Download Unit 4: Geometry - Paramount Unified School District

 ParamountUnifiedSchoolDistrict
EducationalServices
Mathematics7–Unit4
StageOne–DesiredResults
Unit 4: Geometry In this unit, students continue their work with writing and solving equations in order to determine a missing angle measure. In preparation for work on congruence and similarity in Grade 8, they reason about relationships among informal geometric constructions. Students work with three‐dimensional figures, relating them to two‐dimensional figures by examining cross‐sections. Students then build off Grade 6 understandings to solve real‐world and mathematical problems involving area, surface area, and volume of two‐ and three‐dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. They solve real‐
world and mathematical problems involving the area and circumference of a circle. Common Misconceptions: 
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Students may have misconceptions about how to correctly use and read a ruler, compass, and/or protractor. Students may be unsure of which protractor scale to use to measure angles, particularly when the orientation of the angle is not horizontal. Students may confuse the ideas of perimeter and area. Students may interchange the concepts of area, surface area, and volume (including errors with unit labels). Students may have difficulty understanding conservation, i.e. that different three‐dimensional objects or figures can have equivalent volumes. M7, U4 pg. 1
Unit 4 Overview: Geometry Transfer Goals 1) Demonstrate perseverance by making sense of a never‐before‐seen problem, developing a plan, and evaluating a strategy and solution. 2) Effectively communicate orally, in writing, and using models (e.g., concrete, representational, abstract) for a given purpose and audience. 3) Construct viable arguments and critique the reasoning of others using precise mathematical language. Standards Meaning‐Making Draw, construct, and describe geometrical figures and describe the relationships between them. Students will understand that… Students will keep considering… Understandings Essential Questions  How can special angle relationships – supplementary, complementary, vertical, and adjacent – be used to write and solve equations for multi‐step problems?  What are the characteristics of angles and sides that will create geometric shapes, especially triangles?  What are the properties of two‐and three‐dimensional figures?  What is the difference between area and surface area?  How can your understanding about surface area and volume be extended to composite shapes?  How are circumference, diameter, and pi related? 7.G.2 Draw (freehand, with ruler and protractor, and with  Decomposing and rearranging provide a technology) geometric shapes with given conditions. Focus geometric way to both seeing that a on constructing triangles from three measures of angles or measurement formula is the right one and seeing sides, noticing when the conditions determine a unique why it is the right one. triangle, more than one triangle, or not triangle.  Geometric images provide the content in relation 7.G.3 Describe the two‐dimensional figures that result to which properties can be noticed, definitions from slicing three‐dimensional figures, as in plane sections can be made, and invariances can be discerned. of right rectangular prisms and right rectangular pyramids.  Geometric awareness develops through practice in visualizing, diagramming, and constructing. Solve real‐life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Acquisition Knowledge Students will know… 7.G.5 Use facts about supplementary, complementary,  Angle relationships and equations can be used to vertical, and adjacent angles in a multi‐step problem to determine the measure of an unknown angle. write and solve simple equations for an unknown angle in a
 Every triangle has two acute angles. The third angle figure. classifies triangles as acute, right or obtuse. 7.G.6 Solve real‐world and mathematical problems  A triangle can be classified by the number of equal involving area, volume and surface area of two‐ and side measurements – equilateral, isosceles, or three‐dimensional objects composed of triangles, scalene. quadrilaterals, polygons, cubes, and right prisms.  In a triangle, the three angles always add to 180°.  No triangle can have more than one obtuse angle.  The smallest side of a triangle cannot be opposite the largest angle.  The sum of two smaller lengths must be greater than the longest length in order for three given lengths to Skills Students will …  Solve real‐life and mathematical problems involving angle measure including problems requiring writing and solving equations.  Draw, construct, or describe geometric shapes given certain conditions, focusing on triangles.  Describe a two‐dimensional figure resulting from slicing a three‐dimensional figure by a plane.  Solve real‐life and mathematical problems for the area of two‐dimensional objects composed of polygons.  Solve real‐life and mathematical problems for the volume and surface area of three‐dimensional objects composed of right prisms and cubes.  Solve real‐life and mathematical problems for the M7, U4 pg. 2
form a triangle.
