A Learning Trajectory for Shape
Adapted from, Learning and Teaching Early Math, Clements and Sarama
Progression Instructional Tasks Content Core State Standards
0 – 2 “Same Thing” Compares real world Match and Name Shapes – Sits in a circle with children. Using familiar shapes
objects. Example: Says two picture from the Shape Sets in two colors, give each child a shape from one Shape set.
of houses are the same or different. 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 Shape Matcher – Identical your shape. After a correct response is given, follow up by asking how the child
(Comparing) Matches familiar knows his or her shape is a match. The child might offer to fit his or her shape on shapes (circle, square, typical top of your shape to “prove” the match. Have children show their shapes to triangle with the SAME size and 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 orientation. Matches ? to ?. to explore and match shapes later during Work Time.
Shape Matcher – Sizes: Matches
familiar shapes with different sizes.
Shape Matcher – Orientations:
? to ?
3 Shape Recognizer Classifying Circle Time! Have children sit in the best circle they can make. Show and name
recognizes and manes typical a large, flat circle, such as a hula hoop. As you trace the circle with your finger, circle, square and, less often, a show how it is perfectly round; it is a curved line that always curves the same. typical triangle. May physically Ask children to talk about circles they know, such as those found in toys,
rotate shapes in a typical buildings, books, tri-or bicycles, and clothing. Distribute a variety of circles for orientations to mentally match children‟s exploration – rolling, stacking, tracing, and so on. Have children make them to a prototype. circles with their fingers, hands, arms, and mouths. Review a circle‟s attributes:
round and curves the same without breaking.
Example: Names this a square ? Match and Name Shapes, above (Circle Time) includes naming of these shapes. Some children correctly name Do this activity in small groups, as well as in whole groups.
different sizes, shapes, and
orientations of rectangles, but also Mystery, What is in the Mitt? Use an oven mitt, 3 x 5 card, and a pattern block or
call some shapes rectangles that other formed shapes. Glue a shape to the 3 x 5 card. Slide the card into the mitt. look rectangular but are not Put three 3 x 5 cards on the table with different shapes: circle, triangle, rectangles. Example: See below rectangle, or square. (One matches the shape in the oven mitt. Ask the children
what children may name as 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 “rectangles.” ?, ?, 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, ?, , ; (including non-while the others do not look. Then the children use their “eyes” to discover the
shape in the mitt. rectangular parallelograms.
“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 Match and Name Shapes - Follow directions as stated above, but using a wider
(Comparing): Matches a wider variety of shapes from the Shape Sets in different orientations.
variety of shapes with same size Match Blocks - Children match various block shapes to objects in the and orientation. 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.
Shape Matcher – Sizes and
Orientations: Matches a wider Memory Geometry - Place two sets of memory geometry cards face down, variety of shapes with different Sizes each in an array. Cards that do not match are replaced face down; cards that
and orientation. For example: match are kept by the player. Players should name and describe the shapes matches these two 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, Is it or Is it Not? Draw a circle on a surface where the entire class can view it. Ask
and triangle + Classifying: children to name it, and then tell why it is a circle. Draw an ellipse (oval) on the Recognizes some less typical same surface. Ask children what it looks like and then ask them to tell why it is squares and triangles and may not a circle. Draw several other circles and shapes that are not circles, but could
recognize some rectangles, but be mistaken for them, and discuss their differences. Summarizing by reviewing
usually not rhombuses (diamonds). that a circle is perfectly round and consists of a curved lines that always curves Often doesn‟t differentiate the same. Note: Use the computer to “draw” the shapes, print on a legal size
sides/corner. sheet of paper. Then put glue and sand on the lines to help students trace.
