Download Grade 4 - Angle Measure and Plane Figures

Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Getting to the Core of the Common Core: Module 4: Angle Measure and Plane Figures
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Patricia Scavuzzo­Despagni, Ed.D.
Educational Mathematics Curriculum Consulting (EMC2 )
[email protected]
Please make 4 clock appointments each with a different partner for our activities today
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Buddies
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Module Overview
• Overview
• Distribution of Instructional Minutes
• Standards
• Terminology
• Materials
• Lessons (fluency practice, application problem, concept development, student debrief, problem sets, exit tickets, homework)
• Mid­module assessment
• End­of­module assessment
RIGOR
Pedagogical Shifts demanded by the Common Core State Standards
[www.engageNY.org]
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Fluency Practice
Fluency activities serve a variety of purposes:
• Maintenance: Staying sharp on previously learned skills • Preparation: Targeted practice for the current lesson
• Anticipation: Building skills to prepare students for the in‐depth work of future lessons
In fluency work, all students are actively engaged with familiar content. This provides a daily opportunity for continuous improvement and individual success. General categories of fluency activities include:
• Counting exercises • Choral response • Personal marker board activities • Sprints Fluency does not have to be done directly before the math lesson. www.engageny.org
[Grade 4 Module 1]
[Grade 4 Module 1]
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Sprint Directions:
­ Pass out Sprint A face down
­ In 1 minute to do as many problems as you can ­ After 1 minute circle the last problem that you did
­ Read the answers while students call out "YES" and give fist pump if they get it correct
­ Students, write the number you got correct at the top of the page (personal goal)
­ How many got ____ right?
­ Do some kind of movement in between sprints
­ Repeat with Sprint B
­ Stand up if you got more correct on Sprint B than on Sprint A
­ Keep standing if you got ____ more correct
­ Take a moment to go back and correct your mistakes
­ Think about what patterns you noticed in today's Sprint
­ Talk with partner: How did the patterns help you get better at solving the problems
Grade 4 Module 1
Lesson 8 Sprint
Grade 4 Module 1
Lesson 1 Sprint
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Application Problem
• Application involves using relevant conceptual understandings and appropriate strategies even when not prompted to do so.
• Time allotted to application varies, but is commonly 5‐10 minutes of the lesson.
• The Read, Draw, Write (RDW) process is modeled and encouraged through daily problem solving.
www.engageny.org
Grade 4 Module 1
Lesson 12
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Concept Development
• Constitutes the major portion of instruction and generally comprises at least 20 minutes of the total lesson time. • Builds toward new learning through intentional sequencing within the lesson and across the module.
• Often utilizes the deliberate progression from concrete to pictorial to abstract, which compliments and supports an increasingly complex understanding of concepts. • Accompanied by thoughtfully sequenced problem sets and reproducible student sheets.
www.engageny.org
Student Debrief
­ Share and analyze work ­ Check answers with partner or group
­ Answer reflective questions • Includes sample dialogue or suggested lists of questions to invite the reflection and active processing of the totality of the lesson experience.
• Encourages students to articulate the focus of the lesson and the learning that has occurred.
• Promotes mathematical conversation with and among students.
• Allows student work to be shared and analyzed.
• Closes the lesson with daily informal assessment known as Exit Tickets.
www.engageny.org
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Students draw points, lines, line segments, and rays, and identify them in various contexts and familiar figures. They also learn the notation for writing and naming a line, line segment, and ray. Where is the treasure?
Point ­ a precise location on a plane
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Any two rays sharing the same Line Segments, Lines,
and Rays
endpoint create an Angle.
How many angles are in a rectangle?
Where do you see rays in this picture?
Each line segment is a part of a larger ray but we don't have to draw them in. Therefore, line segments meet to form angles and lines meet to form angles. Find points, line segments, lines, rays, and angles in diagrams.
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Interesting point about a point.
