Aaron+M.+&+Yolonda+6-8

​ In this outline the roman numeral is the standard, the letter is the expectation, and the number is the example we addressed in class.
 * Geometry Standard for Grades 6-8**

// **A.** precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties; // 1. Class example: We described what all triangles have and what some triangles have that other ones don't have. · Defined the different types of triangles
 * I. Analyze characteristics and properties of two- and three- dimensional geometric shapes and develop mathematical arguments about geometric relationships ** (These explorations are useful for future investigations and manipulations of their shapes)
 * Acute Triangle: A triangle where each andle is less than 90 degrees
 * Obtuse Triangle: A trianglewith one angle greater than 90 degrees
 * Right Triangle: A triangle that has one right angle
 * Isosceles Triangle: A triangle with 2 equal sides
 * Equilateral Triangle: A triangle where all three sides are equal
 * Scalene Triangle: A triangle with no equal sides

2. We described what features all quadrilaterals have. · Defined the different type of quadrilaterals 3. Tues 2/2 Making posters of reflection and rotational symmetry. We worked on page. 76 4. Thurs 2/4 Discussing and reviewing: posters of reflection and rotational symmetry, how many lines of symmetry do Isosceles, Equilateral, and Scalene triangles have 5. Thurs 2/11 Discussing “House of Quadrilaterals”, and triangles & squares compose polygons (pentagons) 6. Tues 2/16 Fitting triangles in polygons to figure out interior angle sum; using Sketchpad tools to determine triangle properties through construction by using the tutorials 7. Tues 1/26 Understanding that quadrilaterals are composed of triangles; creating Venn diagram showing the relationship between quadrilaterals 8. Thurs 1/21 Can you have a quadrilateral where all of the sides are congruent but none of the angles are congruent using poly strips; we demonstrated how having the same size lengths & changing the angle would have 2 pair of opposite congruent angles or 4 congruent angles; we defined a quadrilateral and triangle 9. Tues 1/19 All angles in a triangle add up to 180 degrees; we defined an Isosceles triangle as having 2 // ** equal ** // sides; we brainstormed properties of quadrilaterals 10. Thurs 1/14 Finalizing triangle properties that all triangles have Vs. some triangles have; define polygon and closed curve; define different type of triangles (right, scalene, equilateral, isosceles); 11. Tues 1/12 Played “Guess my rule” game to sort 2 dimensional shapes into properties; defining concave & convex; categorizing triangles by its properties 12. Tues 3/30 If a shape is made up of line segments, it will always be a polygon, not a circle (pg. 51 #9). Circle definition brainstormed 13. Tues 4/6 Using parallelogram and trapezoid models to create area formula by sliding and rotating parts of the model 14. Tues 4/6 Using circle model to create area & circumference formula by sliding and rotating parts of the model (pg. 219). Also, by gathering and graphing (on a coordinate graph) comparing diameter & circumference to derive, which is their ratio 15. Tues 4/6 Using a circle model and realizing a polygon composed of an infinite number of line segments connected to each other will eventually look like a circle, but never really be a circle 16. Thurs 4/8 Played “What’s my shape?” with 3-D geometric shapes and defined characteristics of geometric solids (flat faces & bases, vertices vs. corner, point, and curved surfaces) 17. Thurs 4/8 Discussed & classified polyhedral Vs. non-polyhedra with a handout and pg. 235

// **B.** understand relationships among the angles, side length, perimeters, areas, and volumes of similar objects; // 1. Defined angle as the space between two rays that meet at a vertex · The measurement of an angle is a degree: one degree is 1/360 of a turn 2.