Part Comparer – Comparing: Says What is in the Mitt? Same game as earlier, now put 5 cards in front of the child two shapes are the same after that are similar to the shape.
matching one side of each. “These Circles and Cans Display several food cans, and discuss their shape. (round)
are the same.” 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 Constructor of Shapes from Parts –use plastic stirrers of various lengths to make shapes they know. Ensure that they Uses manipulatives representing build shapes with correct attributes, such as all sides the same length and all parts of shapes, such as sides, to right angles for squares. All stirrers should be “connected” (touching) at their make a shape that “looks like” a endpoints. Discuss attributes as children build. If children need help, provide a goal shape. May thing of angles as model for them to copy or a drawing on which to place stirrers. Can they a corner which is pointy. choose the correct amount of sizes of stirrers to make a given shape? If children Example: Asked to make a triangle excel, challenge them to get a shape “just right.” Can they place pieces with
with sticks, creates the following: 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) Match Shapes Children match the Shape sets, find the yellow shape that
Looks for differences in attributes, exactly matches the blue shape.
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
4-5 Shape – All Rectangles (Classifying): Shape Hunt: Triangles Tell children to find one or two items in the room with at Kindergarten Critical Area: Students describe their physical Recognizes more rectangle sizes, least one triangle face. For variety, hide Shape Set triangles throughout the world using geometric ideas (e.g., shape, orientation, spatial
shapes, and orientations of room beforehand. Encourage children to count the shape‟s sides and, if relations) and vocabulary. They identify, name, and describe rectangles. possible, show the triangle to an adult, discussing its shape. For example, basic two-dimensional shapes, such as squares, triangles,
triangles have three sides, but the sides are not always the same length. After circles, rectangles, and hexagons, presented in a variety of
discussion, have the child replace the triangle so other children can find it. You ways (e.g., withdifferent sizes and orientations), as well as
may choose to photograph the triangles for a class shape book. three-dimensional shapes suchas cubes, cones, cylinders, and
Side – Parts: Identifies sides as Is it or is it Not (Triangles) – See above for directions, but change side measures, spheres. They use basic shapes and spatialreasoning to model distinct geometric objects. For orientations, and angle measures to provide a variety of triangle shapes. objects in their environment and to construct more
I Spy? example, Asked what this shape is 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, Rectangles and Boxes: Draw large rectangles for the entire class to see, and
examining full shapes, but may trace it, counting each side as you go. Challenge children to draw a rectangle ignores some spatial relationships. 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 “These are the same ;; discuss their shape. Eventually focus on the faces of the boxes, which should Corner (Angle) – parts: Recognizes mostly be rectangles. Talk about the sides and right angles. On large paper, angles as separate geometric place two boxes horizontally and trace their faces. Have children match the objects, at least in the limited boxes to the traced rectangles. Trace more boxes and repeat. Help children context of “corners.” consider other box face shape such as triangles (candy and food storage),
octagons (had and gift boxes), and circles/cylinders (toy and oats containers.)
5 More Shapes (Classifying) Guess My Rule: Tell children to watch carefully as you sort Shape Set Shapes Kindergarten: GEOMETRY
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 Identify and describe shapes (squares, circles, triangles,
round. Sort shapes one at a time, continuing until there are at least two shapes rectangles,hexagons, cubes, cones, cylinders, and spheres).
Recognizes most familiar shapes in each pile. Signal “shhh,” and pick a new shape. With a look of confusion, 1. Describe objects in the environment using names of shapes, and typical examples of other gesture to children to encourage all of them to point quickly to which pile the anddescribe the relative positions of these objects using terms shapes, such as hexagon, rhombus, shape belongs. Place the shape in its pile. such asabove, below, beside, in front of, behind, and next to.
and trapezoid. “Name these After all shapes are sorted, ask children what they think the sorting rule is. 2. Correctly name shapes regardless of their orientations or shapes.” Correctly identifies: Repeat with other shapes and new circles such as: circles vs. squares (same overall size.
orientation), circles vs. triangles, circles vs. rectangles, and triangles vs. 3. Identify shapes as two-dimensional (lying in a plane, “flat”)
rectangles. or threedimensional(“solid”).
Shape Show: Rectangles Show and name large, flat rectangle. Walk your
fingers around its perimeter, describe and exaggerating your actions: short, Analyze, compare, create, and compose shapes.
straiiiight side….turn, long straaiiight side… turn, short straiiight side… turn, long 4. Analyze and compare two- and three-dimensional shapes,
straaiight side…. Stop. Ask children how many sides the rectangle has, and indifferent sizes and orientations, using informal language to
count the sides with them. Ask, “Are the opposite sides the same lengths? How describetheir similarities, differences, parts (e.g., number of
do you know? To model this, take two stirrers that are the same length as one of sides andvertices/“corners”) and other attributes (e.g., having
the parallel sides, compare. “Are they the same length?” “How do you know? sides of equallength).
