5.3.1 Three-Dimensional Figures

Grade: 
5
Subject:
Math
Strand:
Geometry & Measurement
Standard 5.3.1

Describe, classify, and draw representations of three-dimensional figures.

Benchmark: 5.3.1.1 3-D Figures

Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of edges, faces or vertices as well as the types of faces.

Benchmark: 5.3.1.2 Nets

Recognize and draw a net for a three-dimensional figure.

Overview

Big Ideas and Essential Understandings 

Standard 5.3.1 Essential Understandings

Fifth grade students describe and classify 3 dimensional shapes by the number and type of faces and the number of edges and vertices. Students compose and decompose shapes.  They draw various nets that can be folded to create their three dimensional counterparts.  Students problem solve by determining how the individual 2 dimensional polygons can be arranged and folded to create a given 3 dimensional polyhedron, such as cubes, prisms and pyramids. For example, in order to determine if the two-dimensional shape below is a net that can be folded into a cube, students need to pay attention to the number, shape, and relative positions of its faces.

 

All Standard Benchmarks

5.3.3.1  Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of edges, faces or vertices as well as the types of faces.

5.3.3.2  Recognize and draw a net for a three-dimensional figure.

Benchmark Cluster 

Benchmark Group A

5.3.3.1  Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of edges, faces or vertices as well as the types of faces.

5.3.3.2  Recognize and draw a net for a three-dimensional figure.

What students should know and be able to do [at a mastery level] related to these benchmarks:

  • Move fluently between 2-dimensional representations of 3-dimensional shapes.
  • Able to decompose a 3-dimensional shape into its 2-dimensional components.
  • Create nets by correctly arranging the 2-dimensional faces that can be folded into a given 3-dimensional shape.
  • Describe and classify 3-dimensional shapes by their properties: faces, edges, and vertices. For example, all prisms contain 2 congruent parallel shapes connected by rectangular faces, while pyramids have a base face with triangular faces that create an apex.

Work from previous grades that supports this new learning includes:

  • Name, describe, and classify polygons.
  • Identify parallel and perpendicular lines in various contexts and use them to describe geometric shapes.
  • Sketch polygons with a given number of sides or vertices.
  • Describe, compare, and classify 2 and 3-dimensional figures according to number and shape of faces, and the number of sides, edges, and vertices.
  • Identify basic 2 and 3-dimensional shapes.
Correlations 

NCTM Standards

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. 

Grades 3 - 5 Expectations

  • identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes;
  • classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids;
  • investigate, describe, and reason about the results of subdividing, combining, and transforming shapes;
  • explore congruence and similarity;
  • make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.

Use visualization, spatial reasoning, and geometric modeling to solve problems.   

Grades 3-5 Expectations

  • build and draw geometric objects;
  • create and describe mental images of objects, patterns, and paths;
  • identify and build a three-dimensional object from two-dimensional representations of that object;
  • identify and draw a two-dimensional representation of a three-dimensional object;
  • use geometric models to solve problems in other areas of mathematics, such as number and measurement;
  • recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.

Common Core State Standards

Graph points on the coordinate plane to solve real-world and mathematical problems.

5.G.1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

5.G.2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Classify two-dimensional figures into categories based on their properties.

5.G.3. Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

5.G.4. Classify two-dimensional figures in a hierarchy based on properties.

Misconceptions

Student Misconceptions 

Students may think ...

  • squares are not rectangles. Students learn to classify specific quadrilaterals according to properties such as squares having 4 equal sides and 4 equal angles and rectangles having 4 sides with opposite sides equal and equal angles at an early age. Learning names of 4-sided polygons without considering the hierarchy within quadrilaterals first creates classification misconceptions when students are taught about quadrilaterals as the broad category of all 4-sided figures with 4 angles. They need to learn the commonalities among quadrilaterals to understand that certain quadrilaterals have similar properties and therefore, can be subcategories of each other.  All squares are rectangles (sub-category) but not all rectangles are squares.
  • the terms "square" and "cube" are interchangeable names for the same shape. Many students interchange words for two- and three-dimensional shapes. 

Resources

Instructional Notes 

Teacher Notes

Students may need support in further development of previously studied concepts and skills

Emphasize the properties and characteristics of a concept.

Provide many examples and non-examples even if the child is not ready to specifically name the non-example.

Pay close attention to language use.

Challenge understanding and broaden generalizations.

