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
Lesson 7: Quadrilaterals
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:
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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