Emily,+Erika+and+Zack+-+3-5

Geometric Content Standards and Expectations for Grade Levels 3-5
​ ​ = Analyze characteristics and properties of two- and three-dimensional shapes and develop mathematical arguments about geometric relationships: =

attributes of two and three dimensional shapes and develop vocabulary to describe attributes. || * Number of sides that come together to make points of a shape. according to their properties and develop definitions of classes of shapes such as triangles and pyramids. || * Polygon: Simple closed curve made up of line segments. the results of subdividing, combining, and transforming shapes. || * Subdividing shapes: Why does a quadrilateral always add up to 360 degrees? properties and relationships and develop logical arguments to justify conclusions. || Conjectures:
 * Geometry Expectations 3-5 || Class Relationship to Standards/Expectations ||
 * Identify, compare, and analyze
 * Shapes with or without right angles.
 * Concave vs. Convex.
 * Parallel or perpendicular.
 * Symmetric vs. asymmetric.
 * Obtuse vs. acute.
 * Irregular vs. regular shapes.
 * We came up with these by playing a game with different shapes that were cut out and placed in a baggy. It was called "what's my rule?" One person said which shapes were "in" and which shapes were "out" and the other group members had to guess what the rule was based on what shapes were "in" or "out".
 * We used the "House of Quadrilaterals" approach that was presented in class. We were able to look at the relationships given from each of the quadrilaterals. In the sheet the quadrilaterals could only enter into another room if they had either a door or a stairway leading to it. This gave us the assumption that the quadrilaterals could only enter into another room if it had the same characteristics of another quadrilateral.
 * Polyhedra vs. non-polyhedra
 * Faces vs. surfaces
 * Prisms, pyramids, spheres ||
 * Classify two and three dimensional shapes
 * Angle: Two rays that meet at a point.
 * Right angle: 90 degrees.
 * Obtuse angle: an angle that is greater than 90 degrees.
 * Acute angle: an angle that is less than 90 degrees.
 * Interior angle: formed by two adjacent sides of a shape
 * Exterior angle: formed by extending one side of a shape together with an adjacent side.
 * Reflex angle: an angle that measures greater than 180 degrees and less than 360 degrees.
 * Straight angle: an angle that measures 180 degrees.
 * Full rotation: an angle that measures 360 degrees.
 * Angle of rotation:Is the rotation around the point at which a shape is being rotated.
 * Triangle: Three sided polygon.
 * Fact - The sum of the measures of the interior angles of a triangle are equal to 180 degrees.
 * Acute: a triangle where each angle is less than 90 degrees.
 * Right: a triangle that has one right angle.
 * Obtuse: a triangle with one angle greater than 90 degrees.
 * Isosceles: a triangle with two congruent sides.
 * Fact - The base angles of and isosceles triangle are congruent.
 * Equilateral: a triangle with three congruent sides.
 * Fact - Equilateral triangles have three congruent angles.
 * Scalene: a triangle with zero equal sides.
 * Quadrilateral: four sided polygon.
 * Fact - The sum of the measures of the interior angles of a quadrilateral are equal to 360 degrees.
 * 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 adjacent congruent sides.
 * Square: A quadrilateral with four right angles and four congruent sides.
 * Trapezoid: A quadrilateral with exactly one pair of parallel sides.
 * Perimeter: sum of the lengths of the line segments that compose the shape.
 * Area: the space enclosed in the lines of a two dimensional shape.
 * Scale factor: the amount in which you multiply the pre-image in order to get a similar image.
 * Inductive reasoning: is reasoning based on finding a generalization of a pattern.
 * Deductive reasoning: is reasoning based on already established facts.
 * Pythagorean Theorem:The area of the smallest square (a) + the area of the middle square (b) gives us the area of the largest square (c). The squares are coming off of the sides a and b of the triangle in order to find the area of the square coming off of side c squared. We found the square root of the area of square c in order to find the length of side c. a²+b²=c²
 * During class we tested this on acute and obtuse triangles. In testing these different triangles we found that it only worked on right triangles.
 * Acute triangles: a squared + b squared > c squared
 * Obtuse triangles: a squared + b squared < c squared
 * Hypotenuse: the side of a Right triangle opposite of the right angle. The longest side of a right triangle.
 * Circle: a plane curve everywhere equidistant from a given fixed point, the center.
 * Diameter: A straight line going through the center of a circle, connecting two points on the circumference.
 * Circumference: the distance around the edge of the circle.
 * Radius:the distance from the center of the circle to the edge of the circle. it is half of the diameter
 * Polyhedra: solid shapes with all flat face
 * Prism: two parallel congruent faces that are connected by rectangular regoins.
 * Cube: a prism with six congruent faces.
 * Pyramid: one polygonal face and a point not in the plane of this polygon.