Look for the relationships between the number of faces, edges and vertices of a polyhedron and generalize a rule to describe this relationship. Eulers Rule: The relationship of the number of faces, vertices and edges of a polyhedron.

The number of edges is two less than the sum of the number of faces and the number of vertices minus 2.

TIMSS results suggest that students need work with spatial visualization, solving geometric problems, and 3-dimensional geometry.

Early experiences should teach quadrilaterals as a whole versus focusing on a few specific examples for specific examples and attempting to expand their understanding later. For example, teachers should focus on quadrilaterals and then discuss the similarities and difference between specific quadrilaterals.

Use and teach appropriate geometric vocabulary. It is better to introduce new terminology to students rather than subsequently attempt to distinguish between 2 and 3-dimensional shapes. For example, if the lesson being taught focuses on 2-dimensional shapes, students should not be asked to find items in the room, such as a door, as an example of a rectangle.

Dynamic Paper

Need a net for a solid figure?  You can create these and more with the Dynamic Paper tool. Find the images you want, then export as a PDF or as a JPEG image for use in other applications.

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 1-5 rather than 0-4.

For further information on the van Hiele Levels of Geometric Thought, go here

Teachers need to help students move from Gestalt Approach (overall look) to classify shapes to a classification by properties.

This link provides a schematic and explanation of the 4 levels of geometric thinking

  • Good questions, and good listening, will help children make sense of the mathematics, build self-confidence 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)

Instructional Resources 

NCTM Illuminations

Cube Nets - Identifies the eleven possible nets to create a cube. Also, has a net that transforms into a cube.

Dynamic Paper - to create nets.

This is a new tool in Illuminations where teachers can design their own worksheet. In this instance, teachers can choose their own nets for students.

Geometric Solids

This tool allows you to learn about various geometric solids and their properties. You can manipulate and color each shape to explore the number of faces, edges, and vertices, and you can also use this tool to investigate the following question:

For any polyhedron, what is the relationship between the number of faces, vertices, and edges?

Building A Box - nets for a cube

This lesson uses a real-world situation to help develop students' spatial visualization skills and geometric understanding. Emma, a new employee at a box factory, is supposed to make cube‑ shaped jewelry boxes. Students help Emma determine how many different nets are possible and then analyze the resulting cubes.

Schematic of the levels of geometric thinking:  Mind map for the van Hiele Model of Geometric Thought.

Additional Instructional Resources

Anderson, N., Gavin, M., Dailey, J., Stone, W., & Vuolo, J. (2005). Navigating through measurement in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.

Gavin, M., Belkin, L., Soinelli, A., & St. Marie, J. (2001). Navigating through geometry in grades 3-5. 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., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.

New Vocabulary 

New Vocabulary

"Vocabulary literally is the key tool for thinking."

Ruby Payne

net: a two-dimensional shape that can be folded to make or wrap three-dimensional prisms and pyramids.

pyramid: a geometric solid formed by triangles. The faces on a pyramid are trianlges that meet at a point (vertex). They can have different shaped bases and are named by the shapes of their bases.

triangular prism: a solid figure that two triangular faces and three rectangular faces, six vertices and nine edges

rectangular prism: a solid figure that has rectangles and squares for faces. Rectangular prisms are in the shape of a box.

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 non-examples.

Example/Non-example 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

illustration

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.

Professional Learning Communities 

Reflection - Critical Questions regarding the teaching and learning of these benchmarks.

What are the key ideas related to describing and classifying three-dimensional figures at the fifth grade level? How do student misconceptions interfere with mastery of these ideas?

What attributes should fifth graders use when describing and classifying three-dimensional figures?  What common errors do fifth graders make related to these attributes?

What experiences do students need in order to successfully classify three-dimensional figures?

When checking for student understanding related to describing and classifying three-dimensional figures, what do teachers

  • listen for in student conversations?
  • look for in student work?
  • ask during classroom discussions?

 

Additional critical questions:

When examining student work related to describing and classifying three-dimensional figures, 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.

What are the key ideas related to recognizing and drawing nets for three-dimensional figures?  What misconceptions do fifth graders have related to nets for three-dimensional figures?

What experiences do students need in order to successfully recognize and draw the net for a three-dimensional figure?

When examining student work related to recognizing and drawing nets for three-dimensional figures? 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 fifth grade level?

Professional Learning Community Resources

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.

Chapin, S., and Johnson, A. (2006). Math matters: Understanding the math you teach, grades K-8. (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 K-6). Sausalito, CA: Math Solutions.

