Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 73277
What's my degree? Coming full circle
Students will use geoboards to find the measure of the angles within a circle. Students will be able to divide the circle into halves, quarters, and
eighths and through exploration and discussion find the measure of the parts, in degrees.
Subject(s): Mathematics
Grade Level(s): 4
Intended Audience: Educators
Suggested Technology: Computer for Presenter,
Interactive Whiteboard
Instructional Time: 1 Hour(s)
Resource supports reading in content area: Yes
Freely Available: Yes
Keywords: acute angles, straight angles, right angles, halves, eighths, quarters, eights, rotation, degrees
Resource Collection: FCR-STEMLearn Mathematics General
LESSON CONTENT
Lesson Plan Template: Guided or Open Inquiry
Learning Objectives: What will students know and be able to do as a result of this lesson?
Students will recognize that when an angle is decomposed into non-overlapping parts, the whole of the sum is equal to the whole of the parts of the angle, through
the use of exploration. It is recommended that you use one geoboard for every group of three students. If you don't have access to geoboards you may want to use
the template of the circle here.
Students will use manipulatives to make sense of problems and persevere in solving them.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should be familiar with straight, right, and acute angles. The teacher will activate student's prior knowledge through the website
Math is Fun
After reviewing the angles, ask the students:
What is an angle, and how is it formed? Students should respond that an angle is a figure formed by two rays that start at a common vertex.
What is an acute angle? Students should respond that it is an angle whose measure is greater than 0° but less than 90°.
What is a right angle? Students should respond that it is an angle whose measure is exactly 90°.
What is an obtuse angle? Students should respond that it is an angle whose measure is greater than 90° and less than 180°.
What is a straight angle? Students should respond that it is an angle whose measure is exactly 180°.
Guiding Questions: What are the guiding questions for this lesson?
If the sum of a whole circle is 360 °, how can we find the measure of the angles of its parts?
If a circle is divided into two equal parts, what is the measure of each part? How can you prove it? Students should realize that they have to find one of the parts by
using any number of strategies, and that each part equals 180°.
If a circle is divided into four equal parts, what is the measure of each quarter? How can you prove it? Students should realize that each part equals 90°, using any
number of strategies.
page 1 of 4 If a circle is divided into eight equal parts, what is the measure of each eighth? How can you prove it? Students should realize that each part equals 45°, using any
number of strategies.
Some possible student strategies are included here.
Introduction: How will the teacher inform students of the intent of the lesson? How will students understand or develop an
investigable question?
The teacher will begin by activating students' prior knowledge by asking the following questions:
What is a circle? Write down all responses on chart paper. Possible students responses are: a curved figure, two dimensional, has a center point
What do you know about a circle, or what are its properties? For example, is a circle a polygon? Give students a chance to discuss. Write down all responses on the
same chart paper. Students might respond that a circle is not a polygon because it is not made up of straight sides, it is curved. It has a diameter and a radius. You
can find its area and circumference.
What do you think the expression "You have come full circle means?" Give students a chance to respond. They may say "You have come all the way around to where
you started."
Have students stand up and push their chairs in, if possible. Then ask them to turn half a turn to the right, or clockwise.Do this along with the students. To redirect
those students who may have turned less than half a turn say, "You should now be facing the back wall". Ask the students, How many degrees did you turn? The
students should say 180°. Then tell the students to turn half a turn again to the right. Ask the students, How may degrees did you turn that time? Students should state
that they turned 180°. Then ask the students, How many degrees did you turn altogether? How can you prove it? Have students share their strategies. Have them
come to the board and show the mathematical proof for their calculations.
Then ask the students, How many degrees are there in a circle? At this point you are going to probe students until they reach the conclusion that a circle
measures 360°.
Investigate: What will the teacher do to give students an opportunity to develop, try, revise, and implement their own methods to
gather data?
The teacher will prompt the students to use the geoboard to model a circle using a rubber band (see "further recommendations" for suggestions regarding student
use of rubber bands). Have each group show you their circle as they complete them. The teacher will then ask students to take a rubber band and divide the circle
into two equal parts. Give groups time to figure out how to do that. Have each group show you their model.
Ask students, "How would you represent each part of the circle as a fraction?" Walk around to hear the groups' discussions. Strategically select the groups you are
going to have share their strategies. You want to hear students verbalize that the circle is broken up into two parts of a whole, which is written as 1/2 as a fraction.
After two or three minutes have the selected groups share.
Then ask " If the whole circle is 360 °, what is the measure of half the circle?". Give the groups time to discuss. At this time encourage the groups to use their boards
to come up with a solution. Circulate the room to hear the discussions, and select the groups you will have share.
You want to select those groups that realized that they had to divide 360 by 2. Select groups that used different strategies to solve the problem, so that the students
see the different ways they could have solved the same problem.
Have the selected groups share their strategies, and discuss the reasonableness of their answers.
You may prompt the students by saying: "Show me what you mean so I can see it", or " Can you write an equation to represent this situation?".
Students should understand that each half equals 180°, and that they make straight angles.
The teacher will then have the students take their model and divide it into two equal parts again, so that the circle has four equal parts. Have groups show you their
model. Select a student to explain how to describe each part in fraction form.
Now pose the question, "How can you find the measure of each fourth?" Have students discuss in their groups and encourage them to use their boards to show their
strategy. Circulate the room to hear the discussions, and select the groups you will have share.
