will work in pairs to solve three or four real-world math problems that use geometry and coordinate geometry. Examples of questions and scenarios can be found in the Geometry Section of the Internet 4 Classrooms website.

End (5 minutes):The students will write in their journals about what they learned and what they still need help learning. The homework assignment is to fill out the study guide for Tuesday’s test.

Assessment: I will collect the problems and grade them based on understanding.

Research: Internet 4 Classrooms. http://www.internet4classrooms.com/skills_6th_math.htm____________________________________________________________________________

Day 11: Review Day

Teacher-Led Review



Indicator 3: Use multiple classification criteria to classify triangles; e.g., right, scalene triangle.


Goal: The students will review the information they have learned over the last two weeks in order to prepare for the test. Beginning (3-5 minutes):I will walk around and see who completed the study guide as a homework check. I will pass out any remaining projects and/or papers.


Middle (40 minutes): I will lead a game of Jeopardy. The students will be on two teams. The winning team will get two bonus points on their test.

End (5 minutes):The students will have a few minutes to review or ask me any questions before tomorrow’s test.

Assessment: I will observe while we play the review game.____________________________________________________________________________

Day 12: Test Day

Individual Assessment



Indicator 3: Use multiple classification criteria to classify triangles; e.g., right, scalene triangle.


Goal: The students will demonstrate what they have learned over the past two and a half weeks.

Beginning (2 minutes): Any final questions?

Middle (45 minutes): The test

End (if any time left): Students can work on puzzles and other problem solving games.____________________________________________________________________________

3. Assessments:

I have described the assessments throughout the unit overview. The quiz on Day 5 will have approximately 8-10 questions about triangles. It will include a few multiple-choice questions and a few short answer questions. In the first short answer section, the students will draw and label different triangles. In the second short answer section, the students will solve two abstract problems using triangles.

The test will be constructed similarly. It will cover triangles, polygons and coordinate geometry. It will have approximately 25 questions. It will have approximately 10 multiple-choice and 9


short answer questions. The last six questions will be classifications. For each of the six quadrilaterals we studied, the students will have to classify them by their sides and angles. They may draw the Hierarchy Diagram if it is helpful.

4. Rationale:

I am beginning the unit with the basics about triangles including angles and side lengths. It is important to understand what makes a triangle before you can classify it as right, obtuse, etc. I am teaching triangles first because I think it is easier for students if triangles are taught before quadrilaterals. Students can work up from three-sided figures to four-sided figures to more complex polygons. In addition, at Lake, they taught triangles in the middle of polygons, and the students were confused. I am teaching coordinate geometry near the end because I think it is easier to learn about parallel and perpendicular lines of objects after they have learned what those objects are called. My problem solving day is the last day before review because I want the students to use everything they have learned about polygons to solve the problems. Finally, the last day before the test is review so the students can review in class and ask me any questions they have before the test.

With assessments, I am giving my quiz after we look at triangles and before we really start polygons because I want to measure what they have learned about triangles only. The marshmallow project is in place of a quiz on quadrilaterals. I want to see what the students have learned about quadrilaterals before we move on to coordinate geometry. The test is comprehensive, so it makes sense that it is on the last day.

5. Technology

I have mentioned technology throughout as well. I plan to use Geometer’s Sketchpad, a DVD player for the Discovery Channel movie, Youtube for the Math’s Mansion video, the SMART BOARD, electronic geoboards, the internet and computers. The students are welcome to use calculators on appropriate days if they would like.

nicholekurtz.weebly.com€¦ · web viewfor example, page 11 describes squares, rectangles and...

Documents