McKenzie+&+Deanna+PreK-2

Geometry Standards For Grades Pre-K-2

__**​**__ ​ Analyze characteristics and properties of two and three dimensional geometric shapes and develop mathematical arguments about geometric relationships ||< Expectations: //WHY THEY ARE IMPORTANT:// These standards help children to understand shapes and how mathematics applies to shapes. These ideas are important because it helps them to understand things that they have never learned. Additionally, it helps to build on their informal experiences outside of school to more formal experiences in school. By describing things in class and using simpler words, they can understand tough concepts. They can use their own words to help make more sense of something. Kids also learn that there is more to a shape or an idea than just what is in front of them. By taking shapes apart and putting them back together, they learn about symmetry and how different shapes can have things in common. This teaches kids to look for things deeper and to not expect the usual. Kids also learn more about reasoning and how to explain ideas and concepts. These standards are the building blocks for their mathematical education.
 * < Geometry Standard #1
 * recognize, name, build, draw, compare, and sort two- and three-dimensional shapes;
 * describe attributes and parts of two- and three dimensional shapes;
 * investigate and predict the results of putting together and taking apart two- and three-dimensional shapes. ||
 * < How we applied this in class: ||< * **1/12/10:** We named different kinds of triangles and defined terms like concave and convex. We also compared characteristics of different triangles
 * **1/14/10:** We created a list of characteristics that all triangles have and ones that only some triangles have. We also defined a polygon and a simple curve.
 * We learned how to use Geoboard on our calculators to draw triangles and then measure them.
 * **1/19/10:** We placed shapes into their matching category according to their characteristics and properties
 * **1/21/10:** We defined a quadrilateral.
 * **2/2/10:** We defined properties of quadrilaterals and triangles by seperating them into groups according to their symmetry.
 * **2/9/10:** In class we used explore draw on our calculators to draw out different kinds of shapes with the turtle. We had to use the formula for finding the exterior angle of a shape to know how far the turtle had to turn.
 * **2/11/10:** We defined the different types of symmetry: rotation symmetry, reflection symmetry and what a line of symmetry is. We also discussed how to develop what a degree is. These are attributes of shapes.
 * **Project #1:** In project #1, we observed what a shape would look like if we flipped it over one of its sides. It turns out that certain traingles put together make certain quadrilaterals. This is investigating and predicting results of shapes.
 * **2/16/10:** We found out that there is one exterior angle per interior angle and that it is formed by extending one side of a shape together with an adjacent side. This is an attribute of any shape.
 * **3/18/10:** Our class discovered a definition for Similar: Two Geometric shapes are similar if [ALL CORRESPONDING ANGLES] they have the same angle measure, AND are [ALL CORRESPONDING SIDES ARE ]proportional by a scale factor. This is recognizing and comparing shapes and we had alot of practice with shapes that are similar to each other.
 * **4/8/10:** Our groups played a game called "What's my shape?". In this game you would ask questions about the 3D shapes that we were working with (Cylinder, cones, pyramids, prism, sphere, cube) to eliminate shapes and guess which shape was chosen. To play this game you have to know attributes of the shapes and you must recognize each shape. It's possible to learn alot by asking questions too!
 * **4/13/10:** In our course pack we sorted 3-D shapes into categories they fit in; polyhedra, non-polyhedra, prisims, cylinders,pyrimids, or cones. We discussed what characteristics qualify for each different type of 3-D shape. Some shapes could be several different categories or only just one. These are attributes of shapes and recognizing shapes.
 * **4/13/10:** We learned that a Net is a 2-D pattern that can be folded to form a 3D figure. We experimented with making nets and folding net's into shapes. This is investigating and predicting the results of three dimensional shapes and was a great and fun way to work with this standard.
 * **4/20/10:** we played with hollow 3D shapes in class to find out what their volume is. We had them in front of us to determine exactly what volume meant with that shape. We started with a cube and saw that finding the area of the base gives you the area of the first layer of units or cubes that fill the bottom of the shape. Then when you multiply that area by the height, you are taking into account how many layers of that base there are. This idea was applied to the volume of rectangular prisims, and cylinders. This exercise resulted in obtaining equations to find volume for a multiple number of shapes. Very good description here . ||

//MATHEMATICAL IDEAS THEY CONNECT TO:// This standard connects to standard #2. Once kids learn about the different kinds of shapes, they gain more spatial memory. By working with shapes, they gain more familiarity and would be able to remember them if they saw them only for a second. This also helps to make rotations flips or relfections easier. If you dont know much about a shape, it won't be easy to imagine what it would look like in a different orientation.

