KayLynn+L.+&+Kristen+B.++Pre+K-2

Pre- K-2
__**Geometry Standards:**__

__**Standard 1:**__ Analyze characteristics and properties of two and three dimensioal geometric shapes and develop mathmematical arguments about geometric relationships. __**Expectations:**__ recognize, name, build, draw, compare, and sort two and three dimensional shapes; describe attributes and parts of two and three dimensional shapes; __**How we developed these ideas:**__ We used alpha shapes to sort triangles and quadrilaterals into different categories. We classified the different triagnles by angles, and side lengths. This lead to defining the different types of triangles. When moving onto quadrilaterals we stated that putting two triangles together would make a quad. We classified quads by parrallel lines, angles, and congruent sides. -For rotational and reflectional symmetry we showed how a shape could be divided into two equal halves and put back together by using the line of symmetry. When we folded a shape at the line of symmetry we could see that the two halves mirrored each other. In class we drew shapes and were able to guess if they had rotational or reflectional symmetry. Triangle and Quad inequality meets the expectations of building and comparing shapes. In this investigation we were trying to figure out given three (triangle) or four (quad) side lengths could the shape be built. To do this we used polystrips and different combinations of numbers. In the end we decided that you could only form a triangle if the a+b>c. (with a and b being the two smaller lengths.) Than we did the same thing for quads and decided a+b+c>d (with a,b,c being the three smaller lengths.) -When we started working with 3 dimensional shapes we played a "Guess My Shape" type of game. We used 3D shapes and one person had a certain shape picked out in their head that no one else at the table knew. Ther other members at the table could only ask yes or no quesions such as, "does your shape have a square base", "is your shape a polyhedra",in order to eliminate the other shapes and evenyually have the same shape the person had picked in their head. -We played another type of "Guess My Shape game where Nicole flashed a picture quickly on the projector. The students had to try and pick out which shape it was by only using that quick image Nicole flashed. __**How it is useful:**__ These ideas are useful because it allows students to use the same language when talking about certain shapes. This also helps to compare different shapes with eachother and find out what is similar and different between the shapes. One good method to show students would be the venn diagrams. This way students can see how the shapes relate to one another and how some shapes like a square could be classified as other shapes as well like a rectangle. In these lower grade levels you could start by using the example of cars and colors for example if you have a circle for the color blue and a circle for cars. Than show the students a blue car and have them decide where to put it and this would relate to the same as the square being able to fit in different circles. When dealing with the triangle and quadrilateral inequality this is useful because you can talk about how structures are built and if they are sturdy or not. Using polystrips helped to determine in the begining if the shape would be built this is something that students at this grade level could build as well. __**Other Mathematical ideas it connects to:**__ These ideas connect to line symmetry because you need to know about congruent sides in the shape to know if it will have line symmetry. This also connects to angle sums with triangles and quadrilaterals. The inequalities connected to fixed angles and interior and exterior angles. When using the polystrips you could see that buliding a triangle would have fixed angles meaning you couldn't change the angle measures which made the shape more sturdy. However with quads there was no fixed angle. After you created the shape you could move the polystrips and either push them more together or pull them father apart creating infinite number of angles and infinite number of quads given a certain side lengths.

__**Expectations:**__ Describe, name, and interpret relative positions in space and apply ideas about relative position; Describe, name, and interpret direction and distance in navigating space and apply ideas about direction and distance; Find and name locations with simple relationships such as "near to" and in coordinate systems such as maps. __**How we developed these ideas:**__ By using Logo light and Scratch we developed some of these ideas because it helped to show the relationships between certain shape. For example when we used scratch we were able to see how shapes were created correctly with the right amount of turn and angle measures needed. __**How its useful:**__ These activities were useful to develop these ideas because it challanged you to think how a shape would be constructed by certain comands. This would give the position the shape would take on and the direction the sprite would be facing. __**Other Mathematical ideas it connects to****:**__ This ideas also connects to reflection and rotation of shapes because by using scratch we could see the spatial relationships between different shapes and really see the inputs that are needed such as amount of turn and angle measure.
 * Standard 2:** Specify locations and describe spatial relationships using coordinate geometry and other representational systems.

