Ashley+H.+&+Jennie+S.+PreK-2

= Geometry Standards for Grades PRE-K-2 =

= Standard #1: = ==**• Analyze characteristics and properties of two and three dimensional geometric shapes and develop mathematical arguments about geometric relationships. **==

Expectations:
1. Recognize, name, build, draw, compare, and sort two and three dimensional shapes. 2. Describe attributes and parts of two and three dimensional shapes; 3. Investigate and predict the results of putting together and taking apart two and three dimensional shapes.

How we applied it in class:
1. We drew types of triangles and polygons that were flashed for seconds on the overhead projector. We rotated and translated shapes to determine how the points and images are related to one another by comparing the original shape to the image shape. We recognized and sorted polyhedra and non-polyhedra shapes in class after playing "What's My Shape?" game with precut images of these two types of shapes. 2. We sorted two dimensional shapes that were precut into recognizable shapes, such as obtuse vs. acute shapes. We defined a polyhedra as a prism or pyramid shape. We defined non-polyhedra shapes as cylinders, cones, and spheres. 3. We defined line segment, the sides of a triangle, and simplified word terms. (i.e., closed shape was used for polygon definition.) We folded shapes in half to see if they classified as symmetrical. Also we proved how we know a quadilateral interior angles add up to 360 degrees by taking the corners off and making a pie out of them. This would work with proving that a triangles interior angles add up to 180 degrees also. We examined a polyhedra and recognized the rectangle sides and bases on these shapes. Likewise, the non-polyhedra shapes were examined and found no edges because of the circular shape.

Purpose and utility of the activity. It's usefulness.
1. To stimulate our ability to recognize and draw two dimensional shapes. This was a hands on exercise to increase memory retention of the lesson being reviewed compliments visual learners ability to grasp the lesson. We utilized the "What's My Shape?" game to build students spatial reasoning skills. 2. To build on our ability to recognize and sort two dimensional shapes according to like properties. Sorting polyhedra and non-polyhedra shapes reinforces a students geometric vocabulary, and concrete reasoning abilities. 3. To unify the group's definitions of mathematical terms used in the classroom. When we experimented with rotation and translated shapes we were building concrete reasoning skills by determining the degree of rotation and the slide distance of translation for image shapes. Examining the properties of polyhedra and non-polyhedra builds spatial and concrete reasoning skills. Students are able to 'sound off' the similarities and differences of these two categories of solids.

Other mathematical ideas this connects to:
1. Different two dimensional shapes can be crafted out of multi-colored felt, and can be used in elementary grades for students to recognize, name, compare and redraw what the shapes. Shapes can be paper cutouts so students can easily fold the shape in half to prove or disprove its symmetry, or the use of patty paper for students to recognize line segments, rotational and reflection abilities for shapes. Precut images of polyhedra and non-polyhedra shapes can give visual learning reinforcements to written definition knowledge. 2. Previous cut out shapes can be used for sorting them into the game, "Guess My Rule." Such as convex shapes or concave shapes provides greater opportunities for reasoning skill development in analyzing shape properties. 3. Elementary students can discuss simplified definitions to use as a classroom for math terms. While this is a good point, the other purpose of using 'simplified' definitions at the Pre-K-2 level is that the level of rigor or preciseness in //some //  terminology is not a major learning goal at this grade level. Developing more of an informal and intuitive grasp of shapes and related ideas is more important. This can be a significant time to stretch students deductive and inductive reasoning skills if they are allowed to express their mathematical arguments to simplify definitions for their classroom. Team building can result in distinguishing precut polyhedra and non-polyhedra shapes.

= Standard #2 =

Expectations:
1. Describe, name, and interpret relative positions in space and apply ideas about relative position; 2. Describe, name, and interpret direction and distance in navigating space and apply ideas about direction and distance; 3. Find and name locations with simple relationships such as "near to" and in coordinate systems such as maps.