3. Thurs 2/11 Develop a working definition for interior angles of polygons.

4**.** Tues 2/16 Develop formulas for interior angles of polygons (n*180)-360 where n=number of sides 5. Thurs 1/21 Can you have a quadrilateral where all of the sides are congruent but none of the angles are congruent using poly strips; we demonstrated how having the same size lengths & changing the angle would have 2 pair of opposite congruent angles or 4 congruent angles; This results in forming the following quadrilaterals: parallelogram, rectangle, square, kite, rhombus. They have a different quantities of congruent sides, but always have either 2 pair of opposite congruent angles or 4 congruent angles. We defined a quadrilateral and triangle 6. Tues 1/19 All angles in a triangle add up to 180 degrees; we defined an Isosceles triangle as having 2 // **equal** // sides; we brainstormed properties of quadrilaterals. This relates to the sum of all angles that form a straight line. When a line is parallel to one of the sides, the exterior angles and the interior angle of a triangle equal 180 degrees. This is shown by tearing off the tips of the triangle and putting the vertices of each angle together, will form a straight line. 7. Tues 3/16 Define perimeter, area, translations, and rotations 8. Tues 3/9 Drawing shapes for symmetry transformations pg. 133, rotation symmetry pg. 118 9. Thurs 3/25 Through inductive reasoning compared square’s area & side length and realized side length squared=area of a square. Completed the square by drawing triangle to obtain side length, the area off the side of the triangle relates to the Pythagorean theorem. It works only for right triangles and 2 out of 3 side lengths are needed. Hypotenuse definition is brainstormed 10. Tues 3/30 Using “Scratch” to create only similar shapes using one variable pg. 155 #5 11. Tues 3/30 If a shape is made up of line segments, it will always be a polygon, not a circle (pg. 51 #9). Circle definition brainstormed 12. Thurs 4/1 Proving the method of a drawing on developing the Pythagorean theorem pg. 199 #2 13. Thurs 4/1 Develop a formula for a parallelogram, triangle, and trapezoid using area (b*h) and the Pythagorean theorem 14. Tues 4/6 Using a circle model and realizing a polygon composed of an infinite number of line segments connected to each other will eventually look like a circle, but never really be a circle 15. Thurs 4/15 Define surface area formula for a box and area of a triangular prism to figure out the smallest surface area for 24 blocks (pg. 247) and the least amount of surface area for a fixed volume pg. 275 16. Tues 4/20 Discussed what dimensions give the smallest and largest surface area for a rectangular prism pg. 289 17. Tues 4/20 Discussed how to find the volume of a triangular prism V=1/2*l*w*h 18. Tues 4/20 Find the volume of hollow 3D shapes with filler material (cylinder, cone, sphere) 19.Tues 4/20 Comparing volumes of 3D non-polyhedra and a pyramid
 * formed by two adjacent sides of a shape
 * Split the polygon into triangles.
 * Each triangle inside the shape equals 180 degrees
 * add the 180 degrees of each triangle inside the shape.
 * but you have to take in consideration the extra 360 degrees that is formed by all the triangles.
 * the number of sides equal the number of vertices of the triangles.

