Nicole+E.+&+Emily+S.+3-5

Geometry Standard for Grades 3-5 Nicole E. and Emily S. and triangles. Learned about interior angles and symmetrical attributes. We worked in our groups to define our shapes, we went through trial and error and came upon sufficient definitions for triangles and quadrilaterals. -Identifying 3-D Solids: Using blocks, we started talking of dividing 3-d shapes by their attributes like flat faces, or rounded sides. etc. So we then created a 3-D solids Venn Diagram. The big circle is the 3-D Solids, then inside the circle is prisms, pyramids, and truncated pyramids. We then elaborated that by finding the area and volume of many of the shapes using nets and 3-D models. With 3-D models, a student can tactically feel a shape and be able to point out differences other than describing them. ** || equilateral, obtuse, and acute triangles. We tested these with Venn Diagrams. We put triangles in venn diagrams. They were either non-intersecting, overlapping, or had containment. ** ** For example, an equilateral triangle would live inside an isosceles triangle **** -Classified: Quadrilaterals and used the House of Quads to show their relationships. -** **Classifying 3-D shapes: We used the 3-D models to divide them into categories and attributes that can be easily noticed, like a face is a square, or it has 4 flat rectangular faces. With these classifications, we were able to divide 3-D objects into two main catagories, polyhedrals and non-polyhedrals.
 * ~ **Analyze Characteristics and Properties of Two-and Three-Dimensional Geometric Shapes and Develop Mathematical Arguments about Geometric Relationships: ** ||
 * ** Identify, compare, and analyze attributes of two and three dimensional shapes and develop vocabulary to describe the attributes; ** || ** -Identified: different quadrilaterals
 * ** Classify two and three dimensional shapes according to their properties and develop definitions of classes of shapes; such as triangles and pyramids. ** || ** -Classified: isosceles, scalene, right,

Polyhedras: A geometric solid with all flat surfaces.**
 * **Prisms: A solid with two parallel, congruent faces that are connected by rectangular regions.**
 * **Cube: A solid with 6 congruent side**
 * **Pyramid: A solid with one polygonal base and a point not in the plane of the polygon, which we call the "top."**

