RENEE LYNETTE WILLEMSEN-GOODE

swarthmore-rutledge school
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Geometry: A Three Week Unit Plan

Introduction:

While I had been teaching math in the classroom prior to our geometry unit, this unit was the first unit that I introduced and carried through completely. Obviously, I wanted to adhere to the curriculum of the school, but I was interested in working within this curriculum to design a more hands-on experience. The unit which follows is thus closely aligned with the math textbook that serves as the math curriculum for fourth grade. Each lesson follows the instructional goals laid out in each lesson in the textbook and I use the homework assignments, quizzes and tests from the text, so that I could ensure that I was meeting the goals of the curriculum.

Three-Week Objectives:

• Students will be able to categorize polygons by number of sides, side length, angles, and if they have parallel or perpendicular sides.
• Students will be able to describe the differences and similarities between various polygonal forms, using the terminology introduced in the unit.
• Students will be able to define and find perimeter of any polygon; find the area and volume of rectangles and rectangular prisms; and be able to approximate the area and volume of other shapes.

Lesson 1: What is Geometry?

Objective:

Students will be able to write down a meaningful definition of geometry.

Procedure:

Introductory Activity:

Ask the students: “Who’s heard of ‘geometry’?" (Knowing that the students have seen geometry in other classes at the Swarthmore-Rutledge Elementary School, I assume that most of the students will feel comfortable raising their hands.) Next, ask students: "What do you think geometry is? What things are part of geometry?” Allow several students to volunteer answers, and write students’ initial ideas on the board/overhead. Then, pulling ideas from the students' brainstorms, write down a more formal definition of geometry on the board/overhead.

At this time, have all the students take out a piece of loose-leaf and fold it vertically (hot-dog fold). Have students label it with their name and the heading "Geometry Dictionary." Have them label the two columns "word" and "definition." The students will record the definition of geometry in this dictionary. Students will use this dictionary throughout the unit as an introduction to taking and using notes. Examples of Student Work on Dictionary Every word that students put in their dictionary will also be included on a word wall in the classroom. (These words will be introduced in bold-face throughout the lesson plans).

Why Is Geometry Important?

Ask students to think about why geometry is important in the "real world." Write down any ideas that surface in the discussion, making sure students touch on the topics of art, architecture and engineering.

Lesson 2: Exploring Polygons

Objectives:

• Students will be able to identify whether a given shape is a polygon using the properties of polygons.
• Students will be able to identify and name polygons that are triangles, quadrilaterals, pentagons, hexagons, and octagons.
• Students will be able to draw a triangle, quadrilateral, pentagon, hexagon, and octagon.

Materials:

• Overhead with two columns of shapes (polygons and shapes that are not polygons)
• Index cards or small pieces of white paper (4 per student).
• Glue.
• 5 sheets of poster-board or other large paper, labeled Triangles, Quadrilaterals, Pentagons, Hexagons and Octagons, respectively.
• Index cards or paper slips with irregular polygons drawn on them (1 per student).
• 3 worksheet activities - Polygons are Everywhere, Polygon Word Problems, Polygon Root Words

Procedure:

Introductory Activity:

Put the overhead up on a projector. Explain that the two columns of shapes are different from each other in at least one way. Ask students to break up into pairs and discuss what the differences and similarities between the two columns are.

After a couple of minutes, ask students to volunteer what differences and similarities they discovered. (Column B has shapes that have curves and are not closed; all the shapes in column A use only straight lines and are closed). Then ask students what was the same about the figures (They are all flat or two-dimensional). Introduce the terms plane figure and polygon and have students copy definitions for these terms in their math dictionary. Stress that the word polygon actually means “many-angles.” (The idea of word roots will be emphasized throughout the unit). Briefly introduce the concept of an angle as what is formed when two straight lines meet.

Ask students if they know the names for some types of polygons. As students suggest shapes, put their names up on the board and define them. By the end of the conversation, the following terms should be defined and copied into the math dictionary– triangle, quadrilateral, pentagon, hexagon, and octagon. Stress that each of these words can be broken up into two parts that define the word. (For example, quadrilateral means "4 sides." )

Further Exploration:

Pass out four index cards/slips of paper to each student. Have the students draw a polygon on each of these cards. Stress that students may only draw one of the specific types of polygons we have discussed earlier.