// **C.** create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship. // 1. Proposed clarification: When we say equal sides (or angles) we mean congruent. Congruent means having exactly the same shape and size. Equal is having the same value. 2. 3. Thurs 2/4 Class conjectures for reflection symmetry, rotation symmetry, & angle of rotation: a) a triangle can be folded in half and look the same on both sides. This would have line symmetry. b) a non-rectangular parallelogram doesn’t have 4 congruent sides and has rotational symmetry only. c) 360 degrees divided by the number of points gives you the angle of rotation for a regular polygon 4.Thurs 2/11 Discussing “House of Quadrilaterals”; conjecture “how to construct a pentagon” 5.Tues 2/16 How do you get the sum of the measure of the interior angles of a regular hexagon and other polygons? (n*180)-360 where n = the number of sides in the polygon or 180(n+4)+360 or 180(n-2); how to find exterior angles 180-(180 (n-2))/2; How many triangles can you divide a shape into? Is there a limit? The minimal number of triangles in a polygon is 2. This supports our equations stated above. 6.Tues 1/26 Prove that a quadrilateral has 360 degrees 7.Thurs 1/21 conjecture an Isosceles triangle has 3 equal or congruent angles; 3 more conjectures: a) all angles are right angles Vs. contains no congruent angles. This diagram was disjoint, b) every angle is smaller than 180 degrees Vs. contains an obtuse angle. This diagram has an intersection, c) all angles are congruent Vs. all sides are congruent. This diagram has an intersection; Can you have a quadrilateral where all the sides are congruent but none of the angles are congruent 8.Tues 1/19 Prove all angles of a triangle add up to 180 degrees 9.Thurs 1/14 Figure out what all triangles have Vs. what some triangles have as properties 10. Tues 3/23 Inductive & deductive reasoning to show the “Triangle Inequality” (a+b > c) theorem & “Quadrilateral Inequality” (a+b+c > d) theorem 11. Thurs 3/18 Create inductive reasoning about similarity with Mystery Hat Man 12. Thurs 3/25 Through inductive reasoning, compared a square’s area & side length and realized the side length squared=area of the square 13. Thurs 3/25 Completed the square by drawing triangle to obtain side length, the area off the side of the triangle relates to the Pythagorean theorem. It works only for right triangles and 2 out of 3 side lengths are needed. 14. Tues 3/30 Using the Geoboard to test the Pythagorean theorem on different types of triangles to see if it works on any other triangles than right triangles 15. Thurs 4/1 Use 2 different strategies (using Pythagorean theorem or completing the square) to draw a segment with length “the square root of 13” pg. 190 #2. 16. Thurs 4/1 Proving the method by a drawing to develop the Pythagorean theorem

// **A.** use coordinate geometry to represent and examine the properties of geometric shapes; // 1. Thurs 3/25 Find area of figures on coordinate grid pg. 174 2. Thurs 3/25 Through inductive reasoning compared square’s area & side length and realized side length squared=area of a square. Completed the square by drawing triangle to obtain side length, the area off the side of the triangle relates to the Pythagorean theorem. It works only for right triangles and 2 out of 3 side lengths are needed. 3. Tues 3/30 Using the Geoboard to test the Pythagorean theorem on different types of triangles to see if it works on any other triangles than right triangles 4.Thurs 4/1 Use 2 different strategies (using Pythagorean theorem or completing the square) to draw a segment with length “the square root of 13” pg. 190 #2. 5. Tues 4/6 Using parallelogram & trapezoid model to create area formula by sliding and rotating parts of the model 6. Tues 4/6 Using circle model to create area & circumference formula by sliding and rotating parts of the model (pg. 219). Also, by gathering and graphing (on a coordinate graph) comparing diameter & circumference to derive, which is their ratio
 * II. Specify locations and describe spatial relationships using coordinate geometry and other representational systems **

// **B.** use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides //. 1. Thurs 3/25 Find area of figures on coordinate grid pg. 174 2. Thurs 3/25 Draw squares of various areas on coordinate grid pg. 175  3. Tues 3/30 Using the Pythagorean theorem to find side lengths & area of a kite on a coordinate grid pg. 176 #2

**III. Apply transformations and use symmetry to analyze mathematical situations.** (These explorations are important to observe and recognize the effect of their manipulation We were able to form 3D figures out of translating 2D shapes. We are able to create other polygons from simple polygons (like composing quadrilaterals from triangles) from transformations. // **A.** describe sizes, positions, and orientations of shapes under informal transformation such as flips turns, slides, and scaling; // 1. Showed examples of reflection and rotation on special type of triangles · Relationships between triangles and quadrilaterals 2. Shapes that are congruent with angles and/or congruent sides have reflection. 3. Thurs 2/11 Define line & rotational symmetry 4.Tues 2/2 & 2/4 Make posters of reflection & rotational symmetry 5.Thurs 1/21 Describing symmetry in an isosceles triangle   6. Tues 3/9 Drawing shapes for symmetry transformations (pg. 133), line reflections (pg. 113-114), and rotation symmetry (pg. 118) 7. Tues 3/16 Define perimeter, area, translations and rotations brainstorm 8. Thurs 4/1 Develop a formula for a parallelogram (B*H), triangle (1/2 B*H) and trapezoid ((b1+b2/2)*h) using area (b*h) and Pythagorean theorem