Grade 4 Module 4 page 4.A.9
In Lesson 2, students create right angles through a paper folding activity, and identify right angles in their environment by comparison with the right angle they have made. They also draw acute, right, and obtuse angles. This represents their first experience with angle comparison and the idea that one angle's measure can be greater (obtuse) or less (acute) than that of a right angle. Acute Angle Obtuse Angle X
Right Angle Y
Z
Straight Angle 9
Grade 4 ­ Angle Measure and Plane Figures
Right
Acute
Right
Acute
Right
December 02, 2013
Obtuse
Acute
Obtuse
Obtuse
Obtuse
Student Debrief:
Problems c and f are both right angles. Describe their position. Does orientation determine if it is a right, acute, or obtuse angle?
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Angle Shape Sort Activity Cut out shapes and place them on the Sorting Angles Task Sheet
[Common Core Georgia Performance Standards Frameworks Student Edition, Mathematics, Fourth Grade Unit 6 Geometry, Georgia Department of Education, Dr. John D. Barge, State School Superintendent] In Lesson 3, students' knowledge of right angles leads them to identify and define as well as construct perpendicular lines.
Notation:
Important part of lesson:
Draw perpendicular lines on a diagonal. Are these lines perpendicular?
Share your thoughts with your partner.
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Student Debrief:
What was your strategy for drawing the segments perpendicular?
Student Debrief:
In this problem, I only located 8 right angles. How many more right angles are there? What did this problem show you about locating angles on figures?
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Students learn in Lesson 4 that lines that never intersect also have a special relationship and are called parallel. Students use, in conjunction with a straightedge, the right angle template that they creates in Lesson 2 to construct parallel lines. Are these parallel? Why not? I don't see an intersection.
Two lines that never touch no matter how far you extend them are parallel. 13
Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Student Debrief:
How could you use your right angle template to serve as a guide for identifying parallel lines. What pattern did you find in the grids to help you analyze if your lines were in fact parallel. And/Or
Parallel
Intersecting
Intersecting, Perpendicular
Intersecting
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Student Debrief:
True or False: Two segments that don't intersect must be parallel?
In a figure, are opposite sides always parallel?
How do parallel lines differ from perpendicular lines? Physiometry Fluency
Kinesthetic memory is strong memory. Stand up and model figures:
• point (clench fist)
• line segment (arms extended with clenched fists)
• line (arms extended but open hands)
• ray (arms extended one hand in a fist the other hand open)
Model another ray
• right angle
• acute angle
• obtuse angle
• point to the walls that are perpendicular to the wall I am pointing to • point to the walls that are parallel to the wall I am pointing to Remid students that lines and points are not as thick as arms and fists. They are actually infinitely small. 15
Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Assessment of 4.G.1:
2 acute angles and 2 obtuse angles
[North Carolina Department of Public Instruction, Instructional Support Tools for Achieving New Standards, Unpacked Content, Grade 4]
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
A
Students explore the definition of degree measure. Using a circular protractor, students divide the circumference of a circle into 360 equal parts, treating each part as representing 1 degree. Students apply this understanding as they discover that a right angle measures 90 degrees and in turn that the angles they know as acute measure less than 90 and obtuse angles measure more than 90 degrees. The idea that an angle measures the amount of "turning" in a particular direction is explored, giving students the opportunity to recognize familiar angles in varied positions. Student Debrief:
When you listed the benchmark angles, did you notice any numerical patterns?
A full turn is 360 degrees. What could you do to find the degree measure of an angle that takes 10 turns to make a whole turn?
Draw a tape diagram to represent one whole turn and the benchmark angles of Set A. Do the same for Set B. Shade in the region of a 45 degree angle. What fraction of the whole is that? Do the same for your 30 degree angle. What if you shaded in a region defined by a 120 degree angle on your red circle? What fraction of the whole is that? Use your protractor to explain to your partner what a degree is.
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Grade 4 ­ Angle Measure and Plane Figures
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Lesson 5 B
Objective: Measuring an angle with reference to a circle by considering the fraction of the circular arc between the two rays
/Address common angle misunderstandings Adapted from http://lrt.ednet.ns.ca/PD/BLM/table_of_contents.htm
Angles don't always begin on the horizontal 0 degree line. http://www.teacherled.com/resources/anglemeasure/anglemeasureload.html
Using fractional parts of the circle, determine the number of degrees in the following angles. 18
Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
How many degrees in the following acute angles?