How we applied it in class:
1. We used Geometer Sketchpad to create custom tools for specific polygons being studied in class. We played a game "What's My Shape?" in class. We were able to identify examples of Geometric solid faces (flat side) and an edge (where 2 faces meet). The shapes with surfaces (curve) have a radius and vertices of a geometric solid are the points or corners. These shapes are called "solids" whether they are filled or not. 2. We explored angle degrees in Geo-board (on our TI-73) to create different polygons. 3. We practiced angle relationships, interpreted direction and distance for creating shapes in Exp Draw "Logo Light" on the TI-73. The Geo-board is a type of grid that can be used to introduce students to latter math using grids for Algebra problem solving.

Purpose and utility of the activity. It's usefulness:
1. Geometer Sketchpad gives a student hands on experience to visual results and develop their concrete and abstract reasoning abilities. 2. Geo-board provides students with the opportunity to reason the results of direction and distance in creating polygons. 3. Exp Draw "Logo Light" gives students an opportunity to create shapes based on their reasoning abilities and to experience the exterior angle importance of the created shape. "What's My Shape?" game provides students with an opportunity to build their spatial reasoning with one student picking a shape; the other students asking "yes/no" questions to deduce which shape the student has chosen as "their shape."

Other mathematical ideas this connects to:
1. Geometer Sketchpad program can be used for this grade level to develop hand and eye coordination skills in drawing polygons. The "What's My Shape?" game builds teamwork and develops an spatial learning environment for all students whether they are asking the questions or listening to another student ask a question. 2. Geo-board program can be used for this grade level by counting the dots on the screen to create simple shapes. 3. Exp Draw "Logo Light" can be used to introduce students to creating shapes by teacher assisted directions for students unable to make up their own directions. technology is a wonderful tool for allowing students to make and test mathematical conjectures.

= Standard #3 =

Expectations:
1. Recognize and apply slides, flips, and turns. 2. Recognize and create shapes that have symmetry.

How we applied it in class:
1. We had an isosceles triangle on paper and folded the image in half to prove it had symmetry. Also we created a poster from previous shapes into three different groups: reflection symmetry, rotation symmetry, and no symmetry. We used two rubber bands knotted together to show how a shape can be redrawn to a larger size (stretching the figure). We also made a chart of symmetry with the shapes in the bags in our table groups. 2. We brought in examples of symmetry from magazine clippings and posted them in a Venn Diagram that included symmetry shapes. The Venn Diagram overlapped the relationships between images with and without line, reflection, and rotational symmetry. For both expectations 1 and 2, the Geometer Sketchpad project we did teaches the recognition of flips, sides, turns and that shapes have symmetry.

Purpose of the utility and activity. It's usefulness.
1. The symmetry lessons shows students shapes are not created with equal properties. For instance, some shapes have more than one line of symmetry. The rubber band "stretching the figure" exercise proves (although not with 100% to scale accuracy) the students ability to enlarge an image without the use of a angle ruler. 2. The usefulness of this classroom activity is to broaden the students perspective of the different types of symmetry in their everyday life.

Other mathematical ideas this connects to:
1. Students can make shapes out of play dough and patty paper. The play dough shapes can be sliced in half to show symmetry. Patty paper (or tissue paper) can be folded in half to prove shape symmetry. Younger students can cut shapes in half and lay one half on top of the other half to prove symmetry. Students can create polygon shapes out of construction paper and write a note to their parents on the top side of the shape, then fold the shape in half (like a card) to prove the line of symmetry. (These can be given to parents during conferences.) 2. If a teacher has Geometer Sketchpad (or a simplified version) at her/his disposal the students could learn to use it and start to make their own shapes. Also students can use a Geo reflector mirror to show if a shape has reflection symmetry and it will also show how many lines of symmetry it has.



=Standard #4 =

Expectations:
1. Create mental images of geometric shapes using spatial memory and spatial visualization. 2. Recognize and represent shapes from different perspectives. 3. Relate ideas in geometry to ideas in number and measurement. 4. Recognize geometric shapes and structures in the environment and specify their location.