You want to select those groups that realized that they had to divide 360 by 4, or 180 by 2. Select groups that used different strategies to solve the problem, so that
the students see the different ways they could have solved the same problem.
Have the selected groups share their strategies, discuss the reasonableness of their answers.
You may prompt students by saying: " Can anybody explain it in a different way?",or "How are the solutions alike?"
Students should understand that each quarter equals 90°, and form right angles.
The teacher will then have the students divide each quarter into equal parts, to have eight equal parts. The teacher will have each group show their model and
discuss how to write each part as a fraction of the whole, or as eighths. The teacher will then say" Based on our earlier discoveries how can we find the measure of
each eighth?".Have students discuss in their groups, and encourage them to use their boards to show their strategy. Circulate the room to hear the discussions, and
select the groups you will have share.
You want to select those groups that realized that they had to divide 360 by 8, or 90 by 2. Select groups that used different strategies to solve the problem, so that the
students see the different ways they could have solved the same problem.
Have the selected groups share their strategies, discuss the reasonableness of their answers.
You may prompt students by saying: " Do you see a pattern as you worked?", or " How can you test your solution?"
Students should understand that each eighth measures 45 ° and forms an acute angle.
Analyze: How will the teacher help students determine a way to represent, analyze, and interpret the data they collect?
The teacher will then have students take out their math journals and draw and explain how many straight angles are in a circle, how many right angles, and how
many 45° angles. Students must show their mathematical proof using any strategy of their choosing. if students don't have a math journal, students may use line
paper to write their responses. Possible student strategies are included here.
page 2 of 4 Closure: What will the teacher do to bring the lesson to a close? How will the students make sense of the investigation?
The teacher will have three to five students share their journal entries. Then the teacher will ask the whole class the following questions to check for understanding:
How many degrees are there in a whole circle? 360°
How many straight angles are there inside a whole circle, and what is their measure? Two straight angles and they measure 180° each.
How many right angles are there inside a whole circle, and what is their measure? Four right angles and they measure 90° each.
How many eighths are there inside a circle, and what is the measure of one eighth? What type of angle is that? There are 8/8 and each one measures 45 °, which is
an acute angle.
Summative Assessment
Students will use their math journals to explain and demonstrate how a circle is made up of the sum of its' parts. If the students don't have a math journal they may
use lined paper. They will then use their explanations to solve word problems that require the use of the knowledge they gained. The teacher will use the
Assessment.pdf here to assess students' understanding of the concept. Teachers may give students the worksheet to complete, or they may want to project it unto
their interactive board and have students answer the questions using lined paper.
Formative Assessment
The teacher will use guided questioning techniques to assess students' prior knowledge during the lesson, and will probe students through questioning to guide them
through the discovery phase of the lesson. The teacher will use the website Math is Fun to activate the students' prior knowledge.
Feedback to Students
Students will receive feedback through conversations with their peers as well as the class' discussion. The students should be able to either demonstrate, or draw and
verbalize the strategy used to solve each problem. The teacher will guide students responses by asking questions such as: What do we need to find out? What do you
already know that might help you? Why does the answer make sense? Is there another way to reach the same conclusion, can you show me? How do you know your
answer is correct? Is there a more efficient way to solve this?
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Teachers should make sure to strategically place students who may have difficulties with the concept in groups with students that will be patient and make them part
of the discussion.Teachers should also work with the students who had difficulties, in a small group, while the other students work in their math journal. At that time,
the teacher will again use the manipulative to guide the students through their journal entry. The teacher will use fraction circles to show the students the fraction
pieces using small group instruction. The teacher will begin with the 1/2 and show the students that when you put them together they form a whole. An example of
the fractions circles are found here.
The teacher should then do the same for the fourths, and eighths. To help students find the measure of each part, the teacher can use base ten blocks to represent
360. The teacher should then have the student break them up into two equal parts to find each half. The teacher should guide them through regrouping if necessary.
The teacher will continue the same procedure for the fourths, and eighths.
Extensions:
For students who finish early, the teacher can pose the question:
How many degrees would there be in one twelfth of a circle? How is this related to the numbers on a clock?
Students may:
draw a visual representation
write a mathematical proof of their calculations
explain their reasoning
Suggested Technology: Computer for Presenter, Interactive Whiteboard
Special Materials Needed:
Geoboards or template of a circle with a center point(included)
rubber bands (5 per group)
pencil
math journal (lined paper if journals are not utilized)
color pencils
board
dry erase markers
fraction circles for small group instruction (for students who need accomodations)
base ten blocks(for students who may need help with division)
Further Recommendations:
Make sure to go over the rules for cooperative learning before you start your lesson. Place students in groups of three or less to facilitate group discussion. Give each
group five rubber bands only, so that the rubber bands don't become a distraction. Make sure you review what you consider the appropriate use of the rubber bands
before you begin the lesson.
Additional Information/Instructions
page 3 of 4 By Author/Submitter
This lesson is correlated to the following mathematical practice:
1. Make sense of problems and persevere in solving them
4. Model with mathematics
SOURCE AND ACCESS INFORMATION
Contributed by: Anna Guerrero
Name of Author/Source: Anna Guerrero
District/Organization of Contributor(s): Miami-Dade
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.4.MD.3.5:
MAFS.4.MD.3.7:
Description
Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand
concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering
the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns
through 1/360 of a circle is called a “one­degree angle,” and can be used to measure angles.
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure
of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown
angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the
unknown angle measure.
page 4 of 4