Jessica+D.+&+Katey+A.+6-8

Be sure to communicate with each other about this project and how you plan to make it a collaborative effort. You might try trading roles each week or after each class period - editor and creator. Nicole

= Geometry Standard for Grades 6 - 8 =

= Geometry Standard:  Analyze characteristics and properties of two dimensional geometric shapes and develop mathematical arguments about geometric relationships. =

=Expectations used: =  = =
 * =Precisely describe, classify and understand relationships among types of two-dimensional objects using their defining properties =
 * ==How this was used in class: ==
 * ==1.14.2010: We discussed characteristics of triangles; characteristics shared by all triangles and characteristics shared by only some triangles. We came up with a definition for a triangle and polygon. Also discussed the different types of triangles. ==
 * ==1.19-21.2010: Discussed triangles and used Venn Diagrams to compare different types of triangles. After drawing a Venn Diagram for an Acute triangle and an Equilateral triangle, we came to the conclusion that all equilateral triangles will be acute; 3 equal sides, 60 degrees each. We also discussed using equal and congruent interchangeably, and that congruent meant having exactly the same shape and size and equal meant having the same value. We discussed quadrilaterals and completed Venn diagrams just like we for triangles, CP p37. We discussed the question, "Can you have a quadrilateral where all the sides are congruent, but none of the angles are congruent?" We decided the answer was "No," because if you have congruent sides, you're going to have at least two congruent angles. We also came up with necessary and sufficient definitions for a quadrilateral and a triangle. ==
 * ==1.26.2010: We created more Venn diagrams comparing different types of quadrilaterals in class. We compared: kite and parallelogram, kite and rectangle, square and rhombus, and kite and trapezoid. ==
 * ==2.2-2.4.10: We classified triangles, quadrilaterals, and circles into three different groups: Asymmetrical, Reflection Symmetry, and Rotational Symmetry. From the work we did categorizing these shapes we came up with a conjecture that states, "If a shape has more than one line of symmetry it also has rotational symmetry." [WHAT DOES THIS CONJECTURE IMPLY ABOUT THE RELATIONSHIP BETWEEN ROTATIONAL AND REFLECTION SYMMETRY?] ==
 * ==3.25.2010: We started talking about the Pythagorean Theorem and what the definition of a hypotenuse is. We decided that the hypotenuse was the 'side opposite of the right angle'. The interesting thing about this is that the Pythagorean Theorem can only be applied to the right triangle. We determined this through inductive reasoning; taking several different triangles and attempting to find their side lengths using the Pythagorean Theorem. We used the notion of the Pythagorean Theorem to help us find unknown side lengths of triangle and of any line segment we didn't know the length of. We used two different methods of figuring this out. The Pythagorean Theorem and also the idea of the are of a square. If you know the area of a square or rectangle you can work backwards with that formula and get the length of a side. ==

 =How this was used in class: =
 * ==Precisely describe, classify, and understand relationships among types of three-dimensional objects using their defining properties.==

==4.8.2010: We discussed the characteristics of geometric solids. We used the hand-out, "Geometric Solids and Their Parts" as part of our discussion. Flat faces, or bases, are two-dimensional and traceable. Vertices are points or corners and surfaces are curved. A Triangular cross-section is when you cut a shape horizontal and vertical and the two halves are triangles. We also discussed polyhedra and said they are solids with ALL flat faces. Examples of polyhedra are prisms and pyramids. An example of a non-polyhedra is a sphere. ==


 * Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.
 * =How this was used in class:=
 * ==1.26.2010: We discussed the conjecture, "All quadrilateral have an interior angle sum of 360 degrees. We drew a quadrilateral and measured all the angles, and they added up to 360 degrees. We also said that we knew the sum of angle would be 360 degrees because we could see that it could be broken into two triangles. We know a triangle equals 180 degrees, so we knew the quadrilateral was 360 degrees. ==
 * ==2.2.2010: We were asked the following questions: "How do you know you have an angle?" "How can you make that angle?" "What is an angle?" We discussed possible answers with our group. The class discussed the last question and there was a lot of discussion! Several answers, "The point where two lines interest", "a connection of three points", "the space between two intersecting lines", and "two rays that meet at a point." ==
 * ==<span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.11.2010: We discussed what it meant for an angle to be an interior or exterior angle. We developed a definition that explains the relationship between the two angles, in essence you can't have one without the other. This led to our Big Idea that you can only have one exterior angle for every interior angle, and this exterior angle can be found by extending one side of a polygon together with its adjacent side. We used our turtle app on the calculator to aid us in recognizing the relationship between and interior and exterior angle. ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;"><span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.18.2010: We discussed the conceptions of an angle, developed through class discussion. We discussed the different types of angles- right, obtuse, interior, exterior, acute, reflex, and straight, and came up with definitions for each. Another thing we discussed is that "a 'circle' (CP p95 #10) is a full rotation/complete turn. ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">4.1.2010: We discussed area of different 2-D shapes. Although the area may be the same, the units may very. An example of this is when we were talking about the rectangle cut-out. A few students had drawn on the rectangle to figure out the area. One studet drew lines to make the rectangle have an area measuring 6 units squared, another student drew lines to divide the rectangle and gave it an area measuring 24 units squared. The rectangles were exactly the same size and had exactly the same area (area = base x height or length x width), but the unit sizes were different. The area of a parallelogram is also base x height. Base and height are always perpindicular to eachother. If you cut triangle off of a parallelogram, by making a perpindicular line a corner greater than 90 degrees, and connect it to the bottom or top of the parallelogram, the shape you cut off is a triangle and it will fit perfectly on the other side and will make a rectangle. Therefore the area formula of a parallelogram is the same as the area formula of a rectangle. Triangles have an area of 1/2 (base x height). This is because if you put two triangles together you will have either a square or parallelogram, and their area is base x height. You only need to find 1/2 of that area, so that's why you divide it in half. ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">4.20.2010: In class we used hollow 3-d shapes to develop different formulas for volume. Using a filler we discovered that the are connection between the volume of a cylinder, cone and sphere. The filler and hollow shapes would be an excellent tool in a classroom because students can really get an idea of how these volumes relate to each other. We also used this idea with the volume of a cube and a pyramid. We discovered that the volume of a cone is 1/3 of that of a cylinder and that the volume of a sphere is 2/3 of that of a cylinder. ==

= =
 * =<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship =
 * ==<span style="color: #800080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">How this was used in class ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.2-2.4.10: After we completed our charts of Symmetry we each examined the other groups work. When we believed a group had made an error in their classification their was a justification and an understanding of where the error may have been. This led to a deep understanding of non-rectangular parallelograms, if they don't have 4 congruent sides then they have rotational symmetry onl ==
 * 3.18.2010: We discussed similar shapes. Shapes are alike if: angles stay the same, shape is proportionally the same ratio, and if one line segment doubles in length, all others must also. We came to the conclusion that Two geometric shapes are similar if... All corresponding angles have the same measure and all corresponding sides are proportional.
 * 3.18.2010: We discussed similar shapes. Shapes are alike if: angles stay the same, shape is proportionally the same ratio, and if one line segment doubles in length, all others must also. We came to the conclusion that Two geometric shapes are similar if... All corresponding angles have the same measure and all corresponding sides are proportional.
 * 3.18.2010: We discussed similar shapes. Shapes are alike if: angles stay the same, shape is proportionally the same ratio, and if one line segment doubles in length, all others must also. We came to the conclusion that Two geometric shapes are similar if... All corresponding angles have the same measure and all corresponding sides are proportional.
 * 3.18.2010: We discussed similar shapes. Shapes are alike if: angles stay the same, shape is proportionally the same ratio, and if one line segment doubles in length, all others must also. We came to the conclusion that Two geometric shapes are similar if... All corresponding angles have the same measure and all corresponding sides are proportional.
 * 3.23.2010: Our conjecture- If two shapes are similar then the area of the two shapes will be the area of the original shape x the scale factor squared. We began to prove this conjecture by using deductive reasoning to prove two shapes are similar by using the idea of Perimeter - P of original shape: P = l+l+w+w; P = 2l + 2w; P = 2(l + w) then we showed that the P of the similar image is just the Perimeter of the original x scale factor - P = 2(lk) + 2(wk); P = 2(lk + wk); P = 2k(l +w) and putting in the form of our table: 2(l + w) x k.
 * 3.23.2010: Our conjecture- If two shapes are similar then the area of the two shapes will be the area of the original shape x the scale factor squared. We began to prove this conjecture by using deductive reasoning to prove two shapes are similar by using the idea of Perimeter - P of original shape: P = l+l+w+w; P = 2l + 2w; P = 2(l + w) then we showed that the P of the similar image is just the Perimeter of the original x scale factor - P = 2(lk) + 2(wk); P = 2(lk + wk); P = 2k(l +w) and putting in the form of our table: 2(l + w) x k.