When the students are done, collect the cards from the students and redistribute them to the students, so that each student has cards that other students drew. In this mix, also add the pictures of more irregular polygons that the teacher has drawn. Each student should get five cards in all. Have students work with the students in their cluster of desks to label each card with the correct polygon name. Then students should place the shapes, by category, on the appropriate poster-board laid out in the classroom (these will be glued down by the teacher later in the day). Images of these posters

Individual Exploration:

With the remaining time, introduce each of the three worksheet activities and have students work on the activities at their own pace independently.

One activity asks students to search the room for polygons.

The second activity asks students to think of non-mathematical words that begin with the roots tri-, quad-, pent-, hex-, and oct-. Examples of Student Work

The third activity asks students to solve word problems using the properties of different polygons. Examples of Student Work

Evaluation:

Students' ability to draw and classify polygons during the exploration activity; work on the three worksheets; homework.

Homework:

Worksheet 8-2 which accompanies the textbook.

Lesson 3: Equilateral, Isosceles, and Scalene Triangles

Objectives:

• Students will be able to identify triangles as equilateral, isosceles or scalene, given either an image of triangle or the lengths of the three sides.
• Students will be able to write definitions of scalene, isosceles and equilateral.
• Students will be able to to use the properties of these three triangles to solve word problems.

Materials:

Procedure:

Warm-up/Review Activity:

Ask students what the least number of sides a polygon can have is. Tell students they may work with a partner or by themselves to figure this out. After a couple of minutes, ask students to give their answers. Tell students they must defend their answer, either with words, or by drawing a picture on the board.

Tell students they will be studying this polygon, the triangle, more today.

Exploratory Activity:

Provide each student with a ruler and a worksheet with pictures of three different triangles (one equilateral, one isosceles, one scalene). Ask students to measure the sides of each triangle, labeling the picture with the appropriate lengths. Have them write down any observations they have about the differences between the three triangles. Examples of student work

After students have had a few minutes to work on measuring, discuss their findings in the large group. Define scalene, isosceles and equilateral, pointing to which of the triangles fits each label. Have students write definitions for these in their “geometry dictionaries.”

Independent Work:

Spilt students into two even groups. Each group will work at one of two stations for 10 minutes. After approximately 10 minutes, they will switch stations.

Station 1: Students will construct an equilateral, an isosceles and a scalene triangle using precut pieces of straw. Students will then tape these to pieces of construction paper, labeling the type of triangle they made. Sample student work

Station 2: Students will work on two word problem worksheets: The What Triangle am I? worksheet gives students the lengths of the sides of triangles and asks students to classify these triangles. Examples of Student Work. The Triangle Fun worksheet asks students to solve word problems based on the properties of the three triangle types. Examples of Student Work

Evaluation:

Work on word problem worksheets, straw constructions and homework.

Homework:

Practice Sheet 8-3 which accompanies their textbook.

Lesson 4: Classifying triangles by angle

Objectives:

• Students will be able to identify and label acute, obtuse, right and straight angles
• Students will be able to identify and label acute, obtuse, and right triangles
• Students will be able to use the properties of the three triangles in word problems

Materials:

• Geo-boards(1 per student)
• Rubber bands
• Large clock

Procedure:

Warm-up activity:

Ask students the following question: “Yesterday, we sorted triangles by the number of equal sides they have. Can you think of another way to sort triangles?” Remind students that polygons have both angles and sides if they seem stuck.

Exploration:

Ask students to brainstorm what they know about angles. Write student suggestions on the board. Come to a mutually agreeable definition of angle and have students write this down in their “geometry dictionaries.”

Draw two different angles on the board. Ask students if they think that these are angles are the same, citing their reasons why or why not. Discuss that there are ways to measure the size of an angles. There are also ways to classify angles generally. Show the students a right angle by drawing one on the board. Ask students what this shape reminds them of (door corner, etc). Define this angle as a right angle. Tell students that there are 90 degrees in a right angle. (This is not a piece of information I expect students to necessarily fully understand or remember, but I hope that it will be helpful when they do learn how to measure angles.) Then show students an acute and an obtuse angle. For each angle, ask students if the angle is bigger or smaller than a right angle. Ask students to brainstorm mnemonic devices to remember acute and obtuse. Have students record all three angle types in their “geometry dictionaries.”

Finally, demonstrate a straight angle, and define this for the students. Have students record straight angle in their “geometry dictionaries.”Tell students that there are 180 degrees in a straight angle. Ask students if they have ever heard the phrase “do a one-eighty.” Have a student or two demonstrate this concept. Relate this concept to moving 180 degrees by taping an object to a figure clock’s hands, and demonstrate that moving the hands 180 degrees makes the figure change direction.

