2.3.1 Shapes
Overview
Standard 2.3.1 Essential Understandings
Students identify, name, describe, compare and classify two and threedimensional figures/shapes according to number and shape of faces, and the number of sides, edges and vertices (corners). These shapes include squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, rectangular prisms, cones, cylinders and spheres.
All Standard Benchmarks
2.3.1.1 Describe, compare, and classify two and threedimensional figures according to number and shape of faces, and the number of sides, edges and vertices (corners).
2.3.1.2 Identify and name basic two and threedimensional shapes, such as squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, rectangular prisms, cones, cylinders and spheres.
Benchmark Group A
2.3.1.1 Describe, compare, and classify two and threedimensional figures according to number and shape of faces, and the number of sides, edges and vertices (corners).
2.3.1.2 Identify and name basic two and threedimensional shapes, such as squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, rectangular prisms, cones, cylinders and spheres.
What students should know and be able to do [at a mastery level] related to these benchmarks?
 Analyze characteristics and properties of two and threedimensional geometric shapes.
 Recognize and name the parts of two and three dimensional shapes, such as the sides, faces, edges, and vertices.
 Justify classifications of two and threedimensional figures/shapes using geometric vocabulary.
Work from previous grades that supports this new learning includes:
 Describe characteristics of twoand threedimensional objects, such as triangles, squares, rectangles, circles, rectangular prisms, cylinders, cones and spheres.
 Compose (combine) and decompose (take apart) twoand threedimensional figures such as triangles, squares, rectangles, circles, rectangular prisms and cylinders.
 I've Seen That Shape Before. Students learn the names of solid geometric shapes and explore their properties. At various centers, they use physical models of simple solid shapes, including cubes, cones, spheres, rectangular prisms, and triangular prisms.
NCTM Standards
Analyze characteristics and properties of two and threedimensional shapes and develop mathematical arguments about geometric relationships.
PreK2 Expectations
 Recognize, name, build, draw, compare, and sort two and three dimensional shapes.
 Describe attributes and parts of two and threedimensional shapes.
 Investigate and predict the results of putting together and taking apart two and threedimensional shapes. (NCTM, 2000)
Common Core State Standards:
Reason with shapes and their 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.1 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
2.G.2. Partition a rectangle into rows and columns of samesize squares and count to find the total number of them.
2.G.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 the same shape.
Misconceptions
Students may think . . .
 squares are not rectangles
 a change in orientation changes the shape
 a square is only a square if its base is horizontal.
 all triangles sit on a side, i.e. they don't recognize ▷ as a triangle.
 the only triangle is an equilateral triangle.
 shapes have only one label. Not realizing, for example, a square is a parallelogram, a rectangle, and also a rhombus.
 a cube is not a rectangular prism.
Vignette
This classroom vignette takes place midyear in a second grade classroom, near the end of the unit on geometry. The objective of this lesson is to expose the children to irregular pentagons, so they focus on the attributes of shapes, rather than just the appearance. The activity, "The Four Triangle Problem" is from A Collection of Math Lessons Gr. 13, by Marilyn Burns & Bonnie Tank.
Materials: 3" x 3" construction paper squares  two different colors, at least 5 per student of each color
glue  each student or partnership
scissors  each student
6"x9" newsprint  at least 5 per student
3 foot x 5 foot piece of butcher paper
tape
Vocabulary:
congruent parallelogram quadrilateral square
diagonal pentagon rectangle trapezoid
hexagon polygons rhombus triangle
Ms. D.: Today we're going to do a fun exploration called "The Four Triangle Problem." Our objectives for today are to make and classify polygons according to their attributes.
What do you know about triangles?
LaMya: They have three sides.
Colin: They have three straight sides.
Ms. D.: What else?
Elmin: They are a closed shape.
Madisyn: They have three corners.
Kahlid: Corners are called vertices.
Ms. D: Anything else?
No one volunteers.
Ms. D.: We are going to make triangles out of these pieces of paper.