 When more than one triangle can be formed, the 
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circumference and area of circles.
triangles are not the same shape. A unique triangle has a set of conditions that, when constructed, creates only one triangle. Similar to triangles, conditions for a quadrilateral can form a unique shape or more than one quadrilateral. A slice, also known as a plane section, consists of all the points where the plane meets the figure. The location and angle of a slice create planes of different shapes and sizes. A three‐dimensional composite figure is made from two or more solid figures. Composite figures can be decomposed into simple geometric figures. Surface area is the sum of the areas of all faces on a three‐dimensional solid. To find the volume of a three‐dimensional composite object, two or more distinct volumes must be added together or subtracted from each other. The distance around a circle is called the circumference. The circumference of every circle is about 3.14 times as long as the diameter. The area of a circle is the number of square units inside that circle. The area of a circle is related to the area of a rectangle. Key Vocabulary Terms: polygon, triangle (right, acute, obtuse, equilateral, isosceles, scalene), quadrilateral, trapezoid, parallelogram, cube, right‐rectangular prism, right‐rectangular pyramid, square pyramid, cone, cylinder, plane, perpendicular, parallel, base of a three‐
dimensional figure, horizontal slice, vertical slice, area, circumference, pi, circle, radius, diameter, supplementary angles, complementary angles, vertical angles, adjacent angles, linear pairs of angles, volume, surface area
M7, U4 pg. 3
ParamountUnifiedSchoolDistrict
EducationalServices
Mathematics7–Unit4
StageTwo–EvidenceofLearning
Unit 4: Geometry Transfer is a student’s ability to independently apply understanding in a novel or unfamiliar situation. In mathematics, this requires that students use reasoning and strategy, not merely plug in numbers in a familiar‐looking exercise, via a memorized algorithm. Transfer goals highlight the effective uses of understanding, knowledge, and skills we seek in the long run – that is, what we want students to be able to do when they confront new challenges, both in and outside school, beyond the current lessons and unit. These goals were developed so all students can apply their learning to mathematical or real‐world problems while simultaneously engaging in the Standards for Mathematical Practices. In the mathematics classroom, assessment opportunities should reflect student progress towards meeting the transfer goals. With this in mind, the revised PUSD transfer goals are: 1) Demonstrate perseverance by making sense of a never‐before‐seen problem, developing a plan, and evaluating a strategy and solution. 2) Effectively communicate orally, in writing, and using models (e.g., concrete, representational, abstract) for a given purpose and audience. 3) Construct viable arguments and critique the reasoning of others using precise mathematical language. Multiple measures will be used to evaluate student acquisition, meaning‐making, and transfer. Formative and summative assessments play an important role in determining the extent to which students achieve the desired results in stage one. Formative Assessment Summative Assessment Aligning Assessment to Stage One  What constitutes evidence of understanding for this lesson?  What evidence must be collected and assessed, given the desired results defined in stage one?  Through what other evidence during the lesson (e.g. response to questions, observations, journals, etc.) will students demonstrate achievement of the  What is evidence of understanding (as opposed to recall)? desired results?  Through what task(s) will students demonstrate the desired understandings?  How will students reflect upon, self‐assess, and set goals for their future learning? Opportunities
 Inquiry lesson discussions  Unit assessments  Checking for understanding (response boards)  Teacher‐created chapter tests or mid‐unit assessments  Ticket out the door, Cornell note summary, and error analysis  Illustrative Mathematics tasks (https://www.illustrativemathematics.org/)  Illustrative Mathematics tasks (https://www.illustrativemathematics.org/)  Performance tasks  Quizzes M7, U4 pg. 4
The following pages address how a given skill may be assessed. Assessment guidelines, examples and possible question types have been provided to assist teachers in developing formative and summative assessments that reflect the rigor of the standards. These exact examples cannot be used for instruction or assessment, but can be modified by teachers. Skill Standard Assessment Guidelines
Solve real‐life and mathematical problems involving angle measure including problems requiring writing and solving equations. 7.G.5 Students solve real‐life and mathematical problems involving angle measure including problems requiring writing and solving equations. Measures of certain angles in the figure can be shown. Measures of angles shown in the figure should be less than 180. Figures should involve supplementary, complementary, vertical, and/or adjacent angles that contain a missing angle measure. Difficulty can be adjusted:  Angle measures can be whole numbers or decimals to the tenths place. Example Possible
Response Type(s)  Matching Tables Lines XU and WY intersect at point A. Use of Calculator Yes
Based on the diagram, determine whether each statement is true. Select True or False for each statement. (T, T, T)  Matching Tables Lines XU and WY intersect at point A. Based on the diagram, determine whether each statement is true. Select True or False for each statement. (T, T, F) M7, U4 pg. 5
Assessment Guidelines
(cont.) (cont.) Students solve a multi‐step problem involving supplementary, complementary, vertical, and/or adjacent angles. Difficulty can be adjusted:  Angle measures are whole numbers.  Angle measures are decimals.  Angle measures include variables. Students solve an equation for an unknown angle in a figure involving supplementary, complementary, vertical, and/or adjacent angles. Difficulty can be adjusted:  Angle measures are whole numbers.  Angle measures are decimals.  Angle measures include variables. Example Possible
Response Type(s)  Equation/Numeric Consider this figure.