How we applied it in class:
1. We glanced at shapes projected on the board for seconds and then redrew what we saw on our paper. We were expected to recognize a two dimensional shape on the overhead, using our spatial memory and visualization. We explored NETS in class and made a two dimensional shape that could be folded into a 3 dimensional shape for: prisms, cylinders, cones and pyramids. 2. We were given patty paper to measure various angles in our coursepack. The overhead two dimensional shape was then created into a three dimensional shape out of Lego type blocks. 3. We used an angle ruler in class to show an accurate measuring of angles. 4. We brought in the shapes we found in magazines that were geometric shapes.

Purpose of the utility and activity. It's usefulness.
1. The mental images viewed and redrawn develops a students spatial memory and visualization abilities to build a foundation for future geometry tasks. Such as, remembering geometric shapes and visualizing them into a Venn Diagram successfully, in future lessons. The rebuilding technique of a two dimensional shape into a three dimensional shape strengthens a students memory and visualization skills. It is an excellent tool for students to hone their geometric problem solving skills. 2. The purpose of the patty paper was to show creativity in discovering different ways to measure shapes without specific mathematical tools. It also motivated the class to recognize the need for a standard unit of measure and degree. Using Lego type blocks to rebuild a two dimensional shape into a three dimensional shape, reinforces cognitive memory skills and hand and eye coordination. 3. The angle ruler showed the difference of accuracy in solving problems with an accurate tool vs. the hand made tool using the patty paper. 4. The magazine shapes brought geometry to our daily world and their relationship to each other by using a Venn Diagram. The Venn Diagram shows what relationships overlap or are subsets of another property, or may be disjointed from each other.

Other mathematical ideas this connects to:
1. Patty paper can be given to students to help them visualize reflection symmetry in a shape by folding their tracing of a shape on the paper in half. 2. Students can be encouraged to look at their daily world for clues of geometric shapes by bringing them to class. 3. Smile Math program on the TI-73 is a self check program that can be used by students to practice visual reasoning of angle drawings. 4. Angle rulers can be used by students to project many angle positions while advancing their spatial reasoning to solve problems.

= Measurement Standard for Grades Pre-K-2 =

= Standard #1: =

Expectations:
1. Recognize the attributes of length, volume, weight, area, and time. 2. Compare and order objects according to these attributes. 3. Understand how to measure using nonstandard and standard units. 4. Select an appropriate unit and tool for the attribute being measured.

How we applied it in class:
1. We started finding volume by counting how many square units we could fit in a box. We filled 3-D shapes in class with filler substance. 2. We compared the amount of filler used to fill a cylinder to the amount needed to fill other shapes. (i.e., cones, pyramids, spheres) 3. In class we measured circular objects with string to get the diameter and circumference. the ratio of the circumference to the diameter gives you the definition of Pi. While I see the utility in measuring these attributes of objects I'm questioning the appropriateness of developing Pi at this age. The volume in a cone, pyramid or sphere shapes can be compared to the volume in a cylinder shape by pouring their volumes, separately, in the empty cylinder. This measurement shows a student a nonstandard unit of measurement. 4. The unit used for filler was tiny recycled tire pieces poured into shapes to fill them up, and emptied into a larger shape (cube) as a tool to compare volume sizes.

Purpose of the utility and activity. Its usefulness:
1. The usefulness of finding volume by counting how many square units fit in a box is to learn another way to transfer the meaning of volume to children, rather than them thinking of volume as a glass filled with water. 2. Instead of using filler to fill empty shapes, a teacher could demonstrate a box with 12 cans inside to help students recognize the attributes of length and volume. 3. Using measuring devices, other than the traditional rulers, broaden students spatial reasoning to imagine measuring objects with other devices, such as string. 4. When you physically fill a cylinder from the filler of a cone shape, it teaches students the 1/3 visible volume can mean, 1/3 of the bigger shape. Because the cone is 1/3 of the volume of a cylinder shape tested. (With diameters of the same size.) Yes, I can see students of this age really enjoying this activity; it is also something they likely do at home (think sand box or bath tub). Other mathematical ideas this connects to. 1. When students create their own "nets", they can visualize the shape and see how many square units will be needed to fill the volume of the shape. 2. Students can experiment with different size "nets" with congruent volumes, to show them the possibilities of multiple packaging opportunities verses the same volume. 3. Students can be taught that measuring with nonstandard vs. standard measuring tools is acceptable for some measurements. 4. If the students were going to get new classroom carpet and they measured it with their feet, they could estimate the area of the room. However, they could measure it with a standardized tool for an accurate measurement to show students the necessity of the accurate measurement to save on the cost of buying the new carpet, so there wouldn't be any waste or too little purchased.