(shows two 3 x 3 squares).
What do you notice about these papers?
Aleane: They're squares.
Ms. D.: How do you know they are squares?
Aleane: 'Cause they look like squares.
Ms. D.: What attributes does a shape need in order to be a square?
Tonio: They have four sides that are the same.
Ms. D.: How are they the same?
Tonio: They are the same length.
Ms. D.: Good thinking, Tonio. What other attribute is needed to be a square?
Charles: They have to have square corners.
Ms. D.: That's right. All the corners are right angles. What is another name for "right angle?"
Anna: Ninety degree angles.
Ms. D.: Yes, right angles are called "ninetydegree angles." That's a very important concept in geometry. Let's keep that in mind while we work on this problem. Now, I said this was a problem about triangles. How on earth are we going to use these squares in our problem?
Juan: We can make them into triangles.
Ms. D.: How?
Juan: By cutting them.
Ms. D.: How?
Juan: Cut them on the diagonal.
Ms. D.: Would you please show us how to do that?
Juan cuts on the diagonal, but the line is rather crooked.
Juan: Oh, it didn't work right.
Ms. D.: Does anyone have an idea about how to make it easier to cut on the diagonal?
Alfred: He could fold it first.
Ms. D.: Juan, do you want to try it again folding the paper first?
Juan folds first then cuts. It works much better.
Ms. D.: Nice job, Juan. Now, we are going to use those triangles to make polygons. What are polygons?
Jose: They are closed shapes with all straight sides.
Ms. D.: Nice definition, Jose. Now I want everyone to take one square, fold and cut on the diagonal.
Just for practice, we are going to use these two triangles to make some polygons. I have some rules that I want you to follow when we construct our polygons:
1. The triangles must touch along sides  they can't meet at just a point.
2. The sides that are touching have to be the same size and match exactly.
Try making a polygon following those rules, using just your two triangles.
Ms. D. monitors the students as they are working. It only takes about 5 minutes before she has the examples she needs.
Ms. D.: Okay, let's look at some of the shapes we made. Does this shape follow the rules?
Juan: Yes.
Ms. D.: How do you know?
Juan: Because they are touching sides and the sides match.
Ms. D.: What shape is it?
Juan: It's a square. Just like the ones we cut. I just put it back together.
Ms. D.: Well, that certainly works! How about this one?
Anna: It doesn't work because these sides don't match up.
Ms. D.: How about this one?
Madisyn: It's not right because it's only touching at a point.
Ms. D.: And this one?
Elmin: It works and it's another triangle!!
Ms. D.: Nice, Elmin. What different shapes have we not checked yet?
Colin: I have one  it's a parallelogram.
Ms. D.: What makes it a parallelogram?
Colin: It has two sets of parallel sides.
Ms. D.: What do you mean by "parallel sides?"
Colin: It's like if these sides went on forever they would never touch.
Ms. D.: Got it  so a parallelogram has opposite sides that are parallel. Okay, it looks like you understand the rules. Now, let's jump into the four triangle problem. You need to use one square of each color. Fold each square on the diagonal and cut. How many triangles will you have then?
LaMya: Four.
Ms. D.: Yes  two squares, each cut into two triangles each will make four triangles. Now here is the problem: How many different shapes can we make using the four triangles? I want you to work in table groups and try to come up with all the possible shapes. Check with everyone at your table and try to make different shapes.
You know, I have my class do this project every year and I've learned a few tips. Here's a big one  build your shape on a piece of newsprint. Raise your hand so I can check your shape, and then you can glue it on the paper. Don't take it off the paper. Just remove one piece at a time and glue it. Sometimes when you take all the pieces off the paper to start gluing, you might not remember how to make your shape again. So, let's play it safe and remove and glue one piece at a time. Questions?
Ms. D. circulates around the room checking student work. It's advisable to have the teacher check the shapes to make sure the shapes follow the rules. There are fourteen different shapes, but the kids don't need to know that in the beginning. As they've discovered most of them, Ms. D. tells them how many more they need to find.
