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Transcript
Livingston County Schools Seventh (7th) Grade PRE-ALGBRA Unit 4 Math: ACCELERATED Unit Overview Unit 4: Creating, Comparing, and Analyzing Geometric Figures Students use geometric construction, as well as solve real-world and mathematical problems involving area, surface area, and volume of 2 and 3 dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. They draw, construct, and describe geometrical figures and describe the relationships between them. They solve real-life and mathematical problems involving angle measure, area, surface area, and volume. They understand congruence and similarity using physical models, transparencies, or geometry software. They solve problems involving volume of cylinders, cones, and spheres. Length of unit: 39 days KY Core Academic Standard 7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Learning Target K Use ratios and proportions to create scale drawing X Identify corresponding sides of scaled geometric figures X Compute lengths and areas from scale drawings using strategies such as proportions. X Solve problems involving scale drawings of geometric figures using scale factors. R S P Critical Vocabulary Scale Drawings Scale Factors Corresponding Sides Corresponding Angles Congruent Figures X Texts/Resources/Activities Geometric Solids Lab Proportion Proportional Straightedge Protractor Compass Ruler Page 1 of 10 Remember Quantiles, CIITS, resources at www.3x3links.com/network , Critical Areas, and 8 Mathematical Practices!!! 7th Grade PRE-ALGEBRA Unit 4 Parallel Reproduce a scale drawing that is proportional to a given geometric figure using a different scale. 7.G.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. I can state the conditions which create unique triangles, more than one triangle, or no triangle. I can analyze given conditions based on the three measures of angles or lengths of sides of a triangle to determine when there is a unique triangle, more than one triangle, or no triangle. X Perpendicular X X I can construct triangles from three given angle measures to determine when there is a unique triangle, more than one triangle or no triangle using appropriate tools (freehand, rulers, protractors, and technology). X Construct triangles from three given side measures to determine when there is a unique triangle, more than one triangle or no triangle using appropriate tools (freehand, rulers, protractors, X Point Ray Line Line Segment Endpoint Midpoint Supplementary Angles Complementary Angles Adjacent Angles Vertical Angles Acute Angles Obtuse Angles Straight Angles Right Angles Acute triangle Equiangular triangle Right Triangle Obtuse Triangle Scalene Triangle Isosceles Triangle Equilateral Triangle Regular Polygon Cross-Section No-D 1-D Page 2 of 10 Remember Quantiles, CIITS, resources at www.3x3links.com/network , Critical Areas, and 8 Mathematical Practices!!! 7th Grade PRE-ALGEBRA Unit 4 and technology). 7.G.3 Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. I can define slicing as making a cross-section of a 3D figure. X I can describe the twodimensional figures that result from slicing a three-dimensional figure such as a right rectangular prism or pyramid. X 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 circumference and area of a circle. I can identify the parts of a circle including radius, diameter, area, circumference, center, and chord. Analyze three-dimensional shapes by examining two dimensional cross-sections. I can identify and approximate X X π X I can state the formulas for area and circumference of a circle X Given the circumference of a circle, I can find its area. X Given the area of a circle, I can find its circumference. X 2-D 3-D Base Face Lateral Face Length, width, height Cube Pyramid Prism Cone Cylinder Sphere Polygon Quadrilateral Parallelogram Trapezoid Rectangle Rhombus Square Pentagon Hexagon Septagon Octagon Nonagon Decagon Undecagon Dodecagon n-gon Circle Radius Diameter Page 3 of 10 Remember Quantiles, CIITS, resources at www.3x3links.com/network , Critical Areas, and 8 Mathematical Practices!!! 7th Grade PRE-ALGEBRA Unit 4 I can justify that π can be derived from the circumference and diameter of a circle. X π I can apply circumference or area formulas to solve mathematical and real-world problems X I can justify the formulas for area and circumference of a circle and how they relate to π X I can informally derive the relationship between the circumference and area of a circle. 7.G.5 Use facts about supplementary, complementary, vertical, adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. I can identify and recognize types of angles: supplementary, complementary, vertical, and adjacent. I can determine complements and supplements of a given angle. I can determine unknown angle measures by writing and solving Center Chord Tangent X X Circumference Perimeter Area Lateral Surface Area Surface Area Volume Linear Units Square Units Cubic Units Rotation Reflection Translation Prime Notation Image Center of Rotation Direction of Rotation Degree of Rotation Line of Reflection Dilation Scale Factor of Dilation Coordinates of transformed image X Similar X Interior angle of a triangle Exterior angle of a triangle Page 4 of 10 Remember Quantiles, CIITS, resources at www.3x3links.com/network , Critical Areas, and 8 Mathematical Practices!!! 7th Grade PRE-ALGEBRA Unit 4 Remote interior angle of a triangle algebraic equations based on relationships between angles. 7.G.6 Solve real-world and mathematical problems involving area, volume, and surface area of twoand three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. I can state the formulas for area (triangle, rectangle, circle) and volume (right prisms) X I can find surface area (right prisms). X I can use area, surface area, and volume in real-world problems for two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. X I can solve real-world problems involving area, surface area and volume of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. 