 * Non-Polyhedra: solids with not all flat faces, but includes surfaces and spheres.
 * Cylinders: two parallel congruent faces but have circles for a base instead of polygons.
 * Cones: consists of a circular region which is joined to a point not in the plane of the circle.
 * Faces: two dimensional, flat also known as "sides" for a 3-dimensional shape.
 * Bases: the flat surface that the shape is "resting" on and the flat surface that is parallel to it.
 * Volume: how many cubes fit inside of a shape.
 * Surface area: the sum of all the faces and bases added together. ||
 * Investigate, describe, and reason about
 * During class we used our safety compasses to create a perfect circle. Then we chose four points on the edge of the circle and connected the points to one another to create a quadrilateral within the circle. We then divided the quadrilateral into two separate triangles using the property of subdividing. We learned that if you unfold each of the triangles from the angles that are not connected to the line that forms the triangles each triangle create a 180 degree line or straight angle. Then we added the two 180 degrees together to get a total of 360 degrees. We knew to add these together because the two separate triangles were originally part of one whole quadrilateral.
 * We also did an example of this during class when we were just looking at triangles. We used our polystrips to pin together a triangle. I used an equilateral just because It uses the entire length of the polystrip so it leaves no questions like "what about the rest of the strip that wasn't inside of the original triangle?", but you don't have to use and equilateral. I took one angle where two sides were pinned together and unpinned it. Then I unfolded the lines until they created one long connected line, this represents 180 degrees (a straight angle). This demonstration shows us that when the three separate sides of a triangle are unraveled or unfolded they create a single straight line which is always 180 degrees so we know that all triangles must have interior angles that add up to 180 degrees.
 * Combining shapes
 * We know we can combine the corners of a quadrilateral in order to check to see if they all equal 360 degrees. During class we ripped the four corners off of the quadrilateral while still preserving the original angle it had created. Then we combined the four vertices into a pie shape to create a circle. We know a circle (full rotation) is 360 degrees so when these four vertices come together and create the circle,on complete turn, they all equal 360 degrees. This means the total of the interior angles of a quadrilateral is 360 degrees.
 * Transforming shapes
 * During our project one we discovered that we could rotate and reflect triangles. During this rotation/reflection experiment we were able to transform a single triangle into two triangles that created squares, kites, parallelograms, rectangles and rhombuses. We reflected triangles over all three of its sides to create different types of quadrilaterals. We rotated several triangles around constructed midpoints for each of its sides, 180 degrees to transform these original triangles into different types of quadrilaterals. ||
 * Explore congruence and similarity. || * Congruent means having exactly the same shape and size.
 * An example of this would be:
 * A square that is one inch by one inch is exactly the same as another square that is one inch by one inch because they share the same dimensions. [WILL THE DIMENSIONS OF A QUADRILATERAL ALWAYS UNIQUELY DETERMINE ONE SHAPE? THAT IS, IF YOU HADN'T SAID THAT IT WAS A SQUARE, IS GIVING THE DIMENSIONS OF A QUADRILATERAL SUFFICIENT TO PROVE THEY ARE CONGRUENT?]
 * A square that is one inch by one inch is not the same as or congruent to another square that is two inches by two inches because they have different dimensions even though they are the same type of quadrilateral.
 * Similarity
 * Two geometric shapes are similar if they have the same angle measures, the sides are proportional by the same scale factor and the sides grow by scale factor k.
 * An example of this would be:
 * The mystery hat experiment that was done in class (CP 141) showed the relationships between scale factors and the similarities between the preimage and the new images. In the double band experiment we found that the scale factor was 2. The sides of the mystery hat double, or grew by twice the amount, but remained proportional. The angles of the mystery hat had remained the same. We found the same results when trying the triple band with the scale factor being 3, instead of the sides doubling they tripled . The angles of the mystery hat continued to remain the same. When doing this experiment and coming across these discoveries we realized all three of these mystery hats are similar to each other. ||
 * Make and test conjectures about geometric
 * 1) All triangle have interior angles that add up to equal 180 degrees.
 * Triangle ABC is created with AC representing the base of the triangle. If we were to unfold the triangle from the top at vertice B so lines AB and BC are brought down to create one straight line along with line AC we would see that this is a 180 degree line [ANGLE?]. This means that all triangles when unfolded equal 180 degrees, so a triangles interior angles must equal 180 degrees as well.
 * 1) What is the sum of the interior angles for a polygon?
 * Example: When a quadrilateral is drawn out on a piece of paper or on a program like sketchpad on the computer you can subdivide it into two triangles. We already know from conjecture 1 above that a triangle has a sum of the interior angles of 180 degrees. Now we have two triangles inside of this quadrilateral so we have to multiply 180 degrees by two in order to figure out the sum of the interior angles of a quadrilateral, this total is 360 degrees. Two triangle is the least amount of triangles we can split a quadrilateral into without crossing over the lines that are making the triangles. So you have four sides and two triangles inside of the quadrilateral which means it is the total number of sides - 2 to create the least amount of triangles.