__**How we developed these ideas:**__ We first defined line of symmetry and roational symmetry. We used our alpha shapes to depict what form of symmetry each shape had. We found that if the shape was moving around the center point and fitting on top of itself before completing a 360-degree turn then it had rotational symmetry. If we were able to fold the shape at the line of symmetry and the two halves mirrored each other perfectly then we knew it had reflectional symmetry. We also found that some shapes were asymmetrical which means that they did not have rotational or reflectional symmetry. We also found that some shapes have both reflectional and rotational symmetry.Than we made a poster that had several different triangles and quadrilaterals on it placed in the right categrory. We than drew the lines of symmetry onto the shapes and told the center of roation and angle of roation for the roatational symmetrical shapes. Patty paper is also helpful to determine if you really can fold a shape and have line symmetry, if you needed a more percise drawing geometry sketchpad would be helpful and show the lines of reflection and measurments. Translations also fit into this standard we developed these by looking at a translation on geometry sketchpad. This helped to show that the points move the same distance along the vector or sliding line. Next we discussed the importance if you could flip, slide, or turn a shape and it would end up back on the original shape in the same direction(looks the exact same) the shapes would be congruent with one another. In our third writting assinment we discovered this more where we had to measure the original shape compared to the vector and reflect or rotate the shape so that it stayed congruent with the original. This also used symetry and rotation to help make the shapes congruent, using mathmatical evidance such as a ruler and protractor. __**How it is useful:**__ This is useful because if a shape has symmetry you know more of the properties of the shape for example if the sides are congruent. You can also make a quadrilateral out of triangles if you know if a shape has rotational symmetry. Being able to rotate or slide a shape is useful because it allows for you see the relation between the two shapes and how the shapes are mathmatically similar to one another. __**Other Mathematical ideas it connects to:**__ This also connects to creating quadrilaterals out of triangles. Angle of rotation connects to interrior angles and exterior angles, the conjecture we came up in class about this was that the interrior angle and the exterior angle (on the same corner) would add up to 360 degrees which would be a complete turn and when rotating a shape if you had your angle of rotation 360 degrees you would land back on the original shape. The rotating or sliding of a shape connects to developing area formulas. For certain shapes like the parallelogram we cut a right triangle off the end and slid it to the opposite side to make a rectangle. These methods worked for other shapes as well.
 * __Standard 3:__** Apply transformations and use symmetry to analyze mathematical situations
 * __Expectations:__** recognize and apply slides, flips, and turns; recognize and create shapes that have symmetry.

__**Expectations**:__ create mental images of geometric shapes using spatial memory and spatial visualization; recognize and represent shapes from different perspectives; relate ideas in geometry to ideas in number and measurment; recognize geometric shapes and structures in the environment and specify their location. __**How we developed these ideas**:__ We used alpha shapes and flashed the shape onto the screen for a short second and than the class had to remember how the shape was created and draw it on thier own paper. Also playing the rule game with both triangles and quadrilaterals was a visual representation. Another activity is having the commads for scratch and having to draw the shape it would make. As a class we were asked to bring in pictures of shapes from everyday life. We found what type of symmetry each shape had and placed it on the poster board in the front if the room under it's correct form of symmetry. This activity allowed us to see that things in our everyday life, such as stop signs or hearts, can have either rotational or reflective symmetry, both rotational and reflective symmetry, or no symmetry at all (asymmetrical) we showed this relationship by using venn diagrams this also helped to show that some shapes have both rotational and reflective symmetry. The phythagorean theorem helped develop the idea of relating ideas in geometry to ideas in number and measurements. The geometry side of this was what triangles the phytagorean theorem would actually work for where the mathmatical side was how this theorem was going to work. We started this disscussion by trying to prove A squared plus B squared equalled C squared. As a class we knew that C should be the longest side and would be called the hypotenouse. We than talked about the only triangle that had a hypotenouse were right triangles becasue it was the side opposite the right angle. We started by drawing a line segment on dot paper and having to "finish the square" in order to find the length of one side we had to find the square area and than square root that number and it would give us one side length. We than drew a right triangle with the same line segment we used in the first method and drew boxes off each side of the triangle. we noticed that the area of the smallest square plus area of middle square gives you the area of the largest square. We than created a right triangle and just used the formula A squared plus B squared equals C squared. After you have all your numbers you have to square root C and the number you got for C. This will give you the side of the hypotenouse. In class we talked about three dimensional shapes and their nets. A net is a 2-D pattern that can be turned into a 3-D shape. Below is an example of a net for a rectangle:
 * __Standard 4:__** Use visualization,spatial reasoning, and geometric modeling to solve problems.