= Standard #2 =

Expectations:
<span style="font-family: 'Times New Roman',Times,serif; font-size: 141%;">1. Measure with multiple copies of units of the same size, such as paper clips laid end to end. 2. 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. 3. Use tools to measure. 4. Develop common referents for measures to make comparisons and estimates.

<span style="color: #00ff00; font-family: 'Times New Roman',Times,serif; font-size: 130%;">How we applied it in class:
<span style="font-family: 'Times New Roman',Times,serif; font-size: 130%;">1. We measured volume with multiple tire pieces of the same size to fill empty shapes, (i.e., cubes, cylinders, cones, pyramids, spheres.) 2. We used the repetition of the multiple tire pieces to measure the volume of an empty cylinder shape by comparing the number of times the filler within a cone shape had to be refilled in order to fill the cylinder shape completely when emptied into the cylinder. 3. We have been using rulers (inches, cm, mm), measurement tape, angle rulers, and protractors to measure circle diameters, raids and circumferences. We used rulers to measure side lengths of triangles, and the area of parallelograms (by lxw). We discovered the diameter, radius, circumference and areas of circular objects by using these measurement tools. In Project 2, we measured a lot of different shapes with fixed perimeters in order to make sure all the shapes were congruent. 4. When students can visualize volume by pouring from one shape into a larger shape, they can estimate the volume of the larger shape in comparison to the unit of measure used. This builds students spatial estimates on how much volume the larger shape holds.

<span style="color: #0000ff; font-family: 'Times New Roman',Times,serif; font-size: 130%;">Purpose of the utility and activity. It's usefulness.
1. The purpose of this activity is to show students that objects can be measured by other things, not just by using a ruler. A papers straight edge can be used to measure a 90 degree angle. Measuring volume this way, transfers ideas to students with something that they can visualize and become more comfortable with. (Rather than just giving them a hand written formula for finding volume.) 2. By using one single unit of measurement students are able to have a common measurement which will encourage equal charts and group projects. 3. Without standard measurement tools we would not be able to have exact measurements for area, volume, length, and height. 4. This activity builds students spatial and concrete reasoning skills so they are able to make comparisons and estimates regarding measurements.

Other mathematical ideas this connects to.
1. Students can measure the height of their desk by using their hands. (i.e., the height of a horse is measured in "hands.") This reinforces measurements without a standard ruler. 2. Students can measure the length of their classroom by using their feet as a measuring tool. This activity will show students how to estimate the size of their room, and the variance created by using different sized measurement tools (their feet). 3. Students can experiment with their own ways to create shortcuts to solve their own math formulas, but need to be introduced to standard measurement tools for the importance of exact measurements. For instance, an airplane will need to be made with precise measurements to be able to fly safely. 4. Students are consistently using spatial reasoning regarding volume throughout their lives. Such as, pouring milk or juice from a larger container into a smaller container. They have to estimate how much of the liquid in the larger container will fit in the smaller unit of measurement, and stop pouring it in, before it creates an overflow.

<span style="color: #800080; font-family: 'Times New Roman',Times,serif; font-size: 13px; line-height: 19px;">Very clearly organized summary here. Good identification of the big ideas, didn't see much on similarity (which could be connected to Geo standard 1 or 3) but then again, the formal development of this concept is usually for later grades. But connections to this might also be appropriate in younger grades. Students will likely come across many similar objects in real life. Very good connections and descriptions of importance.