5.3.2 Area, Surface Area & Volume

Grade: 
5
Subject:
Math
Strand:
Geometry & Measurement
Standard 5.3.2

Determine the area of triangles and quadrilaterals; determine the surface area and volume of rectangular prisms in various contexts.

Benchmark: 5.3.2.1 Area

Develop and use formulas to determine the area of triangles, parallelograms and figures that can be decomposed into triangles.

Benchmark: 5.3.2.2 Strategies for Volume & Surface Area

Use various tools and strategies to measure the volume and surface area of objects that are shaped like rectangular prisms.

For example: Use a net or decompose the surface into rectangles.

Another example: Measure the volume of a cereal box by using a ruler to measure its height, width and length, or by filling it with cereal and then emptying the cereal into containers of known volume.

Benchmark: 5.3.2.3 Cubes & Volume

Understand that the volume of a three-dimensional figure can be found by counting the total number of same-sized cubic units that fill a shape without gaps or overlaps. Use cubic units to label volume measurements.

For example: Use cubes to find the volume of a small box.

Benchmark: 5.3.2.4 Formulas for Volume of Prism

Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into layers of unit cubes.

Overview

Big Ideas and Essential Understandings 
Standard 5.3.2 Essential Understandings

Measurement at the Fifth Grade level involves the area of triangles, the surface area of rectangular prisms, and the volume of rectangular prisms. Students recognize that the area of a two-dimensional shape can be found by decomposing the shape into other shapes whose area can be more easily determined and then finding the sum of the areas of those parts. This understanding is used to develop a formula for finding the area of a triangle as well as trapezoids, parallelograms, rhombuses, and other figures that can be decomposed into triangles and rectangles.

Fifth graders use their knowledge of nets for three-dimensional figures to find surface area. They move from inefficient strategies of finding the area of each individual face of a prism to looking for congruent faces to help determine a formula for the surface of a rectangular prism. They measure necessary attributes of three-dimensional shapes and use surface area formulas to solve problems.

Fifth graders recognize volume as an attribute of a three-dimensional figure. They understand that volume can be found by counting the total number of same-sized units needed to fill the space without gaps or overlaps. They understand that the standard unit for measuring volume is a cube. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They develop formulas for finding the volume of a rectangular prism by using an understanding of how multiplication can be used to count the number of cubes needed to fill a rectangular prism. As students develop the formula for the volume of a rectangular prism, they need to understand how the area of the base, the number of layers of cubes needed to fill the prism, and multiplication can be used together to count the total number of cubes needed to fill the prism.

All Standard Benchmarks

5.3.2.1
Develop and use formulas to determine the area of triangles, parallelograms and figures that can be decomposed into triangles.
5.3.2.2
Use various tools and strategies to measure the volume and surface area of objects that are shaped like rectangular prisms. For example: Use a net or decompose the surface into rectangles. Another example: Measure the volume of a cereal box by using a ruler to measure its height, width and length, or by filling it with cereal and then emptying the cereal into containers of known volume.
5.3.2.3
Understand that the volume of a three-dimensional figure can be found by counting the total number of same-sized cubic units that fill a shape without gaps or overlaps. Use cubic units to label volume measurements. For example: Use cubes to find the volume of a small box.
5.2.3.4
Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify why base area B and height h are multiplied to find the        volume of a rectangular prism by breaking the prism into layers of unit cubes.

Benchmark Cluster 

5.3.2.1
Develop and use formulas to determine the area of triangles, parallelograms and figures that can be decomposed into triangles.
5.3.2.2
Use various tools and strategies to measure the volume and surface area of objects that are shaped like rectangular prisms. For example: Use a net or decompose the surface into rectangles. Another example: Measure the volume of a cereal box by using a ruler to measure its height, width and length, or by filling it with cereal and then emptying the cereal into containers of known volume.
5.3.2.3
Understand that the volume of a three-dimensional figure can be found by counting the total number of same-sized cubic units that fill a shape without gaps or overlaps. Use cubic units to label volume measurements. For example: Use cubes to find the volume of a small box.
5.2.3.4
Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify why base area B and height h are multiplied to find the        volume of a rectangular prism by breaking the prism into layers of unit cubes.