Ms. D.: Hi, LaMya  Let's look at your shape. Does it have four triangles?
LaMya: Yes.
Ms. D.: Are the touching sides the same length?
LaMya: Yes.
Ms. D.: Is it the same as any other shapes on your table?
LaMya: No.
Ms. D.: How about this one?
LaMya: It's different.
Ms. D.: Try flipping it.
LaMya: Oh, no  it matches.
Ms. D.: Yes  they're congruent. That means they're the same size and shape.
LaMya: Yeah, but they are different colors.
Ms. D.; You're right, they are. But  we are only looking at the shape, not the combinations of colors. Boys and girls  LaMya made an important discovery.
LaMya: You have to try to flip or turn the shapes to see if they match something else on your table. And don't pay attention to the colors  look only at the shape. If it matches, they're . . . I forgot.
Ms. D.: They're congruent. Congruent means same size and shape.
LaMya : Congruent.
Ms. D. continues to circulate checking the shapes, and helping the children determine congruency. As she works with the children, she reminds them of the geometric transformations: turns (rotations) and flips (reflections). (This activity doesn't involve any slides.)
As the class period winds down, she reminds the children to put their names on the back of their shapes and clean up the scraps.
Ms. D.: Let's review our goal for the lesson and see if we attained it. We were going to make and classify shapes according to their attributes.
Jose: We made different shapes  but we didn't classify them.
Ms. D: That's right  we'll classify them tomorrow.
During the next class period, the children sort and name the polygons they have created.
Ms. D.: Okay, Class  get out the shapes we made yesterday. We are going to sort our polygons. Work with your table group to sort your shapes.
Ms. D. circulates around the room as the kids are working on the shape sorts.
Ms. D.: Tonio, how did your group decide to sort the shapes.
Tonio: Well, we sorted them by the number of sides they have.
Ms. D.: Tell me about this shape.
Ms. D. knows this shape is confusing.
Tonio: It has six sides.
Ms. D.: Show me how you counted the sides.
Tonio: One, two, three, four, five, six (counting the long side as "two").
Ms. D.: Aleane  do you agree that this shape has six sides?
Aleane: No, because that is just one side  even though there are two triangles.
Ms. D.: Good point, Aleane. Let's put that shape with the other fivesided shapes.
When the table groups have finished sorting their shapes, Ms. D. calls the class back together.
Ms D.: Okay, class. Let's explore the shapes we made. What different shapes did you find?
Madisyn: We found a square.
Ms. D.: What makes it a square?
Madisyn: Well, it has four sides and all the sides are the same size.
Ms. D.: What other attribute does a square need?
Elmin: It has to have all square corners!
Ms. D.: That's right, Elmin. What's another name for square corners?
Elmin: Ninetydegrees.
Ms. D.: What is another name for this shape?
Anna: Quadrilateral.
Ms. D.: How do you know?
Anna: Because it has four sides.
Ms. D.: Okay, let's make a category on our poster for quadrilaterals. We can put square by the shape as well.
What other quadrilaterals or quadrangles do you have?
Madisyn: I have a rectangle.
Ms. D.: What makes it a rectangle?
Madisyn: It has four sides and four square corners.
Ms. D.: Our other shape has four sides and four square corners. Does that make it a rectangle?
Jose: Yes  a square is a special kind of rectangle.
Ms. D: Where should we put the rectangle on our poster?
Charles: It's a quadrilateral. We can label it a rectangle, though.
Juan: I have a parallelogram  that's a quadrangle, too!
Ms. D.: Come on up and put it on the poster. Any other shapes for this category?
LaMya: I have a trapezoid. I know because it has only one pair of parallel sides.
Ms. D.: Put it on the poster.
LaMya: Can I label it "trapezoid?"
Ms. D.: Go for it.
Emma: I have another quadrilateral, but I don't know if it has another name.
