A Learning Trajectory for Shape Adapted from, Learning and Teaching Early Math, Clements and Sarama Age 0–2 Developmental Progression Instructional Tasks “Same Thing” Compares real world objects. Example: Says two picture of houses are the same or different. Shape Matcher – Identical (Comparing) Matches familiar shapes (circle, square, typical triangle with the SAME size and orientation. Matches □ to □. Match and Name Shapes – Sits in a circle with children. Using familiar shapes from the Shape Sets in two colors, give each child a shape from one Shape set. Choose a shape from the other Shape set, which is a different color but exactly matches a child’s shape. Ask children to name who has an exact match for your shape. After a correct response is given, follow up by asking how the child knows his or her shape is a match. The child might offer to fit his or her shape on top of your shape to “prove” the match. Have children show their shapes to others seated near them, naming the shape whenever they can. Observe and assist as needed. Repeat once or twice. Afterward, tell children they will be able to explore and match shapes later during Work Time. Shape Matcher – Sizes: Matches familiar shapes with different sizes. Matches □to □. Shape Matcher – Orientations: Matches ► to ◄ 3 Shape Recognizer Classifying recognizes and manes typical circle, square and, less often, a typical triangle. May physically rotate shapes in a typical orientations to mentally match them to a prototype. □ Example: Names this a square Some children correctly name different sizes, shapes, and orientations of rectangles, but also call some shapes rectangles that look rectangular but are not rectangles. Example: See below Circle Time! Have children sit in the best circle they can make. Show and name a large, flat circle, such as a hula hoop. As you trace the circle with your finger, show how it is perfectly round; it is a curved line that always curves the same. Ask children to talk about circles they know, such as those found in toys, buildings, books, tri-or bicycles, and clothing. Distribute a variety of circles for children’s exploration – rolling, stacking, tracing, and so on. Have children make circles with their fingers, hands, arms, and mouths. Review a circle’s attributes: round and curves the same without breaking. Match and Name Shapes, above (Circle Time) includes naming of these shapes. Do this activity in small groups, as well as in whole groups. Mystery, What is in the Mitt? Use an oven mitt, 3 x 5 card, and a pattern block or other formed shapes. Glue a shape to the 3 x 5 card. Slide the card into the mitt. Put three 3 x 5 cards on the table with different shapes: circle, triangle, rectangle, or square. (One matches the shape in the oven mitt. Ask the children Content Core State Standards what children may name as “rectangles.” , ◊, □, █, (including non-rectangular parallelograms. to let their fingers be their “eyes” to see the shape in the mitten. The children slide their hand into the mitt and talk about what their “eyes” feel. When they think they know the shape, have them verbally name the shape or match it with other shapes on the table. The child can then slide to card out to check. Variation: More than one child can play in a group. One child hides the shape, while the others do not look. Then the children use their “eyes” to discover the shape in the mitt. “Similar” Comparer: Judges two shapes the same if they are more visually similar than different. Example: “These are the same. They are pointy at the top.” ◊, ⌂, ∆ 3-4 Shape Matcher – More Shapes (Comparing): Matches a wider variety of shapes with same size and orientation. Shape Matcher – Sizes and Orientations: Matches a wider variety of shapes with different Sizes and orientation. For example: matches these two shapes. Match and Name Shapes - Follow directions as stated above, but using a wider variety of shapes from the Shape Sets in different orientations. Match Blocks - Children match various block shapes to objects in the classroom. Have different block shapes in front of you with all the children in the circle around you. Shoe one block, ask children what things in the classroom are the same shape. Talk children through any incorrect responses, such as choosing something triangular by saying it has the shape of a quarter circle. Memory Geometry - Place two sets of memory geometry cards face down, each in an array. Cards that do not match are replaced face down; cards that match are kept by the player. Players should name and describe the shapes together. Use new shape cards that feature additional shapes. Shape Matcher – Combinations: Matches combinations of shapes to each other. For example: matches these two shapes. 