Stacy+and+Hannah's+Page+-+Grades+3-5

​​​​​​​​Geometry Standards: **Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes** >> How do all of the above connect to the expectation?
 * 1. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.**
 * Sides and angles of geometric shapes begin to differentiate one particular shape from another particular shape. We found this by looking at many different shapes and developing different methods on how to categorize the shapes.
 * Angles of shapes can be convex (push out) or concave (cave in).
 * Differences of line terms-
 * Lines- a collection of points that go on forever, they have no starting point or finish point;
 * Line segments- a collection of points that have a starting point and a finish point;
 * Rays- a collection of points that have a starting point but no endpoint.
 * Terminal Ray- the ray that is rotated to make the angle
 * Initial Ray- the original ray that is the base of the angle
 * Definition of Angles-
 * 1. Formed by two rays, line segments, or lines at a shared end point called the vertex.
 * 2. The area between two sides of a shape or line segments, where the meet or come together
 * 3. The turn represented by two rays.
 * What is the biggest angle?
 * A: Angles can go on forever with the turn idea in mind.
 * Measurement of Angles:
 * They are measured in degrees.
 * A degree can be looked at as 1/360th of a complete turn.
 * A reflex angle is an angle between 180 and 360 degrees
 * An exterior angle is the angle that is made if a line is extended out from one side of the angle of a polygon. It is the angle that shares a vertex and side with the interior angle of the polygon
 * An exterior angle can be found on the interior of a shape
 * An exterior angle can be found with the formula 360/n, where n is the number of sides of the regular polygon being examined
 * Vertex angle- the angle of a shape where two sides meet
 * Can be found in a regular polygon with the formula 180-(360/n), where n is the number of the sides of the regular polygon
 * Scale factor- two shapes after big dilated will have a common scale factor or ratio that they've been adjusted too. For example consider two squares one square with sides lengths of 2 inches long and another with side lengths of 4 inches long. The scale factor of the 4 inch sided square compared to the 2 inch sided square would be 2 because the side lengths of the 4 inch one are double that of the 2 inch one.
 * Three-Dimensional Shapes characteristics- found after attempting to classify "bucket-full of shapes" shapes into various categories and homework over 3-D shapes
 * Base-the "bottom' face of a 3-D shape
 * Face- a flat 2-d surface of the 3-D shape
 * Edge- where two faces of a 3-D shape meet

**Classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids**
 * Polygons(working definition)- a simple closed curve with straight sides
 * Triangle definitions
 * a polygon with three sides
 * The class came to this definition after stripping the first attempt of all unnecessary information from a definition such as it has three angles and the sum of all angles is 180°.
 * Triangles can be made if the sum of the 2 shorter sides is greater than the longest side on the triangle.
 * Types of Triangles
 * Right triangle- a triangle that has one right angle and has two acute angles
 * Obtuse triangle- a triangle that has one angle that is greater than 90°
 * Acute triangle- a triangle with 3 acute angles
 * Equilateral triangle- a triangle that has three sides of equal length
 * Isosceles triangle- a triangle with only two equal sides
 * Scalene triangle- a triangle with no equal length sides
 * Classifying Triangles
 * Classified by Angles
 * Working definition focuses on the measure of one angle, or multiple angles that classifies the triangle
 * Obtuse triangle
 * Acute triangle
 * Right triangle
 * Classified by Sides
 * Working definition focuses on the sides of shapes and their lengths, which classifies the triangle
 * Equilateral triangle
 * Isosceles triangle
 * Scalene triangle
 * Some triangles can be classified with both angles and sides
 * When the characteristic that the working definition focuses on (i.e. side) effects the other classification (i.e. angle) and there is only one possibly outcome.
 * Equilateral triangle: Because all sides are equal in length, all interior angles will be equal as well.
 * Scalene triangle: Because all sides are different in length, all interior angles will be different as well.
 * Quadrilaterals
 * Quadrilaterals can be made if the sum of the 3 smallest sides is greater than the 4th side, which is the longest side as well.
 * Types of Quadrilaterals
 * Quadrilateral- A polygon with four sides
 * Parallelogram- a quadrilateral that has two sets of parallel sides
 * Rhombus- A parallelogram with four congruent sides
 * Rectangle- a parallelogram with all sides and angles congruent
 * Trapezoid- a quadrilateral with one set of parallel sides
 * Kite- a quadrilateral with two pairs of congruent adjacent sides
 * Circle
 * A collection of points the same distance from the center
 * The distance of those points form the center is known as the radius
 * Regular polygons
 * A regular polygon is a polygon that have all side lengths congruent, all central angles congruent and all vertex angles congruent. They have both reflection symmetry and rotational symmetry
 * **Three-Dimensional Shapes-found after working in groups attempting to identify and classify various 3-D shapes and receiving a glossary sheet for assistance**
 * Cube- 3-D shape with six congruent faces
 * Cylinder- 3-D with two opposite congruent circular faces and a lateral surface that looks like a rectangle that has been wrapped around the the congruent circular faces
 * Prism- 3-D shape with top and bottom that are congruent polygons and parallelogram shaped lateral faces
 * Pyramid- 3-D shape that has one polygonal base and triangular lateral sides that meet at a common vertex