We developed similarity by using our rubber bands to redraw the mystery man to make him larger. All of the sides got larger by a certain ratio and all of the angles stayed the same.** ||   - **Using geometers sketchpad, we were able to twist, turn, and flip shapes to prove our thoughts of shapes, and to prove their attributes. We also used other shapes to create the shapes.** - **On Geo Sketchpad we reflected a shape over one of its sides to make other shapes. We also rotated shapes along the midpoint of its sides.** ||  ||   || Line Symmetry: When a shape has a line of symmetry that when you reflect one side of the line the reflection would be congruent to the other side. We worked with line symmetry by folding shapes we had drawn on pieces of patty paper and when we would fold it and both sides lined up perfectly we knew we found a line of symmetry.** ||  || -**Also using Geometers sketchpad we created shapes by drawing them and using other shapes to form them.** -All of these examples give the children tactile and visual ways of learning which can broaden their minds and expand their mathematical thinking.** || mathematics, such as numbers and measurement.** ||  ||  Measurement Standards for Grades 3-5 Understand Measurable Attributes of Objects and the Units, Systems, and Processes of Measurement ** and select the appropriate type of unit for measuring each attribute. ** || **Use an angle ruler to find out the size of angles. Also constructed our own angle measuring tools by folding patty paper.** -**3-D objects: We found out surface area of a 3-D object by testing it out on many different occasions, like counting the boxes in the nets, adding the areas of each side. So from the latter hypothesis, we developed a formula for surface area for a polyhedral, that pretty much narrowed all of the tests. 2(length x width)+2(height x width)+2(height x length). For volume; we figured out the formula is length x height x width.** -**Using the geometric solids, we were able to jump into the world of volume by filling up the geometric solids with "fillers." Doing this we came upon conjectured statements about volumes of many cylindrical, and curved shapes. We conjectured that the volume of a cylinder was (Area of a circle) multiplied by the height. We knew area of a circle is (pi multiplied by radius squared). We found that it took three cones with the same base size as the cylinder to fill up the cylinder. So with that we said 1/3(area of a circle) multiplied by the height would get the volume of a cone. With a sphere we saw that the contents of a hemisphere had the same filling as the cone. So two hemispheres is a sphere, so that would be two cones, which is 2/3 of the cylinder. So our conjectured volume equation for a sphere is 2/3(area of circle) multiplied by the height of the sphere which we said was the diameter**. || with standard units in the customary and metric systems ** || **We discussed the differences in using different units. One day when finding the area of various shapes, Nicole told us to show how many units were in a rectangle. There were many different size units but they all covered the same area because our rectangles were all the same size. We could have verified this fact by actually doing some simply unit conversions such as from centimeters to meters. ** || and area when the shape is changed in some way. ** || **Project #2: In this project we tested the effect of perimeter on area. We started out by finding a perimeter of an irregular object. It was hypothesized that the perimeter, if made into a square, would have the same area as the irregular object. So we tested this by using string and push-pins. We found the area of the square, but then to prove the hypothesis right or wrong, we traced the irregular object onto graph paper, then using the knowledge of area on graph paper, we took away units not used by the irregular object. We found out that perimeter and area are very different. Perimeter is the length around the shape, and area is the space inside. One can have a set perimeter, but the area can change drastically. Further on in the project we were given a garden with a fixed perimeter, and we had to find the area of which would create the most garden space. We found that with a fixed perimeter, that the area is greatest with a square shaped garden. We then extended that by testing it on scratch, and realized that a circle has the most area, so to make the most area, once would want to have a shape be a circle, or close to it.** ||
 * Non Polyhdras: A solid that does not have all flat surfaces.**
 * **Cylinder: A solid with two parallel congruent circular faces.**
 * **Cone: A solid which consists of a circular region which is joined to a point not on the plane of the circle.** ||
 * ** Investigate, describe, and reason about the results of sub-dividing, combining and transforming shapes. ** || **If you draw a diagonal though a quadrilateral you will create 2 triangles.** **We then went into subdividing shapes to find interior and exterior angle measurements. We found that subdiving the shape into the minimal number of triangles, and use the equation 180(n-2) [with n being the number of sides your shape has] you can find the sum of the interior angles.** **It all stems from diving the shape into triangles.** ||
 * ** Explore congruence and similarity ** || **Congruency is when you have two things that are exactly the same in size and shape.
 * ** Make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions. ** || **For the first couple of weeks, we argued about the definitions our class would use for the certain shapes. Now we are testing the attributes of the shapes, like congruence and symmetry. We all learned geometry differently, so each of us brought our own ideas. Our ideas meshed, and sometimes contradicted, but we always came to an ending agreement.** ||
 * Specify Locations and Describe Spatial Relationships Using Coordinate Geometry and Other Representational System **
 * **Describe Location and movement using common language and geometric vocabulary ** || **By using geometer's sketchpad we discovered the ways you can rotate and flip images. We also saw how changing certain attributes of the shape affects or doesn't affect certain properties of the shape. Like changing the lengths of parts of a square to make a rectangle. Or changing the angle measurements of a square to create a rhombus. These changes give a visual aid to describe shapes and their characteristics to young children.** ||
 * **Make and use coordinate systems to specify locations and to describe paths; ** || **-Using two intersecting lines, we created our own coordinate system. With these intersecting lines we reflected a shape over both lines. With the two reflections we came to the conclusion that two reflections equal a rotation. Usually a coordinate system is defined when an origin has been identified but I see how this might be connected; some other groups connected the scratch work to coordinate systems because of the specifications about locations and description of paths. ** ||
 * **Find the Distance between points along horizontal and vertical lines of a coordinate system ** || **On geo board we create different shapes to determine there side lengths in order to classify shapes. We specifically did this for triangles to see whether our triangles to test out the lengths of our sides.** ||
 * Apply Transformations and Use Symmetry to Analyze Mathematical Situations**
 * **Predict and describe the results of sliding, flipping, and turning two-dimensional shapes. ** || **Conjecture: the shapes that only fit in the reflection category only have one line of symmetry. If they have two** **or** **more they can be rotational and reflection symmetry.**
 * - We tested this with the shapes by creating a venn diagram. reflection vs. rotational. In the center all of the shapes had more than one line of symmetry and the shapes with one line of symmetry were in the reflection only space because they did no have rotational symmetry. The only shape that was in rotation only had rotational symmetry of 180 degrees but no lines of symmetry, that shape was the rectangular parallelogram.**
 * **Describe a motion or a series of motions that will show that two shapes are congruent. ** || **-When you can flip, slide, or reflect a shape onto another shape and they line up perfectly then those shapes are congruent.** ||  ||   ||
 * **Identify and describe line and rotational symmetry in two and three dimensional shapes and designs ** || **Rotational Symmetry: Being able to turn a shape less than 360 degrees and make the exact same shape. When a shape has rotational symmetry it has a center of rotation which is the point the shape is turning from. Knowing the center of rotation is helpful in finding the degree you have to turn in order to make the same shape, or the degree of rotation.
 * Use Visualization Spatial Reasoning and Geometric Modeling to solve problems **
 * **Build and Draw Geometric Objects** || **Using Scratch and predicting what will be constructed when given directions.**
 * -Using nets of 3-D objects, we can fold them in certain ways to form that 3-D object.
 * **Create and describe mental images of objects, patterns, and paths.** || **-Predicting what will be constructed when given scratch directions.** ||
 * **Identify and build a three-dimensional object from a two-dimensional representation of that object** || **-First we developed ideas for different 3-D objects, after that we used grids to draw nets for a cube, we then tested other ways of folding the cube different ways using different net designs. After testing out our theories we tried making nets for other 3-D shapes.** ||
 * **Identify and build a two-dimensional representation of a three-dimensional object** || **Being given an object and then being asked to make a net for that object.** ||
 * **Use geometric models to solve problems in other areas of
 * **Recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.** || **When talking about surface area of a 3-D object, the least surface area of a rectangular prism was a cube. We can attach this to everyday life in the way of packaging. To send a package, you want to use the least space, and a box shaped like a cube does that.** ||
 * 
 * **Understand such attributes as length,area, weight, volume, and size of angle,
 * **Understand the need for measuring with standard units and become familiar
 * **<span style="color: #000000; display: block; font-family: Arial,Helvetica,sans-serif; text-align: left;">Carry out simple unit conversions, such as from centimeters to meters, within a system of measurement ** ||  ||
 * **<span style="color: #000000; display: block; font-family: Arial,Helvetica,sans-serif; text-align: left;">Understand that measurements are approximations and understand how differences in units affect precision ** || **When writing out answers or equations for a circle it is appropriate to use the pi symbol. When giving an answer that is rounded you need to say "approximately" instead of "equals." Doing this we can actually give an acceptable answer, that doesn't cut off decimal points. Pi is a very large decimal and to type or write out every number to be exact is impossible. For math it is correct to use a pi symbol or square root in your answer but for real life application rounding decimals is necessary but you need to still know it is an approximation.** ||
 * **<span style="color: #000000; display: block; font-family: Arial,Helvetica,sans-serif; text-align: left;">Explore what happens to measurements of a two-dimensional shape such as its perimeter