Note the circle is divided into 36 parts
Common Angle Misunderstandings:
1. Students may believe that the length of an arc or that the area of the wedge determines the measure of a given angle
Lesson 6 Concept Development Note: Use clocks of difference sizes as another visual. No matter how big or small a clock may be, it takes the same amount of time to go from 12:00 to 12:15. 19
Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Common Angle Misunderstandings:
2. Students may believe that a wide angle with short sides may be smaller than a narrow angle with long sides. Students need to realize that the length of the rays do not determine angle measure.
Lesson 6 Practice Sheet Student Debrief:
How are rulers and protractors alike? How are they different?
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
(Measure angles in various positions using various protractors)
Look at the different protractors in front of you. What do you notice?
Guess my Angle Common Core Georgia Performance Standards Frameworks Student Edition, Mathematics, Fourth Grade Unit 7 Measurement, Georgia Department of Education, Dr. John D. Barge, State School Superintendent, May 2012. Alien Angles
http://www.mathplayground.com/alienangles.html
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Guess my Angle Students measure angles to the nearest degree and construct angles of a given measure. Fluency Practice:
This fluency prepares students for the unknown angle problems.
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Student Debrief:
Which were the most challenging angles to draw? Explain.
Explain to your partner how to measure an angle greater than 1800 using a 1800 protractor? 23
Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Sir Cumference and the Great Knight of Angleland
by Cindy Neuschwander Common Core Georgia Performance Standards Frameworks Student Edition, Mathematics, Fourth Grade Unit 7 Measurement, Georgia Department of Education, Dr. John D. Barge, State School Superintendent, May 2012. Students further explore angle measure as an amount of turning. This provides a link to Grade 3 work with fractions, as students reason that a ¼ turn is a right angle and measures 900, a ½ turn measures 1800, and a ¾ turn measures 2700. Student Debrief:
Why is there confusion with turning 900 but not with turning 1800 or 3600 ? Why is there more than one answer for Problem 7? How can the terms clockwise and counterclockwise be used in this problem?
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Assessment of 4.MD.5 and 4.MD.6:
1. A water sprinkler rotates one­degree at each interval. If the sprinkler rotates a total of 100 degrees, how many one­degree turns has the sprinkler made?
100 one­degree turns 2. A lawn sprinkler rotates 65 degrees and then pauses. It then rotates an additional 25 degrees. What is the total degree of the water sprinkler rotation? To cover a full 360 degrees, how many times will the water sprinkler need to be moved?
90 degrees 4 times
3. If the water sprinkler rotates a total of 25 degrees then pauses. How many 25 degree cycles will it go through for the rotation to be at least 90 degrees? four 25 degree cycles because then it would cover 100 degrees, three 25 degree cycles would only cover 75 degrees
[North Carolina Department of Public Instruction, Instructional Support Tools for Achieving New Standards, Unpacked Content, Grade 4]
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Students use concrete examples to discover the additive nature of angle measure. Working with pattern blocks, they see that the measures of all the angles at a point, with no overlaps or gaps, add up to 3600, and they use this fact to find the measure of the pattern blocks' angles. 600 + 600 + 600 + 600 +600 + 600 = 3600
6 x 60 = 360
900 + 900 + 900 + 900 = 3600
How would you find the measurement of angle ABC?
Students use what they know about the additive nature of the angle measure to reason about the relationships between pairs of adjacent angles. Students discov er that the measures of two angles on a straight line add up to 1800 (supplementary angles) and the measures of two angles meeting to form a right angle add up to 900 (complementary angles). Do Now:
Write a sentence
600
300
300
1500
Then move on to Problem 3:
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Students extend their learning by detrmining the measures of unknown angles for adjacent angles that add up to 3600. Additionally through their work with angles on a line, students go on to discover that vertical angles have teh same measure. 90 + 120 + x = 360
90 + 120 = 210
360 – 210 = x
x = 150
Assessment of 4.MD.7:
Example 1: Find x
x
25 + x = 90
x = 65o
Example 2:
25 + x + 20 = 90
45 + x = 90
x = 45
Example 3:
Joey knows that when a clock's hands are exactly at 12 and 1, the angle formed by the clock's hands measures 300. What is the measure of the angle of the angle formed when a clock's hands are exactly on the 12 and 4?