8.G.1 a b c Verify experimentally the properties of rotations, reflections, and translations: Transversal Parallel Lines Interior Angles Exterior Angles Alternate Exterior Angles Alternate Interior Angles Corresponding Angles Adjacent Angles Supplementary Angles Vertical Angles X Define & identify rotations, reflections, and translations. X Identify corresponding sides & corresponding angles. X Page 5 of 10 Remember Quantiles, CIITS, resources at www.3x3links.com/network , Critical Areas, and 8 Mathematical Practices!!! 7th Grade PRE-ALGEBRA Unit 4 a. Lines are taken to lines, and line segments to line segments of the same length. Understand prime notation to describe an image after a translation, reflection, or rotation. X b. Angles are taken to angles of the same measure. Identify center of rotation. X Identify direction and degree of rotation. X Identify line of reflection. X c. Parallel lines are taken to parallel lines. Use physical models, transparencies, or geometry software to verify the properties of rotations, reflections, and translations (ie. Lines are taken to lines and line segments to line segments of the same length, angles are taken to angles of the same measure, & parallel lines are taken to parallel lines.) 8.G.2 Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of X Define congruency. X Identify symbols for congruency. X Apply the concept of congruency to write congruent X Page 6 of 10 Remember Quantiles, CIITS, resources at www.3x3links.com/network , Critical Areas, and 8 Mathematical Practices!!! 7th Grade PRE-ALGEBRA Unit 4 rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. statements. X Reason that a 2-D figure is congruent to another if the second can be obtained by a sequence of rotations, reflections, translation. Describe the sequence of rotations, reflections, translations that exhibits the congruence between 2-D figures using words. 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. X Define dilations as a reduction or enlargement of a figure. X Identify scale factor of the dilation. X Describe the effects of dilations, translations, rotations, & reflections on 2-D figures using coordinates. 8.G.4 Understand that a twodimensional figure is similar to another if the second can be obtained Define similar figures as corresponding angles are congruent and corresponding sides are proportional. X X Page 7 of 10 Remember Quantiles, CIITS, resources at www.3x3links.com/network , Critical Areas, and 8 Mathematical Practices!!! 7th Grade PRE-ALGEBRA Unit 4 from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. 8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same Recognize symbol for similar. X Apply the concept of similarity to write similarity statements. X Reason that a 2-D figure is similar to another if the second can be obtained by a sequence of rotations, reflections, translation, or dilation. X Describe the sequence of rotations, reflections, translations, or dilations that exhibits the similarity between 2D figures using words and/or symbols. X Define similar triangles X Define and identify transversals X Identify angles created when parallel line is cut by transversal (alternate interior, alternate exterior, corresponding, vertical, adjacent, etc.) Justify that the sum of interior angles equals 180. (For example, arrange three copies X X Page 8 of 10 Remember Quantiles, CIITS, resources at www.3x3links.com/network , Critical Areas, and 8 Mathematical Practices!!! 7th Grade PRE-ALGEBRA Unit 4 triangle so that the three angles appear to form a line, and give an argument in terms of transversals why this is so. 8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. of the same triangle so that the three angles appear to form a line.) X Justify that the exterior angle of a triangle is equal to the sum of the two remote interior angles. Use Angle-Angle Criterion to prove similarity among triangles. (Give an argument in terms of transversals why this is so.) Identify and define vocabulary: cone, cylinder, sphere, radius, diameter, circumference, area, volume, pi, base, height Know formulas for volume of cones, cylinders, and spheres. X X X Compare the volume of cones, cylinders, and spheres. X Determine and apply appropriate volume formulas in order to solve mathematical and real-world problems for the given shape. X Given the volume of a cone, cylinder, or sphere, find the radii, X Page 9 of 10 Remember Quantiles, CIITS, resources at www.3x3links.com/network , Critical Areas, and 8 Mathematical Practices!!! 7th Grade PRE-ALGEBRA Unit 4 height, or approximate for π. Common Assessments Developed (Proposed Assessment Dates): Learning Check # 4 (Last day before Summer Break) HOT Questions: What does area mean? What does volume mean? What are two approximations for pi? How would you decide when to use each? What is the exact and approximate circumference of a circle with a diameter of 7 meters? …10 feet? Show all work. Find the circumference of a circle with radius 4 centimeters. John used the formula C=2 π r. Susy used the formula C= correct? Explain. πd. Who was Solve C= πd for pi. Use your solution to define pi. What does that mean? How would you use the solution to graphical represent pi? Compare and contrast pyramids, prisms, cones, and cylinders. Make a polygon flowchart/venn diagram. The answer is two-dimensional. What are the questions? Compare the volume of cones, cylinders, and spheres. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Is the statement sometimes true, always true, or never true? Justify your answer. a) Vertical angles are congruent. b) Alternate interior angles are congruent. Page 10 of 10 Remember Quantiles, CIITS, resources at www.3x3links.com/network , Critical Areas, and 8 Mathematical Practices!!! 7th Grade PRE-ALGEBRA Unit 4