Write the measure of (56) Use of Calculator Yes
C
V
Y
Standard ∠
Skill , in degrees.  Matching Tables The base of a hexagon lies on ray AB as shown.
Based on the diagram, determine whether each equation is true. Select True or False for each statement. (T, T, F) M7, U4 pg. 6
Skill Draw, construct, or describe geometric shapes given certain conditions, focusing on triangles. Standard Assessment Guidelines
Example 7.G.2 Students identify the type of triangles that can be constructed based on given information about the sides and angles of the triangle. Information given may either:  Describe a unique triangle  Be insufficient to determine a unique triangle (more than one triangle)  Be inconsistent with the triangle inequality of the triangle angle sum theorem (no triangle) Difficulty can be adjusted:  Triangles may be right, acute, obtuse, equilateral, isosceles, or scalene.  Angle measures and unit lengths are composed of differing positive rational numbers.  Inclusion of extraneous information. Students generate geometric shapes based on given conditions. Conditions should determine a unique polygon, and measurements should be positive integers reasonable for display in the workspace provided. Difficulty can be adjusted:  Figures may consist of triangles (right, acute, obtuse, equilateral, isosceles, scalene), quadrilaterals, trapezoids or parallelograms, or combinations of the above.  Combinations of given side lengths and angles.  Inclusion of extraneous information. A triangle has a 45° angle, a 60° angle, and a side 3 centimeters in length. Select True or False for each statement about this type of triangle. Possible
Response Type(s)  Matching Tables Use of Calculator No
(F, T, T) Draw a triangle with a 90° angle, a side with a length of 7 units, and a side with a length of 4 units. Each square on the grid is 1 unit in length. Solution:  Graphing M7, U4 pg. 7
Skill Standard Assessment Guidelines
Describe a two‐
dimensional figure resulting from slicing a three‐
dimensional figure by a plane. 7.G.3 Students identify a two‐dimensional figure that results from slicing a given three‐
dimensional figure by a plane. Difficulty can be adjusted:  Three‐dimensional figures can be right‐
rectangular prisms or right‐rectangular pyramids.  Slices may be horizontal planes, vertical planes, or planes that are not parallel to a face.  Increasing number of possible answer choices. Solve real‐life and mathematical problems for the area of two‐
dimensional objects composed of polygons. 7.G.6 Students identify the area of two‐
dimensional figures composed of triangles, quadrilaterals, and/or other polygons for both real‐life and mathematical problems. Dimensions of figures can be explicitly labeled or indicated by a grid. Difficulty can be adjusted:  Dimensions can include whole numbers, decimals, and fractions including mixed numbers or any combination.  Figures can be composed of triangles, quadrilaterals, or polygons. Example Possible
Response Type(s)  Multiple Choice, Multiple Correct Response Use of Calculator No
Select all figures that can be formed by a vertical slice perpendicular to the base of the square pyramid. A. Isosceles trapezoid B. Line segment C. Square D. Triangle (A, B, D) This is the floor plan of Julie’s bedroom.