4 Shape Recognizer – Circle, Square, and triangle + Classifying: Recognizes some less typical squares and triangles and may recognize some rectangles, but Is it or Is it Not? Draw a circle on a surface where the entire class can view it. Ask children to name it, and then tell why it is a circle. Draw an ellipse (oval) on the same surface. Ask children what it looks like and then ask them to tell why it is not a circle. Draw several other circles and shapes that are not circles, but could be mistaken for them, and discuss their differences. Summarizing by reviewing usually not rhombuses (diamonds). Often doesn’t differentiate sides/corner. Part Comparer – Comparing: Says two shapes are the same after matching one side of each. “These are the same.” Constructor of Shapes from Parts – Uses manipulatives representing parts of shapes, such as sides, to make a shape that “looks like” a goal shape. May thing of angles as a corner which is pointy. Example: Asked to make a triangle with sticks, creates the following: 4-5 that a circle is perfectly round and consists of a curved lines that always curves the same. Note: Use the computer to “draw” the shapes, print on a legal size sheet of paper. Then put glue and sand on the lines to help students trace. What is in the Mitt? Same game as earlier, now put 5 cards in front of the child that are similar to the shape. Circles and Cans Display several food cans, and discuss their shape. (round) with children. Shift focus to the bottom and top, collectively the bases, of each can. Ask, “What shape do you see?” Have children point out the attributes to support the circle name. (Perfectly round, the curves are all the same.) Have traced papers handy, or have the children trace some of the larger cans. Mix up all the tracings, then have the children match them to the cans. For children that are uncertain of their choice, have them place the cans directly on the traced circle to check. Make up a station with more cans and different sized circles to match. Shape show: (See these directions below in the Rectangle section below.) Build Shapes/Straw Shapes: In a small group lesson with the teacher, children use plastic stirrers of various lengths to make shapes they know. Ensure that they build shapes with correct attributes, such as all sides the same length and all right angles for squares. All stirrers should be “connected” (touching) at their endpoints. Discuss attributes as children build. If children need help, provide a model for them to copy or a drawing on which to place stirrers. Can they choose the correct amount of sizes of stirrers to make a given shape? If children excel, challenge them to get a shape “just right.” Can they place pieces with little trial and error? Straw Shapes: Triangles In a free-choice center, children use plastic stirrers to make triangles and/or to create pictures and designs that include triangles. Some Attributes (Comparing) Looks for differences in attributes, but may examine only part of the shape. “These are the same” (Indicating the top halves of the shape are similar by laying them on top of each other.) Match Shapes Children match the Shape sets, find the yellow shape that exactly matches the blue shape. Shape – All Rectangles (Classifying): Recognizes more rectangle sizes, shapes, and orientations of rectangles. Shape Hunt: Triangles Tell children to find one or two items in the room with at least one triangle face. For variety, hide Shape Set triangles throughout the room beforehand. Encourage children to count the shape’s sides and, if possible, show the triangle to an adult, discussing its shape. For example, triangles have three sides, but the sides are not always the same length. After discussion, have the child replace the triangle so other children can find it. You may choose to photograph the triangles for a class shape book. Kindergarten Critical Area: Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., withdifferent sizes and orientations), as well as three-dimensional shapes suchas cubes, cones, cylinders, and Side – Parts: Identifies sides as distinct geometric objects. For example, Asked what this shape is Is it or is it Not (Triangles) – See above for directions, but change side measures, orientations, and angle measures to provide a variety of triangle shapes. I Spy? spheres. They use basic shapes and spatialreasoning to model objects in their environment and to construct more complex shapes. The student says it is a quadrilateral, (Or has four sides) after counting each, running fingers along the length of each side.) Most Attribute (Comparing): Looks for differences in attributes, examining full shapes, but may ignores some spatial relationships. “These are the same Corner (Angle) – parts: Recognizes angles as separate geometric objects, at least in the limited context of “corners.” 