Pre+K-2+Lynda+Hart+&+Corrin+Eggelston

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


 * Standard 1 **

__** Analyze characteristics and properties of two and three dimensioanl geometric shapes and develop mathematical arguments about geometric relationships: **__ Expectations


 * **//Recognizes, 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 //**

Triangles: // First, in order to get a better understanding of two dimensional shapes we started off by playing a game called, "Sorting activity". After playing the game we came together as a whole and thought of ways we could classify two dimensional shapes. As a class we came up with several different ways of how we might put certain triangles into groups. After we established this we went onto classify what a polygon is. A polygon is a simple closed curve made up of line segments. A simple closed curve is a line segment that can be traced with the same starting and stopping point without crossing itself or retracing any part of the curve. We also played the game ,"guess my rule". This showed us what kind of triangle that person might be thinking of without actually showing us what type of triangle it is.

Next, we had to make a list of what all triangles have in common and what some triangles have in common with each other. Those were placed into two different categories. First, we had to classify what a triangle is, that is a three sided polygon. All triangles have: three sides, 180 degrees, three angles, no parallel lines, base and height and at least two acute angles. Some triangles have: acute angles, right angle, obtuse angle, scalene, isosceles, equilateral, perpendicular lines, asymmetric and symmetric, and might be proportional to each other. Once we had these two groups we made our class definitions of the triangles. They are as follows: __Acute Triangle-__ each angles less than 90 degrees, __Obtuse Triangle-__ A Triangle that has a angle greater than 90 degrees, __Right Triangle-__ A triangle that has a 90 angle, __Equilateral Triangle-__ three congruent sides and three congruent angles, __Scalene Triangle- A__ triangle with no congruent sides, __Isosceles Triangle- A__ triangle with two congruent sides.

After we agreed upon these definitions, we had to classify what a angle is, and how we might convince someone that when the angles are added up it equals to be 180 degrees. An angle is the space between two intersecting rays at the corner. The way we can convince somebody that all the angles add up to be 180 degrees is to cut the angles out of the triangle and place them together to equal 180 degrees- a straight line. We also proved this to be true by using the geoboard application on our calculator. We also proved that we cannot have more than one obtuse angle in a triangle. If we were to have more than one obtuse angle this would not connect our triangle. This was proved in the Writing Assignment number one. Once, the class had the discussion on what an angle is, we were than asked to play with smile math (a game on the calculator) to guess what kind of angle was being offered, this gave us a sense on how big the angle measurement was, how small it was, or how spot on we were. This also showed us how big the angle would look like. All these things led us into placing the triangles into Venn Diagrams. This would show us what certain triangles had in common, if they had nothing in common, or if they only some things in common. This also showed us what kind of triangles would sit in the Venn Diagrams. //

Quadrilaterals: //After we had finished with the triangles we went on to discuss quadrilaterals. A quadrilateral is a four sided polygon. We continued this conversation just like we had with the triangles. We asked what some quadrilaterals have, what the quadrilaterals had in common. After we made up our list we came up with definitions for regular quadrilaterals, and then figured out what Venn Diagram they belonged too. They are as follows: __Parallelogram:__ a quadrilateral with two pairs of opposite congruent sides, __Rectangle:__ a quadrilateral with four right angles, __Rhombus:__ a quadrilateral with four congruent sides, __Kite:__ a quadrilateral with two pairs of non overlapping adjacent congruent sides, __Square:__ a quadrilateral with four right angles and four congruent sides,__Trapezoid:__ a quadrilateral with exactly one pair of parallel sides.//

//**Rotation and Reflection​ Symmetry** We showed this by taking apart and putting back together by using the line of symmetry. The line of symmetry is what divides the shape in half so that the other half (after divided) looks like the other half. It can be placed right on top of the other half. When taking apart the quadrilaterals it makes two triangles, different types of triangles depending on what type of quadrilateral it is. The same thing happens with the triangles when you place them together it makes a quadrilateral. This also depends on the type of triangle. For example, putting two right triangles together make a square as long as the triangle is isosceles. //

This connects to work done in our project one by rotating and reflecting symmetry by right isosceles. When reflected, it make a square or another right isoscles triangle. When rotated 180 degrees, it created a square.
 * By drawing shapes (types of triangles and quadrilaterals), we were able to predict what if they have rotational or reflective symmetry.**

In summary of triangles, when reflected or rotated, it is useful to create a quadrilateral. Which helped us prove that when a quadrilateral has a line of symmetry, it creates two special types of triangles. For example, when bisecting a square, we have two right isosceles triangles. Other mathematical ideas these are connected to daily objects such as in construction as a contractors measuring tool measures angles for a door, angles for a roof pitch or a floor/wall dimensions.