[[image:webkit-fake-url://A2CCDEAB-9A82-4A4D-884A-637CC8966613/rectangular_prism_net.gif caption="rectangular_prism_net.gif"]]
__**How it is useful:**__ This is useful because you wouldn't need a computer everytime you could see in your mind the shape that is being created and manipulate it to tilt or turn. Having a quick mental image gives you the ability of problem solving as well to figure out what you might not have seen. The phytagorean theorem is useful because when you need to know a side length and you have two of them its easier to plug the number is to the formula rather than taking the time to draw boxes and find the area of the boxes each time. Would also be useful if you are building something and need to know the length of one side of the triangle when you can't find the area and square root it. Yes, it can also be used to find a line segment of a particular length (i.e., use the formula in reverse). I think it is good to discuss the Pythagorean theorem in relation to this standard, but is this content appropriate for K-2? Typically it is not introduced until middle school but I think the problem solving strategies of visualization are definitely an important aspect of this expectation.). What about the 3D perspectives of cube buildings? __**Other Mathematical ideas it connects to:**__ This would connect to spatial locations by knowing what direction certain shapes were facing or how there angles differed. Phytagorean theorem connects to the idea of building and drawing shapes, because you need to know the side lengths of a triangle in order to build it. __**Measurement Standards:**__

__**Standard 1:**__ Understand measurable attributes of objects and the units, systems, and processes of measurement. __**Expectations:**__ recognize the attributes of length, volume, weight, area and time; compare and order objects according to these attributes; understand how to measure using nonstandard and standard units; select an apppropriate unit and tool for the attribute being measured. __**How we developed these ideas:**__ Placing alpha shapes into categories based on length. Using patty paper to measure an angle is nonstandard measuring. Dealing with mathmatical similarity used the expectations of recognizing attributes and comparing objects. We developed this expectation by dealing with the mystery man. First we traced him using a double band rubber band and measured using a standard measuring tool (ruler) to find the side lengths and areas of the original figure and the new mystery man. Seeings how the side lengths doubled and the area quadripuled, also the angles stayed the same. This proved that the shape was indeed similar. The definition of similarity involves corresponding angles congruent and corresponding sides proportional. As a class we decided that for a shape to be similar the ratio between the side lengths would have to be the same and the ratio between the areas would have to be the same the area should change by the scale factor squared. We also developed the ideas of area. First we discussed what area was and that it was a covering of the space in between the lines. Than we discussed how the units would be squared. The first formula we came up with was a square because all the sides were the same it was easy to figure out the formula would be side length squared. From there we moved onto the rectangle and decided it was like a square but with different side lengths so base times height was the formula. Then we changed the other shapes to make them look like shapes we already knew the formulas to by cutting or rotating them. For examle the parallelogram we made a right triangle on one end and cut that off and slid it to the opposite side so the shape became a rectangle.The circle took a little more work because we started out with a circle but wanted a rectangluar soccer field. we noticed that the "height" is actually the radius of a circle and that the "base" is really half the circumference. After that we measured lots of circumferences and diameters to realize the relationship between the two was pi. This lead to the formula of a circle to be (pi)(r)squared.

