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Ricardo Richards Elementary 2013-2014
Day
Mathematics
mon
Standards:
Objectives:
Procedures:
Assessment:
Presidents Day No School
1 of 5
Day
Mathematics
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Standards: 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into
rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and
mathematical problems.
Objectives: Understand some important properties of polygons and recognize polygonal shapes both in and out of the classroom.
Shapes and Designs, Investigation 3, Problem 3.1 　Completed Angle Sums or Regular Polygons 　 Mathematical Goals • Find
angle sums of regular polygons. • Determine relationships between the number of sides and the angle sum of a regular
polygon.
Vocabulary: angle sum Technology: ExamView CD-ROM, Materials: Student notebooks, Overhead projector, Overhead Shapes Set,
Shapes Set, Blank Transparencies, Transparency markers, Angle rulers
National Standards NAEPG1d, NAEPG2d, NAEPG3c, NAEPG3f, NAEPA1a
1. MATHEMATICS BACKGROUND 　 Use the Math Background to help you understand the mathematics being taught in this unit. 　
Mathematics Background Prentice Hall Connected Math Web Site
2. LAUNCH 　 Remind students what a regular polygon is. • We call the sum of the interior angles of a polygon the "angle
sum." Arrange six regular polygons on the overhead or refer to the textbook. • Which polygon has angles that appear to
be the smallest? • Which polygon has angles that appear to be the largest? Make sure that students see that the size of
the interior angles increases as the number of sides increases. Demonstrate at the overhead projector how the sizes of the
angles compare by placing one shape on top of another. Arrange students in groups of 2 and 3. 　 Transparency 3.1A Regular
Polygons Transparency 3.1B Table Transparency 1.1E Shapes Set
3. EXPLORE 　 During this time, students begin by organizing the data they collected in Problem 2.3. They should continue
measuring and recording measurements of a regular pentagon and octagon. For students who see the patterns quickly, ask them to
make a general rule. Some may even be ready to use symbols. Have the students use their rule to find the angle sums for a
polygon with seven, nine, and ten sides. 　 Labsheet Shapes Set
4. SUMMARIZE Accept and record all student answers on the chart. Some students may disagree with the measurements others give.
When the chart is complete, you may want to question the data. • Why do we have different answers when we all measured
the same angles? How might we resolve the angle measures we disagree on? • Look at all the answers that are now recorded
on the chart. Are there any that don’t seem reasonable? Remove numbers from the chart when students have given a
mathematical reason for eliminating them. • What patterns do you notice in the way the size of the angles is increasing?
What patterns do you notice in the way the size of the angle sum is increasing? 　 Continue with the discussion until the class
arrives at the correct measurements. See the extended Summarize for more details and questions. Put three different-sized
equilateral triangles on the overhead and ask about their angle measures and then about the side lengths. Repeat this for one
or two other polygons. Point out that two polygons with the same number of sides and equal corresponding angles are not
necessarily identical.
　 　 5. ASSessment Core 1–2 Other Connections 15–16 (angle ruler needed for 15); Extensions 21 Adapted For
suggestions about adapting Exercise 2 and other ACE exercises, see the CMP Special Needs Handbook. 　 3ACE Exercise 2 Adapted
Version 3ACE Exercise 2 (Alternative) Adapted Version Answers to ACE and Mathematical Reflections
Gr 6 Notes Reading Plus Schedule. Desir8:00am Wednesday 1:15 pm Tuesday Edwards 8:00 am Thursday 1:15pm Wednesday Pelle 8:00
am Tuesday 1:15pMThursday
Intervention: Scoring High pgs47-49
2 of 5
Day
Mathematics
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Standards: 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into
rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and
mathematical problems.
Objectives: Understand some important properties of polygons and recognize polygonal shapes both in and out of the classroom.