5 More Shapes (Classifying) █ Recognizes most familiar shapes and typical examples of other shapes, such as hexagon, rhombus, and trapezoid. “Name these shapes.” Correctly identifies: Rectangles and Boxes: Draw large rectangles for the entire class to see, and trace it, counting each side as you go. Challenge children to draw a rectangle in the air as you count, reminding them that each side should be straight. Show a variety of boxes to children, such as toothpaste, pasta, and cereal boxes, and discuss their shape. Eventually focus on the faces of the boxes, which should mostly be rectangles. Talk about the sides and right angles. On large paper, place two boxes horizontally and trace their faces. Have children match the boxes to the traced rectangles. Trace more boxes and repeat. Help children consider other box face shape such as triangles (candy and food storage), octagons (had and gift boxes), and circles/cylinders (toy and oats containers.) Guess My Rule: Tell children to watch carefully as you sort Shape Set Shapes into piles based on something that makes them alike. Ask the children to silently guess your sorting rule, such as circles versus squares of four-sided shapes versus round. Sort shapes one at a time, continuing until there are at least two shapes in each pile. Signal “shhh,” and pick a new shape. With a look of confusion, gesture to children to encourage all of them to point quickly to which pile the shape belongs. Place the shape in its pile. After all shapes are sorted, ask children what they think the sorting rule is. Repeat with other shapes and new circles such as: circles vs. squares (same orientation), circles vs. triangles, circles vs. rectangles, and triangles vs. rectangles. Shape Show: Rectangles Show and name large, flat rectangle. Walk your fingers around its perimeter, describe and exaggerating your actions: short, straiiiight side….turn, long straaiiight side… turn, short straiiight side… turn, long straaiight side…. Stop. Ask children how many sides the rectangle has, and count the sides with them. Ask, “Are the opposite sides the same lengths? How do you know? To model this, take two stirrers that are the same length as one of the parallel sides, compare. “Are they the same length?” “How do you know? Then place the stirrers that are the same length as one pair of sides for the horizontal on the rectangle. Repeat the pairs for the vertical parallel sides. Now look at the right angle. Give students corners of paper with the edges darkened, have them show you the capital letter L. Illustrate right angles, talk about the angle-like an upper case L in a doorway. Make uppercase Ls with Kindergarten: GEOMETRY Identify and describe shapes (squares, circles, triangles, rectangles,hexagons, cubes, cones, cylinders, and spheres). 1. Describe objects in the environment using names of shapes, anddescribe the relative positions of these objects using terms such asabove, below, beside, in front of, behind, and next to. 2. Correctly name shapes regardless of their orientations or overall size. 3. Identify shapes as two-dimensional (lying in a plane, “flat”) or threedimensional(“solid”). Analyze, compare, create, and compose shapes. 4. Analyze and compare two- and three-dimensional shapes, indifferent sizes and orientations, using informal language to describetheir similarities, differences, parts (e.g., number of sides andvertices/“corners”) and other attributes (e.g., having sides of equallength). 5. Model shapes in the world by building shapes from components (e.g.,sticks and clay balls) and drawing shapes. 6. Compose simple shapes to form larger shapes. For example, “Can youjoin these two triangles with full sides touching to make a rectangle?” children using thumbs and index fingers. Fit your L on the angles of the rectangle. Ask children what they have at home that are rectangles. Find different examples of rectangles in class, use corner guide to prove a right angle. Have some non-rectangular parallelograms available, too. Then have children walk around a large, flat rectangle, such as a rug. Once seated have children draw rectangles in the air. Rectangles and Boxes, (See –Age 4, for instructions.) Shape Step- Make shapes on the floor with masking or colored tape or chalk shapes outdoors. Tell children to step on a certain class of shapes (e.g. rhombuses). Have a group of five children step on the rhombuses. Ask the rest of the class to watch carefully to make sure the group steps on them all. Whenever possible, ask children to explain why the shape they stepped on was the correct shape. (How do you know that was a rhombus?” Repeat the activity until all the groups have stepped on shapes. Geometry Snapshot: Show a simple configuration of shapes for just 2 seconds, student match that configuration to four choices from memory. (Imagery.) What is in the Mitt? At this stage, follow the directions but do not provide any cards with matching images. Rather ask the student to use their fingers to “see” what the shape is. Ask the child to name the shape, then explain why? If this is a station, provide a sheet of paper with many shapes, the child will circle it, color the sides one color, and the angles another color. Use any of the previous instructional tasks, include shapes appropriate for this level. 