 * The formula for this is: 180 (n-2)
 * 1) Alternate Interior Angles
 * This is another way of proving that all triangles have an interior angle sum of 180 degrees all of the time.
 * [|Alternate Interior Angles.gsp]
 * 1) Concave and Convex Quadrilaterals
 * All quadrilateral that are convex have interior angles that are smaller than 180 degrees. [INTERESTING CONJECTURE, HOW DO YOU KNOW THIS?]

= Specify locations and describe spatial relationships using coordinate geometry and other representational systems: =
 * All quadrilateral that are concave have one interior angle that is greater than 180 degrees.
 * We know this because around one vertice there is a total of 360 degrees (imagine a small circle enclosing this vertice) so there is a concave shape that means the shape has two sides that point inward for example, an inverted kite. This vertice has an exterior angle that is less than 180 degrees and the interior angle is very large, greater than a straight line which is 180 degrees so it has to be more than 180 degrees. A way we can check this is to place a piece of clean edged paper down on one side of the two inverted segments. One side will always be hidden or wrapped all the way around beyond the side you are measuring. This indicates an angle that is larger than 180 degrees.
 * 1) Isosceles triangles always have 2 congruent angles.
 * Triangle KLM is an isosceles right triangle. This is the easiest example to understand why an isosceles triangle always has two equal angles. An isosceles triangle is defined as a triangle with two equal sides. When a triangle has two equal sides that means they are equidistant from the third side so they connect to that side at the exact same angle. When you have a right isosceles triangle you have one angle that is 90 degrees and two other angles that are congruent to each other because the two sides are congruent to each other. In order to find those angles you take the sum of the interior angles which is 180 degrees and subtract 90 degrees from it, you end up with 90 degrees remaining. These two angles must be congruent so you divide 90 degrees by 2 and you end up with 45 degrees which is the interior angle measurement for the two congruent angles that are missing.
 * 1) Venn Diagram Relationships
 * During class we used venn diagrams to discuss/show the relationships between two different polygons.
 * One relationship example is a kite vs. a parallelogram.
 * A kite is defined as a quadrilateral with two pairs of adjacent congruent sides. A parallelogram is defined as a quadrilateral with two pairs of opposite congruent sides. These would both be on the outside portions of an interlocking venn diagram. The center of this diagram would hold only the shapes that can by definition be both a kite and a parallelogram. A square would live in this center portion because by definition it is a quadrilateral with four right angles and four congruent sides. A kite and a parallelogram would have to have four congruent sides and four right angles in order for them to be considered a square. This means that no matter what a square will always be considered a kite and a parallelogram based on its definition. A rhombus also lives within the center of this diagram because its definition is a quadrilateral with four congruent sides. This is a stem off of a square and we examined this in class by using our polystrips which began as squares and were tilted but still had four congruent sides. A kite and a parallelogram must have four congruent sides in order for them to be considered a rhombus. A kite can be a parallelogram but it must pass through a transition stage, which is the center of the diagram, becoming a square or a rhombus before it actually becomes the parallelogram. A parallelogram can also become a kite through this transition stage. A square and a rhombus are the only two quadrilaterals that fall within the means of a kite and parallelograms definition. A rectangle is a quadrilateral with four right angles. A rectangle can fall under the definition of a parallelogram but would not be able to fall under the definition of a kite without transitioning through the square and staying a square. If it doesn't have four congruent sides then it will never be able to become a kite because it wouldn't have two pairs of adjacent congruent sides. A trapezoid is a quadrilateral with exactly one pair of parallel lines. Based on all of our definitions stated above it does not fit into any of them so this quadrilateral cannot be a part of our venn diagram. These explanations stated above proves that the square and the rhombus are the only two quadrilaterals that are allowed to live in the center of our diagram.
 * 1) When the midpoints of the sides of a quadrilateral are connected, they will always make a parallelogram.
 * [[file:Midpoints of quad makes parallelogram.gsp]]
 * The above link is an interactive diagram that proves any quadrilateral connected at its midpoints will always create a parallelogram on the inside of the shape.
 * This works because the midpoints never change, they are always equidistant from the vertices on every side. When all of these midpoints are connected on the inside of the shape they will always be equidistant from each other because the midpoints of the outer shape never change even when the length of the side changes. This creates two pairs of opposite congruent sides all of the time and when the quadrilateral is a square. This creates four congruent sides because a square is equidistant all the way around from its center or from midpoint to midpoint. A square cannot be cut across its center to connect midpoints because it would subdivide the square into two triangles and this distance is greater than the other midpoint/line relationships.