Procedures
Shapes and Designs, Investigation 3, Problem 3.2 　Completed Angle Sums of Any Polygon 　 Mathematical Goals • Develop
informal arguments for conjectures about the relationship between the number of sides and the angle sum of any polygon. •
Find angle sums of any polygon. Technology: ExamView CD-ROM,
Materials: Student notebooks, Overhead projector, Overhead Shapes Set, Shapes Set, Construction paper, Scissors
National Standards NAEPG1d, NAEPG2d, NAEPG3c, NAEPG3f, NAEPA1a 1. MATHEMATICS BACKGROUND 　 Targeted Resources 　 Use the
Math Background to help you understand the mathematics being taught in this unit. 　 Mathematics Background Prentice Hall
Connected Math Web Site 2. LAUNCH (10minutes) 　 Targeted Resources 　 Ask students whether or not they think the angle sum
formula or pattern for regular polygons will hold for polygons in general. • Do you think the angle sum of any triangle
is 180°? How can we check? Draw a triangle on a sheet of paper or transparency and label each angle 1, 2, and 3. After
cutting out the triangle, tear (or cut) off all three angles and arrange the angles around a point on another sheet of paper
or on the overhead. • What do you observe about the sum of the angles of the triangle? Repeat the experiment with
different shaped triangles. • Based on your experiments, what is the angle sum of any triangle? • What if this
experiment was repeated for a quadrilateral? • Can you predict the sum of the angles of the quadrilateral? Draw a
quadrilateral on a sheet of paper or transparency and label each angle 1, 2, 3, and 4. After cutting out the quadrilateral,
tear (or cut) off all four angles and arrange the angles around a point or on the overhead. • Based on the picture, what
is the sum of angles 1, 2, 3, and 4? How do you know? • Make a conjecture about the angle sum of any quadrilateral.
• Can you predict the sum of the angles of a pentagon? A hexagon? Any N-sided polygon? Describe Cody and Tia’s
methods for finding the angle sum of a polygon. Arrange the students into groups of 2 and 3. 　 Transparency 3.2 Getting Ready
Transparency 1.1E Shapes Set 3. EXPLORE (20minutes) 　 Targeted Resources 　 Students should begin exploring Tia’s
method. You may need to remind students that the angle sum of every triangle is 180º, and that the sum of the angles
around a point is 360º. For students having a hard time seeing that the sum of the angles of a polygon are equal to the
sum of the angles in the (N – 2) triangles, you can suggest numbering the angles of the triangles in Tia’s method.
For students who see the patterns quickly, ask them to make a general rule. Ask how their answers compare to their answers to
Problem3.1. 　 Labsheet 3.2 Angle Sums Labsheet Shapes Set 4. SUMMARIZE (15minutes) 　 Targeted Resources 　 Have someone
explain each method. • Are the angle sums of these polygons the same as the angle sums for regular polygons with the same
number of sides? • What about the measure of each interior angle? Ask the class: • What is the angle sum of a 12sided polygon? A 100-sided polygon? What is the angle sum of any polygon with N sides?
　 　 5. ASSIGNMENT GUIDE 　 Targeted
Resources 　 Core 3–6, 9 Other Applications 7, 8, 10; Connections 17; Extensions 22–24; unassigned choices from
previous problems Adapted For suggestions about adapting ACE exercises, see the CMP Special Needs Handbook. 　 Answers to ACE
and Mathematical Reflections
Gr 6 Notes Reading Plus Schedule. Desir8:00am Wednesday 1:15 pm Tuesday Edwards 8:00 am Thursday 1:15pm Wednesday Pelle 8:00
am Tuesday 1:15pMThursday
Intervention: Scoring High pg50-51
3 of 5
Day
Mathematics
thu
Standards: Standards: 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by
composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving
real-world and mathematical problems.
Objectives: Understand some important properties of polygons and recognize polygonal shapes both in and out of the classroom.
Procedures:
Shapes and Designs, Investigation 3, Problem 3.3 　Completed Back to the Bees! 　 Mathematical Goal • Decide which regular
polygons will tile by themselves or in combinations using information about interior angles. Technology: ExamView CD-ROM,
Materials: Student notebooks, Overhead projector, Overhead Shapes Set, Shapes Set
National Standards NAEPG1d, NAEPG2d, NAEPG3c, NAEPG3f, NAEPA1a
1. MATHEMATICS BACKGROUND Use the Math Background to help you understand the mathematics being taught in this unit. 　
Mathematics Background Prentice Hall Connected Math Web Site
2. LAUNCH 　 Students should have an idea about which regular polygons will tile and which will not. If needed, remind them
about the discussions from Investigation 1. • Which of the regular polygon shapes (Shapes A–F) did we learn would
tile a surface—fit together so that there are no gaps or overlaps—by themselves? Students should recall that the
triangle, square, and hexagon all tile. • How can we tell for sure that a shape, like these hexagons, fits exactly around
each point in a tiling? We know the fit looks good, but how can we use mathematics to tell for sure? Have students work
together in groups of 2–4. 　 Transparency 1.1E Shapes Set
3. EXPLORE 　 As you move from group to group, ask questions about angles to help students focus on them as a consideration in
forming a tiling. • How much turn must there be to completely surround a vertex point? • How many degrees are in the
angles of the polygon you are investigating? • What would we expect the angle sum around a point to be? When students
have completed the exploration of the regular polygons, they should move to the combinations of regular polygons. Going
Further Ask students to explore whether any quadrilateral will tile.
