Geometric Thinking Mrs. Ellis is designing square swimming pools. Each pool has a square center that is the area of the water. Mrs. Ellis uses blue tiles to represent the water. Around each pool there is a border of white tiles. Here are pictures of the three smallest square pools that she can design with blue tiles for the water and white tiles for the border.
Sorting and Counting For each square pool, sort the tiles into blue tiles for the water and white tiles for the border. Count how many tiles are in each pile. Are there more blue tiles than white tiles?
Pattern in the Blue Squares Build each of the three blue squares. How many blue tiles are in each square? Build the next-biggest square that you can make out of the blue tiles. Then build the next. Count the squares in each. What patterns do you see? What is a square? Blue square 1 Blue square 2 Blue square 3
T:Why do you call it a square? Whats a square to you? G:A block. T:If it were longer down like this, would it still be a square? G:No, it would turn into a rectangle. T:So what makes it a square? G:That its not as far down as a rectangle. T:Is there anything else about the sides? How long is this side? G:Three squares. T:How long is this side? G:Three squares. T:So what is a square?
G:Can I try something? Im putting out three to see if I can scramble them around and make a square. [Grace is working with three unit squares and trying to build a square. Notice that the teacher was trying to draw Graces attention to the equality of the sides. But, not unexpectedly, she became interested instead in the number 3 and its relationship to the square.] T:A square out of three? [Grace notices that she will need four squares to make a square and builds it.] T:How can you be sure its a square? G:You can, because all the sides are the same length.
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Color the Shapes Color all the triangles green. Color all the rectangles red. Color all the circles blue. Are there some shapes you havent colored?
Shapes by grade K: squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres 1 st : rectangles, squares, trapezoids, triangles, half- circles, and quarter-circles 2 nd : triangles, quadrilaterals, pentagons, hexagons, and cubes 3 rd : rhombuses 4 th : parallelogram is implied by classifying figures based on parallel lines
Shapes All Around Us K.G.1 Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
van Hiele Levels of Geometric Thought Level 0: Visualization Students recognize and name figures based on the global, visual characteristics of the shape. Students at this level are able to make measurements and even talk about the properties of shapes, but these properties are not abstracted from the shape at hand. It is the appearance of a shape that defines it for a student. A square is a square because it looks like a square. Other visual characteristics may include pointy, fat, sort of dented in. Classification of shapes at this level is based on whether they look alike or different. from Van de Walle and Lovin, 2006
van Hiele Levels of Geometric Thought Level 1: Analysis Students are able to consider all shapes within a class rather than a single shape. By focusing on a class of shapes, students are able to think about what makes a rectangle a rectangle (four sides, opposite sides parallel, opposite sides equal, four right angles, etc.) Irrelevant features (e.g. orientation or size) fall into the background. Students begin to appreciate that a collection of shapes goes together because of its properties. from Van de Walle and Lovin, 2006 = K.G.2 Correctly name shapes regardless of their orientations or overall size.
Attributes Give each student a sheet of shapes. Have them brainstorm ways to describe the shapes. Record their responses on chart paper. Guide students to look for ways other than color and size when describing the shapes such as by number of sides, number of corners, or no corners.
Describe this shape Show a shape. Ask students to say something about what the shape looks like. Write their descriptions on the board. If needed, guide them to be specific: How many sides? How many corners? Show two squares of different color. Ask, are these both squares? Why are they squares? Can squares come in different colors?
Attributes 1.G.1 Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); for a wide variety of shapes; build and draw shapes to possess defining attributes. 2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. (Sizes are compared directly or visually, not compared by measuring.)
van Hiele Levels of Geometric Thought Level 2: Informal Deduction Students are able to develop relationships between and among properties of shapes. They recognize sub-classes of properties: If all 4 angles are right angles, it is a rectangle. Squares have 4 right angles, so squares must be rectangles. from Van de Walle and Lovin, 2006
Shape sorts 1. Take any shape. Tell one or two things you find interesting about the shape. 2. Choose two shapes. Find something alike and something different about the two shapes. 3. The group selects one shape and places it in the center of the workspace. Find all other shapes that are like this shape according to the same rule. 4. Do a second sort with the same target shape but using a different property.
Shape sorts 4. Groups share their sorting rules with the class and show examples. Everyone draw a new shape that will also fit in the group according to the same rule. 5. Do a secret sort by selecting about 5 shapes that fit a secret rule, leaving some similar shapes in the pile. Others find similar shapes and try to guess the rule.
Match the Rule Georgia Common Core first grade workbook p. 36
Attributes In your small group, look at the pictures of the shapes on your page. List all the attributes you can find that all the shapes share. Sides Corners (Diagonals and symmetry are introduced in 4 th gr.) The rhombus has 4 sides 4 corners opposite sides are the same length
Orientation Draw a triangle. Does it look like one of these triangles? Is this a triangle? How do you know? If we turn this triangle, is it still a triangle?