 * 1) Making a equilateral triangle out of three congruent connected circles.
 * We used patty paper during class and sketchpad in the computer lab to create this diagram. First you draw one circle and then draw a second circle with its midpoint starting on the first circles outer edge. The outer edge of the second circle crosses through the midpoint of the first circle which proves that their radii are the same. Then you draw a third circle with its midpoint connected to the point where the first and second circles' outer edges meet. This third circle will then have the same radius as the other two as well. Then you connect the vertices that the center of these three congruent circles have created through their intersection and you will have an equilateral triangle. We know this because all of their radii are the congruent to each other.
 * 1) How to find rotational symmetry.
 * A shape has rotational symmetry if you can turn it in its center of rotation (midpoint) less than 360 degrees and it lands exactly on top of itself again.
 * In order to find rotational symmetry you use the formula: 360/n with n representing the number of sides.
 * We know this because around the midpoint of any shape there is an invisible 360 degree turn and when you divide the number of sides into 360 you end up with the break down of each exterior angle. This is how much the shape has to sweep around the outside of its original format in order to land back on top of itself.
 * 1) A parallelogram is the only quadrilateral that has only rotational symmetry.
 * A parallelogram that is non-rectangular and does not have 4 congruent sides cannot have reflectional symmetry because it cannot mirror itself. All other quadrilaterals have atleast one line of symmetry which means it can be rotated atleast once less than 360 degrees in order to land back on top of itself again
 * 1) How to find the measurement of an exterior angle.
 * An exterior angle is formed by extending one side of a shape, creating a 180 degree line, and it still having an adjacent side. This angle created by this extended line is considered the exterior angle.
 * There is only one exterior angle per interior angle.
 * 1) The sum of the two shorter sides of a triangle must be greater than the length of the longest side.
 * During class we used the random integer function in our calculator to give us three different lengths. While using the random integer function we used the numbers 1, for the smallest side length, 20, for the largest side length, and 3, for the number of sides. These numbers were in parenthesis.Once we had three different lengths we used polystrips to build a triangle using these lengths. If the sum of the two shorter sides, or the two smaller lengths, were equal or less than the largest side, the longest length, the polystrips would not connect and form a triangle.This called the Triangle inequality theorem. If a+b=c
 * 1) The sum of the three smaller sides of a Quadrilateral must be greater than the largest side.
 * We used the same process as above when finding the Triangle inequality theorem. Instead of using three different lengths we used four different lengths to find the Quadrilateral inequality theorem. You can build more than one quadrilateral with with the same side lengths. The second quadrilateral can be a concave or convex quadrilateral. ||


 * Geometry Expectations 3-5 || Class Relationship to Standards/Expectations ||
 * Describe location and movement using common language and geometric vocabulary. || # Rotational symmetry
 * This is showing how a shape can move about its center and land back on top of itself turning less than 360 degrees.
 * 1) Parallelogram inside of a quadrilateral
 * We discovered this on sketchpad during our project one. When we create a quadrilateral and mark the four sides' midpoints and connect them we will always end up with a parallelogram inside of the quadrilateral. No matter how much the quadrilateral is moved around we will always have a parallelogram inside of the quadrilateral.
 * 1) Smilemath with angles being created.
 * We used smilemath in class and outside of class to discover how it worked. A line would move away from another line that shared a vertex with the first line and it would create an angle. We had to guess what the measurement of that angle was based on how far the line had moved.
 * 1) In the program Exploredraw we made a turtle move so many steps to a specific location. We used this to create all different types of shapes.
 * 2) In the program Scratch we made the sprite move so many steps to a specific location to create a shape.
 * 3) WHY ARE THESE IDEAS USEFUL? WHAT OTHER IDEAS ARE THEY CONNECTED TO? ||
 * Make and use coordinate systems to specify locations and to describe paths. || Exploredraw is a program located on our TI-73 calculator. We used this application in class to expand our knowledge base to figure out the angle measurements and distances for shapes such as regular polygons. The Logo light program that we used in Exploredraw has a set of commands that we used in order to guide the turtle to create the shapes on the screen. There were many analyzed decisions we made using this program trying to find the angle measures needed to create the regular polygons. For example we used Exploredraw to figure out how to make a polygon such as a hexagon. Command: Repeat 6 (FD 10 LT 60)

We also started working with another program called Scratch. We used this program as well to describe paths and locations to make certain shapes. We made many more analyzed thoughts and decisions using this program. This program challenged us to think of the path that the sprite would use to create the shapes we needed to make. We created such shapes as squares, equilateral triangles, parallelograms, pentagons, hexagons, octagons, and circles. An example to create a regular octagon Command: Pen down->Repeat 8->__Move 50 steps->Turn 45 degrees__.The underlined section would be repeated the eight times that the command tells the repeated section to repeat. Another big idea we accomplished through this program was to create a command to make a circle. We found that the easiest way to create a circle would be to create a 360gon which would allow the sprite to use a one step and turn technique. Considering that a circle has 360 points that make up a circle. So our command was Pen down->Repeat 360->__Move 1 step->Turn 1 degree__. Again as stated before, the underlined section would be repeated the 360 times that the command states it to repeat.

