Transcript NYS COMMON CORE MATHEMATICS CURRICULUM

NYS COMMON CORE MATHEMATICS CURRICULUM
Geometry
Grade 7 – Module 6
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Session Agenda
• Module Overview
• Focus on Topics and Lessons
• Module in Review
• Implementation Discussions
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Session Objectives
• Gain an understanding of the content enveloped in Module 6.
• Understand what was, what is, and why the change.
• See how geometry concepts from here forward are starting to take shape.
• Gain the knowledge necessary to implement with fidelity.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Curriculum Overview of A Story of Ratios
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NYS COMMON CORE MATHEMATICS CURRICULUM
Grade 7 Module 6 Overview
Let’s look at:
• Foundational Standards (p.5)
• Focus Standards (p.4)
• Mathematical Practice Standards (p.5-6)
• Table of Contents and Narrative
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Topic A: Unknown Angles
7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles
in a multi-step problem to write and solve simple equations for an unknown
angle in a figure.
• Prerequisites: G4 M4 and G7 M3
• Goal is to build fluency in these difficult problems
• Build recognition and use of structure MP.7
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Complementary and Supplementary Angles
The measures of two supplementary angles are in the ratio of 2: 3. Find
the two angles.
Lesson 1 / Example 1
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Solving for Unknown Angles Using Equations
Two lines meet as the common vertex of two rays. In a complete sentence,
describe the relevant angle relationships in the diagram. Set up and solve
an equation to find the value of 𝑥. Find the measurements of ∠𝐴𝑂𝐵 and
∠𝐵𝑂𝐶.
Lesson 2 / Exercise 4
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Solving for Unknown Angles Using Equations
Two lines meet at the common vertex of two rays. In complete sentences,
describe the relevant angle relationships in the diagram. Set up and solve
an equation to find the value of 𝑥. Find the measurements of angles ∠𝐵𝐴𝐶
and ∠𝐵𝐴𝐻.
Lesson 3 / Example 3
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Solving for Unknown Angles Using Equations
Two lines meet at a point. Set up and solve an equation to find the value of
𝑥. Find the measurement of one of the vertical angles.
Lesson 3 / Example 4
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Solving for Unknown Angles Using Equations
2
The measurement of an angle is the measurement of its supplement.
3
Find the measurement of the angle.
Lesson 4 / Example 3
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Topic B: Constructing Triangles
7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric
shapes with given conditions. Focus on constructing triangles from three
measures of angles or sides, noticing when the conditions determine a unique
triangle, more than one triangle, or no triangle.
New Terminology
New Tools
New Approach
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Correspondence
Two identical triangles are shown below. Give a correspondence that
matches equal sides and equal angles.
Lesson 5 / Example 2
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Drawing Geometric Shapes
Use a ruler, protractor, and compass to complete the following:
1. 3 segments: 4 𝑐𝑚, 7.2 𝑐𝑚, 12.8 𝑐𝑚
2. Complementary angles, one of 35°
3. Vertical angles, one of 125°
4. 3 circles: 𝑟 = 2 𝑐𝑚, 𝑟 = 4 𝑐𝑚, 𝑟 = 6 𝑐𝑚
5. 3 adjacent angles: 25°, 90°, 50°
6. Rectangle 𝐴𝐵𝐶𝐷, 𝐴𝐵 = 8 𝑐𝑚, 𝐵𝐶 = 3 𝑐𝑚
7. Segment: 5 𝑐𝑚, cut a segment 5 𝑐𝑚 using your compass
8. Segment AB; Circle with center A and radius AB, and circle with center B
and radius BA.
Lesson 6 / Exploratory Challenge
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Drawing Geometric Shapes
Use a ruler, protractor, and compass to complete the following:
Lesson 6 / Exploratory Challenge
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Drawing Geometric Shapes
Use a ruler, protractor, and compass to complete the following:
Lesson 6 / Exploratory Challenge
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Drawing Parallelograms
Use a setsquare, ruler and protractor to construct parallelogram 𝐷𝐸𝐹𝐺 so
the measurement of ∠𝐷 = 40°, 𝐷𝐸 = 3 𝑐𝑚, the measurement of ∠𝐸 =
140°, and the altitude to 𝐷𝐸 is 5 𝑐𝑚.