6 classifying: Names most common shapes, including rhombuses, without making mistakes such as calling ovals circles. Recognizes (at least) right angles, so distinguishes between a rectangle and a parallelogram without right angles. Trapezoids and Rhombuses: Use pattern blocks to draw a variety of these two shapes. Then ask students to make a picture using the shapes. Class Mascot is confused! Have the stuffed animal or cartoon figure get “mixed-up” and name shapes wrong. Encourage students to give clues of attributes to explain why he named the wrong figure and what he should consider to name it correctly. (Especially with the square, rhombus, quadrilateral, and parallelogram.) 1st Grade: Critical Area Students compose and decompose plane or solid figures (e.g., puttwo triangles together to make a quadrilateral) and build understandingof part-whole relationships as well as the properties of the original andcomposite shapes. As they combine shapes, they recognize them fromdifferent perspectives and orientations, describe their geometric attributes,and determine how they are alike and different, to develop the backgroundfor measurement and for initial understandings of properties such ascongruence and symmetry. 1st Grade – GEOMETRY Reason with shapes and their attributes. 1. Distinguish between defining attributes (e.g., triangles are closed andthree-sided) versus non-defining attributes (e.g., color, orientation,overall size); build and draw shapes to possess defining attributes. 2. Compose two-dimensional shapes (rectangles, squares, trapezoids,triangles, half-circles, and quarter-circles) or threedimensional shapes(cubes, right rectangular prisms, right circular cones, and right circularcylinders) to create a composite shape, and compose new shapes from the composite shape. 3. Partition circles and rectangles into two and four equal shares, describethe shares using the words halves, fourths, and quarters, and use thephrases 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. 7 Angle - Parts: Can recognize and describe contexts in which angle knowledge is relevant, including corners (can discuss “sharper” angles, (e.g., crossings – scissors, Parts of Shapes – classifying: Identifies shapes in terms of their components. “No matter how skinny this triangle looks, it is still a triangle because it has 3 sides and 3 angles.” Congruence– Comparing: Determines congruency by comparing all attributes and all spatial relationships. “The two shapes are the same shape and the same size after comparing every one of their sides and angles. Congruence – Comparing Moves and places objects on top of each other to determine congruence. Constructor of Shapes from parts – Exact: Representing: Uses manipulatives representing parts or shapes, such as sides of angles “connectors” to make a shape that Students do not need to learn formal names such as “right rectangular prism.” 2nd Grade Critical Area: Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about decomposing and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes, students develop afoundation for understanding area, volume, congruence, similarity, andsymmetry in later grades. Patty paper – tracing. Trace the first shape, color each angle a different color. Color the corresponding angles in the 2nd shape. (Let the students decide which ones match.) Then put the patty paper on the 2nd shape and line up the angles. Making Triangles Applet, Illuminations. Virtual GEOBOARD: http://www.nctm.org/standards/content.aspx?id=25007 2nd Grade – GEOMETRY Reason with shapes and their attributes. 1. Recognize and draw shapes having specified attributes, such as a givennumber of angles or a given number of equal faces. Identify triangles,quadrilaterals, pentagons, hexagons, and cubes. 2. Partition a rectangle into rows and columns of same-size squares andcount to find the total number of them. 3. Partition circles and rectangles into two, three, or four equal shares,describe the shares using the words halves, thirds, half of, a third of,etc., and describe the whole as two halves, three thirds, four fourths.Recognize that equal shares of identical wholes need not have thesame shape. 