Jose: It's a parallelogram.
Emma: But it doesn't look like Madisyn's parallelogram. Her's is skinny.
Jose: But it has two pairs of parallel sides, so it has to be a parallelogram.
Ms. D.: What do you think, class?
Juan: Parallelograms can look different as long as they have four sides and the opposite sides are parallel.
Ms. D.: Are you okay with labeling your shape a parallelogram, Emma?
Emma: Yep  can I put it on the poster?
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Ms. D: Sure. What about the square and the rectangle? They have two sets of parallel sides.
Elmin: Does that make them parallelograms then?
Ms. D.: What do you think, class?
Jose: I think they are parallelograms because they follow the rules. I know that shapes can have different names.
Madisyn: Yeah, I think they're parallelograms and rectangles and quadrilaterals and quadrangles.
Ms. D: It really helps to know the attributes of a shape, doesn't it? It would be really hard to recognize all the different types of shapes if we didn't know the rules for each shape.
Alfred: I have a triangle. See  it has three sides!
Ms. D.: Okay, Alfred. Come on up and put it on the poster. We'll label that section "Triangle."
Any more triangles?
How about different shapes?
Ahlam: I have one.
Ms. D.: How many sides does it have?
Ahlam: It looks like an arrow.
Ms. D.: Yes, it does. Hold it up and trace the sides while we count them.
Class: One, two, three, four, five, six
Ms. D.: What do we call a shape with six sides?
Emma: Well, a hexagon has six sides, but that doesn't look like a hexagon.
Ms. D.: So we classify shapes just by how they look?
Aleane: No, we use their att. . . attributes.
Ms. D.: What are the attributes of a hexagon?
Jose: A polygon with six sides. So, that's a hexagon.
Ms. D.: Come on up and put it on the poster, Ahlam. We'll label that section . . .
Jose: Hexagons!
LaMya: I have another hexagon.
Ms. D.: Count the sides for us. Great! Put it on the poster.
Nou: I have another one. See  one, two, three, four, five, six!
Ms. D.: Nice  put it on the poster.
Kearia: I think this will work.
Isaac: Here's one. I'll put it on the poster.
Sophie: I have one, too.
Ms. D.: Any more?
Colin: Here's one. See  it has six sides.
Ms. D.: You're right  it does have six sides. See if you can match it up to any of the other shapes on our poster. Try turning or flipping it.
Colin: It matches this one.
Ms. D.: When something is the same size and shape we say it's "congruent." Everyone say, "Congruent." Does anyone have any other shapes that look different from the shapes on our poster, but are really congruent?
A few children come up and check their shapes for congruency.
Emma: I have one.
Aleane: That only has five sides! Those two triangles are on the same side!
Ms. D.: Please show us what you are talking about, Aleane.
Ms. D.: What do we call shapes that have five sides?
Jose: Pentagons! And I have another one!
Ms. D.: Jose, would you please show us how you counted the sides?
Jose: One, two, three, four, five.
Ms. D.: Thanks, Jose. Please put it up on the poster.
Let's look at our shape poster.
What are some things you learned from this exploration?
Madisyn: There are different kinds of hexagons.
Ms. D.: What do you mean?
Madisyn: Well, they all look different, but they're still hexagons.
Ms. D.: How can you tell when a polygon is a hexagon?
Madisyn: When it has six sides.
Ms. D.: Yes, it's important to know the attributes of shapes. What else have we learned?
Tonio: It was hard to find all the different shapes.
Ms. D.: What made it hard?
Tonio: It's hard to tell when you're making the same shape as somebody else.
Ms. D.: What did you do to help figure it out?
Tonio : Aleane helped me compare some of the shapes.
Ms. D.: You did a good job of sticking to it and figuring it out, Tonio. That's called perseverance and I'm proud of you.
It's almost time to wrap things up. What were our objectives for this activity?
LaMya: To make and classify shapes according to their attributes. And we did it!!