 Equation/Numeric Yes
Write the amount of carpet, in square feet, needed to completely cover Julie’s bedroom floor. (46 ¾) The figure shown is created by joining three rectangles.
 Equation/Numeric This figure is a square pyramid. Write the area of the figure, in square centimeters. Round to the nearest hundredth. (46.75) M7, U4 pg. 8
Skill Standard Assessment Guidelines
Solve real‐life and mathematical problems for the volume and surface area of three‐
dimensional objects composed of right prisms and cubes. 7.G.6 Students identify the surface area or volume of three‐dimensional objects. Difficulty can be adjusted:  Dimensions can include whole numbers, decimals, and fractions including mixed numbers or any combination.  Figures can be composed of cubes and right prisms or any combination. Example The figure shows a set of concrete stairs to be built.
Possible
Response Type(s)  Equation/Numeric Use of Calculator Yes
Write the amount of concrete, in cubic feet, needed to build the stairs. Round your answer to the nearest hundredth. (5.82) The figure shown is created by joining two right rectangular prisms.  Equation/Numeric Write the volume of the figure, in cubic centimeters. (72) The figure shows the dimensions for a package to be shipped.
 Equation/Numeric Write the minimum amount of wrapping paper, in square inches, needed to cover the package. Round your answer to the nearest whole inch. (174) M7, U4 pg. 9
Skill Standard Assessment Guidelines
Example Solve real‐life and mathematical problems for the circumference and area of circles. 7.G.4 Students give the area of circles for mathematical problems. Difficulty can be adjusted:  Radius is a whole number, decimal, fraction, including mixed numbers.  Diameter is a whole number, decimal, fraction, including mixed numbers.  Number of computational steps.  Partial areas or circumferences. Students give the area of circles for real‐life problems. Difficulty can be adjusted:  Radius is a whole number, decimal, fraction, including mixed numbers.  Diameter is a whole number, decimal, fraction, including mixed numbers.  Number of computational steps.  Partial areas or circumferences. The radius of a circle is 7.5 centimeters.
Write the area of the circle, in square centimeters. Round your answer to the nearest hundredth. (176.63 – 176.79) Possible
Response Type(s)  Equation/Numeric A circular table top has a radius of 3 feet.
Write the area, in square feet, of the table top. Round your answer to the nearest hundredth. (28.26 – 28.305)  Equation/Numeric Jill buys two circular pizzas.
The small pizza has The medium pizza has an 8‐inch diameter. a 12‐inch diameter.  Equation/Numeric Use of Calculator Yes
How much greater, in square inches, is the area of the medium pizza than the small pizza? Round your answer to the nearest hundredth. (62.80 – 62.90) M7, U4 pg. 10
Skill Standard Assessment Guidelines
(cont.) (cont.) Students give the circumference of a circle.
Difficulty can be adjusted:  Radius is a whole number, decimal, fraction, including mixed numbers.  Diameter is a whole number, decimal, fraction, including mixed numbers.  Number of computational steps.  Partial areas or circumferences. Example Students give a fractional part of the area of a circle for both real‐life and mathematical problems. Difficulty can be adjusted:  Radius is a whole number, decimal, fraction, including mixed numbers.  Diameter is a whole number, decimal, fraction, including mixed numbers.  Number of computational steps.  Partial areas or circumferences. The radius of a circle is 7 centimeters.
Write the circumference of the circle, in centimeters. Round your answer to the nearest hundredth. (43.96 – 44.03) Possible
Response Type(s)  Equation/Numeric The circumference of a circle is 31.4 inches.