Translation: It rotates two and three dimensional shapes. No, rotations rotate or turn the shapes. Translations only slide. This slides the figure matching each point to an image point. We developed this by a book example on course pack p. 138, this is used with the doubling of a rubber band and connects to enlarging a figure. This is connected to scale transformations (similar figures, not translations).

Transformation: This produces a copy or image of an original figure in a new position. We developed this by reflecting a polygon, on coursepack page 113. This also helped us with translation. This was used to help us find the line of symmetry when reflecting a figure.

Triangle Inequality Theorem: By looking at polystrips and using random numbers from in this case 1-20, we made several varieties of triangles. This was useful by visually seeing that the two similar numbers need to add up to greater than the third side so that the triangle was solid and not fall in on itself. It connects to the triangle inequility theorem. If the two smaller sides are smaller than the greatest it will not connect. If they are equal, it will create a straight line. (a+b>c).

Pythagorean Theorem: a2+b2=c2. The hypothenuse equals the side opposite the right angle. This only works for right triangles. We figured this out by actually checking it out on obtuse triangles and acute triangles. When we did it with the acute triangle we noticed that it is a2+b2 c2. This content is likely not appropriate for this grade band. Not sure how it fits with this standard.

3D Shapes: Face- flat surface, bases are the bottoms, vertex is where a point meets.

polyhedra- has all flat surfaces. A nonpolyhedra is everything else. Anything with non flat surfaces.

Prism: 3d shape- Two parallel, congruent faces that are connected by rectangular regions.

Pyramids: A pyramid consists of one polygonal face and a point not in the plane of this polygon, which we call as its top.

Cylinder: Two parallel and congruent faces.

Cone: a circular region which is joined to a point not in the plane of a circle.

Cube: it has all lateral faces, all the sides are congruent, 3d prism where all are squares.

Nets: these are outlines of what the actual figure looks like, if you were to dissect a 3d image or a 2d image.

__**Specify locations and describe spatial relationships using coordinate geometry and other representational systems;**__ For shapes, we had to learn how to interpret each and apply our ideas as to how each was proportional to another. We had to measure and create ways to measure distances using interior and exterior angles. By using tools such as patty paper, we can figure out what a degree is. A degree is 1/360 of a circle. By using tools of technology such as geometer sketchpad, we learned how to move angles, reflect and rotate all types of triangles and quadrilaterals. We found that our ideas were useful in daily life by being able to take a piece of paper and have the ability to measure something. These ideas are connected to daily life looking at a skyscraper and noticing that they require many measurements and angles to build.
 * Standard 2**
 * Expectations **
 * **//Describe, name and interpret relative positions in space and apply ideas. //**
 * **//Describe, name and interpret direction and distance in navigating spaces and apply ideas about directions and distances. //**
 * **//Find and name locations with simple relationships such as a "near to" and in coordinate systems such as maps. //**
 * 

Interior angles gives us the angles that are inside a shape. Exterior angles gives us the angles that are outside the shape that makes up the 360 degrees.** The formula that is used to calculate interior angles is: 180n- 360. The formula to calculate the exterior angle of a regular polygon is 360 divided by n. Where n is the number of sides the figure has. The first formula is true for any polygon with n sides and the second formula is only true for regular polygons. These formulas will help a construction guys to help build a house, or a garage, or even help manufactures build cars, couches, or even help judge when you are turning in any vehicle.

Translations- This will help us judge how near we are to sliding a couch in a room when trying to rearrange furniture. We developed a theory that all points stay in the same orientation when moved to a new location. This connects to the idea of insuring that things will be able to fit into a new spot.

Transformation: This helps us find the line of symmetry when we reflect a figure or created a new image in a different spot. It helps us explore the associations between reflection, rotation, and translation. No need to duplicate information from above. Simply reference that there are connections between the standards. Triangle Inequility Theorem: We developed an idea from using the polystrips. We randomly picked 3 numbers out of the calculator using randint. With these three number we had to see if they would connect, not connect, or form a straight line. What we noticed that is if a+b>c they create a triangle, if a+b>c, the triangle will not connect. If A+b=c, it will create a staight line. We also noticed that this also will work for quadrilaterals. If a+b+c>d it will form the quadrialteral. If a+b+c<d, it will not connect, also if a+b+c=d it will create a straight line.