Ms. D.: Talk at your tables about the attributes you used to classify your shapes.
The students responded with number of sides and parallel sides. Some students shared other attributes of the shapes but these were not used as the students classified the shapes they had constructed from four triangles.
Resources
Teacher Notes
 Students may need support in further development of previously studied concepts and skills.
 Students may give simple verbal labels like square or triangle, and not necessarily understand the concepts.
 Students need to see a variety of examples of the same geometric shape. For example, they should see triangles with a variety of positions and angle sizes.
 Second graders need to see a variety of nonexamples of a geometric shape.
 Encourage students to share why a shape does or does not belong to a shape category.
 Provide handson exploration to encourage students to explore shapes and their attributes. This may include using geoboards, pattern blocks, and geometric solid shapes. This can extend to common household products, such as cereal boxes, soup cans, and party hats. Kids enjoy creating a classroom "Shapes Museum" out of these items and sorting them by their threedimensional shapes.
 Children may miscount the number of faces, edges, and vertices by missing one or by counting one twice. Pieces of tape placed on faces, edges, and vertices as students count will avoid over or undercounting.
 The van Hiele Levels of Geometric Thought.
Adapted from: Van de Walle J. (2006) and Van de Walle, J. & Lovin, L. (2006).
The van Hiele Levels of Geometric Thought is a hierarchy describing the way that students learn to reason about shapes and other geometric ideas. There are five levels in the hierarchy that are deemed to be sequential. Students need to move through each prior level before moving on to the next. Student thinking about particular concepts will likely be at different levels at any given time. The levels describe how we think and what types of geometric ideas we think about, rather than how much knowledge we have. Movement through the levels depends on the types and amount of experiences students have with geometry. Instruction that takes place at a level higher than the students' functional level will be ineffective. Many adults remain at level 1 even though they have had a geometry course in high school. With appropriate experiences, however, students can reach level 2 in elementary school.
Level 0  Visualization
The objects of thought at level 0 are shapes and what they "look like."
Students may be able to talk about the properties of the shapes, but the properties are not thought about explicitly. They characterize individual shapes based on appearance. "It is a square because it looks like a square." At this level, students think shapes "change" or have different properties when rotated or rearranged.
The products of thought at level 0 are classes or groupings of shapes that seem to be alike.
Level 1  Analysis
The objects of thought at level 1 are classes of shapes rather than individual shapes.
Students are able to think of properties (number of sides, angles, parallel sides, etc.) of a shape rather than focusing on the appearance of a shape. A student operating at this level might list all the properties the student knows about a shape, but not discern which properties are necessary and which are sufficient to identify the shape. "It is a square because it has square corners and the sides are the same." Though students see properties of shapes, they cannot make generalizations about how different shapes relate to one another. Students at this level will not see the relationship of a square to a rectangle.
The products of thought at level 1 are the properties of shapes.
Level 2  Informal Deduction or Abstraction
The objects of thought at level 2 are the properties of shapes.
Students develop relationships between and among properties. Shapes can be classified using minimal characteristics. "Rectangles are parallelograms with a right angle." Students at level 2 will be able to follow an informal deductive argument about shapes and their properties. While many adults remain at level 0 or level 1, with appropriate experiences, most students could reach level 2 by the end the elementary grades.
The products of thought at level 2 are relationships among properties of geometric objects.
Level 3  Deduction
The objects of thought at level 3 are relationships among properties of geometric objects.
Students work with abstract statements about geometric properties and make conclusions based on logic. This is the level of a traditional high school geometry course.
The products of thought at level 3 are deductive axiomatic systems for geometry.
Level 4  Rigor
The objects of thought at level 4 are deductive axiomatic systems for geometry.
Students operating at this level focus on axiomatic systems, not just deductions within the system. This is usually at a level of a college geometry course.
The products of thought at level 4 are comparisons and contrasts among different axiomatic systems of geometry.
Note: In some literature the van Hiele Levels of Geometric Thought are labeled 15 rather than 04.
