Debbie Ferry
Macomb ISD
Mathematics Consultant / Carol Nowakowski
Retired
Mathematics Consultant
K-4 Project Coordinator / Marie Copeland
Warren Consolidated
Macomb MSTC
5-8 Project Coordinator
2004 Project Contributors
David Andrews
Chippewa Valley Schools / William Ashton
Fraser Public Schools / Lynn Bieszki
Chippewa Valley Schools
Sharon Chriss
Romeo Schools / Kimberly DeShon
Anchor Bay School District / Barbara Diliegghio
Retired, Math Consultant
Kimberly Dolan
Anchor Bay School District / Jodi Giraud
L’Anse Creuse Schools / Julie Hessell
Romeo Schools
Amy Holloway
Clintondale Schools / Barbara Lipinski
Anchor Bay School District / Linda Mayle
Romeo Schools
Therese Miekstyn
Chippewa Valley Schools / James Navetta
Chippewa Valley Schools / Gene Ogden
Anchor Bay School District
Rebecca Phillion
Richmond Comm. Schools / Charlene Pitrucelle
Anchor Bay School District / Shirley Starman
Van Dyke Public Schools
Ronald Studley
Anchor Bay School District
2005 and 2006 Session/Module Developers
Carol Nowakowski
Retired, Math Consultant / Deb Barnett
Lake Shore Public Schools / Luann Murray
Genesee ISD
Kathy Albrecht
Retired, Math Consultant / Jo-Anne Schimmelpfenneg
Retired, Math Consultant / Marie Copeland
Warren Consolidated
Terri Faitel
Trenton Public Schools / Debbie Ferry
Macomb ISD
Fifth Grade Session 5 GLCEs & Instructional Sequence
Grade 5: Know the meaning of angles, and solve problems
G.TR.05.01 / M.GS.05.02 / G.GS.05.03 / G.GS.05.04Associate an angle with a certain amount of turning; know that angles are measured in degrees; understand that 90°, 180°, 270°, and 360° are associated, respectively, with 1/4, 1/2, 3/4, and full turns. / Measure angles with a protractor, and classify them as acute, right, obtuse, or straight. / Identify and name angles on a straight line and vertical angles. / Find unknown angles in problems involving angles on a straight line, angles surrounding a point, and vertical angles.
Grade 5: Solve problems about geometric shapes
G.GS.05.05 / G.GS.05.06 / G.GS.05.07Know that angles on a straight line add up to 180° and angles surrounding a point add up to 360°; justify informally by “surrounding” a point with angles. / Understand why the sum of the interior angles of a triangle is 180° and the sum of the interior angles of a quadrilateral is 360°; and use these properties to solve problems. / Find unknown angles using the properties of: triangles, including right, isosceles, and equilateral triangles; parallelograms, including rectangles and rhombuses; and trapezoids.
Instructional Sequence:
Grade 5 - Session 5 – Angles and Triangles – Participant Packet
Page - 9 - Revised 10.15.07
TIPS / COMMON MISCONCEPTIONS· When students disagree with each other, encourage them to justify their reasoning to each other or the class. Often they are correct in their thinking but they are not answering the question given.
· Solving problems without giving students time to think about them often leads to “fragile knowledge”.
· It is important to take some time to be sure students understand similarity and congruency.
· Children need to work from the Conceptual (hands-on) stage to the Pictorial (drawing) before being presented with the Abstract (symbolic).
· The purpose of the questions asked on the worksheets is not always for a grade but for discovery, reasoning, justifying, and communication practice. / · The size an angle (including those found in triangles, quadrilaterals, etc) is determined by the amount of opening not the length of the sides.
· A common error is distinguishing between: adjacent angles on a straight line, adjacent angles not on a straight line, and non-adjacent angles.
Fifth Grade Participant Packet Session 5
Focus on: Geometry: Angles and Triangles
Name of Activity / Description of Activity / Materials/Handouts / Key Tips for Presenter /I. INTRODUCTION TO SESSION 5 & GLCEs /
· Discuss GLCEs found in packet.
· Compare the geometry strand across the grades.· In groups, participants should discuss the connections seen between the levels: introductory and development stages. / · Session 5 participant packets
· Overhead projector/Elmo / · Students need a firm foundation from which to build, grow and develop.
· Working through the stages: Conceptual (hands-on), Pictorial (drawing), and Abstract (symbolic) will lead students making connections.
II. The van Hiele Levels of Geometric
Thought / · Explain how the van Hiele model moves learners through five levels of geometric understanding (0-4)
· Level 1 (Analysis): Students analyze component parts of the figures (opposite angles of parallelograms are congruent) but interrelations between figures and properties cannot be explained)
· Level 2 Informal Deduction: Students establish interrelationships of properties within figures (in a quadrilateral, opposite sides being parallel necessitates opposite angles being congruent) and among figures (a square is a rectangle because it has all the properties of a rectangle). Informal proofs can be followed but students do not see how the logical order could be altered nor do they see how to construct a proof starting from different or unfamiliar premises.