// **B.** examine the congruence, similarity, and line or rotational symmetry of objects using transformations. // 1. Rotational symmetry: A shape has rotational symmetry if you can turn it on its center of rotation less than 360 degrees and it lands exactly on "top" itself again. 2. Reflective symmetry: A shape that when divided in half is exactly the same. We derived this notion from our Geosketch pad usage during our first project of reflecting triangles to form quadrilaterals. 3. Tues 3/9 Drawing shapes for symmetry transformations (pg. 133), line reflections (pg. 113-114), and rotation symmetry (pg. 118) 4. Tues 3/16 Define perimeter, area, translations and rotations brainstorm 5. Thurs 3/18 Examine similarity with Mystery Hat Man

// **A**. draw geometric objects with specified properties, such as side lengths or angle measures; // 1. 2. 3.Tues 2/2 Make posters of reflection & rotational symmetry
 * IV. Use visualization, spatial reasoning, and geometric modeling to solve problems ** (These activities are useful in communicating math concepts effectively. For example, deriving volume formulas for certain 3D figures based on previous derived figure manipulation from a different figure. After filling a cylinder with "filler material", a cone uses only a third of the "filler material". From that, we were able to determine that the volume of a cone will be a third of the volume of a cylinder. Our cylinder volume was considered our "whole" and was our foundation to base other non-polyhedra volume formulas from.

4.Thurs 1/28 Usisng Sketchpad tools to determine triangle properties from tutorials

5.Tues 1/26 Draw an equilateral triangle by drawing 3 circles; draw Venn diagram showing relationship between quadrilateral 6.Thurs 1/21 Drawing an isosceles triangle to show symmetry and congruence; drawing diagrams of conjectures: a) all angles are right angles Vs. contains no congruent angles. This diagram was disjoint, b) every angle is smaller than 180 degrees Vs. contains an obtuse angle. This diagram has an intersection, c) all angles are congruent Vs. all sides are congruent. This diagram has an intersection; 7. Tues 1/19 Drawing a picture to show if you put all 3 angles together in a straight line they fit together perfectly; introduce the Venn diagram concept; using Geoboard to draw a triangle 8. Tues 1/12 Draw shapes from the projector  9. Tues 3/9 Drawing shapes for symmetry transformations (pg. 133), line reflections (pg. 113-114), and rotation symmetry (pg. 118) 10. Thurs 3/18 Drawing Mystery Hat Man 11. Thurs 3/25 Drawing squares of various areas on coordinate grid pg. 175 12. Tues 3/30 Using the Geoboard to test the Pythagorean theorem on different types of triangles to see if it works on any other triangles than right triangles 13. Thurs 4/1 Use 2 different strategies (using Pythagorean theorem or completing the square) to draw a segment with length “the square root of 13” pg. 190 #2. 14. Thurs 4/8 Activity using and drawing silhouettes or 2-D representations of different views of 3-D shapes pg. 245-246

// **B**. use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area and volume; // 1. Thurs 4/8 Activity using and drawing silhouettes or 2-D representations of different views of 3-D shapes pg. 245-246 2. Tues 4/13 Using a net (2-D) to fold into a 3-D figure to explain volume (formula for a rectangular prism) pg. 265 3. Tues 4/20 Discussed what dimensions give the smallest and largest surface area for a rectangular prism pg. 289

// **C**. use visual tools such as networks to represent and solve problems // ; 1. Using the rods to describe what an angle is and what to determine the angle to be. // 1. Tues 4/6 Using parallelogram & trapezoid model to create area formula by sliding and rotating parts of the model 2. Tues 4/6 Using circle model to create area (pie*r squared) & circumference (D*pie) formula by sliding and rotating parts of the model (pg. 219). Also, by gathering and graphing (on a coordinate graph) comparing diameter & circumference to derive, which is their ratio 3. Tues 4/6 Using a circle model and realizing a polygon composed of an infinite number of line segments connected to each other will eventually look like a circle, but never really be a circle 4. Tues 4/13 Using a net (2-D) to fold into a 3-D figure to explain volume (formula for a rectangular prism) pg. 265 5. Thurs 4/15 Define a surface area formula for a box and area of a triangular prism to figure out the smallest surface area for 24 blocks (pg. 274) and the least amount of surface area for a fixed volume (pg. 275)
 * D**. use geometric models to represent and explain numerical and algebraic relationships; //