Pythagorean Theorem: a2+b2=c2. The hypothenuse equals the side opposite the right angle. This only works for right triangles. We figured this out by actually checking it out on obtuse triangles and acute triangles. When we did it with the acute triangle we noticed that it is a2+b2<c2. We also noticed that when we tested it out on the obtuse triangles we noticed that a2+b2> c2.

3D Shapes: Face- flat surface, bases are the bottoms, vertex is where a point meets.

Nets: these are outlines of what the actual figure looks like, if you were to dissect a 3d image or a 2d image.


 * Standard 3**

<span style="color: #000080; display: block; font-family: 'Times New Roman',Times,serif; font-size: 110%; text-align: left;">**__Apply transformations and use symmetry to annalyze mathematical situations__**; **<span style="color: #800000; display: block; font-family: 'Times New Roman',Times,serif; font-size: 150%; text-align: center;"> Expectations **
 * **//<span style="color: #800080; font-family: 'Times New Roman',Times,serif;">Recognize and apply sides, flips, and turns. //**
 * **//<span style="color: #800080; font-family: 'Times New Roman',Times,serif;">Recognize and create shapes that have symmetry. //**

//Rotational Symmetry- is the shape moving around a certain point and still landing in the same spot. Reflection symmetry- is taking the shape and flipping it in the opposite direction and it still looks the same. Shapes that are rotated will look the same and shapes that are reflected will be flipped or mirror images of each other. We did examples of this with the alpha shapes. It also should sit on top of itself. The other one that we talked about is the asymmetrical- this would have neither rotational or reflection symmetry. Although some shapes have rotational and reflection symmetry.//

//As a class we took a look at our alpha shapes and went through and found out what ones have symmetry and what ones did not. Then we went onto establish what ones have rotational, reflection, neither, or both.


 * Triangles:** reflection: r,n,v rotational: n asymmetrical: b, h, i
 * Quadrilaterals**: rotation: k, o, u, q, w reflection: t, g, s, w, u, q asymmetrical: c, a, e//

This is connected and useful to create mental images of geometric shapes using spatial memory and spatial visualization. We can use activities such as rotation and reflection symmetry to build the idea of the 360 degree angles.

Translation: We drew a line though the same point or pre image, then drew a perpendicular line through it. This helped us line up where the next figure was going to be slid. It connects to the idea of knowing that this figure is able to be congruent by either flipping, sliding, or turning.

Transformation: We created a new image off of the original image, from there we created a line of symmetry/ reflection symmetry. This helped us show that the figure is equidistant from the original figure. This helps us with the knowledge of knowing that when creating a new image in a new spot that it is associated with reflection, rotation, and translation.

Pythagorean Theorem: a2+b2=c2. The hypothenuse equals the side opposite the right angle. This only works for right triangles. We figured this out by actually checking it out on obtuse triangles and acute triangles. When we did it with the acute triangle we noticed that it is a2+b2<c2. We also noticed that when we tested it out on the obtuse triangles we noticed that a2+b2> c2.

<span style="color: #851e1e; display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: center;">**Standard 4**

<span style="color: #000080; display: block; font-family: 'Times New Roman',Times,serif; font-size: 110%; text-align: left;">**__Use visualiation, spatial reasoning, and geometric modeling to solve problems:__**
 * <span style="color: #800000; display: block; font-family: 'Times New Roman',Times,serif; font-size: 150%; text-align: center;">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 measurement.
 * Recognize geometric shapes and structures in the environment and specify their location.

//To begin with this we viewed an image quickly and reproducing it on paper, this was an idea of what we thought the image looked like.

Once we had the image on the piece of paper, the teacher asked us what we looked at in order to get a visualization of the image that was being produced. We discussed the angles that we seen, along with the number of sides, or even what it looked like. Then we went onto discuss why all the angles added up to 180 degrees in a triangle and 360 degrees in a quadrilateral.//

By using techniques such as patty paper, we can fold this to form different angles. We can then see how the angles reflect and show symmetry and/or reflection symmetry for each shape. The class participated in bringing in magazine pictures of everyday life.

Objective is to have the students see math in their everyday life. When posted on board, we did see how everyday life can be symmetrical, asymmetrical,reflective, rotational or various shapes and sizes.