8+ is completely correct, based on knowledge of components and relationships. Angle – Parts Represents various angle contexts as two lines, explicitly including the reference line (horizontal or vertical for slope; a “line of “sight” for turn contexts) and at least implicitly, the size of the angle as the rotations between these lines (May still maintain misconceptions about angle measure, such as relating angle size to the lengths of side’s distance between endpoints and my not apply these understanding to multiple context. Congruence – Comparing Refers to geometric properties and explains with transformations. “These must be congruent, because they have equal sides, all square corners, and I can move them on top of each other exactly. Shape Class – Classifying Use properties explicitly. Can see the invariants in the changes of state or shape, must maintaining the shapes’ properties. “I put the shapes with opposite sides parallel over here, and those with four sides but not both pairs of sides parallel over there.” Property Class – Classifying Use class membership for shapes (e.g., to sort or consider shapes similar) explicitly based on properties, including angle measure. Is aware of restrictions of transformations and also of the definition Investigating Shapes (Triangles Lessons) http://illuminations.nctm.org/LessonDetail.aspx?ID=L84 3rd Grade – Critical Area Students describe, analyze, and compare properties of two dimensional shapes. They compare and classify shapes by their sides and angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole. 3rd Grade – GEOMETRY Patty paper – tracing. Trace the first shape, color each angle a different color. Color the corresponding angles in the 2nd shape. (Let the students decide which ones match.) Then put the patty paper on the 2nd shape and line up the angles. Making Triangles Applet, Illuminations. Virtual GEOBOARD: http://www.nctm.org/standards/content.aspx?id=25007 Previous instructional strategies listed above, but with appropriate levels and information. - Guess my Rule - Shape Step - I Spy - Our Mascot is Confused? (Help the mixed – up mascot name the proper shape by identifying the properties.) Shape Tool, Illuminations applet: http://illuminations.nctm.org/activitydetail.aspx?id=35 - This tool allows you to create any geometric shape imaginable. Squares, triangles, rhombi, trapezoids and hexagons can be created, colored, enlarged, shrunk, rotated, reflected, sliced, and glued together. Which Shape Could it Be? Slowly reveal a shape from behind a screen At each ‘step ask children what class of shape could it be and how certain they are. Reason with shapes and their attributes. 1. Understand that shapes in different categories (e.g., rhombuses,rectangles, and others) may share attributes (e.g., having four sides),and that the shared attributes can define a larger category (e.g.,quadrilaterals). Recognize rhombuses, rectangles, and squares asexamples of quadrilaterals, and draw examples of quadrilaterals thatdo not belong to any of these subcategories. 2. Partition shapes into parts with equal areas. Express the area of eachpart as a unit fraction of the whole. For example, partition a shape into 4parts with equal area, and describe the area of each part as 1/4 of the areaof the shape. 4th Grade Critical Area: Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional shapes, students deepen their understanding of properties of twodimensional objects and the use of them to solve problems involving symmetry. 4th Grade GEOMETRY Draw and identify lines and angles, and classify shapes by propertiesof their lines and angles. 1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. 2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. 3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. Shape Sorter: Illuminations, A rhombus has four equal sides. A rectangle has four right angles. But a square has four equal sides and four right angles. Using a Venn diagram, the relationship would look like this: What other relationships can be described using Venn diagrams? Use this tool to explore many different geometric properties and shapes. Teacher led use of Venn Diagrams to think about how the shapes are alike and how they are different. 5th Grade – GEOMERY Graph points on the coordinate plane to solve real-world and mathematical problems. 1. Use a pair of perpendicular number lines, called axes, to define acoordinate system, with the intersection of the lines (the origin)arranged to coincide with the 0 on each line and a given point inthe plane located by using an ordered pair of numbers, called itscoordinates. Understand that the first number indicates how far totravel from the origin in the direction of one axis, and the secondnumber indicates how far to travel in the direction of the secondaxis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis andy-coordinate). 2. Represent real world and mathematical problems by graphing pointsin the first quadrant of the coordinate plane, and interpret coordinatevalues of points in the context of the situation. Classify two-dimensional figures into categories based on theirproperties. 3. Understand that attributes belonging to a category of twodimensionalfigures also belong to all subcategories of that category.For example, all rectangles have four right angles and squares arerectangles, so all squares have four right angles. 4. Classify two-dimensional figures in a hierarchy based on properties.