// **E.** recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life. // 1. Brought in pictures out of a magazine to describe which shape has reflective symmetry, and which one has rotational symmetry, and which one has both. This standard is related to the third standard about applying transformations because it makes it a tangible reality. 2. Thurs 2/4 Bring in pictures of everyday life in Venn diagram 3. Tues 3/9 Looked at 2008-2009 M.E.A.P. scores 4. Tues 3/16 Looked at visual examples of Ethan’s (nephew) geometry knowledge 5. Tues 4/13 Constructed a Venn diagram of Polyhedra & Non-Polyhedra with everyday life objects and pictures

**Measurement Standard for grades 6-8

// V. Understanding measurable attributes of objects and the units, systems, and processes of measurement //** (This is an important foundation of knowledge for future usage of application; Knowing the "what" helps you to understand the "why". In Bloom's taxonomy knowledge is the lowest level. For example, by knowing the foundation of degrees, we will be able to comprehend its application in polygon angle measurements or polygon characteristics pertaining to degrees. 1. Talked about degrees in regards to the measure of an angle. This allows for consistency amongst everybody.  // **B.** understand relationships among units and convert from one unit to another within the same system; // 1. // **C.** understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume. //  1. We measure angles in degrees. This is appropriate for children because it makes something abstract more concrete. They know what a "turn" is and we relate that to 360 degrees. Then 1 degree is 1/360 degrees or a small part of a turn. This also allows for consistency amongst everybody. Radians and Pie are too abstract for small children. 2. Tues 3/23 Brainstorm ideas about area 3. Thurs 3/25 Find figure’s area on coordinate grids pg. 174 4. Tues 3/30 Define area of rectangle (base*height), which is filled with square units 5. Tues 3/30 Using “Scratch” to create only similar shapes using one variable pg. 155 #5 6. Thurs 4/15 Define surface area for box and area of a triangular prism to understand the units for surface area and volume
 * A.** // understand both metric and customary systems of measurement; //

// ** VI. Apply appropriate techniques, tools, and formulas to determine measurements ** // **.** (These activities are useful for problem solving application) //**A.** use common benchmarks to select appropriate methods for estimating measurements;// 1. used patty paper to estimate the degree of different angles //. // // 2. // Thurs 3/25 Find area of figures on coordinate grids pg. 174 3. Using the rods to describe what an angle is and what to determine the angle to be.

//**B.** select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision;// 1 . Tues 3/23 Find area within figures pg. 174   2. Thurs 3/18 Drawing Mystery Hat Man with a scale factor of 2K & 3K 3. Tues 3/9 Drawing shapes for symmetry transformations (pg. 133), line reflections (pg. 113-114), and rotation symmetry (pg. 118) using angle rulers & patty paper 4. Thurs 3/25 Find area of figures on coordinate grids pg. 174 5. Thurs 3/25 Through inductive reasoning compared square’s area & side length and realized side length squared=area of a square. Completed the square by drawing triangle to obtain side length, the area off the side of the triangle relates to the Pythagorean theorem. It works only for right triangles and 2 out of 3 side lengths are needed. 6. Tues 3/30 Using “Scratch” to create only similar shapes using one variable pg. 155 #5 7. Tues 3/30 Using the Pythagorean theorem to find side lengths & area of a kite on a coordinate grid pg. 176 #2 8. Tues 3/30 Using the Geoboard to test the Pythagorean theorem on different types of triangles to see if it works on any other triangles than right triangles 9. Thurs 4/1 Use 2 different strategies (using Pythagorean theorem or completing the square) to draw a segment with length “the square root of 13” pg. 190 #2. 10. Tues 4/6 Using parallelogram & trapezoid model to create area formula by sliding and rotating parts of the model 11. Tues 4/6 Using circle model to create area & circumference formula by sliding and rotating parts of the model (pg. 219). Also, by gathering and graphing (on a coordinate graph) comparing diameter & circumference to derive, which is their ratio 12. Tues 4/6 Using a circle model and realizing a polygon composed of an infinite number of line segments connected to each other will eventually look like a circle, but never really be a circle 13. Thurs 4/15 Define a surface area formula for a box and area of a triangular prism to figure out the smallest surface area for 24 blocks (pg. 274) and the least amount of surface area for a fixed volume (pg. 275)

