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A Learning Trajectory for Shape
Adapted from, Learning and Teaching Early Math, Clements and Sarama
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.
□to □.
Shape Matcher – Orientations:
► to ◄
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
◊, □, █,
(including non-rectangular
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.”
◊, ⌂,
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.
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
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:
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
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
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.”
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.
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
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
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.
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
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
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:
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.
is completely correct, based on
knowledge of components and
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
Investigating Shapes (Triangles Lessons)
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
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:
Previous instructional strategies listed above, but with appropriate levels and
- 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:
- 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
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
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
Classify two-dimensional figures into categories based on
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
4. Classify two-dimensional figures in a hierarchy based on