Which One Doesnt Belong? The Teacher will place attribute blocks in a brown paper bag. One student will come to the front of the room and grab a handful of blocks from the bag. The student will show the blocks to the class, describe the blocks and decide which one(s) does not belong. The teacher will want to model this prior to the students completing in front of the class. The student will place the blocks back in the bag and another student will repeat. Complete this activity several times until the students have grasped the concept of which one does not belong. Georgia Common Core first grade workbook p. 24
Build a Shape Read The Greedy Triangle (on wiki)on wiki Ask students if they can make shapes with their bodies and a piece of yarn. Use straws, pipe cleaners, or other manipulatives to create a triangle, rectangle, square and trapezoid. Model how you connect the straws and pipe cleaners to create a shape (sample below). Read The Greedy Triangle again. Have students create the shapes as you come to each shape in the book. Georgia Common Core first grade workbook p. 29 K.G.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
Build a cube It is often difficult for students to visualize as it requires a coordination of both two and three-dimensional shapes. Activities which require students to think about, manipulate, or transform a shape mentally will contribute to students overall visualization skills.
Making Shapes National Library of Virtual Manipulatives What can students learn from using Geoboards?
Composing Shapes K.G.6 Compose simple shapes to form larger shapes. For example, can you join these two triangles with full sides touching to make a rectangle? 1.G.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter- circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as right rectangular prism.)
Composing Shapes 1.G.2 Compose two-dimensional shapes or three- dimensional shapes to create a composite shape, and compose new shapes from the composite shape.
Decomposing Shapes Display a large square to the students and ask, What will happen if I cut this shape straight down the middle? What shapes will be created? Why do we call these two shapes rectangles, not squares?
Decomposing Shapes Emphasize the lines that students cut have to be straight horizontal, vertical, or diagonal and then demonstrate these to the students. Example cuts should include ones that are not just straight through the middle; instead the teacher should snip off one corner demonstrating a small cut. This will show students their cuts can be of various lengths. Take turns having one student demonstrate a cut and then other students model the same cut.
Decomposing Shapes Give each student a set of shapes (p. 36) Tell them to cut out each shape and see what shapes can be made by making one cut. Have the students glue their pieces down puzzle style. Have each student share how they cut one of their shapes and identify the new shapes they made. Georgia Common Core first grade workbook p. 33
Differentiation Extension Ask students, What kind of shapes would be created by making two cuts? Allow students to explore with combining three shapes to create a new shape. Intervention Allow students who may be having a difficult time describing or making the shapes extra time with pattern blocks as a model. Students could also use tangram pieces if they are having difficulty with the cuts.
Fractions of a shape Part I Gather students in a common area. Hold up one sheet of paper. Tell students that the paper represents a cake that four students won at the fair and then fold it unevenly. Tear off three small pieces to give to the three random students and then give the one big piece to a fourth student. Ask, Is this fair? Why do you say that? What should I do to make it fair? Invite further discussion with students about situations where they have had to share things such as cookies, candy or toys, and listen for them to verbalize the importance of making sure everyone gets a fair share.
Fractions of a shape Part II Read A Fair Bear Share by Stuart J. Murphy or similar book on fractions. After the story, remind students of the cake scenario you discussed before reading. Ask Is there a way to cut the cake so it will be fair? Allow students to share ideas. Give each student 3 sheets of construction paper that are the same size. Tell the students that these represent 3 whole cakes. Have them label one of the sheets with the number one (because it represents one whole cake). It should also be labeled one whole. Next, tell the students they are going to share the second cake (piece of construction paper) with one friend. Tell them to fold the paper in a way that it will create two equal pieces. Keep in mind some students may fold their paper vertically, horizontally or diagonally. Allow all representations to be shared and discussed. Ask questions such as: Are these two representations of ½ the same size? How do you know?
Fractions of a shape For the third sheet of construction paper, tell the students they are going to share this cake with 3 friends and fold it in a way that creates four equal pieces. Some students may fold it vertically (like a fan) or vertically and horizontally (making a grid). Allow both representations to be shared and discussed. The discussion for ¼ should be similar to the one you had related to ½. Label each part of the third cake with both the fraction ¼ and the words fourths and quarter. Make sure to ask students What is happening to our pieces as we add more folds to the paper? Why is this happening? What if we shared this cake with ten people, would we get more or less cake? How do you know? Which is bigger ½ or ¼? (Or ask in this way, Which is larger, one half or one quarter?) Can you prove it?
Fraction Fill In Game Students will work with a partner to play Fraction Fill In to develop proficiency with fractions. To use spinner, put a paperclip in the middle. Hold it in place with the tip of the pencil. Have the student thump the paper clip to spin and see where it lands. pp. 55-56
Fractions from Composing What simple fractions can be made from pattern blocks? 1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
Videos Clever Car http://www.youtube.com/watch?v=4idPhHinb_g
Classify, Sort and Graph K.MD.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. 2 APPLES4 ORANGES5 BANANAS
Classify, Sort and Graph 1.MD.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another. 2 APPLES4 ORANGES5 BANANAS
1.MD.3 Tell and write time in hours and half-hours using analog and digital clocks. This is an introduction to clocks and how we use them to tell time. Student look at make-believe clocks that are set to hours or half-hours and learn the pattern of 12:30, 1:00, 1:30, etc.