Write the radius of the circle, in inches. Round your answer to the nearest whole number. (5)  Equation/Numeric A corner shelf has a radius of 10.5 inches and represents ¼ of a circle, as shown.  Equation/Numeric Write the area of the shelf, in square inches. Round your answer to the nearest hundredth. (86.55 – 86.68) The circumference of the circle is approximately 100.48 centimeters. The shaded region is 3/10 of the whole circle. Write the area of the shaded region, in square centimeters. Round your answer to the nearest hundredth. (240.77 – 241.15) Use of Calculator Yes
 Equation/Numeric M7, U4 pg. 11
ParamountUnifiedSchoolDistrict
EducationalServices
Mathematics7–Unit4
StageThree–LearningExperiences&Instruction
Unit 4: Geometry Prior to planning for instruction, it is important for teachers to understand the progression of learning and how the current unit of instruction connects to previous and future courses. Teachers should consider: What prior learning do the standards and skills build upon? How does this unit connect to essential understandings of later content? How can assessing prior knowledge help in planning effective instruction? What is the role of activating prior knowledge in inquiry? Looking Back Looking Ahead In Mathematics 6, students: In Mathematics 8, students will:  Solved for the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; applied these techniques in the context of solving real‐world and mathematical problems.  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real‐world and mathematical problems.  Solved for volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths.  Applied the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths to solve real‐world and mathematical problems.  Distinguish between rational and irrational numbers.  Use rational approximations of irrational numbers to compare sizes of irrational numbers.  Drew polygons in the coordinate plane given coordinates for the vertices and used coordinates to find the length of a side in the context of real‐world and mathematical problems.  Represented three‐dimensional figures using nets made up of rectangles and triangles, and used the nets to find the surface area in the context of real‐world and mathematical problems. M7, U4 pg. 12
Mathematics7
Timeframe: Feb. 14 – Mar. 31
Unit 4: Geometry Course Textbook: McGraw Hill, California Math
Transfer Goals: 1) Demonstrate perseverance by making sense of a never‐before‐seen problem, developing a plan, and evaluating a strategy and solution. 2) Effectively communicate orally, in writing, and using models (e.g., concrete, representational, abstract) for a given purpose and audience. 3) Construct viable arguments and critique the reasoning of others using precise mathematical language. Understandings: Essential Questions:
 How can special angle relationships – supplementary, complementary, vertical, and adjacent – be  Decomposing and rearranging provide a geometric way to both used to write and solve equations for multi‐step problems? seeing that a measurement formula is the right one and seeing why 
What are the characteristics of angles and sides that will create geometric shapes, especially it is the right one. triangles?  Geometric images provide the content in relation to which  What are the properties of two‐and three‐dimensional figures? properties can be noticed, definitions can be made, and invariances  What is the difference between area and surface area? can be discerned.  How can your understanding about surface area and volume be extended to composite shapes?  Geometric awareness develops through practice in visualizing,  How are circumference, diameter, and pi related? diagramming, and constructing. Time & Skills Learning Goals
Possible Lessons & Knowledge Focus Questions
Teacher Notes
Content Activities 4 days Activate Prior Knowledge – possible activity: Geometry Tools Angle Solve real‐life and Reason about angle How can you use angle Angle relationships and Students may Inquiry Task: Use a Measures mathematical relationships. relationships to create and benefit from straight edge to construct equations can be used to problems determine the measure of an solve equations to thinking about a a large X. Measure all Identify angle involving angle determine an unknown for pan balance to angles. What relationships unknown angle. relationships, including measure including an angle figure? help set up and do you notice about the vertical, adjacent, vertical angles problems solve equations. angle measurements? complementary, and adjacent angles requiring writing 7‐1 Classify Angles supplementary. and solving ** Additional TE pp. 535 – 542 Represent angle teacher‐created equations. (examples 2‐3) (calculator allowed) relationships with practice for For now, the idea of two angles being equal should writing and equations. focus on the fact each angle has the same degree solving equations Use angle relationships to measurement as the other. Refrain from using the is needed. create and solve equations term “congruent.” to determine an unknown for an angle figure. Verify solutions and reason about angle measures in the context of the problem. M7, U4 pg. 13
Time & Content 4 days (cont.) Skills (cont.) Learning Goals
(cont.) Possible Lessons & Activities Inquiry Task: Construct each angle on a piece of patty paper: 30°, 60°, 120°. Make combinations of angle pairs (join at vertex) and document your observations. 7‐2 Complementary and Supplementary Angles TE pp. 543 – 550 Use angle relationships to Inquiry Task A: Multi‐Step create and solve equations Angle Problems to determine an unknown for an angle figure. Knowledge Focus Questions
Teacher Notes
(cont.) (cont.)
