Brittany+and+Jenna's+page+for+grades+3-5

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 * There are four goals for the geometry standard and two goals for the measurement standard that are common across all grade bands:**

The Expectations for 3rd-5th Grades include:

=** __Geometry__ **= = =

-BIG IDEA FROM CLASS: starting to identify shapes and if they are a polygon and starting to develop vocabulary dealing with 2- 3- dimensional shapes and angles. How is this done? -TYPES of LINES: Line (<-->)Go on forever. Line Segment(---)Piece of a line, has 2 endpoints. Ray (--->) -CHARACTERISTICS of an angle: at any vertex there are at least 2 angles possible. -TYPES of TRIANGLES: obtuse, acute, isosceles, equilateral, scalene. -TYPES of QUADRILATERALS: kite, square, rhombus, trapezoid, rectangle, parallelogram. -what is a CIRCLE: a collection of points all the same distance away from a center.
 * **Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships**
 * A.** Identify, compare, and analyze attributes of two- and three- dimensional shapes and develop vocabulary to describe the attributes:

-(idea from class): Definition:a triangle is a polygon that has 3 sides. .-Properties of a triangle: The sum of all the interior angles is 180 degrees. -Look at classifying triangles: obtuse, acute, isosceles, equilateral, scalene. -DEFINITION of a Triangle: A polygon that has three sides and three interior angles. Interior sum of all angles in a triange is 180 degrees. -DEFINITION of an Angle: 1. formed by two line segments (rays) at a shared endpoint, called the vertex. 2. The area between two sides of a shape or line segments where they meet or where they come together. 3. The turn that is represented by two rays. -DEFINITION of a Polygon: a simple closed curve with straight lines. -DEFINITION of a Quadrilateral: Polygon with 4 sides and 4 angles. -DEFINITION of a Rhombus: is a quadrilateral with all four sides congruent. KITE: a quad with 2 pairs of congruent adjacent sides. SQUARE: a parallelogram with all sides and angles congruent. A rhombus with all angles congruent. RECTANGLE: a parallelogram with 2 / 4 right congruent angles. RHOMBUS: a parallelogram with 4 congruent sides. TRAPEZOID: a quadrilateral with one set of parallel sides. PARALLELOGRAM: a quadrilateral that has 2 sets of parallel sides.
 * B.** Classify two- and three- dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids:


 * C.** Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes:
 * When a square is subdivided by its diagonal, two triangles can be created.
 * A rectangle can be formed from a parallelogram by cutting the region formed by the parallelogram and then translating the cut-off piece and rearranging it so that the parallelogram visually matches a rectangle. This can lead to conjectures about the relationship between the areas and rectangles with the same base and height.
 * What about the tangram?

-CONGRUENCY is a term used to represent an idea of "equal" in geometry. -CONGRUENCY: a congruent shape would be a shape that matches when you fold it. The shape would match and be the same as the other side of the shape when cut in half. When performing an isometry (translation, reflection, rotation), the resulting shape is congruent to the original shape. Similar, the the math world, means that two shapes are not exactly alike but the have some of the same properties that make them alike. Refine based on what you now know or put below with more formal ideas. When performing dilations (stretching or shrinking of a shape), the resulting shape is NOT congruent to the original shape, but it is similar to the original shape. //Conjecture for similiarity:// Two shapes are similar if: 1. Their shape is preserved 2. Angles are congruent 3. Corresponding sides are scaled to the given factor //Conjecture for congruency:// Two shapes are congruent if: 1. Corresponding side lengths are congruent 2. All corresponding angle measurements are congruent 3. If we transformed the shape by performing an isometry: translation, rotation, or reflection.
 * D.** Explore congruence and similarity:
 * Congruency:**
 * Similarity**:
 * E.** Make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions:


 * **Specify locations and describe spatial relationships using coordinate geometry and other representational systems.**

-We can describe movement by using a coordinate system or synthetically (i.e. visually on paper or using a plane of some other sort). This is used when transforming shapes by reflecting, rotating, or translating a shape.
 * A.** Describe location and movement using common language and geometric vocabulary:

-We can use the coordinate system to help us locate the coordinate points of each shape. By finding the points on the coordinate system, we will be able to figure out exactly how many spaces, or how far, the shape moved. This works for all the transformations: rotation, translation, and reflection.
 * B.** Make and use coordinate systems to specify locations and to describe paths:

-The distance between points along the horizontal and vertical lines of a coordinate system can be found using two numbers to name points on the coordinate grid, creating a coordinate pair. The first number represents the X value and the second the Y value. The distance between these two points can be found by starting at one point and moving in the direction of the next, along the coordinate plane either left or right and/or up or down.
 * C.** Find the distance between points along horizontal and vertical lines of a coordinate system:
 * ** Apply transformations and use symmetry to analyze mathematical situations **

-Using Sketchpad, you can predict what a shape will look like when it is slid, flipped, or turned from the original shape and then a separate page can be created where there are reflections of the original shape and the shapes can transformed to make new shapes. -Tools to help determine if a shape is congruent would be a protractor, patty paper, mira, or ruler. For example: (x,y)=(x,-y) means that (4,-2) becomes (4,2)**
 * A.** Predict and describe the results of sliding, flipping, and turning two- dimensional shapes:
 * When performing a reflection over the x-axis, (x,y) =(x,-y)
 * When performing a reflection over the y-axis, (x,y) =(-x, y)
 * When translating (sliding) vertically by (d) units, (x,y) =(x, y + b)
 * When translating (sliding) horizontally by (d) units, (x,y) = (x + b, y)
 * When rotating the shape 360 degrees, (x,y) = ( x, y)
 * When rotating the shape 180 degrees, (x,y) = (-x, -y)
 * When rotating the shape 270 degrees, (x,y) =( y, -x)
 * When rotating the shape 90 degrees, (x,y) = (-y, x)
 * The negative signs mean that the previous number for that coordinate becomes the opposite of what it was originally.

B. **Describe a motion or a series of motions that will show that two shapes are congruent:** -You can rotate a shape from a center -point and see if it has rotational symmetry. -You can use: -Rotational, Reflection or Transitional Symmetry to help find out if a shape is congruent. -For reflection symmetry, you can take a shape and cut it in half and if those two halves match up to one another then it is congruent. -For rotation symmetry, you can turn the shape NOT a full turn (all the way around) and it will match up with the original shape. -Rotational=(TURN) -Reflection=(FLIP) -Transitional=(SLIDE) -Tools to help determine if a shape is congruent would be, protractor, patty paper, mira, ruler. Symmetry and transformations are two different ideas. Here it wanted a series of motions vs determining if the shape had symmetry or not. C. **Identify and describe line and rotational symmetry in two- and three- dimensional shapes and designs:** -If a shape has line symmetric symmetry, you can fold/cut in half and its sides will match up... or mirror each other. -If a shape has rotational symmetry, an image of the shape can be rotated about a center point and the image will match up exactly with the original shape, without going a full 360 degrees. Use visualization, spatial reasoning, and geometric modeling to solve problems
 * Patty paper can be used by tracing the original shape and then turning the patty paper about the center point until it matches up exactly with the original shape.
 * A tangram can be used to show how one large shape can be divided into smaller, different shapes and used to create new shapes with the pieces.
 * The application, Building Perspective on the calculator, can be used to help use spatial reasoning and visualization.
 * The game, Tetris, can be used to help with spatial orientation because the shapes must be turned in order to fit together.
 * 3-D cubes help to create shapes/objects and help students see the different views of the shape

A. **Build and draw new geometric objects:** -Use geoboard to draw shapes on your calculator. -Use Sketchpad to draw shapes on computers. -Use Scratch to draw shapes, which helps explore exterior angles.

B. **Create and describe mental images of objects, patterns and paths: -**You can look around your classroom and see different angles. Like 90 degrees= Ceiling tiles, chalkboard, etc... What about nets?

C. **Identify and build a three- dimensional object from two- dimensional representations of that object:** -By looking at 2-D silhouettes of the front, top, back, and side views of shapes, we can build 3-D shapes with 3-D cubes.

D. **Identify and build a two- dimensional representation of a three- dimensional object:** -By looking at 3-D models/shapes, we can create 2-D silhouettes of the top, front, back, and side views.