=<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;"> Geometry Standard: Apply transformations and use symmetry to analyze mathematical situations =

=<span style="color: #000080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Expectations Used: =
 * =<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides or scaling. =
 * ==<span style="color: #800080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">How this was used in class ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.2-4.2010: We discussed reflection symmetry, rotational symmetry, and assymetrical and came up with definitions and examples for each. We created a Venn diagram to help us see the relationship between shapes. One circle was reflection symmetry and the other was rotation symmetry. We also came to the conclusion that 360 degrees divided by the number of points gives you the angle of rotation, for a quadrilateral [REGULAR POLYGON?]. ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.2-2.4.10: Again this came about when we were classifying polygons into symmetrical categories. When we classified rotational symmetry we also had to include the angle of rotation. When we began the discussion on angle of rotation we talked about how many times a polygon needed to rotate in order to come back to its original position. This is important because the number of rotations is directly linked to the angle of rotation. ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">3.16.10 Today we discussed big ideas related to translations. These big ideas were 1) The pre-image must be an exact replica of its original image (same orientation as well) 2) The lines of translation are parallel to each other and are congruent in length. 3) The shape can be translated to any other point on the same plane. 4) The translation will create a 3-D image when all the original points are connected to their prime points. ==


 * =<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Examine the congruence, similarity, and the line or rotational symmetry of objects using transformation =
 * ==<span style="color: #800080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">How this was used in class ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.2-2.4.10: We used patty paper to draw out shapes and examine their lines of and rotational symmetry. We did this by folding the paper and examining where the polygon folds over itself exactly. ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.11.2010: We finished developing the ideas for our definitions of rotational and reflection symmetry as well as a line of symmetry. The definitions that we created in class can be found in our Working Definitions page on this same Wiki site. ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">We talked about rotational symmetry about a point. The rotational symmetry has a direction of rotation and all the points of the pre-image move about the point of rotation at the same rate.. that is the all stay in the same orientation as they travel in rotation. This connects to our idea of rotational symmetry of an object... as the object rotates about the point or within itself they alaways end up coming around to their original orientation. That is what it means to have rotational symmetry. ==

=<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;"> Geometry Standard: Use visualization, spatial reasoning and geometric modeling to solve problems. = = = =<span style="color: #000080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Expectations Used: =
 * =<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Draw geometric objects with specified properties, such as side lengths or angle measures =
 * ==<span style="color: #800080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">How this was used in class ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.2-2.4.10: As a class we brainstormed how we would construct a draggable right triangle using GeoSketch. We used GeoSketchpad to create different shapes. For example, we created different types of triangles- scalene, right, obtuse, etc. ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.11.2010: We used our Turtle App on our calculators to draw various polygons such as triangles and pentagons. We had to carefully calculate the measure of the exterior angle in order to have our turtle turn in the right direction to get an accurate representation of the shape we were trying to draw. ==