(cont.)
complementary angles supplementary angles linear pairs of angles Problems for student practice: p. 542, #26 p. 548, #12, 14 p. 550, #25 6 days Draw, construct, or describe Classify & geometric shapes Construct given certain Shapes conditions. Verify solutions and reason about angle measures in the context of the problem. Inquiry using GeoGebra:
Review types of triangles.
Classify Triangles by Sides Discover how the and Angles classification of a triangle https://www.geogebra.org
changes when the side /m/V39WMWrx and/or angle measurements change. 7‐3 Triangles TE pp. 555‐562 Discover not just any set of three angle measures will make a triangle. TRIANGLE
a° + b° + c° = 180° Determine a rule for identifying whether 3 angle measures will form a triangle or not. Construct triangles and LearnZillion: Describe the type(s) of triangle(s) describe the type of formed by analyzing the triangle formed by analyzing the conditions of conditions of the triangle constructions the triangle. Code: 4MB8PU5 Every triangle has two acute angles. The third angle classifies triangles as acute, right, or obtuse. A triangle can be classified by the number of equal side measurements – equilateral, isosceles, or scalene. In a triangle, the three angles always add to 180°. No triangle can have more than one obtuse angle. How do angles determine triangle classification? How do sides determine triangle classification? How can you tell if given angle measures for a triangle will indeed make a triangle? Practice for classifying triangles by angles (4.G.2) http://www.math‐
play.com/classifying
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triangles/classifying
‐triangles.html Can any side be opposite any angle? Explain. If you are given two angle The smallest side of a triangle measures, can you classify cannot be opposite the largest the triangle by angles? By angle. sides? Explain. acute, right, obtuse triangles How can knowing the equilateral, isosceles, scalene angles of a triangle help you triangles classify the triangle by the sides? LearnZillion practice exercises can be used for independent practice. M7, U4 pg. 14
Time & Content 6 days (cont.) Skills (cont.) Learning Goals
Possible Lessons & Activities Discover not just any set of Inquiry using Annenberg
Learner: Linkage‐Strip three lengths will make a Constructions for Triangles triangle. http://www.learner.org/co
Determine a rule for urses/learningmath/geom
identifying whether 3 etry/session2/part_b/inde
lengths will form a triangle x.html or not. OR Knowledge Focus Questions
Teacher Notes
The sum of two smaller lengths must be greater than the longest length in order for three given lengths to form a triangle. Can you make a triangle with any three side lengths? Why or why not? How do you determine whether a triangle can be formed given three side lengths? ** Additional teacher‐created practice needed. Construct a triangle given three conditions. Measure side lengths and angle measures. Construct triangles based on given conditions. Discover that based on given conditions:  more than one triangle can be drawn  only one unique triangle can be drawn  no triangle can be drawn Determine whether a unique triangle can be created with three given conditions. Inquiry using GeoGebra: Triangle Inequality Exploration https://www.geogebra.org
/m/pHUfRu7V TRIANGLE
short side + short side > longest side NOT A TRIANGLE short side + short side ≤ longest side OR Inquiry Lab: Create Triangles TE pp. 551‐552 Inquiry using Annenberg Learner: Different Triangles http://www.learner.org/co
urses/learningmath/geom
etry/session2/part_a/inde
x.html Inquiry Task B: Construct Triangles Based on Given Conditions When more than one triangle can be formed, the triangles are not the same shape. Scale drawings of a triangle have equal corresponding angle measurements, but corresponding side lengths that are proportional. A unique triangle has a set of conditions that, when constructed, creates only one triangle. What makes a triangle unique? How can you tell if a set of conditions for a triangle forms a unique triangle? More than one triangle? No triangle? ** Additional teacher‐created practice needed. condition unique triangle M7, U4 pg. 15
Time & Content 6 days (cont.) Skills Learning Goals
Possible Lessons & Activities Determine how a set of Inquiry using Annenberg conditions for a Learner: Linkage‐Strip quadrilateral forms a Constructions for Triangles unique shape or more than http://www.learner.org/co
urses/learningmath/geom
one quadrilateral. etry/session2/part_b/cons
tructing.html Knowledge Focus Questions
Teacher Notes