Specify locations and describe spatial relationships using coordinate geometry and other representational systems || Expectations: //WHY THEY ARE IMPORTANT:// Using different types of technology is not only useful in mathematics, but it is also useful in other areas. Caculators are used in drafting, economics, chemistry, physics and a lot of other classes that students will eventually have to take. Getting them started with technology at an early age prepares them for the future. Developing relative positions helps improve thinking and reasoning. Being able to imagine something that's not already there improves how you think about things. If you are able to imagine something in your head and be able to describe it, you are more likely to solve problems and reason about those problems efficiently.
 * Geometry Standard #2
 * describe, name, and interpret relitive 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 applied this in class || * We used a computer program called geometers sketch pad to create different shapes that we discussed in class and to complete project #1.
 * We used the geo-board application on our caculators to create different triangles and quadrilaterals. then we were able to measure their angles and side lengths.
 * We used a computer program called scratch. With this program we needed to learn how to position shapes that we created, we also needed to understand the distance we wanted our shape to cover and we used exterior angles to create our shapes.
 * **3/16/10:** We discovered ideas about relative postion and distance. We did this by learning that if you create a circle on an image, each point is dependant on that circle. Corresponding points follow along the same circle, they cannot move from the path.
 * **4/15/10:** We found that the volume of a 3D shape is how many cubes can fit inside of a shape or the space it takes up. As a class, we referred to volume as amount of space.
 * **4/20/10:** in class we worked with 3D shapes and the space inside of them. By finding the area of the base, you are finding the space that the bottom layer of cubes inside th shape take up. Then by multiplying by the height, you are finding the space that all of the layers of cubes inside the shape take //up.// ||

//MATHEMATICAL IDEAS THEY CONNECT TO:// Finding locations in coordinate mathematics is connected to finding locations on a map. Using latitude and longitude coordinates are a similar idea to this. Using the program scratch relates to angle measurements and side lengths. Geometers sketch pad help to connect different ideas like symmetry, rotations, and what would happed to the interior angles if the shape were changed.

Apply transformations and use symmetry to analyze mathematical situations || Expectations: > **2/4/10:** We discussed symmetry and which shapes have symmetry. Everybody brought in pictures of shapes with different forms of symmetry and we placed them in the correct categories. We also discussed how to figure out a shapes degree of rotation symmetry. //WHY THEY ARE IMPORTANT:// If you know a shape has reflection or rotational symmetry, you can build other shapes from just one shape. For example if you want to make a square and you only have a right triangle, you would know that a triangle reflected over its hypotenuse will give you a square. Knowing if a shape has symmetry and what you can do with a shape helps develop a better understanding of geometry. Learning about shapes and their symmetry carries on into other math problems and even everyday life. Good connection to the math we experience outside of the classroom.
 * Geometry Standard
 * 3
 * recognize and apply slides, flips, and turns;
 * recognize and create shapes that have symmetry. ||
 * How we apply this in class || * **2/2/10:** We drew quadrilaterals and triangles in categories they belonged in; Rotational Symmetry, Reflection Symmetry, and Assymetric (meaning has no symmetry).
 * **Project #1:** In the project we worked with different shapes and their symmetry. We observed what would happen if you flipped a shape a few different ways. We also had to use lines of symmetry to create different shapes. We found that some shapes had symmetry. We worked with rotations and reflections and what the resulting image would look like.
 * **3/9/10:** we were looking at a star and each of its points we labled. If the star was rotated so that it would match up with the original shape, we saw which points matched up. We then learned that most rotations are counterclockwise. Then we were given a flag and had to draw what it would look like if it was rotated 60 degrees counterclockwise. This helped us to understand flips and turns.
 * **3/16/10:** We came up with the conjecture that the pre-image and the original image stay the same no matter if you flip, slide or turn it. We also created shapes with symmetry, using Translation. All points move in the same direction the same distance, this creates a 3D shape when drawing in the vectors or translation lines.
 * **Project #2:** part of the project was to find the area of a hexagon. A hexagon has symmetry and if you fold it in half, you will have two congruent trapezoids relected over a line of symmetry. To make fiding the area easier, all you do is find the area of one trapezoid and then multiply that by 2. We used what we learned about symmetry multiple times to make problems easier to complete. ||