For further information on the van Hiele Levels of Geometric Thought see this page.
 Dynamic Paper
Need a set of pattern blocks where all shapes have oneinch sides? You can create these and more with the Dynamic Paper tool. Place the image you want, then export it as a PDF or as a JPEG image for use in other applications.
See this page.
 Good questions, and good listening, will help children make sense of the mathematics, build selfconfidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if...?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work? Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part ...
Adapted from They're Counting on Us, California Mathematics Council, 1995)
Additional Instructional Resources
Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in Grade: 2 Teaching with Curriculum Focal Points. Reston, VA: National Council of Teachers of Mathematics.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J. A., & Lovin, L H. (2006) Teaching studentcentered mathematics grades K3. Boston: Pearson Education.
New Vocabulary
rectangular prism  a box with faces that are all rectangles
"Vocabulary literally is the key tool for thinking."
Ruby Payne
Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions. Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.
Learning vocabulary in the mathematics classroom is contingent upon the following:
Integration: Connecting new vocabulary to prior knowledge and previously learned vocabulary. The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.
Repetition: Using the word or concept many times during the learning process and connecting the word or concept with its meaning. The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.
Meaningful Multiple and varied opportunities to use the words in context. These
Use: opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems. Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.
Strategies for vocabulary development
Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.
Mathematics Word Bank: Each unit of study should have word banks visible during instruction. Words and corresponding definitions are added to the word bank as the need arises. Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.
Labeled pictures and charts: Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.
Frayer Model: The Frayer Model connects words, definitions, examples and nonexamples.
Example/Nonexample Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words. Teachers should use these during the instructional process to engage student in thinking about the meaning of words.
Vocabulary Strips: Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.
word  definition  illustratio 
Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.
Additional Resources for Vocabulary Development
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Reflection  Critical Questions regarding the teaching and learning of these benchmarks
What are the key ideas related to identify, naming, describing, comparing and classifying two and threedimensional figures/shapes at the second grade level? How do student misconceptions interfere with mastery of these ideas?
What experiences do students need in order to identify, naming, describing, comparing and classifying two and threedimensional figures/shapes successfully?
What attributes might second graders use when classifying two and threedimensional figures/shapes?
Examine student work related to a task involving identify, naming, describing, comparing and/or classifying two and threedimensional figures/shapes. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.
How can teachers assess student learning related to these benchmarks?
How are these benchmarks related to other benchmarks at the second grade level?
Professional Learning Community Resources
Bamberger, H., Oberdorf, C., & SchultzFerrell, K. (2010). Math misconceptions prekgrade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Chapin, S. and Johnson, A. (2006). Math matters, understanding the math you teach, grades k8. (2nd ed.). Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K6). Sausalito, CA: Math Solutions.
Hyde, Arthur. (2006) Comprehending math adapting reading strategies to teach mathematics, K6. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k8 classrooms! Thousand Oaks, CA.: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k2. Sausalito, CA: Math Solutions.
Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k8. Portsmouth, NH: Heinemann.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Additional Resources
Bender, W. (2009). Differentiating math instructionStrategies that work for k8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k2.Sausalito, CA: Math Solutions.
Murray, M. & Jorgensen, J. (2007). The differentiated math classroomA guide for teachers k8. Portsmouth, NH: Heinemann.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.
Van de Walle, J. & Lovin, L. (2006). Teaching studentcentered mathematics grades k3. Boston, MA: Pearson Education.
Additional Resources
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k8. (2nd ed.). Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
For Administrators
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
References
Bamberger, H., Oberdorf, C., & SchultzFerrell, K. (2010). Math misconceptions prekgrade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Bender, W. (2009). Differentiating math instruction: Strategies that work for k8 classrooms! Thousand Oaks, CA: Corwin Press.
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k2. Sausalito, CA: Math Solutions Publications.
Burns, Marilyn. (2007). About teaching mathematics: A k8 resource (3rd ed.). Sausalito, CA: Math Solutions Publications.