Lesson 7 / Exercise 1
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Drawing Triangles
A triangle is to be drawn given the following conditions: the measurements
of two angles are 30° and 60°, and the length of a side is 10 𝑐𝑚. Note that
where each of these measurements is positioned is not fixed.
Lesson 8 / Exploratory Challenge
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Three Sides Condition
Two Sides and the Included Angle Condition
4.
△ 𝑋𝑌𝑍 has side lengths 𝑋𝑌 = 2.5 cm, 𝑋𝑍 = 4 cm, and ∠𝑋 = 120°. Draw
△ 𝑋′𝑌′𝑍′ under the same conditions. Use your compass and protractor to
draw the sides of △ 𝑋′𝑌′𝑍′. Use your ruler to measure side lengths. Leave
all construction marks as evidence of your work, and label all side and
angle measurements.
Under what condition is △ 𝑋′𝑌′𝐶′ drawn? Compare the triangle you drew
to two of your peers’ triangles. Are the triangles identical? Did the
condition determine a unique triangle? Use your construction to explain
why.
Lesson 9 / Exploratory Challenge 4
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Two Angles and a Given Side
3.
A triangle △ 𝐽𝐾𝐿 has angles ∠𝐽 = 60° and ∠𝐿 = 25° and side 𝐾𝐿 = 5 cm.
Draw triangle △ 𝐽′𝐾′𝐿′ under the same condition. Leave all construction
marks as evidence of your work, and label all side and angle
measurements.
Under what condition is △ 𝐽′𝐾′𝐿′ drawn? Compare the triangle you drew
to two of your peers’ triangles. Are the triangles identical? Did the
condition determine a unique triangle? Use your construction to explain
why.
Lesson 10 / Exploratory Challenge 3
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Conditions on Measurements that Determine a
Triangle
Are there conditions on the lengths of the sides of a triangle?
Are there conditions on the measures of the angles in a triangle?
Lesson 11 / Exploratory Challenge 1&2
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Conditions on Measurements that Determine a
Triangle
Is there a “Three Angles” condition for unique triangle?
Lesson 11 / Exploratory Challenge 2d
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Two Sides and a Non-Included Angle
Use your tools to draw △ 𝐴𝐵𝐶 provide 𝐴𝐵 = 5, 𝐵𝐶 = 3 𝑐𝑚, and ∠𝐴 = 30°.
• What makes this drawing challenging?
• Use a ruler and compass to locate 𝐶.
Lesson 12 / Exploratory Challenge 1
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Two Sides and a Non-Included Angle
Use your tools to draw △ 𝐷𝐸𝐹 provide 𝐷𝐸 = 5, 𝐸𝐹 = 3 𝑐𝑚, and ∠𝐹 = 90°.
Lesson 12 / Exploratory Challenge 2
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Two Sides and a Non-Included Angle
Use your tools to draw △ 𝐽𝐾𝐿 provide 𝐾𝐿 = 8, 𝐾𝐽 = 4 𝑐𝑚, and ∠𝐽 = 120°.
Conclusions based on the results of Exploratory Challenges 1-3?
Lesson 12 / Exploratory Challenge 3
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NYS COMMON CORE MATHEMATICS CURRICULUM
Checking for Identical Triangles
Lesson 14 / Examples 1-2
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Using Unique Triangles to Solve Real World and
Mathematical Problems
Quadrilateral 𝐴𝐵𝐶𝐷 is a model of a kite. Diagonals 𝐴𝐵 and 𝐶𝐷 represent
the sticks that help keep the kite rigid.
Lesson 15 / Exercise 2
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NYS COMMON CORE MATHEMATICS CURRICULUM
Mid-Module Assessment
• Assessment
• Progression Toward Mastery
Rubric
• Exemplary Student Responses
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Topic C: Slicing Solids
7.G.A.3 Describe the two-dimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right
rectangular pyramids.
