3.3.1 Parallel/Perpendicular Lines & Polygons
Overview
Standard 3.3.1 Essential Understandings
Third graders identify parallel and perpendicular lines. They use these lines to describe geometric shapes such as right triangles, rectangles including squares, parallelograms, trapezoids and rhombuses (rhombi). Third graders draw polygons with a given number of sides and vertices including pentagons, hexagons and octagons.
Benchmark Group A
3.3.1.1 Identify parallel and perpendicular lines in various contexts, and use them to describe and create geometric shapes, such as right triangles, rectangles, parallelograms and trapezoids
3.3.1.2 Sketch polygons with a given number of sides or vertices (corners), such as pentagons, hexagons and octagons.
What students should know and be able to do [at a mastery level] related to these benchmarks:
 Identify perpendicular and parallel lines in various contexts.
 Use parallel lines to describe shapes including parallelograms, rectangles including squares, trapezoids and rhombuses (rhombi).
 Use perpendicular lines to describe shapes including right triangles and rectangles including squares.
 Understand the right angle symbol (square in a corner).
 Identify a figure by the number of sides.
 Draw polygons when given the number of sides or vertices.
Work from previous grades that supports this new learning includes:
 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.
NCTM Standards
Analyze characteristics and properties of twoand threedimensional geometric shapes and develop mathematical arguments about geometric relationships.
 Identify, compare, and analyze attributes of two and threedimensional shapes and develop vocabulary to describe the attributes.
 Classify two and threedimensional 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.
Common Core State Standards
Reason with shapes and their attributes.
3.G.1. Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
3.G.2. Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.
Misconceptions
Students may think...
 parallel and perpendicular lines are the same or not know which is which
 irregular shaped figures can't be pentagons, hexagons, octagons, etc. For example, students may think this is not an octagon.
 a change in orientation means a change in shape.
Resources
Teacher Notes
 Students may need support in further development of previously studied concepts and skills.
 Students need to see a variety of examples of the same geometric shape including regular and irregular pentagons, hexagons and octagons.
 Third 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.
 The vanHiele Levels of Geometric Thought: adapted from: Van deWalle, J. (2006); and Van deWalle, 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 thehierarchy 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 labelled 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 images you want, then export it as a PDF or as a JPEG image for use in other applications.
 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)
NCTM Illuminations
 Using the Shape Cutter, students construct two dimensional shapes.
Additional Instructional Resources
Gavin, M., Belkin, L., Soinelli, A., & St. Marie, J. (2001). Navigating through Geometry in Grades 35. 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., & Lovin, L. (2006). Teaching studentcentered mathematics grades k3. Boston, MA: Pearson Education.
Van de Walle, J., & Lovin, L. (2006). Teaching studentcentered mathematics grades 35. Boston, MA: Pearson Education.
New Vocabulary
"Vocabulary literally is the key tool for thinking."
Ruby Payne
parallel  two or more lines that are and always remain an equal distance apart
perpendicular  lines that come together at a 90 degree angle
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  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.
Reflection  Critical Questions regarding the teaching and learning of these benchmarks:
What are the key ideas related to identifying parallel and perpendicular lines? How do student misconceptions interfere with mastery of these ideas?
What experiences do students need in order to identify parallel and perpendicular lines successfully?
When checking for student understanding of parallel and perpendicular lines at the third grade level, what should teachers
 listen for in student conversations?
 look for in student work?
 ask during classroom discussions?
Examine student work related to a task involving the identification of parallel and perpendicular lines. 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 sketching polygons with a given number of sides or vertices? How do student misconceptions interfere with mastery of these ideas?
What common errors do third graders make when sketching polygons with a given number of sides or vertices?
When checking student understanding of sketching polygons with a given number of sides or vertices, what should teachers
 listen for in student conversations?
 look for in student work?
 ask during classroom discussions?
Examine student work related to a task involving sketching polygons with a given number of sides or vertices. 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 third 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. (2^{nd} 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.