E. **Use geometric models to solve problems in other areas of mathematics, such as number and measurement:** -Cut out shapes of triangles, circles, rectangles and squares to see how they can measured/put together to form other shapes. -Take a piece of computer paper and fold it into a square and cut and fold multiple times to make a puzzle from the shapes. See how they can fit together to make one big sheet.

F. **Recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life:** -How the ceiling tiles are parallel, perpendicular. The chalkboard is at a 90 degree angle. You can use geometry in everyday life, everything that you look at is at some degree or angle.

Understand measurable attributes of objects and the units, systems, and processes of measurement:

A. **Understand such attributes as length, area, weight, volume and size of angle and select the appropriate type of unit for measuring each attribute:** -Length is used to measure the side of a shape. Looking for the "area between" the two lines of a shape, or where they meet or come together is a step you must use to find the angle measurement, and then determine that degrees are used to find the size of the angle. -Angles are measured in degrees.

B. **Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems:** -Angles are measured using degrees. Degrees are made especially for measuring angles. This is to help make sure the measurements are accurate. How did we address the Need for using standard units with measuring the angle? C. **Carry out simple unit conversions, such as from centimeters to meters. within a system of measurement:** -Unit conversions at the 3rd-5th grade level are simple, such as expressing 150 centimeters to 1.5 meters or 3 feet as 36 inches. But did we do this? You are to include activities that we did in class and connect them to these standards. If we didn't do it, we didn't do it. D. **Understand that measurements are approximations and understand how the differences in units affect precision: - ** The larger the unit of measure being used, the less precise the measurement will be. For example, is if we measured the classroom trashcan in feet, the number would be less accurate than if we used inches or centimeters because larger units of measure require more rounding, which in return causes more room for error.

E. **Explore what happens to measurements of two- dimensional shapes such as its perimeter and area when the shape is changed in some way:** When the shape is scaled to a given scale factor, the perimeter and the area change. How? Example: When calculating the perimeter and area of a given square, we can use x and y to represent the side measures. ORIGINAL SQUARE: Perimeter: to find the perimeter of the original square, we could use 2x+2y. Area: to find the area of the original square, we could use xy. If we scaled used a scale factor of 2, going from the original to the new shape: Perimeter: the perimeter could be found by 4x+4y Area: to find the area of the new shape could be found by 4xy. What about the project we just finished? Apply appropriate techniques, tools and formulas to determine measurements:

A. **Develop strategies for estimating the perimeters, areas, and volumes of irregular shapes:** -Using regular shapes, we can draw these regular shapes around the irregular shape to help us calculate the area. This can also be found by covering it with transparent grid paper and counting units or by breaking it apart into regular shapes that can be measured. -Using a string, or some other flexible object, similar to what was used for "the blob", we can place the string around the outside of the irregular shape so that it fits exactly and then measure the length of the string What will this measure? -For volume, an estimation can be made by visually imagining how much space the object/shape occupies. in cubic units B. **Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles:** -Area is measured in square units. Area in the space that fills up something (the covering of the floor). -Angles are measured in degrees. -Temperature is measured in degrees. -Weight can be measured in grams, kilograms, ounces, pounds, etc...depending on the object being weighed. -Time can be measured in seconds, minutes, etc -Volume is measured in milliliters, liters, ounces -Length can be the measurement of a line or the "length" of a side. You can measure length in feet, inches, meters, centimeters etc.

C. **Select and use benchmarks to estimate measurements:** -Benchmarks that can be used are: paper clips, pieces of paper, length of pinky finger (or any finger), length of a pencil, paces

D. **Develop, understand, and use formulas to find the area of rectangles and related triangles and parallelograms:** How to find area: Triangles: 1/2 base x height Trapezoids: (B1+B2)h/2 or 1/2(B1+B2)h *h=height Parallelogram: Base x height Rectangle: Base x height or length x width

E. Develop strategies to determine the surface areas and volumes of rectangular solids:** -Surface area can be found using "nets" which are two dimensional patterns that show the faces of a three dimensional shape, laid out flat so that all of the units of the faces are visible. -Rubix (Rubik) cubes are good for showing surface area because they have colored faces that show the surface area.