**Investigate, describe, and reason about the results of subdividing, combining and transforming shapes** > Figures that have been transformed, whether synthetically or on a coordinate plan are always congruent what about tangrams? How did we use this idea above to find areas? **Explore congruence and similarity** **Make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions**
 * Transformations include slides (translations), flips (reflections) and turns (rotations) - Class groups were given different types of transformations to explore and explain to the rest of the class What were the explanations?
 * Congruent shapes- shapes that match or are the same but can be in different orientations or positions.
 * Similar objects are those who's overall shape has been preserved, corresponding angles are congruent and corresponding sides lengths are scaled to the same given factor
 * All congruent shapes are similar, but not all similar shapes are congruent.- For example two squares are congruent therefore they are also similar because their overall shape has been preserved, corresponding angles are congruent and corresponding sides lengths are scaled to the same given factor. However two squares one that is 2 inches by 2 inches and another is 4 inches by 4 inches aren't congruent because the shapes don't have the same side lengths but they're similar because their overall shape has been preserved, corresponding angles are congruent and corresponding sides lengths are scaled to the same given factor. What other ideas did we test in class? Developing area formulas? Finding the sum of side lengths of triangles and quads?

**Describe location and movement using common language and geometric vocabulary** Think about common language: slides, flips, turns, stretching, shrinking
 * 2. Specify locations and describe spatial relationships using coordinate geometry and other representational systems.**
 * When sliding or translating shapes the direction and the distance are needed in order for the translation to be accurate and to ensure that the final shape is still congruent to the original shape
 * Isometries are considered to transformations because the shape remains congruent
 * Dilations are when manipulations to the shape leave the shape similar but not congruent

**Make and use coordinate systems to specify locations and describe paths**


 * On coordinate systems locations are described by the distances in relation to the x and y axis, or (x,y). The "x" means the location is what ever distance, positive or negative, from center point zero on the x-axis. The "y" means the location is whatever distance, positive or negative, from center point zero on the y-axis. Where the two coordinates intersect is the location.
 * Paths in coordinate systems can be horizontal across the x-axis, vertical along the y-axis, or some combinations there of which may be at angle between horizontal or vertical
 * To describe the path to translate a shape a pattern was found in class that to move a shape horizontally a variable should be added to the x-coordinate but not the y-coordinate (x+a,y); to move a shape vertically a variable should be added to the y-coordinate but not the x-coordinate (x,y+a); to move a shape along the line, or a similar direction as the line y=x, a diagonal, the variable should be added to both the x- and y- coordinates (x+a, y+a)
 * Find the distance between points along horizontal and vertical lines of a coordinate system **
 * Distances between points along horizontal and vertical lines of a coordinate system can be found by locating one of the points and then locating the other point and then comparing their horizontal or vertical coordinates. Along either line the distance is usually in units of the coordinates (by ones or tens, etc.)