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We had a chart in the front of the room where we were able to classify shapes on the chart into groups of polyhedras and non-polyhedras, we also had sub categories to place the shapes in such as pyramids, prisms, etc.======

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In class we determined the formulas to find out the areas of different shapes, the perimeter of different shapes, the circumference of different objects, and the radii of different shapes, We cut out shapes and used them to see how they related to other shapes in order to find area formulas for each of them.======

In class we were able to figure out how to maximize and minimize the surface area by using wooden blocks.
__**How it is useful:**__ This is useful because it shows students that not all measurment needs to be done with an actual tool. Also compares shapes by length to categorize and sort what you might need. The mathmatical similarity is useful becasue it allows you to compare shapes in a greater aspect. Knowing the area formulas is useful because it shows students how a formula was thought of and how you can take apart shapes to better understand the area formulas. This also helps to know how much of something it would take to cover a surface. __**Other Mathematical ideas it connects to:**__ This connects to recognizing shapes and angles, and describing the attributes of a shape. The students would be describing the shape based on side lengths when placing them in categories. Measuring the angles can only be done once they understand what an angle is and be able to recognize it. Being mathmaticaly similar leads to finding areas of different shapes. This connects because in order to find out if the shape is actually similar you would need to know what the area is increasing by so you would need to know how to develop the area formulas. Area connects to the idea of volume because you are looking at how much it takes to fill an area. It also links to the expectation in standard one in geometry about putting together and taking apart shapes.

__**Standard 2:**__ Apply appropriate techniqes,tools, and formulas to determine mearuements __**Expectations:**__ measure with multiple copies of units of the same size, such as a paper clips laid end to end; use repetition of a single unit to measure something larger that the unit, for instance measureing the length of a room with a single meterstick; use tools to measure; develop common referents for measures to make cmparisons and estimates. __**How we developled these ideas:**__ Using sketchpad we were able to use a formal method of measuring the angles and side lengthgs of the shapes that were created in this program.

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In class we took empty shapes, such as cubes, cylinders, cones, shperes, and measured the volume of them by filling them with special filling that we had. We compared the volume of the different shapes by comparing the number of times the filler within a cone shape had to be refilled in order to fill the cylinder shape completely when emptied into the cylinder. By pouring the filler from one shape into a larger shape it allows students so be able to see and estimate the volume of the shapes in comparison to each other. It helps the students develop skill in seeing how much larger the volume of the larger shape is compared to the volume of the smaller shape.======

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Other Mathematical ideas it connects to: This connects to standared measurment and comparing and sorting shapes because it helps to show students that objects can be measured by other things, not just by using a ruler.======

GENERAL COMMENTS: VERY WELL DONE AND EASY TO FOLLOW.
 * MAJOR CONCEPT**S- ALL MAJOR BIG IDEAS SEEM TO BE INCLUDED. THE USE OF PARTICULAR REPRESENTATIONS SUCH AS VENN DIAGRAMS TO ANALYZE RELATIONSHIPS AMONG SHAPES WITH PARTICULAR PROPERTIES COULD HAVE BEEN DRAWN OUT MORE EXPLICITLY. The big ideas of the course were well covered, but some of the content included here may not be appropriate for K-2.
 * DEVELOPMENT, IMPORTANCE, AND CONNECTIONS OF CONCEPTS**- VERY NICE; GOOD TO FIND BROAD WAYS THESE CONCEPTS ARE RELATED TO EACH OTHER AND USEFULNESS OF EACH IDEA. GIVE ONE OR TWO MORE SENTENCES ABOUT THESE CONNECTIONS. Very good connections to other concepts within geometry and in describing the usefulness and importance of studying these ideas.
 * CONNECTIONS TO PSSM**- GENERALLY GOOD. See comment above.