Using the large clock, give students sample angles to classify as a group.

Independent Work:

Have students use geo-boards to demonstrate that they can build each of the following: acute, obtuse, right and straight. Students may have trouble making an obtuse angle at first, by instead producing a right angle. Show students that they can check for a right angle by holding a corner of a piece of loose-leaf up to an angle. If they match up, the angle is right.

Further Exploration:

Remind students that the whole reason we learned about these angles was because we were going to sort triangles. Tell students we have three types of triangles: acute, obtuse and right. Define these for the students.

Ask the students two challenge problems: 1. Can a triangle have more than one obtuse angle? 2. Can a triangle have more than one right angle? Discuss as a group, having volunteers come to the board to try to draw these triangles.

Independent Work:

Have students make right, acute and obtuse triangles on geo-boards.

Evaluation:

Students' ability to make triangles and angles on the geo-board and tell their teacher what angles and triangles they have created.

Homework:

Practice 8-4 from the textbook (classifying triangles by angles).

Tetrominoes! worksheet (setup for tomorrow's lesson)

Lesson 5: Congruent and Similar Figures

Objectives:

• Students will be able to recognize a slide, flip and turn of a shape
• Students will be able to draw a slide, flip and turn of a shape
• Students will be able to identify similar and congruent shapes

Materials:

• Copies of the following four worksheets
• Pre-made tetrominoes made of colored tiles for the overhead projector (2 of each type)
• Precut similar cardboard polygons (5 similar polygons in a set)
• Sets of Tangrams
• Pattern Blocks
• Construction Paper
• Glue
• Scissors
• Completed Tetrominoes! worksheet.

Procedure:

Introductory Activity:

Ask students to tell which different tetrominoes they found. Have students show these using colored tiles on the overhead projector. Record each unique shape that the students supply. When two students supply two congruent shapes, ask students if these they believe these shapes are the same or not. After taking a few comments, demonstrate to the students (or have a student demonstrate) that the two shapes are the same by taking two pre-made tetrominoes and showing how to flip or rotate them to get them to look the same.

Exploration:

Define congruence. Explain the concepts of turn and flip to the students, using the different pre-made tetrominoes. Then introduce the notion of slide. Discuss how these three concepts are relevant to the computer game “Tetris,” and demonstrate how these concepts play out by playing a mock game of Tetris using the pre-made shapes on the overhead projector. Ask students what each of the five shapes looked like when flipped, turned or slid.

Finally, explain the concept of similarity. Ask students to distinguish why similarity and congruence are different.

Independent Work:

Introduce the activities on the following four worksheets.

Show Me the Slide, Flip and Turn, asks students to cut out four congruent polygons, and glue these to a sheet of paper, such that each piece shows a turn, slide or flip of the original piece. Demonstrate the an easy way for students to get 4 congruent shapes (by folding the paper first). Examples of Student Work

In Homage to a Polygon, students will look at a web-site with the work of Josef Albers. He created many paintings called "Homage to a Square, which simply consist of a series of similar squares. Students will build their own Homage to a Polygon using a polygon of their choosing. Students can use precut pieces of oak tag to trace their shapes so they will truly be similar. Examples of Student Work

Similarity and Congruence in Tangrams asks students to find congruent and similar shapes with a set of tangrams.

Similarity and Congruence in Blocks asks students to build similar and congruent shapes with blocks. Examples of Student Work

Have students rotate at their own speed through the four activities sheets.

Evaluation:

Student work on independent activities.

Homework:

Practice Sheets 8-5 and 8-6 from the textbook.

Quiz Time – 8-A quiz from the textbook

Lesson 6: Perpendicular, Intersecting and Parallel Lines

Objectives:

• Students will be able to identify lines that are perpendicular, parallel and intersecting.
• Students will be able to describe the difference between line and line segment.

Materials:

• Geo-boards and rubber bands

Procedure:

Introductory Activity:

Show students a makeshift, hand-drawn map of New York City on the overhead, showing where Miss Willemsen-Goode lives (This is my hometown, and I thought it might make an interesting hook for the students). Draw a dot representing your house. Ask students what corner you live on. Explain that this is also the intersection that you live on, or that the two streets intersect. Using the map, introduce the concept of lines and verteces.

Exploration:

Define parallel, perpendicular and intersecting lines, continuing to use the map as a framework. Ask students to find examples of the lines in the classroom and share their thoughts about them.