 * 3. Apply transformations and use symmetry to analyze mathematical situations.**
 * Predict and describe the results of sliding, flipping and turning two- dimensional shapes **
 * Transformations can be done on a coordinate plane or synthetically (without coordinate plane)
 * Transformation categories:
 * Slides-translations
 * The manipulation to the x- and y- coordinate can only be in the form of addition or subtraction. If the either or both are multiplied or divided by a number, then the shape will become bigger or smaller, but not translate.
 * When translating points or shapes on a coordinate plane the same operations have to made to the coordinate. Examples: The same operations have to made to every x-coordinate in order for the coordinate to move horizontally. The same operations have to be made to every y-coordinate in order for the coordinate to move vertically. To move along the line y=x, the same operations have to done to both the x- and y-coordinate.
 * When sliding a shape, you do not change the shapes orientation. You must slide and rotate the shape in order for the orientation to be changed.
 * Turns-rotations
 * When rotating a shape around a specific point all points of the shape, and therefore the shape, are moved in a circular direction, specified by degrees.
 * When rotating a shape the shape stays congruent to the original shape
 * A rotation can also be "made" by completing two reflections, the angle of rotation is twice the angle of between the two lines that were used for reflection
 * Flips-reflections
 * When a shape is reflected over a line, the original points and it's primes (the reflected points) are the same distance from the line of reflection
 * Tools to determine reflections include patty paper, angle rulers, protractors, and rulers
 * When reflecting shapes and/or points on a coordinate plane certain relationships between the line of reflection and the resulting reflected point coordinates
 * Reflecting over x-axis originals: (x,y)-- reflected points (x,-y)
 * Reflecting over y-axis: originals (x,y)-- reflected points (-x,y)
 * Reflecting over the line y=x: originals (x,y)-- reflected points (y,x) *the values switch not the actual coordinates


 * Describe a motion or a series of motions that will show that two shapes are congruent**
 * Two shapes can be shown to be congruent in a number of ways. At the most basic level one could use patty paper and trace one of the shapes being compared onto the paper and then laying the patty paper over the shape and if the two shapes are identical then or one isn't sticking out from the other they could be considered congruent. On a more advanced level using tools that measure length and angles one could measure all the angles and side lengths of one shape and see if they match the corresponding sides and corners of the other shape.

4. **Use visualization, spatial reasoning, and geometric modeling to solve problems.**
 * Identify and describe line and rotational symmetry in two- and three- dimensional shapes and designs**
 * Reflectional symmetry- if a shape was divided/cut into halves then the two halves would be identical mirror images
 * Rotational symmetry- If a shape was rotated around its center then it could be rotated less than a whole rotation and the original shape would be identical to the rotated figure.
 * Building Geometric Shapes using Geometer’s Sketchpad Using GSP we’re able to build triangles, isosceles and equilateral from circles. We’d draw a circle then place two segments that originated in the center of the circle and went to the perimeter of the circle and then connected the two points where our original two segments met the outside of the circle to form a triangle. Depending on the distance between where the two segments met the outside of the circle it could form an isosceles or equilateral triangle. Without the measure function of GPS, we couldn’t know for sure if we had built an equilateral triangle so we had to draw two circles that overlapped with the same radius
 * nets? silhouettes? Building perspective app?

**
 * Create and describe mental images of objects, patterns and paths
 * Looking around the room and identifying all the shapes within the walls, tables, chairs. Could be various circles, squares, rectangle. Further examine the potential angles created. All mainly 90 degrees.

Identify and build a three-dimensional object from two-dimensional representations of that object
 * The students can build three-dimensional shapes from looking at pictures of the front, side, and top views of the potential shape in a two-dimensional view. This will give them a visualization and knowing that it is a three-dimensional shape it will help them to work with the spatial representation of that shape. Refer to what we did in class vs what could be done with the students. Trying to determine all connections from 1510 to these expectations.

**Identify and build a two-dimensional representation of a three-dimensional object**
 * After finding a three dimensional object to build a two-dimensional representation takes spatial visualization. A cube for example is a combination of six congruent square faces so when say drawing a three-dimensional figure the number of faces is key to remember. The length and shape of the faces is also important to remember when building a two-dimensional representation. The view from which the three-dimensional shape is drawn is important too, because a two-dimensional representation can't show all the faces of a three-dimensional object. Using special computer applets two-dimensional representations can be drawn, as well as by hand.


 * Use geometric models to solve problems in other areas of mathematics, such as number and measurement**
 * Using different shapes, either found around the room or cut out of paper, you can use them to measure up to each other, or put them together to form other shapes.
 * In class, we folded up a piece of paper and ripped it to created different shapes. This gave us a visualization of how other shapes fit into various shapes, and how you can use, say, a triangle to determine the are of a square.
 * Recognize geometric ideas and relationships and apply then to other disciplines and to problems that arise in the classroom or in everyday life**
 * Right angles are used commonly in construction or building because of the stable structure they supply. When looking around our classroom or building it is difficult to find angles that aren't right angles
 * In snowboarding the number of turns that the person makes while flying through the air is referred to in number degrees for example a 360 for one complete rotation or a 720 for two complete rotations.
 * Find area and perimeter is especially important and useful in everyday life. For example knowing how much pasture you can fence in with a certain amount of fencing (perimeter), knowing how much carpet is needed to cover a room (area), knowing the size of your bike tire to calculate the distance it travels in one rotation (circumference), or figuring the area of a pizza to know how much pepperoni will fit on it (area of a circle).