// **C.** develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes; // 1. Tues 3/30 Define area of rectangle (base*height), which is filled with square units 2. Thurs 4/1 Develop a formula for a parallelogram, triangle and trapezoid using area (b*h) and Pythagorean theorem 3. Tues 4/6 Using parallelogram & trapezoid model to create area formula by sliding and rotating parts of the model 4. Tues 4/6 Using circle model to create area & circumference formula by sliding and rotating parts of the model (pg. 219). Also, by gathering and graphing (on a coordinate graph) comparing diameter & circumference to derive, which is their ratio 5. Tues 4/20 Discussed how to find the volume of a triangular prism

// **D.** develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders; // 1. Tues 4/13 Using a net (2-D) to fold into a 3-D figure to explain volume (formula for a rectangular prism) pg. 265   2 . Thurs 4/15 Define a surface area formula for a box and area of a triangular prism to figure out the smallest surface area for 24 blocks (pg. 274) and the least amount of surface area for a fixed volume (pg. 275) 3. Tues 4/20 Discussed what dimensions give the smallest & largest surface area for a rectangular prism pg. 289 4.Tues 4/20 Find the volume fo hollow 3D shapes with filler material (cylinder, cone, sphere) 5. Tues 4/20 Comparing volumes of 3D non-polyhedra and pyramids

// **E.** solve problems involving scale factors, using ratio and proportion; // 1. Thurs 3/18 Drawing Mystery Hat Man with a scale factor of 2K & 3K 2. Tues 3/23 Triangle scale factor “K” related to perimeter & area summary 3. Tues 3/30 Using “Scratch” to create only similar shapes using one variable pg. 155 #5

// **F.** solve simple problems involving rates and derived measurements for such attributes as velocity and density //. 1. Overall, these activities, discussions, and explorations are important and useful. The knowledge attained from them can be applied to future math classes as well as life. The incorporation of technology is useful in teaching because children need rigorous, high engagement activities. Also, since our world is driven by technology, it is necessary for it to merge with math knowledge. Many careers are based on geometry such as tailors, and floor tilers are benefited from utilizing the beauty of geometry.

GENERAL COMMENTS: -MANY OF THE IDEAS WERE WELL SUPPORTED AND DEVELOPED; MORE EVIDENCE CAN BE GIVEN IN SOME CASES TO SUPPORT THE DEVELOPMENT OF IMPORTANT CONCEPTS AND IEAS. WHEN MAKING CONNECTIONS BETWEEN STANDARDS AND TO OTHER AREAS OF MATHEMATICS, PLEASE GIVEN AN EXAMPLE OR CLEAR DESCRIPTION TO MAKE THE POINT MORE CLEAR. - GOOD TO INCLUDE A BROAD STATEMENT AT THE BEGINNING OF STANDARDS WHEN DESCRIBING WHY THEY ARE USEFUL, MOST OF THEM NEED MORE ELABORATION, A SENTENCE OR TWO OR AN EXAMPLE. - NOT ALL EXAMPLES INCLUDED ARE EXPLICITLY CLEAR WHY THEY MEET THE PARTICULAR EXPECTATION THEY ARE INCLUDED UNDER. TO HELP WITH THIS, TRY TO USE THE WORDING OF THE STANDARDS IN WRITING YOUR RATIONALE FOR HOW THE IDEAS WE COVER IN CLASS ARE RELATED TO THOSE SUGGESTED IN THE STANDARDS.