Individual Exploration:

Have students work on "Problem Solving 8-7," a worksheet which accompanies the text and asks students to use a map and find parallel, perpendicular and intersecting streets. When students are feeling stuck or are finished, go over the answers as a group. Have early finishers work from their textbooks on page 347, numbers 5-11. Have students check their work with the teacher when done.

Review – Day 2

(Many students had to leave in the middle of the previous lesson for music lessons and several students were absent, so I did a 30 minute review of this material the next day).

With all students in a large group, have students define parallel, perpendicular and intersecting, based on the work done the previous day. Rewrite definitions for these terms. Have groups of students volunteer to act out these terms, by standing in two lines that are parallel, that intersect or that are perpendicular.

Then, have students make parallel, intersecting and perpendicular lines on a geo-board. Students could finish up the page from the text if they did not complete it yesterday.

Evaluation:

Work on the worksheet and ability to make the shapes on the geo-board

Homework:

Worksheet "Practice 8-7" from the textbook

Objectives:

• Students will be able to identify parallelograms, squares, rectangles, rhombii and trapezoids
• Students will be able to identify multiple names for shapes when relevant (squares are rectangles, etc.)
• Students will be able to identify the properties of these five shapes

Materials:

• 5 geo-boards displaying 3 examples of a given quadrilateral (either rhombus, square, rectangle, trapezoid, or parallelogram)
• Rubber bands
• Tangram sets
• Computers with Internet access
• Copies of a worksheet about tangrams and about multiple meanings.

Procedure:

Quick review:

Ask students that the SIX types of triangles are. Link this to the idea to the fact that there are also many types of quadrilaterals. Remind students that the a triangle can have two names (i.e., it could be both acute and equilateral). Quadrilaterals can also have multiple names.

Exploration:

Have students in four groups (based in their table location) and assign each group one of the following quadrilaterals: parallelogram, square, rectangle, trapezoid. Hand out the geo-board with the appropriate shape to each group. Ask students to classify, as a group, what is special about their quadrilateral. Tell students to pay attention to side length, parallel sides and anything special about the angles. Have students generate a list of characteristics about their shape to present to the class.

As the groups present what they discovered, write up the characteristics of each quadrilateral on the board, having students copy these into their math dictionaries. Then go over the relationships between the various shapes (i.e., a square is a special kind of parallelogram, and a special type of rhombus). Draw a Venn Diagram showing the “world of quadrilaterals” to show the students how they are all related.

Finally, introduce the rhombus to the whole class, by projecting the geo-board examples using an overhead projector. Have students generate where this shape fits in on the Venn Diagram, then redraw this. Make sure to mention that there are quadrilaterals that are not special types. Have students draw some examples and tell where these would fit in. (During the course of discussion, students gave these shapes the name "wild quadrilateral" which would be a great way to introduce this type of quadrilateral, especially given the construct of "World of Quadrilateral," which I had greatly personified.)

Independent Work:

Have students work on the following three activities independently.

Worksheet in which students must fill in the blank with the name of a quadrilateral to make sentences true (i.e. All rectangles are ________.) Examples of student work

Worksheet which asks students to build various quadrilaterals from tangram sets. Examples of student work

Students may also use an Internet-based tangram simulation and attempt the puzzles in which the build a square and parallelogram out of all seven pieces.

Link to a response to using this Internet-based simulation.

Review:

Bringing the students back together, do a quick spot check of whether or not they can classify quadrilaterals, by giving them various quadrilaterals to classify.

Evaluation:

Student answers to spot check questions at the end of class; homework.

Homework:

Practice 8-8 from textbook

Lesson 8: Line symmetry

Goals:

• Students will be able to determine whether a given image or object has line symmetry.
• Students will be able to draw in lines of symmetry on images that do have line symmetry.

Materials:

• precut construction paper quadrilaterals (square, rectangle, rhombus, trapezoid and parallelogram)
• pattern blocks
• Symmetry in Alphabet worksheet
• Two or Three computers set up for students to explore this website on Symmetry

Procedure:

Quick Review/Setup:

Have students draw a square, rectangle, parallelogram, trapezoid and rhombus in their notebooks. Have five volunteers draw a picture of one of these shapes each on the board. Review what features make each shape unique.

Exploration:

Then, using precut construction paper quadrilaterals, ask the students: “If I cut out these shapes on a piece of construction paper, which could I fold in half with no pieces jutting out? Which fold evenly?” Have students predict which will fold evenly, then test their predictions by actually folding the paper. Ask students if there is more than one way to fold these shapes evenly. Finally, introduce the idea of line symmetry, relating it to the folds of the shapes.