Fosnot, C., & Dolk, M. (2002). Young mathematicians at work: Multiplication and division. Portsmouth, NH: Heinemann.

Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.

Otto, A., Caldwell, J., Wallus Hancock, S., & Zbiek, R.(2011). Developing essential understanding of multiplication and division for teaching mathematics in grades 3 - 5. Reston, VA.: National Council of Teachers of Mathematics.

Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.

Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

Schielack, J. (2009). Focus in grade 3, teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

References 

Anderson, N., Gavin, M., Dailey, J., Stone, W., & Vuolo, J. (2005). Navigating through measurement in grades 3-5. Reston, VA: NCTM.

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding.  Portsmouth, NH: Heinemann.

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Burns, Marilyn. (2007). About teaching mathematics:  A k-8 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 K-8. (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 K-6). Sausalito, CA: Math Solutions.

Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. 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 pre-k-grade 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, K-6. 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 k-8. 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 K-5. 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., Bay-Williams, 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 student-centered mathematics grades K-3. Boston, MA: Pearson Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.

Assessment

Which net creates a cylinder?

Solution: A. 

Benchmark:  5.3.3.2                                                                           

MCA III Item Sampler

How many edges does a hexagonal prism have?

A. 6

B. 8

C. 12

D. 18

Solution: D. 18

Benchmark 5.3.1.1                                                                             

MCA III Item Sampler

Solution:  D

Benchmark: 5.3.1.1                                                                            

TIMSS Released Item

Differentiation

Struggling Learners 

Students need opportunities to move from 2-dimensional nets to 3-dimensional polygons and vice versa.  For example, students could be provided with a series of nets and their 3 dimensional counter parts.  They predict which net can be matched with each 3-d model, and then fold the nets to justify their answers. Students should have multiple opportunities to construct and decompose 3d models with materials and online computer programs. Vocabulary should be developed while doing these activities. This is a perfect opportunity to use a word wall in mathematics.

Concrete - Representational - Abstract Instructional Approach

(Adapted from The Access Center: Improving Access for All K-8 Students,

The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based 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 on-task.   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. 

Additional Resources

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5. Sausalito, CA: Math Solutions.

Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. 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., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.

English Language Learners 
  • Word banks need to be part of the student learning environment in every mathematics unit of study. Refer to these throughout instruction.
  • 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:

I know this is a ________________ because ________________________________.

This is not a net for a _________________ because __________________________.

This is a net for 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 3-5. Sausalito, CA: Math Solutions Publications.

Extending the Learning 

Students could investigate special classification systems such as the properties of semi-regular or Archimedean solids, which are solids with identical vertices but composed of more than one kind of regular polygonal region, versus perfect solids that are composed of only one regular polygon with identical vertices. In addition, they may compare and contrast their special classification system with other classification systems being investigated by the class.

Additional Resources:

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5. Sausalito, CA: Math Solutions.

Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.

Parents/Admin

Classroom Observation 

Administrative/Peer Classroom Observation

Students are ...

Teachers are ...

studying models of polyhedra.

modeling various poyhedra and asking students to name and describe each of the models according to the number of vertices, edges, faces and face types.

sorting and classifying plolyhedra.

explaining that building a net for a polyhedron means breaking down the polyhedron into its 2-dimensional shapes.  Also explaining that the opening activity  contains vital information needed to draw a net for a polyhedron but that there is one more important piece and that is to determine the order of how to connect the faces in order to rebuild the polyhedron. Asking the students to partner up and look at the information they recorded and try different drawings of nets to rebuild the polyhedron. They are to test the nets by drawing the net on 1inch grid paper, and cutting out the net and folding the net along the edges to see if it rebuilds the polyhedron.. Students are to include all drawings tried and tell whether or not it built a polyhedron.

comparing and contrasting polyhedra

 

drawing nets, cutting nets out and folding them to see if they have rebuilt a polyhedron.

circulating around the room, listening to conversations, asking clarifying questions, and recording work she wants to highlight during discussion of the investigation.

 

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 real-world 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.

 

Additional Resources

For Mathematics Coaches

Chapin, S. and Johnson, A. (2006).  Math matters: Understanding the math you teach: Grades k-8. (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.

Parents 

A mathematics handbooks to be used as a home reference:

Great Source Education Writing Group. (2004). Math at hand: a mathematics handbook.  Wilmington, MA:  Great Source Education Group, Inc.

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

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 self-confidence 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.