We predicted what turning the sprite 45 degrees would do compared to the original shape using squares. We found that the square would be titled 45 degrees from the original starting point of the sprite from the first square. || = Apply transformations and use symmetry to analyze mathematical situations: = = Use visualization, spacial reasoning, and geometric modeling to solve problems: = Scratch guessing and minor assessment #3 reflection + reflection = 125degree rotation around point E(center point) ||
 * Find the distance between points along horizontal and vertical lines of a coordinate system. ||  ||
 * Geometry Expectations 3-5 || Class Relationship to Standards/Expectations ||
 * Predict and describe the results of sliding, flipping, and turning two dimensional shapes || * Rotational symmetry: A shape has rotational symmetry if you can turn it in its center of rotation (midpoint) less than 360 degrees and it lands exactly on top of itself again.
 * Reflection symmetry: A shape has reflection symmetry if it has a line that divides a shape in two equal halves that mirror each other.
 * During class we made a venn diagram of shapes that we cut out of magazines. The venn diagram was two interlocking circles that had reflectional symmetry, rotational symmetry and the center was when a quadrilateral had both. We came to the conclusion that the only quadrilateral that fit into the only rotational side was a parallelogram. This parallelogram can not have four congruent sides and has to be non-rectangular. Every other quadrilateral that has rotational symmetry also has reflectional symmetry.
 * During our project one on sketchpad we learned how to flip triangles along the side that we marked the mirror. Then we reflected the original shape over that mirrored side and we were able to create all types of special quadrilaterals except for a trapezoid. A trapezoid requires a combination of two separate unequal triangles so mirroring one will never create this shape.
 * We also rotated triangles to be able to create different types of quadrilaterals. We started by creating the midpoint on one side of the triangle and then marked its center. We then proceeded to rotate it around that midpoint 180 degrees. We discovered this rotation will always create a parallelogram. We know this because the length of the sides are not changing and we are rotating around a midpoint on any given line which will always turn the original triangle upside down while keeping the rotation sides touching. ||
 * Describe a motion or a series of motions that will show that two shapes are congruent || We used sketchpad to mirror or reflect images. We specifically mirrored triangles over each of its sides to create a quadrilateral. All of these triangles that were mirrored ended up with an exact replica of itself including congruent sides and angles. So, when an object is mirrored or reflected it is congruent with the original shape so there are two congruent shapes. ||
 * Identify and describe line and rotational symmetry in two and three dimensional shapes and designs || Two Dimensional Shapes:
 * Rotational symmetry:
 * During class we were given a piece of poster paper and plastic polygons. Our directions were to separate them into categories of only rotational symmetry, only reflectional symmetry and both rotational and reflectional symmetry. After a class discussion, we realized that every shape that had reflectional symmetry had rotational symmetry as well. We also discovered that a parallelogram has only rotational symmetry.
 * During class we made a venn diagram of shapes that we cut out of magazines. The venn diagram was two interlocking circles that had reflectional symmetry, rotational symmetry and the center was when a quadrilateral had both. We came to the conclusion that the only quadrilateral that fit into the only rotational side was a parallelogram. This parallelogram can not have four congruent sides and has to be non-rectangular. Every other quadrilateral that has rotational symmetry also has reflectional symmetry.
 * We also rotated triangles to be able to create different types of quadrilaterals. We started by creating the midpoint on one side of the triangle and then marked its center. We then proceeded to rotate it around that midpoint 180 degrees. We discovered this rotation will always create a parallelogram. We know this because the length of the sides are not changing and we are rotating around a midpoint on any given line which will always turn the original triangle upside down while keeping the rotation sides touching. ||
 * Geometry Expectations 3-5 || Class Relationship to Standards/Expectations ||
 * Build and draw geometric objects || # We used Exploredraw to create geometric shapes. We guided the turtle by giving it certain commands to create whatever shape we wanted it to make.
 * 1) We used Sketchpad to create many different types of shapes by using the tool bar to build any shape we wanted.
 * 2) We used Scratch to create different types of shapes by giving sprite a list of commands.
 * 3) We used our safety compass in class to draw circles and other shapes that required a straight edge.