4. SUMMARIZEStudents should be able to explain why there are only three regular polygons that tile using angle measure as part
of their argument. They should also be able to explain why certain combinations of regular polygons work using angle measures.
• Why do you think your design forms a tiling with no gaps or overlaps? There are nine combinations of regular polygons
that will tile (note the numbers in parentheses refer to the polygon by side number): 2 octagons and 1 square (8-8-4) 1
square, 1 hexagon, and 1 dodecagon (4-6-12) 4 triangles and 1 hexagon (3-3-3-3-6) 3 triangles and 2 squares (4-3-4-3-3) 1
triangle, 2 squares, and 1 hexagon (4-3-4-6) 1 triangle and 2 dodecagons (3-12-12) 3 triangles and 2 squares (4-3-3-3-4) 2
triangles and 2 hexagon (3-6-6-3) 2 triangles and 2 hexagons (3-6-3-6) Note there are two arrangements with triangles and
squares, but depending on the arrangement they produce different patterns.
　 　
5. ASSESSMENT; Core 11, 12 Other Connections 18, 19; Extensions 25; unassigned choices from previous problems
Gr 6 Notes Reading Plus Schedule. Desir8:00am Wednesday 1:15 pm Tuesday Edwards 8:00 am Thursday 1:15pm Wednesday Pelle 8:00
am Tuesday 1:15pMThursday
Intervention: Scoring High pg52-53
4 of 5
Day
Mathematics
fri
Standards: Standards: 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by
composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving
real-world and mathematical problems.
Objectives: Understand some important properties of polygons and recognize polygonal shapes both in and out of the classroom.
Procedures:Shapes and Designs, Investigation 3, Problem 3.4 　Completed Exterior Angles of Polygons 　 Mathematical Goal •
Explore the sum of the exterior angles of a polygon. Vocabulary: interior angle, exterior angle Technology: ExamView CD-ROM,
Materials: Student notebooks, Overhead projector, Overhead Shapes Set, Shapes Set, Angle rulers
National Standards NAEPG1d, NAEPG2d, NAEPG3c, NAEPG3f, NAEPA1a 1. MATHEMATICS BACKGROUND Use the Math Background to help you
understand the mathematics being taught in this unit. 　 Mathematics Background Prentice Hall Connected Math Web Site
2. LAUNCH Put up several regular polygons on the overhead. • What pattern do you see in the sizes of the interior angles
as the number of sides increases? • Will they ever equal or be greater than 180º? • What happens to the shape
of the polygon as the interior angle measures increase? Show the class an example of an exterior and interior angle of a
polygon. These two angles come in pairs. Students might note that the sum of their measures is 180º. Remind the class
that there are two sets of exterior angles depending on how you extend the sides of a polygon. The important thing is that the
sides have to be extended in the same direction—either all clockwise or all counterclockwise. Ask the class if any of
them have done any skateboarding. You might ask them how angles or the language of angles is used in skateboarding. Tell the
class about the skateboarder who is skating around a garden in the shape of a polygon. You might demonstrate this with a
polygon. • As she turns the first corner, what angle of turn does she make? • Your challenge is to find how many
degrees the skateboarder skates through as she skates around a park shaped like a polygon. The skateboarder is going
counterclockwise around the park. Students can work in groups of 2–4.
3. EXPLORE Be sure that students have a consistent set of exterior angles. For those students that finish early, you can ask
them to explore the sum of the exterior angles if the skateboarder skates in the opposite direction. There is no difference in
the sum. Going Further Suggest that students try a square or another regular polygon. Some might want to draw an irregular
polygon and measure the exterior angles. 　 　
4. SUMMARIZE;Go over the answers. Ask the class to explain why the sum of 360° makes sense. Put up some regular polygons.
Ask for the measure of each interior angle. Then ask for the sum of the exterior angles. Put up an irregular polygon and label
the measures of each interior angle. Ask for the sum of the exterior angles. Repeat the above activities for the other set of
exterior angles.
　 　 5. ASSESSMENT Core 13, 14 Other Connections 20; unassigned choices from previous problems
Gr 6 Notes Reading Plus Schedule. Desir8:00am Wednesday 1:15 pm Tuesday Edwards 8:00 am Thursday 1:15pm Wednesday Pelle 8:00
am Tuesday 1:15pMThursday
Intervention: Scoring High pg54-57
Grade 6
Week of February 17, 2014
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