Then place the stirrers that are the same length as one pair of sides for the 5. Model shapes in the world by building shapes from
horizontal on the rectangle. Repeat the pairs for the vertical parallel sides. components (e.g.,sticks and clay balls) and drawing shapes.
Now look at the right angle. Give students corners of paper with the edges 6. Compose simple shapes to form larger shapes. For
darkened, have them show you the capital letter L. Illustrate right angles, talk example, “Can youjoin these two triangles with full sides
about the angle-like an upper case L in a doorway. Make uppercase Ls with 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.
st6 classifying: Names most common Trapezoids and Rhombuses: Use pattern blocks to draw a variety of these two 1 Grade: Critical Area
shapes, including rhombuses, shapes. Then ask students to make a picture using the shapes. Students compose and decompose plane or solid figures
without making mistakes such as (e.g., puttwo triangles together to make a quadrilateral) and
calling ovals circles. Recognizes (at build understandingof part-whole relationships as well as the Class Mascot is confused! Have the stuffed animal or cartoon figure get
least) right angles, so distinguishes properties of the original andcomposite shapes. As they “mixed-up” and name shapes wrong. Encourage students to give clues of
between a rectangle and a combine shapes, they recognize them fromdifferent attributes to explain why he named the wrong figure and what he should
parallelogram without right angles. perspectives and orientations, describe their geometric consider to name it correctly. (Especially with the square, rhombus,
attributes,and determine how they are alike and different, to quadrilateral, and parallelogram.) develop the backgroundfor measurement and for initial
understandings of properties such ascongruence and
st1 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 three-
dimensional 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.
Students do not need to learn formal names such as “right
nd7 Angle - Parts: Grade Critical Area: Students describe and analyze 2
Can recognize and describe shapes by examining their sides and angles. Students contexts in which angle knowledge investigate, describe, and reason about decomposing and is relevant, including corners (can combining shapes to make other shapes. Through building, discuss “sharper” angles, (e.g., drawing, and analyzing two- and three-dimensional shapes, crossings – scissors, students develop afoundation for understanding area, Parts of Shapes – classifying: volume, congruence, similarity, andsymmetry in later grades.
Identifies shapes in terms of their ndcomponents. “No matter how 2 Grade – GEOMETRY
skinny this triangle looks, it is still a Reason with shapes and their attributes.
triangle because it has 3 sides and 3 1. Recognize and draw shapes having specified attributes, angles.” such as a givennumber of angles or a given number of equal Congruence– Comparing: faces. Identify triangles,quadrilaterals, pentagons, hexagons, Determines congruency by and cubes.
comparing all attributes and all 2. Partition a rectangle into rows and columns of same-size
spatial relationships. “The two Patty paper – tracing. Trace the first shape, color each angle a different color. squares andcount to find the total number of them. ndshapes are the same shape and the Color the corresponding angles in the 2 shape. (Let the students decide which 3. Partition circles and rectangles into two, three, or four equal ndsame size after comparing every ones match.) Then put the patty paper on the 2 shape and line up the angles. shares,describe the shares using the words halves, thirds, half
one of their sides and angles. of, a third of,etc., and describe the whole as two halves, three Congruence – Comparing Making Triangles Applet, Illuminations. Virtual GEOBOARD: thirds, four fourths.Recognize that equal shares of identical Moves and places objects on top of http://www.nctm.org/standards/content.aspx?id=25007 wholes need not have thesame shape.
each other to determine
Constructor of Shapes from parts –
Exact: Representing: Uses
manipulatives representing parts or
shapes, such as sides of angles
“connectors” to make a shape that
is completely correct, based on
knowledge of components and
rd8+ Angle – Parts Grade – Critical Area 3
Represents various angle contexts Students describe, analyze, and compare properties of two
as two lines, explicitly including the Investigating Shapes (Triangles Lessons) dimensional shapes. They compare and classify shapes by reference line (horizontal or vertical http://illuminations.nctm.org/LessonDetail.aspx?ID=L84 their sides and angles, and connect these with definitions of for slope; a “line of “sight” for turn shapes. Students also relate their fraction work to geometry by contexts) and at least implicitly, the expressing the area of part of a shape as a unit fraction of the size of the angle as the rotations whole.