 * 4) We also used Geoboard on our TI-73 calculators to build and draw geometric objects on a peg board grid. This application gave us the abilities to construct quadrilaterals and polygons and gave us the ability to measure angles, and side lengths that we have used so far in class. ||
 * Create and describe mental images of objects, patterns, and paths || During our homework, using scratch, we had to mentally figure out how an object would look if we repeated the same commands 8 consecutive times. After figuring it out on pencil and paper using my head I tried it on scratch and I had guessed correctly!
 * Identify and build a three dimensional object from two dimensional representations of that object || A three dimensional box can be created by drawing a two dimensional rectangle on a piece of paper. After the first one is drawn, draw another one above it and slightly to the left or right depending on your transformation lines. Once both are drawn, connect the corresponding corners with each other and you will begin to see a three dimensional rectangular box. These slide transformations can be done with all sorts of shapes in order to create prisms or cylinders. ||
 * Identify and build a two dimensional representation of a three dimensional object || What we have covered so far is the creation of nets. These are unfolded two dimensional shapes that can be cut out and created into geometric shapes. During class we were challenged to find how many different nets could be made without repetition in order to create a geometric cube. We found there were a total of eleven different nets ||
 * Use geometric models to solve problems in other areas of mathematics, such as number and measurements || In order to find the volume of a three dimensional shape you must know the shapes dimensions. These numbers allow you to use a formula to measure the specific amount of cubed units that can fit within the shape. The formula for a box is length x width x height. The formula for a cylinder is pie x radius squared x height. ||
 * Recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life || Being able to find the area, height, length,width, or perimeter for a shape is very helpful in everyday life. This can help with a number of things from rearranging a room to building a swing set. Find the surface area and volume of shapes can come in handy when trying to packing an item or ship an item. This comes in handy because you can find what dimensions will give you the least amount of surface area for the packing for the volume that you have. All of these things help in the classrooms and tie into other subjects that the children would be taking. ||

Measurement Content Standards and Expectations for Grade Levels 3-5
= Understand measurable attributes of objects and the units, systems, and processes of measurement: = ​
 * Measurement Expectations 3-5 || Class Relationship to Standards/Expectations ||
 * Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute || We used the Geoboard on our TI-73 calculators to build and draw geometric objects on a peg board grid here as well. This application gave us the abilities to construct quadrilaterals and polygons and gave us the ability to measure angles, and side lengths that we have used so far in class.

We were given patty paper in class and asked to fold the paper in a way to find which of three angle measurements were the largest and to order them from smallest to largest. We found that the base of the patty paper could represent 180 degrees because it was a straight line. With this thinking, we realized that if we could fold the paper 180 times that we could come up with the all the degrees to measure the angles. With our folds we found enough angles to measure the three angles. Not with absolute measures but enough to comprehend which angles were largest in size. We had degree measurements for 0, 22.5, 45, 66.5, 90, 113.5, 135, 157.5, and 180 degrees.

Area is used to measure how many units surround shape. Surface area is used to measure how many units are on the outside of the object. Surface area can be measured the same way as area by taking the three dimensional shape apart and creating it into a net. Then you use the formula length x width for the entire two dimensional net in order to find the surface area of the entire three dimensional shape. Area is measured in square units.

Volume and Surface Area to be done by Zach! While doing our homework, we started to dive into finding surface area, not just speaking of these two in general but how we can use them in the contexts of other situations such as packaging, or gift wrapping. We were given different problems to try and find out what boxes would have the lowest or highest surface areas. In doing so, we discovered by drawing boxes on grid paper, using cube manipulative's, and inquiry thinking. We looked at trying to fit 24 blocks into the shape of a rectangular prism. We had to find all of the ways that we could package a rectangular prism with 24 cubes. In doing this we found many different packaging strategies for the 24 cubes. Although we had to go further and state which package allowed for the least material so we calculated which had the least surface area. We noticed first that shapes such as a 1x24x1 combination would give the largest surface area due to the spreading out of the cubes. We ended up coming to a conclusion that the more compacted the cubes were into a cube like shape the less surface area it would create making packages more efficient. The 1x24x1 gave a surface area of 98 units cubed. The formula for the surface area calculation was 2(L x W) +2(W x H) +2(L x H). We used the same calculations to find that the combination of 4x3x2 gave the smallest surface area and therefore the best option for packaging. When we calculated 4x3x2 with the formula for surface area we found that the surface area was 52 units cubed. So we learned through inquiry approach that the more compacted the prism the lower the surface area will be and the more spread out the prism is the higher the surface area will be.

We went even further through the homework to find that we could find the lowest surface area easier by factoring down the cubes. We went through certain problems where we needed to find the best package fitting for 27 cubes. We factored down 27 cubes into 9x3, and then decided to factor 9 into 3x3. We ended up with 3x3x3 and found that the surface area would end up at 54 units cubed. If we would have used 1x27x1 we would have had a surface area of 82 units cubed. This solidifies further our idea of the compacted cubes and helps us see that we can factor the number we need to find packaging space for to make best selection for a cost efficient packaging strategy.