· Refer to the handout for further explanation of the levels / · van Heile handout / · Some models use different numbering systems (1-5)
· Most students in 5th grade are at level 1 moving toward level 2.
· The levels of geometric reasoning are sequential
· Students must pass through all prior levels to arrive at any specific level
· The levels are not age-dependent in the way Piaget described development
· Rich geometric experiences have the greatest influence on advancement through the levels.
· Language and Instruction at a higher level than the level of the student may inhibit learning.
· How do the GLCE’s fit into the van Hiele model? They provide specific expectations for students to move from one level to another
III. INTRODUCTION TO ANGLES AS AN AMOUNT OF TURN
G. TR.05.01
Associate an angle with a certain amount of turning; know that angles are measured in degrees; understand that 90°, 180°, 270°, & 360° are associated, respectively, with ¼, ½, ¾ and full turns.
G.GS.05.02b
Classify angles as acute, right, obtuse, or straight. / · DISCOVERY
o Students demonstrate kind of moves made on a skateboard.
o Discuss and relate to starting at 0, turning 180° & 360°.
o Introduce 180° as a STRAIGHT angle.
· ANGLE WHEEL
o Use 2 different colors of circles cut from the die-cut (Ellison) machine. Fold each circle into fourths, pinch to identify the center. Cut on a radius (from one edge to the center) on both circle and fit together.
o With partners, practice forming various angles (RIGHT, OBTUSE (greater than right) & ACUTE (less than right) – compare for differences in sizes. Are all obtuse angles the same size? Are all right angles the same size? etc. Have students recall what a right triangle is. Use the corner of an index card as a “right angle tool” to compare and verify the type of angles formed.
o Students show fractional parts (½ , ¼, etc.) of the “angle wheel” and relate to the number of degrees and type of angle formed. Complete the table on Worksheet #1.
· Discuss ¼ and ½ turns. (Use the “right angle tool” to demonstrate and verify.) What is special about these turns? Are all ¼ turns the same size? Are all ½ turns the same size?
· Compare the times on a clock to their fractional parts. What times on the analog clock represents ¼ turn? ½ turns?
· Predict the number of degrees in a and turn. Verify and explain.
· Introduce the naming of angles – draw an angle and label a point on each ray and the endpoint of each ray (the vertex).
· Complete Worksheet #2 on triangle types.
ASSESSMENT:
· Draw and label various types of angles using their correct name, # of degrees (using fourths), and their corresponding times on a clock.
· With a straightedge or ruler, draw several examples of RIGHT, OBTUSE, ACUTE, and STRAIGHT angles. Describe each.
· Explain how fourths can be used to determine the angle measurements of 180°, 270°, and 360°. / · Skateboarding or Snowboard videos can be found at http://www.go211.com/
· 2 same-sized circles of different colors – the 4” die-cut circles work well, use either construction paper or cardstock
· Scissors
· Index card as a right angle tool
· Worksheet #1
· Worksheet #2 / · The goal is for students to experience that an angle is the amount of turn.
· Connect ¼ turn as a right angle and a 90° angle ( ½ as a 180° angle).
· Right angle – measures 90° & is ¼ turn
· Acute angle – measures between 0° and 90° & is less than ¼ turn
· Obtuse angle – measures between 90° and 180° & is more than ¼ turn, but less than ½ turn
· Straight angle – measures 180° & is ½ turn
· Full turn, turn, measures 360°
· turn is 270°
· When drawing a right angle, always use the square corner symbol/indicator.
· Make connections between students’ previous knowledge to the other areas of mathematics, the real world, etc.
· It is important to ask students “how they know”, “explain their reasoning”, “generalize”, “show”, “verify”, etc.
· Three ways to name an angle:
If only one angle made at the vertex then the angle may be named by its vertex letter – always a capital letter
A number written in the interior of the angle
If more than one angle then three points must be used – a point on one ray, the vertex and a point on the other ray, i.e.
IV. TANGRAMS & ANGLES
/ · Working with a partner or in small groups, find a Tangram piece with a 90° interior angle. Use this to estimate the angle measurements of the angles of the other Tangram pieces. Fill in Worksheet #3. / · Tangram sets for each group· Worksheet #3 / · The use of Tangrams allows the students to discover the number of degrees in some basic angles without using a protractor
· There are 3 different-sized angles (45°, 90° and 135°) within the square, triangles & parallelogram of the Tangram.