Translations ( scale transformations relate objects that are mathematically similar ): Developed the idea of all points moving the same distance and same direction. We used a double knotted rubber band to translate a picture from one paper to another. This made it larger but the lines were greater and the angles stayed the same. This would be useful to transfer a object/diagram from one spot to another. The idea connects to enlarging one object while keeping the line lengths the same proportional distance.

We also tried enlarging the same picture with a three knotted rubber band and noticed that it had the same effects as the double rubber band. The angles also stayed the same as well as the lines of the Mystery Hat Man growing in size. We noticed that it had a scale factor of (k) which is 2 x's and 3 x's the original amount of the Mystery Hat Man. This helped us develop on how it was related and how it affected everything around it. Everything gets enlarged at some point, but the angles still stay the same. It connects to enlarging clothes, shoes, or xerox copies.

Transformation: Developed the idea of producing the image in a new position. It helps us see that there is a line of symmetry when creating a new copied image. It also helps with the idea of translation, because you can see that the points should all line up exactly the same amount as in the line of symmetry, if not then your line of symmetry is in the wrong position.

Triangle Inequility Theorem: We developed an idea from using the polystrips. We randomly picked 3 numbers out of the calculator using randint. With these three number we had to see if they would connect, not connect, or form a straight line. What we noticed that is if a+b>c they create a triangle, if a+b>c, the triangle will not connect. If A+b=c, it will create a staight line. We also noticed that this also will work for quadrilaterals. If a+b+c>d it will form the quadrialteral. If a+b+c<d, it will not connect, also if a+b+c=d it will create a straight line.

Pythagorean Theorem: a2+b2=c2. The hypothenuse equals the side opposite the right angle. This only works for right triangles. We figured this out by actually checking it out on obtuse triangles and acute triangles. When we did it with the acute triangle we noticed that it is a2+b2<c2. We also noticed that when we tested it out on the obtuse triangles we noticed that a2+b2> c2.

<span style="color: #000080; display: block; font-family: 'Times New Roman',Times,serif; font-size: 110%; text-align: left;">__Understand measurable attributes of objects and the units, systems, and processes of measurement:__ // For measuring with the non standard units, we took a look at the patty paper. This showed us what we could do when trying to measure an angle. This was easy because the corner is ninety degrees. This also helped us to prove that we could use anything with a ninety degree angle to get an measurement, it might not be accurate but we could get a general idea of what an angle measurement might be. [GOOD.] //
 * <span style="color: #8c2222; display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: center;">__Measurement standards__ **<span style="color: #882525; display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: center;">__**Standard 1**__ <span style="color: #000080; display: block; font-family: 'Times New Roman',Times,serif; font-size: 121%; text-align: left;">
 * <span style="color: #800000; display: block; font-family: 'Times New Roman',Times,serif; font-size: 150%; text-align: center;">Expectations **
 * //<span style="color: #800080; font-family: 'Times New Roman',Times,serif;">Recognize the attributes of length, volume, weight, area, and time . //
 * //<span style="color: #800080; font-family: 'Times New Roman',Times,serif;">Compare and order objects according to these attributes . //
 * //<span style="color: #800080; font-family: 'Times New Roman',Times,serif;">Understand how to measure using non standard and standard units. //
 * //<span style="color: #800080; font-family: 'Times New Roman',Times,serif;">Select an appropriate unit and tool for the attribute being measured. //

//With the standard units we used a ruler. This gave us an accurate look into how long the line segment is. We also used a protractor, angle ruler, or a safety compass, this helped us measure the angle measurement itself.//

To measure translation we used a rubber band. We then also took and double knotted a rubber band which enlarged the figure 2 times. We also the went on to triple knotting the rubber band which then enlarged the figure 3 times. This is useful because it showed us that we enlarged the figure 2 or 3 times the original figure, and got the same figure, only larger, and the angles stayed the same. We also proved this by tracing the original figure on patty paper and copied it into the other new figures, and realized that the lengths were proportional and the angles stayed the same by the scale factor. When we drew in the original figure and traced it into the new figures, we realized that it was the area formula. This connects to the area and perimeter formulas, which will be developed later on in class.