Slicing 3-Dimensional figures by a plane and the resulting 2-Dimensional slices.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Slicing a Right Rectangular Prism with a Plane
Lesson 16 / Exercises 5-6
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NYS COMMON CORE MATHEMATICS CURRICULUM
Right Rectangular Pyramid
Bla
Lesson 17
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Right Rectangular Pyramid
Lesson 17
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Slicing a Right Rectangular Pyramid with a Plane
Lesson 17 / Example 5
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NYS COMMON CORE MATHEMATICS CURRICULUM
Slicing on an Angle
Lesson 18 / Example 1
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Slicing on an Angle
Lesson 18 / Example 3
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Understanding 3-Dimensional Figures
If slices parallel to the tabletop are taken of this
figure, then what would each slice look like?
Given the level slices that you determined, how
many unit cubes are in the figure?
Lesson 19 / Exercise 2
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Topic D: Problems Involving Area and Surface Area
7.G.B.6
Solve real-world and mathematical problems involving area, volume, and
surface area of two- and three-dimensional objects composed of triangles,
quadrilaterals, polygons, cubes, and right prisms.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Real World Area Problems
Ten targets are being painted as shown in the following figure. The radius
of the smallest circle is 3 𝑖𝑛., and each successive, larger circle is 3 𝑖𝑛. more
in radius than the circle before it. A “tester” can of red and of white paint
is purchased to paint the target. Each 8 𝑜𝑧. can of paint covers 16 𝑓𝑡 2 . Is
there enough paint of each color to create all ten targets?
Lesson 20 / Example 2
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Mathematical Area Problems
Examine the change in dimension and area of squares as the lengths and
widths are increased by the given amount.
Lesson 21 / Example 1
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Area Problems with Circular Regions
The diameters of four half circles are sides of a square with side lengths of
7 𝑐𝑚. Find the exact area of the shaded region.
Lesson 22 / Problem Set #4
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NYS COMMON CORE MATHEMATICS CURRICULUM
Surface Area
Determine the surface area of the right prism.
Lesson 23 / Exercise 4
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Surface Area
A right rectangular pyramid has a square base with side lengths of 10
inches. The surface area of the pyramid is 260 𝑖𝑛2 . Find the height of the
four lateral triangular faces.
Lesson 24 / Example 3
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Topic E: Problems Involving Volume
7.G.B.6 Solve real-world and mathematical problems involving area, volume, and
surface area of two- and three-dimensional objects composed of triangles,
quadrilaterals, polygons, cubes, and right prisms.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Volume of Right Prisms
Two containers are shaped like right triangular prisms, each with the same
height. The base area of the larger container is 200% more than the base
area of the smaller container. How many times must the smaller container
be filled with water and poured into the larger container in order to fill the
larger container?
Lesson 25 / Example 3
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Volume of Composite 3-Dimensional Objects
Find the volume of the right prism shown whose base is the region
between two right triangles. Use two different strategies.
Lesson 26 / Example 2
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Real-World Volume Problems
Jim wants to know how much his family spends on water for showers.
Water costs $1.50 per 1000 gallons. His family averages 4 showers per day
(total). The average length of a shower is 10 minutes. He places a bucket
in his shower and turns on the water. After one-minute, the bucket has 2.5
gallons of water in it. About how much money does his family spend on
water for showers in a month (30 days)?
Lesson 27 / Exit Ticket
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment
• Assessment
• Progression Toward Mastery
Rubric
• Exemplary Student Responses
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A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Module 6 in Review
Take a few moments to discuss, with those around you, the major
highlights of the Module.
• Likes
• What was and what is
• Concerns
• Major Takeaways
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Key Points
•
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Student transitioning to more complex levels of geometry
Using algebra to solve unknown angle problems
Explore the structure of triangles
Determine the conditions that determine unique triangles
Identify identical triangles and use their relationship to solve problems
Describe the slices of three-dimensional figures
Apply strategies for area, surface area, and volume in solving real world
and mathematical problems
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