30 x 4 = 120 120o
[North Carolina Department of Public Instruction, Instructional Support Tools for Achieving New Standards, Unpacked Content, Grade 4]
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Students recognize lines of symmetry for two­dimenstional figures, identify line­
symmetric figures, and draw lines of symmetry. Given half of a figure and a line of symmerty, they draw the missing half. The topic then builds on students' prior knowledge of two­dimensional figures and allows students time to explore theri properties. Throughout this topic, students use all theri prior knowledge of line and angle measure to classify and construct two­dimensional figures. 28
Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Using Triangles A ­ F, students investigate the triangle cutouts using rulers and protractors. Students record their findings in the Attribute column of the Practice sheet, including measures of sides and angles, as well as other general observations. Students then sort and classify triangles by side length and angle measure. Sorted according to side length:
Isosceles Triangle: B, E, F
(Fold on its line of symmetry)
Sorted according to angle measure:
Right Triangle: D, E
(Fold the other 2 angles into the RT angle)
Equilateral Triangle: A
(Fold on all lines of symmetry) Obtuse Triangle: C, F
Acute Triangle: A, B
Scalene Triangle: C and D
Student Debrief:
How many lines of symmetry can be found in scalene triangles? Equilateral triangles? Isosceles triangles?
Can you determine whether or not a triangle will have a line of symmetry just by knowing whether it is an acute triangle or obtuse triangle? How about scalene or isosceles?
How many acute angles do right triangles have?
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Students apply their understanding of triangle classification as they construct triangles given a set of classifying criteria. • Construct obtuse isosceles triangle
• Construct a right scalene triangle
How many different types of triangles are there? Are there 9?
Sort the entire triangle collection from Lesson 13 into the following categories to find out. A
B
E
D
F
C
Of the nine triangles, two are not possible
• An equilateral right triangle is not possible because equilateral triangles have three acute angles that measure the same. In a right triangle only 1 angle is right and the other two are acute.
• An equilateral obtuse triangle is not possible because in an obtuse triangle one side is longer and in an equilateral triangle all the sides are the same. Quadrilaterals What is mathematically wrong with this puzzle? 30
Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Students explore the definitions of familiar quadrilaterals and reason about their attributes, including angle measure and parallel and perpendicular lines. This work builds on Grade 3 reasoning about the attributes of shapes and lays a foundation for hierarchical classification of two­dimensional figures in Grade 5. triangle
quadrilateral
parallelogram
trapezoid
rectangle
square
rhombus
Use examples and non­examples to help students make an accurate definition.
Even with good examples of what a triangle is, a learner does not have enough information to know what is not a triangle. These are examples of triangles:
These are not examples of triangles:
Specific non­examples help focus attention on details that might be otherwise missed. The three sides must be straight not curves, there can be no extra frills or bows the lines can't hang over they must intersect at their endpoints, the corners cannot be curves, and the figure must be closed. Well selected non­examples help children improve their verbal descriptions. [http://thinkmath.edc.org/index.php/Examples]
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Keep quadrilaterals on the grid lines
Assessment of 4.G.1, 4.G.2, and 4.G.3:
rectangle
2 pairs of parallel sides
opposite sides are equal
4 right angles
parallelogram
2 pairs of parallel sides
No, if it is a right angle than one of the sides is 90o If it is an acute angle then all of the angles are less than 90o
Trapezoid
[North Carolina Department of Public Instruction, Instructional Support Tools for Achieving New Standards, Unpacked Content, Grade 4]
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Grade 4 ­ Angle Measure and Plane Figures
December 02, 2013
Some shapes fall in the overlapping sections of the circles because they have both properties ­ they have at least one pair of parallel sides and they have at least on right angle. [North Carolina Department of Public Instruction, Instructional Support Tools for Achieving New Standards, Unpacked Content, Grade 4]
[North Carolina Department of Public Instruction, Instructional Support Tools for Achieving New Standards, Unpacked Content, Grade 4]
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