//MATHEMATICAL IDEAS THEY CONNECT TO//: Rotations and reflections relate to the development of spatial memory because if you are asked to apply a rotation, you have to imagine what it would look like before you actually begin to solve the problem. Recognizing shapes that have symmetry will help to be able to take apart shapes and put them back together. If you recognize that a hexagon is made up of two trapezoids and that it is symmetrical, finding the area will be a lot easier. Reflections also represent shapes in a different perspective (standard #4). When you reflect an image over a line of symmetry, the resulting image most of the time will not look like the pre-image. This requires a student to create a mental image of the shape from a differnt perspective. Yes, using spatial visualization is very important in geometry. Use visualization, spatial reasoning, and geometric modeling to solve problems || Expectations:
 * Geometry Standard #4
 * 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 measurement;
 * recognize geometric shapes and structures in the environment and specify thier location. ||
 * How we apply this in calss || * **1/12/10:** An image was quickly flashed in front of us and we were expected to draw what we saw.
 * **1/19/10:** We explained why all of the angles in a triangle add up to 180 degrees by taking off each corner and lining them up next to each other so that they create a straight line.
 * **1/26/10**: We explained why all of the angles in a Quadrilateral add up to 360 degrees by making it into two triangles, each triangle = 180 degrees so two triangle together will equal 360. We also compared the angles to a pie, the whole pie equals 360 degrees and the angles are the slices.
 * **2/4/10:** We recognized geometric shapes and structures in the enviroment by talking in class about things we see everyday such as stop signs, doors, and pizza. We kept an eye out and brought some interesting shapes into our next class period.
 * **3/18/10:** We used an interesting method to learn about Triangle inequality. We generated three random numbers with our calculator and then tried to construct the triangle with the numbers as our three sides. Taking notes on which triangle's worked, and which did not would help us to our conclusion. We learned that the sum of the two shorter sides must be larger than the length of the longest side. This exercise helped us to create mental images of what we thought the triangles may look like, we also were able to recognize the shape to tell if it created a triangle or not. It would be interesting to see how PreK-2 grade children work with polystrips. I'm wondering about the appropriateness of this content at this age.
 * **3/23/10:** We were given 4 random numbers for the side lengths of quadrilaterals and had to determine if it would make a proper quadrilateral. We found that the sum of the 3 shortest sides had to be greater than the longest side in order for it to be a quadrilateral. This activity helped us produce mental images of quadrilaterals. It also related quadrilateral sides to actual nmbers. This exercise was similar to the triangle one and we used what we learned from that lesson to help with this one.
 * **4/13/10:** We were asked to create nets for 3-D figures. we used spatial visualization (expectation #1) to predict whether the net would create a prisim or another shape. We had to visualize and recognize shapes to complete this exercise. ||

//WHY THEY ARE IMPORTANT:// These concepts help build mathmatical reasoning. By being able to create mental images, you are able to describe ideas and concepts more thoroughly. Working with spatial memory improves memory function overall. Having a shape flashed in front of us and then having to draw that shape forces us to teach ourselves a more efficient way of memorizing. It also teaches us to look for specific details when trying to memorize things. Being able to find geometric ideas in our environment gives us a reason to want to learn more. Kids always say "when are we ever gonna use this?" Finding things in real life that relate to what you are learning in the classroom shows kids that they will be using mathematics even when they aren't in school. Mathematics is used everyday and that is something we need to learn how to pass on to our students in a fun and exciting way.

//MATHEMATICAL IDEAS THEY CONNECT TO:// Using spatial memory and spatial visualization relates to translations, reflections and rotations (standard #3). In order to complete a translation, reflection or a rotation, you must visualize what the resulting image would look like or where it would be positioned in space (standard #2).

Understand measurable attributes of objects and the units, systems,and processes of measurment || Expectations: //WHY THEY ARE IMPORTANT//: These concepts and ideas are important because measurement doesnt only happen inside of a math classroom. Kids will be using measurement in many different classes besides mathematics.Teaching kids to measure with standard and non standard measurements is important because it makes them think outside the box. Most people are used to just using a measuring tape and being done with the problem. But if there is no measuring tape and only string or paper, they will still know how to solve the problem. Measurement also helps kids understand a different kind of attribute that shapes have live area or volume or surface area. Another reason Non-Standard measurements can be helpful is because what is standard to some people may not be to others. It is always best to have more than one way to solve any problem, and that is something teachers need to pass on to students.
 * Measurement Standard #1
 * 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 appropriate unit and tool for the attribute being measured ||
 * How we applied this in class || * **2/6/10:** We used a piece of patty paper, which is a nonstandard unit, to measure an angle. We could tell if an angle was bigger than another if it was larger than the corner of the paper or if it was larger than half of the corner of paper.
 * **2/18/10:** We came up with working definitions for Reflex angles, Acute angles, straight angles and exterior angles. All of these measure part of a shape in degrees, a standard unit.
 * **3/16/10**: We started to get an idea of what Area and Perimeter were. When we were working with translations, we described Area as enclosed by the lines of a 2-D shape and perimeter as the sum of the length of the line segments that compose that shape. Even though this was an introduction idea, it gives a general understanding of Area and Perimeter and a great starting point. When we translated the "mystery man" from our assignment, we learned the area, perimeter and sides increased but the angles stayed the same.
 * **3/25/10:** We did work with Area and used different ways to find area. Our class formed squares off of right triangles to fnd the area. This led us to the Pythagorean Theroum or A^2 + B^2 = C^2 Good to recognize the 'mathematics on the horizon' but likely students won't explore this idea at this age level.
 * **4/6/10:** Our class discovered Area Formula's for Triangles, Rectangles, Parallelograms, Trapezoids, and Circles. This is a form of measurement in standard units.
 * **4/15/10**: Surface Area is the sum of all the areas of faces and bases added together. To find Surface area, count all boxes inside a net. The concept of counting to find the surface area can help students learn what surface area is and how to find it. This is a measurement technique that may be considered nonstandard.
 * We also learned about Volume and discussed what Volume was. The ideas we came across were how much water is in a glass, LxWxH=V, and the space inside an object. This helps put together an idea of what volume is and how it can be found.
 * **4/15/10**: Made a conjecture: If the dimensions are closest together it will minimize the surface area when there is a set volume.
 * **4/20/10:** we played with hollow 3D shapes in class and used filler material to find the volume of each. We would transfer the filler materail of one shape into another to compare their volumes. In doing this we found volume formulas for rectangular prisims with square bases, rectangular prisims with a rectangular base, cubes, cylinders, spheres, cones, and pyramids. ||