Have students search for symmetry in the room, trying to address any misconceptions students may have about symmetry.

Individual Activities:

Rotate students between these three activities (about 7 minutes at an activity)

Activity 1: Students must find a partner and stand on either side of a table with him/her. Each child will take a turn placing pattern blocks on a table. The first child will place a block. Then the next child will place a block down so that it creates a symmetrical image. Now they should switch roles, so that the first child puts down a block, and the first child find the symmetrical match. The students will repeat this process until they are satisfied with their design. Photographs of student work

Activity 2: Alphabet worksheet: students will look for symmetry in the letters of the alphabet Examples of Student Work

Activity 3: Have students get in small groups at the computers and explore images from a website which has complied many different ways to look at symmetry in the world around us. Have each student write 5 things that they learned from the web-site. Example of student work

Evaluation:

Work on the worksheets, progress on block symmetries, and homework.

Homework:

Practice 8-9 from textbook

QUIZ TIME: 8B from their Textbooks

Lesson 9: Perimeter

Goals:

• Students will be able to write a definition of perimeter.
• Students will be able to determine the perimeter of of any polygon, provided they given the dimensions of each side, or enough information to infer this.

Materials:

• Textbooks
• Yardstick or other large ruler

Procedure:

Introductory Activity:

Introduce with a word problem involving finding the perimeter of an object. (I have a garden that is 8 ft by 10 ft. If I wanted to put a fence around my garden, how many feet of fence would I need?) Ask students how they would find the answer to this problem. Using these initial musings, define perimeter with the students and add it to the "geometry dictionary."

Exploration:

Using the overhead projector or blackboard, draw a few examples of polygons and have students discuss how they could find the perimeter. Then model finding perimeter by having students pick an object in the classroom and working as a class to find its perimeter.

Individual Work:

Have students work on problems from their text book (all problems on page 369). If students finish early, have them work on Enrichment and Problem Solving worksheets from their textbook.

Evaluation:

Student work on problems from textbook.

Homework:

Practice 8-11 from textbook accompaniment.

Lesson 10: Area of rectangles

Objectives:

• Students will be able to find the area of a rectangle, given the length of two of its sides.
• Students will be able to generalize a formula to find the area of a rectangle.
• Students will be able to approximate the area of any shape, given a sheet of graph paper.

Materials:

• Graph paper

Procedure:

Introduce with a word problem about area. (Same problem as for perimeter only asking for how many square feet are in the garden, instead of its perimeter.) Then, explore a definition of area with the students.

Trace any item from the classroom on a piece of overhead graph paper. Ask students how they could find the area, or how many squares are in, this shape. When students suggest counting the squares, model counting the squares with the students. Then have students find their own object in their desk and have them trace it and find the area of one of its faces. As students are finished, record the object the student picked and its area on the board.

Finally, ask students if there is an easier way to figure out the area of a rectangle than adding up the squares. Link area to arrays, which they have sued to solve problems before. Then generalize a formula for finding the area of a rectangle with the students.

Finally, have students work on a page from the textbook pertaining to area, and check student work.

Evaluation:

Work on textbook problems, during individual exploration.

Homework:

Practice sheet 8-12 from the textbook.

Lesson 11: Volume

Objectives:

• Students will be able to define volume
• Students will be able to find the volume of a rectangular prism

Materials:

Procedure:

Start the class off with a word problem similar to those used the last two days. This time, students need to figure out what amount of water would be in a pool with some dimensions. Using a shoe box to help, ask students what shape they might use to measure the amount of space or volume of the pool. Remind students that they used squares for areas. If they need prompting, ask what a three dimensional square is.

Go through, as a class, a series of problems in the textbook that ask students to count the cubes in a 3 by 4 array of cubes, a stack that has two of these arrays on top of one another and one that had three of the arrays on top of one another. See if students can generalize a formula for volume of a prism using this information and their formula for area.

With the students, find a way to find out how many 1' by 1' by 1' balloons would fit in the room. Translate this into a volume problem and measure the room to solve, The students will probably be excited about the number of balloons that can fit.

Finally have students work independently to solve problems from the textbook independently.

Evaluation:

Work during independent time.

Homework:

Practice sheet 8-13 from the textbook

One Last Quiz: 8-C from the Text Book

Test Time: Unit Test From the Text Book

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