Triangle Inequility Theorem: We developed an idea from using the polystrips. We randomly picked 3 numbers out of the calculator using randint. With these three number we had to see if they would connect, not connect, or form a straight line. What we noticed that is if a+b>c they create a triangle, if a+b>c, the triangle will not connect. If A+b=c, it will create a staight line. We also noticed that this also will work for quadrilaterals. If a+b+c>d it will form the quadrialteral. If a+b+c<d, it will not connect, also if a+b+c=d it will create a straight line.

Pythagorean Theorem: a2+b2=c2. The hypothenuse equals the side opposite the right angle. This only works for right triangles. We figured this out by actually checking it out on obtuse triangles and acute triangles. When we did it with the acute triangle we noticed that it is a2+b2<c2. We also noticed that when we tested it out on the obtuse triangles we noticed that a2+b2> c2.

3D shapes- As a class we came up with some ideas of how to find area for 3D images. 2d shapes area formulas: Knowing the formulas is good, but knowing how to derive the formulas and where they originate from is even better. Not sure I would expect first graders to memorize formulas such as these, but instead, they should have an intuitive idea for what area and volume represent (spatially). Square: length x width Rectangle: Length x width Parallelogram: base x height Triangle: Half x base x height Trapezoid: base one plus base two x height all divided by two Circle: pie r squared

3D Area formulas: for a box, its length x width x height Mckenzie's theory was the using the factor tree: example is: 24< 12 and 2. 12< 6 and 2. 6<3 and 2. This will give you the lowest factor for the dimensions of a box. The closer the numbers are we realized that they will give us the less amount of volume that are being used.

Volume: how many cubes can fit inside any given box. The space that is taken up by space.

<span style="color: #842424; display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: center;">**<span style="color: #b02727; display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: center;">Standard 2 ** <span style="color: #000080; display: block; font-family: 'Times New Roman',Times,serif; font-size: 110%; text-align: left;">**<span style="color: #000080; display: block; font-family: 'Times New Roman',Times,serif; font-size: 110%; text-align: left;">__Apply appropriate techniques, tools and formulas to determine measurements:__ **
 * Expectations**

// We used a protractor to measure angle measurements, we also used a ruler to measure the line segment. A safety compass was used to create a perfect circle for the Venn Diagrams. These were all used in measuring angles, and side lengths. This helped us classify where we wanted to place our triangles and quadrilaterals, to show us what belonged where in the Venn diagrams. //
 * //<span style="color: #800080; font-family: 'Times New Roman',Times,serif;">Measure with multiple copies of units of the same size, such as paper clips laid end to end. //
 * //<span style="color: #800080; font-family: 'Times New Roman',Times,serif;">Use repetition of a single unit to measure something larger than the unit, for instance measuring the length of a room with a single meter stick. //
 * //<span style="color: #800080; font-family: 'Times New Roman',Times,serif;">Use tools to measure. //
 * //<span style="color: #800080; font-family: 'Times New Roman',Times,serif;">Develop common referents for measures to make comparisons and estimates. //

We discussed as a class how we could potentially measure our classroom by using a sheet of paper. (possibly multiple copies of each) We would figure out the measurement of the paper and then multiply by how many times that would cover the volume in the room. We used the floor as the example.

The class used our safety compasses to draw three circles the same size so that we could construct an equilateral triangle. This would open the door for us to engage in the sketchpad.

The class did group projects on sketchpad to design and discuss how each shape is made and measurements of each. Sketchpad taught us to understand measurement angles.

Translation- We used patty paper to measure out the lengths of the sides, as well as the angle measurements, a rough idea of what the angles were. This is useful because we know that we don't have to use a protractor or a ruler every time. It connects to the idea of knowing that the sizes can be similar and are proportional to one another.

Transformation: For this we used coursepack pages 115 and 116. On one page we had to draw the figure from the line of symmetry and then we had to create a line of symmetry for another figure. First we measured the distance and then we drew in the line of symmetry which helped us understand where the line should actually be placed.

Volume and surface area: We used pictures of cubes and prisms to figure out the number of faces and volume inside of each cube. This would be useful to figure out the amount of volume inside of the cube as well as the amount of surface area. This connects to how we decide what sizes of packaging boxes are most beneficial to use.

Generally identified the major ideas in the course and made some good connections to the standards. Some information was duplicated throughout the page. It would be clearer to make explicit connections between standards by explaining how they are related instead of just copying the text. Not all content seems appropriate for this grade band. It was encouraging to see the connections to real-world applications and most of the ideas were well articulated in terms of the development of in class work.