 * =<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Use geometric models to represent and explain numerical and algebraic relationships. =
 * ==<span style="color: #800080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">How this was used ==
 * ==<span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.11.2010: In discussing interior and exterior angles of a polygon we touched on the subject of the sum of interior angles of a polygon. We used the idea of a pentagon to begin finding an algebraic equation that would allow us to find the sum of interior angles. When we examined the pentagon we discovered that it could be broken up into 2 shapes, a quad and a triangle. This helped us to see that the sum of the interior angles is 360 (quad) + 180 (triangle) = 540. We also came up with a general equation for finding the angle measure of an exterior angle: 180 degrees - measure of interior angle = exterior angle, we examined different quads and triangles in our exploration. ==
 * ==<span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.16.2010: We delved into the notion that the sum of interior angles could be found for most polygons by breaking the shapes into pieces like triangles. This led us to the equation (number of sides * 180 degrees) - 360 (for the angles formed in the center of the polygon by the triangle pieces) could in most cases give us the sum of the interior angles. As we talked more we began to understand that all of the equations we came up with to find the sum of interior angles were really all equal to each other. We narrowed down as we could and came up with 180 (n - 2). To find the measure of one interior angle we found 180 (n - 2)/n will give us this answer, and further if we manipulate our equation to find an exterior angle into a true algebraic equation it would look like this: 180 degree - (180 degrees(n-2))/n, where n is the number of sides. ==

=<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;"> Measurement Standard: Understand measurable attributes of objects and the units, systems, and processes of measurement. =

=<span style="color: #000080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Expectations Used: =


 * =<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Understand, select, and use units of appropriate size and type to measure angles. =
 * ==<span style="color: #800080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">How this was used ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.2-2.4.10: Everyday we talk about angles and in some cases the angle measure. We know that the unit of measure for angles is degrees and we can find this using various instruments such as a compass. It is also important to also use the various tools available to use on the computer or our calculator to find the measurements of angles because it allows us to become familiar with the tools that our students will be using in the classroom. This kind of knowledge will facilitate our jobs as teachers in aiding students in their mathematical endeavors. ==
 * ==<span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2.11.2010: We furthered our understanding of the degree by talking about everyday objects such as pie. We understood that it would take 360 pieces of pie to make a whole pie. [HOW EXACTLY DOES THIS HELP ONE TO UNDERSTAND WHAT A DEGREE IS?] ==

=<span style="color: #ff0000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Measurement Standard: <span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Apply appropriate techniques, tools, and formulas to determine measurements. = = = =<span style="color: #000080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Expectations Used: =

= =
 * <span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Select and apply techniques and tools to accurately find length, area, volume, and angle meausures to appropriate levels of precision.
 * ==<span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">How this was used in class ==
 * ==<span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">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. ==
 * ==<span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">We have been working on this a great deal in this later half of the semester. We showed how the formula for the are of a circle can be derived from the formula for the area of a triangle. If we cut up the circle into equal parts we find that as those peices get smaller the circle begins to look more and more like a rectangle. We know that the area of a rectangle is b x h. Our base of the cut up circle is half of the circumference (the perimeter of a circle) so our formula now looks like A = 1/2 c x h. The height of the approximated rectangle is really just the radius of the circle and the circumference of a circle is 2pi x r. If we put all these things together we do end up with the area of a circle which is pi x r squared. Coming up with formula in this manner was great becuase you could really start to see the relationship between shapes and their areas. ==
 * ==<span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">We also played around with out cut up shapes of trapeziods and parallelograms and noticed that all of these shapes can manipulated so that you can easily find the area without having to know the formula. A paralleogram we turned into a rectangle by cutting off one of its ends and attaching it to the other angled side and saw that the area of a paralleogram is really just the area formula for a rectangle. It was awsome and a great way for students to learn and observe relationships and how they can be used to uncover more complex relationships. ==
 * <span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.
 * <span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.