between these lines (May still rdmaintain misconceptions about 3 Grade – GEOMETRY
angle measure, such as relating
angle size to the lengths of side‟s Reason with shapes and their attributes.
distance between endpoints and 1. Understand that shapes in different categories (e.g., my not apply these understanding rhombuses,rectangles, and others) may share attributes (e.g., to multiple context. Patty paper – tracing. Trace the first shape, color each angle a different color. having four sides),and that the shared attributes can define a ndCongruence – Comparing Color the corresponding angles in the 2 shape. (Let the students decide which larger category (e.g.,quadrilaterals). Recognize rhombuses, ndRefers to geometric properties and ones match.) Then put the patty paper on the 2 shape and line up the angles. rectangles, and squares asexamples of quadrilaterals, and explains with transformations. draw examples of quadrilaterals thatdo not belong to any of “These must be congruent, Making Triangles Applet, Illuminations. Virtual GEOBOARD: these subcategories.
because they have equal sides, all http://www.nctm.org/standards/content.aspx?id=25007 2. Partition shapes into parts with equal areas. Express the square corners, and I can move Previous instructional strategies listed above, but with appropriate levels and area of eachpart as a unit fraction of the whole. For example,
them on top of each other exactly. information. partition a shape into 4parts with equal area, and describe Shape Class – Classifying Use - Guess my Rule the area of each part as 1/4 of the areaof the shape.
properties explicitly. Can see the - Shape Step thinvariants in the changes of state or - I Spy 4 Grade Critical Area: Students describe, analyze, compare, shape, must maintaining the - Our Mascot is Confused? (Help the mixed – up mascot name the proper and classify two-dimensional shapes. Through building, shapes’ properties. shape by identifying the properties.) drawing, and analyzing two-dimensional shapes,
“I put the shapes with opposite students deepen their understanding of properties of two-
sides parallel over here, and those Shape Tool, Illuminations applet: dimensional objects and the use of them to solve problems with four sides but not both pairs of http://illuminations.nctm.org/activitydetail.aspx?id=35 involving symmetry. thsides parallel over there.” - This tool allows you to create any geometric shape imaginable. Squares, 4 Grade GEOMETRY
Property Class – Classifying triangles, rhombi, trapezoids and hexagons can be created, colored, Draw and identify lines and angles, and classify shapes by
Use class membership for shapes enlarged, shrunk, rotated, reflected, sliced, and glued together. propertiesof their lines and angles.
(e.g., to sort or consider shapes 1. Draw points, lines, line segments, rays, angles (right, acute, similar) explicitly based on obtuse), and perpendicular and parallel lines. Identify these in properties, including angle two-dimensional figures.
measure. Is aware of restrictions of 2. Classify two-dimensional figures based on the presence or transformations and also of the absence of parallel or perpendicular lines, or the presence or definition absence of angles of a specified size. Recognize right
Which Shape Could it Be? Slowly reveal a shape from behind a screen At each triangles as a category, and identify right triangles.
„step ask children what class of shape could it be and how certain they are. 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.
th5 Grade – GEOMERY
Graph points on the coordinate plane to solve real-world and
1. Use a pair of perpendicular number lines, called axes, to Shape Sorter: Illuminations, A rhombus has four equal sides. A rectangle has four
define acoordinate system, with the intersection of the lines right angles. But a square has four equal sides and four right angles. Using a
(the origin)arranged to coincide with the 0 on each line and Venn diagram, the relationship would look like this:
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
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. What other relationships can be described using Venn diagrams? Use this tool to Classify two-dimensional figures into categories based on explore many different geometric properties and shapes. Teacher led use of theirproperties. Venn Diagrams to think about how the shapes are alike and how they are 3. Understand that attributes belonging to a category of two-different. dimensionalfigures 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.