//MATHEMATICAL IDEAS THEY CONNECT TO:// Working with volme of 3D shapes relates to the position in space (standard #2) that a specific shape occupies. This also relates to spatial visualization (standard #4) because a 3D shape is not flat. There is a whole different dimension that needs to be considered. Measurement relates to rotations, reflections and translations (standard #3) because if you dont have a program that does these things for you, you would need to use a ruler, a proractor and/or a safe-t compass. All of these tools require knowledge in measurement.

Apply appropriate techniques,tools, and formulas to determine measurements || Expectations: //WHY THEY ARE IMPORTANT//: Figuring out formulas can be an important process to understanding how those formuals actually result in what you need. There is a difference between memorizing a formula and actually coming up with a formula. If you memorize a formula then there is a good chance you will forget it since it's just a bunch of numbers, but if you actually found the formula and understand why it works, you will understand the formula instead of just know it. When we worked with 3D objects, we weren't just given a bunch of numbers and processes to find the volume. We actually looked at how the shape was composed and figured out ourselves how to find the volume. Once we found the volume of one shape, we could find the volume of the rest. Good point to make here about the importance of developing formulas, not just memorizing them.
 * Measurement Standard #2
 * meausre with multiple copies of units of the same size, such as paper clips laid end to end
 * use repetition of a single unit to measure something larger than that unit, for instance, measuring the length of a room with a singlee meterstick;
 * use tools to measure;
 * develop common referents for measures to make comparisons and estimates. ||
 * How we apply this in class || * **1/26/10:** We used a safe-t compass to draw 3 circles of the same size so that we could construct an equalateral triangle.
 * We learned how to use Geoboard on our calculators to draw and then measure different types of triangles.
 * **1/28/10:** We went to the math lab and worked on the sketchpad to create and measure different geometric figures
 * **2/4/10:** We discussed rotational symmetry and the angle of rotation. We found that if you divide 360 degrees by the number of points the shape has, it gives you the angle of rotation for a regular polygon.
 * **2/6/10:** We used a clear sheet called patty paper to make an instrument that would measure angles, similar to a protractor.
 * .**2/16/10:** We split shapes up into triangles making each side a triangle. We then found the formula for finding the sum of the interior angles of a shape: (N x 180) - 360.
 * We also found that the last formula and the formulas 360 + 180(N - 4) and 180(N - 2) are all equal and they give you the same answer.
 * To find only one angle of a regular polygon: (180(N - 2)) / N
 * To find and exterior angle: 180 - interior angle
 * To find an exterior angle of a regular polygon: 180 - (180(N - 2)) / N.
 * In class we were trying to find what the formula for circumference is. We found different objects that were circular, wrapped string around the outside of the circle and then measured the string. We used string and a measuring tape as a tool. ||

//MATHEMATICAL IDEAS THEY CONNECT TO//: finding formulas and using measurement relates to volume, area and surface area (measurement #1) since each shape has a formula to find each of those attributes. This measurement also relates to rotations and reflections(standard #3) because there are different techniques for using tools in order to apply a rotation or reflection to a shape. The formulas that we learn in geometry can also be applied to the mathematics in other subjects like physics or chemistry.

Generally very good recognition of big mathematical ideas explored this semester. Some ideas were more identifications of things that are appropriate for later grade bands, but it is good to recognize that students will be studying this in the future. It was really good and helpful to see the descriptions of the importance and connections of the standards at the end. Well done.