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Caldera, C. (2005). Houghton Mifflin math and English language learners. Boston, MA: Houghton Mifflin Company.
Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades K8. (2nd ed.). Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K6). Sausalito, CA: Math Solutions.
Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k2. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration prekgrade 2. Reston, VA: National Council of Teachers of Mathematics.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 2 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K6. Portsmouth, NH: Heinemann.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k8. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K5. Sausalito. CA: Math Solutions.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., ... & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J. A., & Lovin, L. H. (2006). Teaching studentcentered mathematics grades K3. Boston, MA: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., Trevino, E., & Zbiek, R. M., (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: a quest for coherence. Reston, VA: NCTM.
Seeley, C. (2009). Faster isn't smartermessages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Small, M. (2009). Good questions: great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., BayWilliams, J. (2010). Elementary and middle school mathematics: teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.
Van de Walle, John A., & Lovin, LouAnn H. (2006) Teaching studentcentered mathematics grades K3. Boston: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: transforming mathematics lessons. Portsmouth, NH: Heinemann.
Assessment
 Circle all the triangles.
Solution:
Benchmark: 2.3.1.1/ 2.3.1.2
 Which figure has 6 faces, 12 edges, and 8 vertices?
B. C. A.
Solution: A
Benchmark: 2.3.1.1, 2.3.1.2
 How many faces does a cube have?
Solution: 6
Benchmark: 2.3.1.1
 What shape are the faces in this pyramid?
A. triangle
B. square
C. rectangle
D. kite
Solution: A. triangle
Benchmark: 2.3.1.1
 Describe a hexagon.
Solution: a closed shape (polygon) with six sides and six vertices.
Benchmark: 2.3.1.1
Differentiation
Struggling Learners
Students may have difficulty identifying plane shapes by their sides and vertices. To support them, provide cutouts of the shapes and help them describe and identify each one. Write the names on each shape and have children keep them for reference.
Matching two and threedimensional figures is a great way for students to begin noticing the attributes of a shape.
Share books with many examples of various shape categories to build their vocabulary and understanding.
Attribute blocks are a good introduction to logical thinking geometrically. These blocks are math manipulatives that have four different features. These are shape, color, size and thickness. Students learn to use four different descriptive words with each block they choose. For example, "I pick a square, blue, big, thick block." This may be quite tricky for students in the beginning and describing only one or two features to begin with is fine. To build logical thinking, challenge students to complete sequences of shapes with one different attribute for each additional shape. Encourage them to explain their sequences using math vocabulary.
Concrete  Representational  Abstract Instructional Approach
(Adapted from The Access Center: Improving Access for All K8 Students)
The ConcreteRepresentationalAbstract Instructional Approach (CRA) is a researchbased instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.
The CRA approach is based on three stages during the learning process:
Concrete  Representational  Abstract
The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts. At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level. Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and ontask. Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.
The Representational Stage is the drawing stage. Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems. They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking. Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.
The Abstract Stage is the symbolic stage. Teachers model mathematical concepts using numbers and mathematical symbols. Operation symbols are used to represent addition, subtraction, multiplication and division. Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding. Moving to the abstract level too quickly causes many student errors. Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.
English Language Learners may confuse the various names for two and threedimensional shapes. Use physical models and create a chart for students to use until they can readily identify and classify shapes.
Almost all plane and solid shape names in English have Spanish cognates. Use them to reinforce their meaning if students are familiar with them. These include triangle (triángulo), rectangle (rectángulo), hexagon (hexágono), trapezoid (trapezoide), vertex (vértice), and vertices (vértices), rectangular prism (prismo rectangular), pyramid (pirámide), cone (cono), cylinder (cilindro), sphere (esfera), and cube (cubo).
To reinforce new vocabulary, play "Guess My Shape" using attribute clues. This game can be used with both twodimensional and threedimensional shapes. Let students make up clues and have a partner guess the shape. For examaple, a student gives a clue, such as "My shape has three vertices." Other students try to guess the shape.