Similar to triangles, conditions for a quadrilateral can form a unique shape or more than one quadrilateral. How can you determine if a set of conditions for a quadrilateral forms a unique shape or more than one quadrilateral? ** Additional teacher‐created practice needed. quadrilateral parallel right angle perpendicular parallelogram rhombus trapezoid polygon 1 day Opportunities for Review: * Review procedural skills * Fluency practice * Open‐ended questions * Independent practice with transfer goals * Illustrative Mathematics tasks (https://www.illustrativemathematics.org/) Review & Assess Formative Assessment 2 days Opportunities for Review: * Return to The Boarder Problem * Return to essential questions * Review procedural skills * Fluency practice * Open‐ended questions * Illustrative Mathematics tasks (https://www.illustrativemathematics.org/) * Independent practice with transfer goals Review & Assess Mid‐Unit Summative Assessment (teacher‐created) 2 days Cross Sections Describe a two‐
dimensional figure resulting from slicing a three‐dimensional figure by a plane. Understand that a two‐
dimensional cross‐section can be formed by slicing a three‐dimensional figure. Describe the figure(s) created by slicing a three‐
dimensional figure. Apply cross‐sections to a real‐world context. Inquiry Task C: Cross Sections A slice, also known as a plane section, consists of all the points where the plane meets the figure. The location and angle of a slice create planes of different shapes and sizes. plane plane section horizontal slice vertical slice angled (or diagonal) slice right‐rectangular pyramid square pyramid What two‐dimensional figures can be made by slicing a prism (and pyramids) by planes? How does slice location and direction impact the resulting plane section? What strategies help you visualize a cross section? Materials:
Tubs of play dough Dental floss M7, U4 pg. 16
Time & Content 1 day Skills Possible Lessons & Activities Knowledge Focus Questions
Teacher Notes
Review the previous unit assessment ‐‐ Activities include error analysis, comparing extended‐constructed responses to exemplars and the rubric, practice 4 days Solve real‐life and mathematical Composite problems for the Figures – area of two‐
Area, dimensional Surface objects composed Area & of polygons. Volume Learning Goals
Understand area of composite shapes. Calculate area to solve a problem given a real‐world context. (calculator allowed) Inquiry Task:
Find the area of the figure below. A three‐dimensional composite figure is made from two or more solid figures. Composite figures can be decomposed into simple geometric figures. composite figure How does decomposing a figure help when determining the area? Surface area is the sum of the
areas of all faces on a three‐
dimensional solid. Each face is made up of two‐
dimensional figures. The shape of each face is analyzed and the area is computed. How does decomposing a three‐dimensional shape help when determining the volume? How can I use surface areas of plane figures to derive formulas for the surface area of solid figures? 8‐3 Area of Composite Figures TE pp. 631 – 638 (examples 2‐4) Solve real‐life and mathematical problems for the volume and surface area of three‐dimensional objects composed of right prisms and cubes. Understand volume and surface area. Calculate volume and surface area to solve a problem given a real‐world context. (calculator allowed) Week of March 6 Integers Fluency Assessment Scores due March 17th. 8‐4 Volume of Prisms
TE pp. 639‐646 (example 2) 8‐6 Surface Area of Prisms TE pp. 665‐672 Inquiry Lab: Composite Figures TE pp. 685‐688 8‐8 Volume and Surface Area of Composite Figures TE pp. 689‐696 (example 1, guided practice 2) To find the volume of a three‐
dimensional composite object, two or more distinct volumes must be added together (if they are joined together) or subtracted from each other (if one is a missing section of the other). Right Angle prisms:
face, base surface area cube right‐rectangular prism M7, U4 pg. 17
Time & Skills Learning Goals
Possible Lessons & Knowledge Focus Questions
Teacher Notes
Content Activities 1 day Opportunities for Review: * Review procedural skills * Fluency practice * Open‐ended questions * Independent practice with transfer goals * Illustrative Mathematics tasks (https://www.illustrativemathematics.org/) Review & Assess Formative Assessment 6 days Circum. And Area of Circles Discover pi. Discover the formula for the circumference of a circle (using diameter and radius). (calculator allowed) Solve real‐life and mathematical problems for the circumference and area of circles. Determine the circumference of a circle. Determine the diameter or radius based on the given circumference of a circle. Solve real‐world and mathematical problems involving circumference of a circle. Illuminations Lesson: The Ratio of Circumference to Diameter http://illuminations.nctm.