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,
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.
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 K5. 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.
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.
O'Connell, S., & SanGiovanni, J. (2011). Mastering the basic math facts multiplication and division strategies, activities & interventions to move students beyond memorization. Portsmouth, NH: Heinemann
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.
Uittenbogaard, W., & Fosnot, C. (2008). Minilessons for early multiplication and division, Grades 34. Portsmouth, NH: Heinemann.
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.
Wickett, M., Ohanian, S., & Burns, M. (2002). Teaching arithmetic: Lessons for introducing division, Grades 34. Sausalito, CA: Math Solutions.
Yeatts, K., Battista, M., Mayberry, S., Thompson, D., & Zawojewski, J. (2004). Navigating through problem solving and reasoning in grade 3. Reston, VA: National Council of Teachers of Mathematics.
Zemelman, S., Daniels, H., & Hyde, A. (2005). Best practices: Today's standards for teaching and learning in America's schools. Portsmouth, NH: Heinemann.
Assessment
NCTM Grades 35 Mathematics Assessment Sampler pg.104106, pg. 109, pg. 76, pg. 78.
 Which of the following letters has two parallel line segments?
(A) A
(B) T
(C) K
(D) N
Solution: (D) N
Benchmark: 3.3.1.1
 Alan says that if a figure has four sides, it must be a rectangle. Gina does not agree. Draw a figure that shows that Gina is correct.
The solution involves identifying the many twodimensional shapes that may share a given characteristic.
Solution: possible answers
Sample A Sample B Sample C
Benchmark: 3.3.1.2
Solution: B. Perpendicular
Benchmark: 3.3.1.1
MCA III Item Sampler
Differentiation
 Uncooked spaghetti noodles can be used to model parallel and perpendicular lines. Ask students to find examples of parallel and perpendicular lines in the classroom.
 Classify shapes by the number of sides or vertices prior to sketching a figure with a given number of sides or vertices.
 Students need experience with both regular and irregular pentagons, hexagons, and octagons.
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.
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 35. 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.
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.
Van de Walle, J., & Lovin, L. (2006). Teaching studentcentered mathematics grades 35. Boston, MA: Pearson Education.
 The words parallel and perpendicular may be confusing for students. These words must be clearly defined and modeled concretely. Uncooked spaghetti noodles can be used to model parallel and perpendicular lines. Ask students to find examples of parallel and perpendicular lines in the classroom.
 Classify shapes by the number of sides or vertices prior to sketching a figure with a given number of sides or vertices.
 Students need experience with both regular and irregular pentagons, hexagons, and octagons.
 Additional words to clarify are points, lines, planes, polygons, right angle, congruent, and similar.
 Word banks need to be part of the student learning environment in every mathematics unit of study.
 Word 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:
Parallel lines are __________________________________________. 
Perpendicular lines are _____________________________________. 
I know this shapes has______________ lines 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.
 Challenge students to draw three intersecting lines, but no parallel line.
 Explore how many sets of perpendicular fold lines can be formed by continuing to fold a piece of paper into halves over and over. Is there a pattern?
 Have students use a local street map and identify parallel and perpendicular lines or draw a map of their neighborhood using parallel and perpendicular lines.
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 35. 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.
Parents/Admin
Administrative/Peer Classroom Observation
Students are...  Teachers are... 
identifying parallel and perpendicular lines and recognizing them in geometric shapes.  providing concrete examples of parallel and perpendicular lines.

drawing polygons given the number of sides or the number of vertices.  providing both regular and irregular examples of polygons. 
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.
Additional Resources
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.
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.
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
Ask your child to find examples of parallel and perpendicular lines at the playground, around the house, or at the grocery store.
Books about geometry to read with your child at home:
 Owen, C. Fiji Facts and Figures. (2005). IL: Weldon Owen Education, Inc.
 Owen, C. Perfect Patterns. (2005). IL: Weldon Owen Education, Inc.
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