 Word banks need to be part of the student learning environment in every mathematics unit of study.
 Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.
 Sentence Frames
Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions. Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.
Sample sentence frames related to these benchmarks:
This is a ______________________ because __________________________________. 
This is not a _______________________ because ______________________________. 
 When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding.
Additional ELL Resources
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k2. Sausalito, CA: Math Solutions Publications.
As an extension, students can construct solid shapes from the faces of a given shape. For example, have students trace the faces of a cube on construction paper or tag board, cut them out and tape the faces together to make a cube. Students can try other solids, such as a rectangular prism or pyramid.
Use Venn Diagrams to sort and classify two and threedimensional shapes.
Parents/Admin
Administrative/Peer Classroom Observation
Students are...  Teachers are... 
identifying and naming shapes found in the realworld.  asking students to describe the attributes of shapes found in the real world. 
describing two and threedimensional shapes according to attributes.  focusing students on the attributes of various shapes, such as a pentagon is a polygon with 5 sides or a cube has six square faces. Encouraging students to construct "nonregular" polygons. 
comparing shapes and describing how they are alike and how they are different.  sharing both examples and nonexamples of two and threedimensional shapes. 
sorting shapes into groups with one or more different attributes and justifying their placements.  providing a variety of two and threedimensional shapes as students sort and classify. 
making various two and three dimensional shapes using attributes 

What should I look for in the mathematics classroom? (Adapted from SciMathMN,1997)
What are students doing?
 Working in groups to make conjectures and solve problems.
 Solving realworld problems, not just practicing a collection of isolated skills.
 Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
 Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
 Recognizing and connecting mathematical ideas.
 Justifying their thinking and explaining different ways to solve a problem.
What are teachers doing?
 Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
 Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
 Connecting new mathematical concepts to previously learned ideas.
 Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
 Selecting appropriate activities and materials to support the learning of every student.
 Working with other teachers to make connections between disciplines to show how math is related to other subjects.
 Using assessments to uncover student thinking in order to guide instruction and assess understanding.
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters, understanding the math you teach, grades k8, 2nd edition. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students Learn: Mathematics in the Classroom. Washington, DC.: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: charting your course. Sausalito, CA: Math Solutions.
West, L., & Staub, F. (2003). Content focused coaching: transforming mathematics lessons.
For Administrators
Burns, M. (Edt).(1998). Leading the way: principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: helping children learn mathematics. Washington, DC.: National Academies Press.
Leinwand, S., (2000). Sensible mathematics: a guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
Seeley, C. (2009). Faster isn’t smartermessages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Parent Resources
Mathematics handbooks to be used as home references:
Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Helping your child learn mathematics
Provides activities for children in preschool through grade 5
What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN
Help Your Children Make Sense of Math
Ask the right questions
In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.
Good questions, and good listening, will help children make sense of the mathematics, build selfconfidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to ...?
Can you make a prediction?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work?
Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part...
Adapted from They're counting on us, California Mathematics Council, 1995
 To enrich your child's realworld experiences with solid shapes, explore your kitchen! It is a great place to find different solid shapes. Have your child collect a variety of items and sort them by shape. Your child can explore how the dimensions of rectangular prisms, cylinders, and other solid shapes can be very different. A tuna can is a cylinder that is short and squat, but a soup can is a taller cylinder. Encourage your child to sort the solid shapes into groups and label them, such as: prisms, cylinders, cones, and spheres. Invite them to describe how the solid shapes are alike and different.
 Your child can build threedimensional shapes with patterns from this link
 To build your child's spatial and geometric sense, invite them to explore the Chinese puzzle known as the Tangram. It is a popular dissection puzzle formed from 7 polygons. The aim of the puzzle is to seamlessly arrange all the geometric pieces to form a variety of figures. You'll find more information and a puzzle to print off and explore at this link
 Enjoy children's books such as Shape Up! Fun with Triangles and other Polygons by David A. Adler.
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