1. Understand measurable attributes of objects and the units, systems, and processes of measurement.**
 * Measurement Standards

**Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute**
 * Length-length is a straight distance that can be measured in feet, inches, centimeters, millimeters, kilometers, etc...
 * Area- is a covering of a flat space, measured in square units such as square feet, square inches, square centimeters, square kilometers, etc...
 * Angles are measured in degrees, a degree can be looked at as 1/360th of a complete turn


 * Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems**
 * Standard units are important for measuring because they are the always the same no matter who uses them or where they are from. If a science experiment isn't measured in standard units and then the experiment results are sent to another scientist the second scientist can't accurately interpret the data because he may not understand the units used by the first scientist. If however the first scientist used standard units any other scientist or person can review the data and understand what it is telling.
 * Standard units in our customary system that we have used in class for measuring length are feet, and inches. Standard metric units that our class has used and become familiar with are millimeters and centimeters.


 * Carry out simple unit conversions such as from centimeters to meters, within a system of measurement**
 * Unit conversions for this grade level come from their knowledge of relationships between units and their understanding of multiplicative situations to make conversions. Their conversions would be expressed in such a way: 150 centimeters as 1.5 meters or 3 feet as 36 inches.
 * Understand that measurements are approximations and understand how difference in units affect precision**
 * Angles are measured in degrees- this is because a common unit of measure is needed for measuring angles so comparison of measurements can be made accurately.
 * Other measurements such as lengths, if they are measured in accurately and then used to figure area not only are the lengths incorrect but also the length. Precision is important for measuring at all times


 * Explore what happens to measurements of a two-dimensional shape such as its perimeter and area when the shape is changed in some way**
 * Dilations effect perimeter and area. When for example a square with side lengths of 2 inches is dilated to a square with side lengths of 4 inches both the perimeter and area change. The original square had a perimeter of 8 inches (4 sides x 2 inches) and an area of 4 square inches (2 inches x 2 inches). The dilated square has a perimeter of 16 inches (4 sides x 4 inches) and an area of 16 square inches (4 inches x 4 inches). The perimeter doubled and the area quadrupled.

**Develop strategies or estimating the perimeters, areas and volumes of irregular shapes** No volumes???
 * 2. Apply appropriate techniques, tools and formulas to determine measurements.**
 * Perimeters of polygons can be found by adding up all the outside side lengths. So a rectangle with dimensions of 3 inches by 7 inches would have a perimeter of 3+3+7+7=20 inches.
 * Areas of polygons can be found with specific formulas depending on the shape.
 * A rectangle can be found with the formula of length x width. A square is like an array that was discussed in Math 1500.
 * A parallelogram can be found with the formula of base x height. This is because a parallelogram was found that when one pointy end was "cut off" and and then placed next to the other pointy end in a complementary manor is makes the shape of a rectangle.
 * A triangle can be found with the formula of 1/2 x base x height. This is because a triangle is like half of a rectangle, two triangle put together in a complementary way can form a rectangle. A single triangle is half a rectangle
 * A trapezoid can be found with the formula of 1/2x (base1 + base2)x height. The trapezoid can resemble a rectangle with triangles put together. Therefore the formula has two different base lengths base1 + base2. The triangles account for the 1/2 and the height is because of its rectangle resemblance.
 * Perimeter aka the circumference of a circle can be found with the formula of pie x diameter. We found this by taking make circles found in everyday like and measuring their diameter and perimeter by hand with tape measures. A correlation of close to three was found between the diameter and perimeter.
 * Area of a circle can be found with the formula of pie x radius squared
 * Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles**
 * Angles can be measured with many tools. People can make their own tools to measure them. However in order for there to be a common unit with which to measure angles most of the time protractors, angle rulers, or other similar devices that measure in degrees are used.
 * Length-length is a straight distance that can be measured in feet, inches, centimeters, millimeters, kilometers, etc...
 * Area- is a covering of a flat space, measured in square units such as square feet, square inches, square centimeters, square kilometers, etc...
 * Angles are measured in degrees, a degree can be looked at as 1/360th of a complete turn