org/lesson.aspx?id=1849 ** Prior to the lesson, ask students to bring in several flat, circular objects that they can measure. 8‐1 Circumference
TE pp. 613‐620 The distance around a circle is called the circumference. The circumference of every circle is about 3.14 times as long as the diameter. The ratio of the circumference to the diameter circumference
(
) of every diameter
circle is a ratio approximated 22
at (or 3.14) and 7
represented by the Greek letter π. C
  , then C   D . D
How is the circumference of a circle similar to (and different from) the perimeter of a quadrilateral? circumference diameter pi, π radius M7, U4 pg. 18
Time & Content 6 days (cont.) Skills (cont.) Learning Goals
Possible Lessons & Knowledge Focus Questions
Teacher Notes
Activities Discover the formula for How is finding the area of a Inquiry Task: Using Radius The area of a circle is the area of a circle. Squares to Find the Area of number of square units inside circle similar to (and different from) finding the a Circle (see below) that circle. area of a quadrilateral? OR The area of a circle is related to the area of a rectangle. A Illuminations Lesson: circle can be cut into wedges and rearranged into a Discovering the Area Additional activities (simulations):
parallelogram. The more Formula for Circles wedges, the closer the figure http://illuminations.nctm.
Inquiry using GeoGebra: Circle to Parallelogram Area org/Lesson.aspx?id=1852 is to a rectangle. The two relationship lengths of the rectangle are https://www.geogebra.org/m/yNjqMQ9G ** Use the flat, circular made from the edge of the objects students brought circle (circumference) and the Inquiry using GeoGebra: Circle Area (By Peeling!) in. heights are both radii. The https://www.geogebra.org/m/WFbyhq9d area of the circle can be determined by multiplying the length (half of the circumference, 2πr, is πr) by the width (r). A = πr2 is used to find the area of every circle. area square units Inquiry Task: Using Radius Squares to Find the Area of a Circle Provide each student with the same‐sized circle and several radius squares (example shows a circle with a radius of 5 units, so each radius square must be a 5 x 5 grid). Ask: Predict ‐‐ How many radius squares do you think will be needed to cover the circle? Students then cut and glue radius squares to cover the circle in a way that allows them to determine how many radius squares are needed. Students will discover it takes approximately 3.14 radius squares to cover the circle. Ask: How many radius squares did it take to cover the circle? (approximately 3.14 radius squares ‐‐ WOW!!!) Ask: What do the squares represent? (the area of the circle) Ask: Estimate the area of the circle? (take all estimates and ask how estimated) (should mention using 5 x 5 radius squares) Ask: Is there a way to find the area of the circle without counting all of the squares? Discovery: finding the area of a circle is similar to finding the area of a quadrilateral when using radius squares. The idea that area is length x width for a quadrilateral can be extended to length x width x pi for circles). M7, U4 pg. 19
Time & Content 6 days (cont.) Skills (cont.) Learning Goals
Determine the area of a circle. Solve real‐world and mathematical problems involving area of a circle. Possible Lessons & Activities 8‐2 Area of Circles
TE pp. 623‐630 (examples 1‐2, 4‐5) 8‐3 Area of Composite Figures TE pp. 631 – 638 (example 1) Knowledge Given the radius or diameter of a circle, we can find its area. Focus Questions
Teacher Notes
Is the area of a portion of a circle proportional to the area of a full circle? Explain. semicircle 1 day Independent practice with transfer goals [e.g., Illustrative Mathematics tasks (https://www.illustrativemathematics.org/), fluency practice, open‐ended questions] 5 days Opportunities for Review: * Return to essential questions * Review procedural skills * Fluency practice * Open‐ended questions Review * Illustrative Mathematics tasks (https://www.illustrativemathematics.org/) * Independent practice with transfer goals & Assess March 27‐31 End‐of‐Unit Summative Assessment M7, U4 pg. 20