length, area, volume, weight, time, temperature, and the size of angles.** || **When measuring angles we use an angle ruler or a protractor the measure we look for are the degrees. We tried our own angle measure tool out of a piece of patty paper.
 * <span style="color: #ed4a07; display: block; font-family: 'Comic Sans MS',cursive; text-align: center;">Apply appropriate techniques, tools, and formulas to determine measurements **
 * **Develop strategies for estimating the perimeters areas and volumes of irregular shapes.** || **To find the area of an irregular shape it is helpful to create a rectangle that you know the area of around the shape and then subtract the area of the rectangle that the irregular shape does not take up. To find the perimeter of an irregular shape you can measure how much string you need to make it go around the shape.** ||
 * **Select and apply appropriate standard units and tools to measure

To measure length we were given rulers to measure millimeters, centimeters or inches. Meters, Yards, and even miles and kilometers can be used to measure distance. You choose your unit depending on the length you wish to measure. It would be ridiculous to measure a room in inches.

Area is measured in square units

volume we measure in cubic units** || Triangles: 1/2 base x height Parallelograms: Length x Height** || -We found this out by testing out different theories. We came upon this one knowing that there we always going to be 2 faces of the same size on a rectangular solid. So each face has different side lengths, so we used height=h, length=l, and width=w. This then gives us the area covering the shape. -Finding the volume was more of a challenge. We said that in a rectangular solid, you take the bottom layer's area (length times width) we called that one "stack." Then you see that the height is how many stacks are in the shape all together. So you multiply the bottom layer (area of the base) by the height of the object. We later figured out a simpler way to phrase it other than length times width times height, is to say area of the base times the height. That will equal the volume of a rectangular solid, and many other prisms.** ||
 * **Select and use benchmarks to estimate measurements** ||  ||
 * **__Develop__, Understand, and use formulas to find the area of rectangles and related triangles and parallelograms.** || **rectangle: Length x Height
 * **Develop strategies to determine the surface areas and volumes of rectangular solids** || **Surface area of a rectangular solid = 2(length x height) + 2(height x width) + 2(width x length)

GENERAL COMMENTS: <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; color: #800080;">Good to tie in the project work and in class work. Connections between standards were not always made explicit but usefulness of activities was usually evident in the descriptions. It is clear that many of the standards were addressed and I hope that this page serves as a good reference for your future use. Nice work.
 * MAJOR CONCEPT****S**- SOME IDEAS ON ANGLES AND THE DEVELOPMENT OF MEASURING ANGLES SEEMED TO BE MISSING. GOOD RECOGNITION OF THE BIG IDEA OF COMPARING AND CLASSIFYING SHAPES BASED ON THEIR PROPERTIES, INCLUDING SYMMETRY.
 * DEVELOPMENT, IMPORTANCE, AND CONNECTIONS OF CONCEPTS**- IN SOME OF THE EXPECTATIONS, MERE DESCRIPTIONS OR DEFINITIONS OF THE IDEAS WERE GIVEN INSTEAD OF HOW THEY IDEAS WERE DEVELOPED. THE IMPORTANCE AND CONNECTIONS OF THESE IDEAS WAS ABSENT. TO IMPROVE ON THIS FOR NEXT TIME, CONSIDER EACH STANDARD AS A WHOLE, THINK ABOUT OTHER STANDARDS THESE IDEAS ARE RELATED TO INCLUDING OTHER GEOMETRY AND MEASUREMENT STANDARDS, REASONING AND PROOF, AND EVEN NUMBER AND OPERATIONS OR ALGEBRA. THESE CONNECTIONS NEED TO BE MADE EXPLICIT (ALONG WITH THE USEFULNESS).
 * CONNECTIONS TO PSSM**- MOST MAIN IDEAS CLEARLY CONNECTED TO STANDARDS, BESIDES SOME ON MEASURING ANGLES AND SPATIAL VISUALIZATION INCLUDING DIRECTION AND DISTANCE USING VARIOUS TOOLS.