Grade 5 - Session 5 – Angles and Triangles – Participant Packet
Page - 9 - Revised 10.15.07
Name of Activity / Description of Activity / Materials/Handouts / Key Tips for PresenterV. ANGLE MEASURES
G.GS.05.02
Measure angles with a protractor, and classify them as acute, right, obtuse, or straight. / · Read: Sir Cumference and the Great Knight of Angleland by Cindy Neuschwander – the chronicles of Sir Cumference’s son, Radius, in a quest to earn his knighthood by rescuing a king. The circular medallion (a protractor) given to Radius by his father and his mother, Lady Di of Ameter, aid him in examining every angle along the way
· Model how to use the circular protractor (Worksheet #4) using the “friendly” (approximated) number of degrees of 30°, 45°, 60°, 90°, 120°...). Stress the placement of the protractor center on the vertex. Then aligning the horizontal line on one ray of the angle you move along the protractor scale from 0° until you reach the second ray of the angle. The amount of turn that you moved is the angle measurement in degrees.
· Worksheet #5 provides practice on drawing and measuring angles with a circular protractor.
o Draw an angle. What type of angle did you draw? Record it in the table.
o Estimate the measure of the angle and record your estimate in the table. Is your estimate reasonable for the type of angle it is?
o Use the transparency of the circular protractor from Worksheet #4 to find the degree measurement. Record in the table.
o How close was your estimate? Find the difference between the actual and your estimated value. Record the difference in the table.
o Continue practicing by drawing various “friendly” angles and checking with the circular protractor. / · Sir Cumference and the Great Knight of Angleland by Cindy Neuschwander
ISBN:
157091169X
· Transparency of Worksheet #4 (the protractors)
· Worksheet #5 / · When 1st finding the number of degrees in an angle, use “friendly” angle measures.
· The Circular Protractor is a tool for measuring angles more than 180° easily.
· It is important for students to estimate angle measures so they can eventually check their measurements.
Grade 5 - Session 5 – Angles and Triangles – Participant Packet
Page - 9 - Revised 10.15.07
Name of Activity / Description of Activity / Materials/Handouts / Key Tips for PresenterASSESSMENT:
· Given an angle measurement draw an angle of that amount of turn.
· Given angles use protractors to measure the amount of turn – to the closest 30°.
VI. SEMI-CIRCULAR PROTRACTOR CONNECTION
G.GS.05.02
Measure angles with a protractor, and classify them as acute, right, obtuse, or straight. / Modeling: Teacher models each step.
· Read: Hamster Champs by Stuart J. Murphy With a few blocks, a board, and a protractor to measure the angles, the hamster champs have built a ramp that lets them fly high! But will this stunt be good enough to outwit Hector the cat?
Model how to use the semi-circular protractor (Worksheet #4) using the “friendly” (approximated) number of degrees of 30°, 45°, 60°, 90°, 120°...). Stress the placement of the protractor center point (the triangle) on the vertex. Then aligning the horizontal line on one ray of the angle you move along the protractor scale from 0° until you reach the second ray of the angle. The amount of turn that you moved is the angle measurement in degrees. There are two numbers at this position. One amount will be for an acute angle and one for an obtuse angle (the measures of two adjacent angles that form a line sum to 180°.) What kind of angle is it? Which amount of turn is reasonable for this type of angle? / · Hamster Champs by Stuart J. Murphy
ISBN: 0060557737
· Straight edge or ruler
· Unlined paper
· Worksheet #4
· Worksheet #6 / · Learning HOW to read the protractor is more important than measuring to the nearest degree.
Grade 5 - Session 5 – Angles and Triangles – Participant Packet
Page - 9 - Revised 10.15.07
Name of Activity / Description of Activity / Materials/Handouts / Key Tips for Presenter· Worksheet #6 provides practice on drawing and measuring angles with a semi-circular protractor.
o Draw an angle. What type of angle did you draw? Record it in the table. Estimate the measure of the angle and record your estimate in the table. Is your estimate reasonable for the type of angle it is?
o Use the transparency of the semi-circular protractor from Worksheet #3 to find the degree measurement. Record the measurement in the table.
o How close was your estimate? Find the difference between the actual and your estimated value. Record the difference in the table.
o Continue practicing by drawing approximations of the various “friendly” angles and checking with the semi-circular protractor.
o Compare and contrast the measurements of the angles. Are there any angles equal in measure? Can you find any special sums? Write about the angle relationships that you have found.
ASSESSMENT:
· Measure and/or draw various angles.
· Explain how to read and use a protractor.
· Illustrate and explain how to check angle measurements.
· Draw and explain how to form a 270° and 360° angle. / · The accuracy of an angle measurement is unimportant at this point, except to recognize the approximate number of degrees 30°, 45°…, and note that the number of degrees in each angle is the same whether we begin from the right or left.
· Each pair of Vertical angles will have equal measures.
· The special sums that you are looking for from the compare and contrast is that adjacent angles formed by intersecting lines add to 180° and that all four angles add to 360°.
Grade 5 - Session 5 – Angles and Triangles – Participant Packet