4.3.3 Geometric Transformations
Use translations, reflections and rotations to establish congruency and understand symmetrics.
Apply translations (slides) to figures.
Apply reflections (flips) to figures by reflecting over vertical or horizontal lines and relate reflections to lines of symmetry.
Apply rotations (turns) of 90˚ clockwise or counterclockwise.
Recognize that translations, reflections and rotations preserve congruency and use them to show that two figures are congruent.
Overview
Standard 4.3.3 Essential Understandings/Ideas
Fourth graders build on their informal understanding of slides, flips and turns as they construct definitions of the terms translation, reflection and rotation. They will use these mathematical terms and their applications to show that two shapes are congruent.
Fourth graders find lines of symmetry and relate reflections to lines of symmetry.
All Standard Benchmarks
4.3.3.1 Apply translations (slides) to figures.
4.3.3.2 Apply reflections (flips) to figures by reflecting over vertical or horizontal lines and relate reflections to lines of symmetry.
4.3.3.3 Apply rotations (turns) of 90˚ clockwise or counterclockwise.
4.3.3.4 Recognize that translations, reflections and rotations preserve congruency and use them to show that two figures are congruent.
Benchmark Group A
4.3.3.1 Apply translations (slides) to figures.
4.3.3.2 Apply reflections (flips) to figures by reflecting over vertical or horizontal lines and relate reflections to lines of symmetry.
4.3.3.3 Apply rotations (turns) of 90˚ clockwise or counterclockwise.
4.3.3.4 Recognize that translations, reflections and rotations preserve congruency and use them to show that two figures are congruent.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Draw a translation (slide), reflection (flip), and 90˚ rotation (turn) of a given figure.
- Identify how two figures are related through a translation, reflection, or 90˚ rotation (clockwise or counterclockwise).
- Identify lines of symmetry.
- Draw figures that have a line of symmetry.
- Show that two shapes are congruent by using translation, reflection or rotation.
Work from previous grades that supports this new learning includes:
- An informal understanding of what it means to slide, flip or turn an object.
- An informal understanding of symmetry.
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.
Apply transformations and use symmetry to analyze mathematical situations
Grades 3-5 Expectations
- predict and describe the results of sliding, flipping, and turning two-dimensional shapes;
- describe a motion or a series of motions that will show that two shapes are congruent;
- identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs.
Common Core State Standards
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
4.G.3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
Misconceptions
Students may think...
- the terms translation, reflection and rotation are interchangeable.
- clockwise and counterclockwise are the same or interchangeable terms.
- the diagonal of any rectangle is a line of symmetry.
- a figure can have only one line of symmetry.
- a line of symmetry can only be a vertical line.
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- Use a variety of manipulatives such as pattern blocks, interlocking cubes, cut out alphabet letters, or other shape cut outs (hearts, rectangles) to have students practice:
- rotating shapes both clockwise and counterclockwise;
- reflecting shapes over horizontal, vertical and other lines;
- showing congruency of two shapes by rotating, reflecting or translating the first shape to show how it exactly matches the second shape.
- Students need hands-on experiences while connecting the vocabulary of transformations to the "actions" of sliding (translation), flipping (reflection) and turning (rotation).
- Connect the idea of the line of symmetry and a reflection line by having students explain how they are alike.
- Students should predict the results of a given translation, reflection or rotation.
- 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)
NCTM Illuminations/Figure This/Other Resources
- Lesson Name: Archimedes' Puzzle
Lesson Objective: The Stomachion is an ancient tangram-type puzzle. Believed by some to have been created by Archimedes, it consists of 14 pieces cut from a square. The pieces can be rearranged to form other interesting shapes. In this lesson, students learn about the history of the Stomachion, use the pieces to create other figures, learn about symmetry and transformations, and investigate the areas of the pieces.
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
"Vocabulary literally is the key tool for thinking."
Ruby Payne
clockwise: moving in the same direction as the hands on an analog clock, rotating to the right around the face of the clock.
counter clockwise: moving in the opposite direction as the hands on an analog clock, rotating to the left around the face of the clock.
congruent: two plane or solid figures are congruent if one can be obtained from the other using rotations, reflections, and translations.
line of reflection: the line that an image is reflected over.
line of symmetry: the line that divides a figure so that each half is a reflection of the other.
reflection: a flip of a figure over a line.
rotation: a turn around a specified point.
translation: a slide, often described with a specific direction and distance.
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.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks.
What are the key ideas related to translations, reflections, rotations and symmetry at the fourth grade level? How do student misconceptions interfere with mastery of these ideas?
What type of transformations (translations, reflections, and rotations) are readily understood and described by students?
What type of transformations (translations, reflections, and rotations) are the most difficult for students to understand and describe?
How would you know a student understands translations? reflections? rotations? symmetry?
What are some examples of transformation (translations, reflections and rotations) or symmetry activities that could be used in an instructional setting with fourth graders?
When checking for student understanding, what should teachers
- listen for in student conversations?
- look for in student work?
- ask during classroom discussions?
Examine student work related to a translations, reflections, rotations or symmetry. What evidence do you need to say a student is proficient? Using three pieces of 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 fourth grade level?
Materials - suggested articles and books for book study with PLC:
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.
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.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
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. (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 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.
Wickett, M., Kharas, K., & Burns, M. (2002). Grades 3-5 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications
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.
Wickett, M., Kharas, K., & Burns, M. (2002). Grades 3-5 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.
Assessment
- Which shows a 90˚ counterclockwise rotation of the figure?