See below for more explanations on this standard. "Apply appropriate techniques, tools, and formulas to determine measurements: select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles." || = Apply appropriate techniques, tools, and formulas to determine measurements: = In order to find the perimeters of irregular shapes we used the Pythagorean theorem. We started by sectioning off the irregular shape into right triangles and then used, a squared + b squared = c squared, when c equals the hypotenuse, in order to find the length of c. Once we found the lengths of all the sides of the irregular shape we added them together and the total was our perimeter. When finding the area of a trapezoid we used a fair share method. We took the first base + the second base and divided by 2 to find the average. We need to find the average of the bases in order to distribute equally because the first base is longer than the second base. We then need to multiply by the height of the trapezoid. The entire formula for the area of a trapezoid is ((base 1 + base 2) / 2)*h || Area can be measured using grid paper. It is measured in square units, the units can be any of the above stated in length. Volume is found by using a ruler or grid paper to find the length, width, height, and radius of different three dimensional shapes. Then it is computed by using the appropriate formula for the different shapes. Volume is measured in units cubed where the units can be any of the above stated in length. Size of angles is measured by using a protractor or an angle ruler. Angles are measured in degrees. If you do not have a protractor you can use a straight edged clean corner or a piece of notebook paper, patty paper or printer paper. That is a 90 degree angle, if you fold it exactly in half you get a 45 degree angle and then if you continue to fold it in half you will continue to get a degree half of the previous degree. This can be handy but it will not alway give you an exact angle measurement, it will just get you a very good educated guess. We have not gone over weight, time or temperature in class yet. || In order to find the area of related triangles you have to cut the original triangle in half at a 90 degree angle so that it creates two congruent triangles. One congruent triangle can be reflected over is connecting side and then rotated about the midpoint of the hypotenuse 180 degrees so that it creates a parallelogram. During this process we cut the base of the original triangle in half so the base of the parallelogram is half of the original base. The height of the triangle has not changed during the transformation. This is why the formula for a triangle is 1/2bxh. A parallelogram can be transformed into a rectangle in order to find its area as well. Starting from each obtuse angle of the parallelogram, draw a line at a 90 degree angle, perpendicular to the opposite congruent side. The angles will be congruent with each new triangle created inside of the parallelogram because of the opposite congruent angles due to opposite congruent parallel sides. Slide one right triangle over so the hypotenuses' are connected. This creates a rectangle which is connected to the rectangle that was made in the center of the original parallelogram when the right triangles on the ends were created. Now there is one big rectangle with the same base and height as the original parallelogram, we know this because all we did is slide one triangle with part of the length cut off to the other side so it reconnected with the same length on the other side of the polygon. Now we can use the formula bxh in order to find the area of a parallelogram. || When figuring out the least amount of surface area used for a specific volume we found that the dimensions closest to being a cube would work best. This is helpful because we can figure out which dimensions would save surface area which in turn would save money on packaging in the everyday world. We found this in class using different nets that we built into three dimensional shapes from our course packs. Each three dimensional shape that we cut out from the course pack had different dimensions. We found that the dimensions that were closest to being cubed provided the least amount of surface area and the dimensions that were the furthest apart were going to provide the most amount of surface area. To find the volume of rectangular solids we used blocks. In class we had nets drawn in our course packs and had to use blocks in order to build these nets. Using the blocks helped us figure out what the volume is of a rectangular solid and how to find it. The formula that was discovered in class for volume is l*w*h. In other words the volume formula for a rectangular solid is the area of the "bottom layer" x the height. This gives us the area of all of the "layers" or how many cubes will fit within the solid. ||
 * Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems ||  ||
 * Carry out simple unit conversions, such as from centimeters to meters, within a system of measurement || Not Applicable ||
 * Understand that measurements are approximations and understand how differences in units affect precision ||  ||
 * Explore what happens to measurements of a two dimensional shape such as its perimeter and area when the shape is changed in some way. || In project number two we were asked to figure out if we traced string around the geometric shape if we could stretch it into a shape on graph paper to find the area of the original shape. We found that the perimeter would b approximately the same but the area would be changed due to the area mass that was stretched. We decided to cut out the blob figure and trace it on the graph paper and approximate the area. We found that the two different options gave much different areas so we found the differences in the area are changed if the shape or perimeter is moved. The perimeter will stay the same but the area can change. ||
 * Measurement Expectations 3-5 || Class Relationship to Standards/Expectations ||
 * Develop strategies for estimating the perimeters, areas, and volumes of irregular shapes || During class we looked at irregular shapes and the way we found to find the actual, non-estimated, area of this shape was to draw a box around the entire shape. The boxes height would be from the tallest point of the irregular shape to bottom or base of the irregular shape. The width would go from the widest point from left to right. We counted all of the whole boxes inside of the irregular shape and then were left with non-whole boxes. Instead of guessing, we connected cubes together by drawing small boxes around them in order to find the area of a cube originated from a diagonal line. Example: if the diagonal line was starting in one corner of a cube and connecting to the opposite corner of a cube, three cubes down and two cubes over, we would draw a box around the line with dimensions 2x3. We knew the area of the 2x3 box was 6 and we could see that the diagonal line was cutting the box in half so we knew the area of half the box was 3square units. We did this throughout the entire irregular shape until we found the entire area of all the non-whole boxes and added all of the separate areas together in order to find the complete area of the irregular shape.
 * Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles || Length is measured by using a ruler. It can be measured in meters, feet, inches, centimeters, millimeters or standard units.
 * Select and use benchmarks to estimate measurements || In class when learning the volume formulas for cones, cylinders, circles, and spheres we used benchmarks to estimate the measurements to find formulas. When find the formula for the volume of a cone we filled an empty cone with "filler" or little plastic squares. We know that these "fillers" will not cover the entire space inside of the cone due to air and the shape of the little plastic squares compared to the surface of the cone. We would fill the cone up with the "filler" and then pour the "filler" from the cone into the cylinder. We did this process three times until the cylinder was as full as possible with "filler". From this we figured out that the volume of a cone is 1/3pie(r squared)*height. We did the same strategy to figure out the volume formulas for a circle and a sphere, instead of comparing them to the volume of a cylinder we compared them to the volume of a cone. From this strategy we found that we had to fill up the sphere times in order to fill the cone one time. give us a completed and simplified formula of Volume= 4/3pie(r cubed) ||
 * Develop, understand, and use formulas to find the area of rectangles and related triangles and parallelograms || The area of a rectangle is base x height. Base and height are always perpendicular to each other so we know that we can use that formula for finding the area of a rectangle because it has four right angles. The base of a rectangle measures how many congruent squares can fit right to left filling the entire length. The height of a rectangle measures how many congruent squares are stacked on top of each other from top to bottom. Then you multiply length x width to calculate the entire amount of square boxes that can fit within the perimeter of the rectangle, this is the total area.
 * Develop strategies to determine the surface areas and volumes of rectangular solids || To find the surface area of a rectangular shape we used nets. When using the nets we split them (the nets) up into different sections, which sections were faces and which sections were bases. We then found the area of each face and each base by doing length * width, height * width, and length * height. After finding the areas to the faces and bases we added all of the sums together. In turn we found the total surface area of the net, or the three dimensional shape. The formula we discovered for surface area was 2lw+2hw+2hl. There needs to be a 2 in front of the lw, hw, and lh because you have two faces of each dimension that needs to be accounted for. The 2 in the equation is making the equation simpler instead of having it be lw+lw+hl+hl+hw+hw which can make it draw out and confusing. If the three dimensional shape is a square box the surface area formula is 2(area of base)+ 4(area of the faces). This is because the two bases are going to be congruent with each other and the four different faces are going to be congruent with each other due to the definition of a square.

GENERAL COMMENTS: YOU'VE RECORDED A LOT OF WORK HERE. VERY THOROUGH, ALMOST TOO MUCH DETAIL IN SOME PLACES. FOCUS ON BEING CONCISE WITHOUT REPETITION IN MULTIPLE SECTIONS. THE WAY TO ACCOUNT FOR THE REPETITION IS TO EXPLICITLY NOTE THE CONNECTIONS BETWEEN STANDARDS AND EXPECTATIONS. THE IMPORTANCE OR USEFULNESS OF CONCEPTS WAS ONLY SOMETIMES MENTIONED. TRY TO STRIKE A GREATER BALANCE BTWEEN THE DEVELOPMENT OF IDEAS, WHY THEY ARE IMPORTANT, AND HOW THEY ARE CONNECTED TO OTHER MATHEMATICS. YOU EMPHASIZED SOME GOOD CONNECTIONS TO REASONING AND PROOF IN THE STATEMENTS OF CONJECTURES AND GAVE GOOD EXAMPLES TO SUPPORT YOUR REASONING. CONNECTIONS TO PSSM EXPECTATIONS WERE TYPICALLY CLEAR, BUT SOMETIMES OBSCURED BY LENGTHY DESCRIPTIONS. FOR NEXT TIME, FOCUS ON CLARITY OF BIG IDEAS (OVER DETAILS), CONCISENESS (INCLUDING RELATION OF IDEAS TO EXPECTATIONS AND UTILITY), AND CONNECTIONS (TO OTHER CONTENT OR STANDARDS). WELL DONE.