↑
A. ↑ B.←
C.→ D.↓
Solution: B. ←
Benchmark: 4.3.3.3 and 4.3.3.4
MCA III
- A shape is shown.
Which shows a translation of the shape?
A. B.
C. D.
Solution: B.
Benchmark: 4.3.3.1 and 4.3.3.4
MCA III
- Which figure shows a line of symmetry?
Solution: B.
Benchmark: 4.3.3.2
- Ron draws a trapezoid, then rotates it 90 degrees.
Which statement is true about the two trapezoids?
A. They are congruent because all trapezoids are congruent.
B They are congruent because rotating a trapezoid does not change its size and shape.
C They are not congruent because rotating the trapezoid changes its side lengths.
D. They are not congruent because rotating the trapezoid changes its angle measures.
Solution: B.
Benchmark: 4.3.3.3 and 4.3.3.4
MCA III Item Sampler
Differentiation
Students who struggle with these benchmarks typically are confusing the terms with the action. They know how to turn a shape, but "clockwise and counterclockwise rotation" might throw them off.
Slides, Flips, and Turns (adapted from 1997 MN Mathematics Framework)
The purpose of this activity is to have children illustrate slides, flips, and turns as they move them- selves and objects for understanding the properties of geometric figures. For this activity you may first have the children themselves feel the action of sliding, flipping, and turning. Then have them use stuffed animals or 3 x 5 cards to demonstrate moves. If using cards, they can draw a picture on one side showing themselves lying face up and on the other side a picture lying face down. Have them repeat the drawings on a second card so each child has two cards.
- Slides
Have the children lie on the floor and ask them to give their personal interpretation of a slide.
How would you show a slide?
Some may slide forward, others backward, and still others sideways. Discuss which slides are easiest.
- Flips
(Mathematical note: What is called a flip in this informal way is really a 180° rotation in space in geometrical terms. A geometric flip, or reflection, is done on a plane figure and may later be modeled using a mirror as the line of reflection or shapes on tracing paper folded along a line, traced, and unfolded. For now, the everyday usage of "flip" gives a sense of the type of movement.)
Have the children lie on the floor and give their interpretation of a flip. You might suggest they flip about their left sides, their right sides, and about their feet Are some motions easier to do than others? If your head is pointing at me to start, where is it pointing after a flip? (For a right or left flip, the head will be pointing the same way; for a head or feet flip, the head will be pointing in the opposite direction.)
Is a somersault a flip?
- Turns
Have the children lie on the floor and demonstrate a turn. The amount of the turn is arbitrary at this point.
How would you show a turn? If moving "back to stomach" is a flip and not a turn, what does a turn look like? Are your bodies pointing in the same direction before and after a turn? (No, the direction is different for all turns except a complete turn.)
- Talk About It
Ask the children to sit in a large circle and demonstrate the different possibilities for a slide, a flip, and a turn, using their cards or stuffed animals.
Discuss how all slides are alike.
(You point in the same direction; you stay on your back or stomach.)
Discuss how all flips are alike.
(You move from stomach to back or from back to stomach but may not always point in the same direction.)
Discuss how all turns are alike.
(You stay either on your back or on your stomach. You usually point in a different direction.)
Ask the children to describe the motion that would take them from a start (initial position) to a finish (final position). Use the cards to illustrate many situations that require one move.
Repeat, using situations that require two moves or three moves.
Discuss various questions such as: If you start on your back, would you be on your back or on your stomach after two flips? (On the back) How would you be lying after three flips? (On the stomach) How would you be lying after a slide, a flip, and a slide? (On the stomach)
- About this Activity
Because young children are constantly receiving information through their senses, their spatial capabilities usually exceed their numerical skills. Building informally upon the geometry observed in the world around them should lead to experiences investigating and discussing geometric concepts in different contexts and exploring the child's ideas for validity.
- Where do we go from here?
Repetition, discussion, and variations of successful experiences are important to young children. Explore various combinations of moves and have students make up and follow each other's instructions combining moves. Have them try to predict outcomes of combinations.
Continue to explore slides, flips, and turns with a variety of geometric solids.
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.
English Language Learners may confuse the terms with the action they represent. They know how to turn a shape, but "clockwise and counterclockwise rotation" might confuse them. Connecting mathematical language to the actions of sliding, flipping and turning will help students as they are asked to communicate their understanding.
- 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:
A translation is ___________________________________________________________. |
A reflection is ____________________________________________________________. |
A rotation is _____________________________________________________________. |
This is a reflection because ________________________________________________. |
This is a translation because ________________________________________________. |
This is a rotation 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.
Students should explore the results of translations, reflections and rotations when using free form shapes and irregular shapes.
Students should be asked to show the "starting position" when given the ending position and the actions that were used.
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
Students are... | Teachers are... |
identifying and practicing clockwise and counter clockwise rotations, reflections across both horizontal and vertical lines and translations of figures. | clarifying the difference between clockwise and counterclockwise rotations and reflections across vertical and horizontal lines. Encouraging students to use appropriate mathematical vocabulary when describing transformations. |
applying reflections, rotations, and translations to show figures are congruent. | asking students to show two shapes are congruent. |
identifying how a shape has been moved (reflected, rotated, and translated) from its original position to its current position by moving the shape and without moving the shape. | providing opportunities to observe and visualize how a shape changes position when it is rotated, reflected or translated. |
identifying lines of symmetry. | providing opportunities for the exploration of figures that have a line or lines of symmetry. Helping students identify similarities between a line of reflection and a line of symmetry. |
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.
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 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.