What students should know and be able to do [at a mastery level] related to these benchmarks   
  • Understand, develop and use formulas to determine the area of triangles, trapezoids, parallelograms and other figures that can be decomposed into triangles.
  • Find the area of any polygon by breaking the polygon into triangles or rectangles. Find the area of each triangle or rectangle, and then add the areas together to find the area of the polygon.
  • Measure the volume of rectangular prisms.
  • Measure the surface area of rectangular prisms - strategy may include decomposing the prism into its net.
  • Find the volume of rectangular prisms by finding the area of the base and then counting the number of layers of cubes needed to fill the prism. Use multiplication to find the number of cubes needed to fill the prism.
  • Understand, develop and use a formula to determine volume.
Work from previous grades that supports this new learning includes:
  • Understand that area of two-dimensional figures can be found by counting the total number of same size square units that cover a shape without gaps or overlaps
  • Justify why length and width are multiplied to find the area of a rectangle by breaking the rectangle into one unit by one unit squares and viewing these as grouped into rows and columns
  • Find the areas of geometric figures and real-world objects that can be divided into rectangular shapes.
  • Use square units to label measurements.
  • Recognize and draw a net for a three-dimensional figure (current grade level).
Correlations 

NCTM Standards 

Understand measurable attributes of objects and the units, systems, and processes of measurement

Grades 3-5 Expectations

  • understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute;
  • understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems
  • carry out simple unit conversions, such as from centimeters to meters, within a system of measurement;
  • understand that measurements are approximations and how differences in units affect precision;
  • explore what happens to measurements of a two-dimensional shape such as its perimeter and area when the shape is changed in some way.
Apply appropriate techniques, tools, and formulas to determine measurements

Grades 3-5 Expectations

  • develop strategies for estimating the perimeters, areas, and volumes of irregular shapes;
  • select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles;
  • select and use benchmarks to estimate measurements;
  • develop, understand, and use formulas to find the area of rectangles and related triangles and parallelograms;
  • develop strategies to determine the surface areas and volumes of rectangular solids.

Common Core State Standards

Convert like measurement units within a given measurement system.

5.MD.1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition.

5.MD.3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

  • 5.MD.3a. A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.
  • 5.MD.3b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

5.MD.4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

5.MD.5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

  • 5.MD.5a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
  • 5.MD.5b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
  • 5.MD.5c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Misconceptions

Student Misconceptions 
Students may think . . .
  • units of measurement for perimeter, area and volume are interchangeable.
  • area and perimeter are the same or confuse one for the other.
  • the only way to find area of a given shape is length x width.
  • volume can be found by determining the area of a single face on a prism and multiply that by the total number of faces.
  • area and volume are the result of plugging numbers into a formula. They do not understand the concepts of area and volume.

Resources

Instructional Notes 
Teacher Notes
  • Students may need support in further development of previously studied concepts and skills.
  • "Most researchers agree there are four components of measuring:

1.  conservation (objects maintain their same size and shape when measured),
2.  transitivity (two objects can be compared in terms of a measurable quality, using another object),
3.  units (the type of units used to measure an object depends on the attribute being measured), and
4.  unit iteration (the units must be repeated, or iterated, in order to determine the measure of an object" (Chapin & Johnson, 2006).

  • To measure an attribute of an object with understanding, students should complete three steps:

1.       Decide on the attribute (length) to be measured.
2.       Select an appropriate unit for measuring that attribute (length). 
3.       Count the number of units needed to accurately represent the attribute (length) being measured.

  • Developing a formula does not mean inserting numbers and calculating an answer. Finding the area of a triangle needs to be linked to what students know about finding the area of a rectangle. This will lead to developing the formula for finding the area of a triangle which can then be used when finding the area of a parallelogram, trapezoid, rhombus and other figures.
  • Understanding the formula for finding the volume of a rectangular prism as V = Bh (as well as V = lwh) will enable students as they develop formulas for finding the volume of any prism or cylinder in sixth grade.
  • Students need to visualize concepts and relationships when finding surface area and volume. This is best done using some sort of manipulative or model. Ask students to solve problems set in familiar, meaningful situations and to examine the reasonableness of their results.
  • Teachers need to model problems using manipulatives and/or a sketch of the problem situation to help students make sense of the measurement they are asked to find as well as an appropriate unit of measurement.
  • Students need to understand the difference between units used when finding length, area and volume.
  • Teachers need to listen to student thinking as they are decomposing prisms when finding  surface area in order to address any misconceptions.
Questioning

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

Cubes - Exploration with volume in rectangular prisms

Fill a box with cubes, rows of cubes, or layers of cubes. The number of unit cubes needed to fill the entire box is known as the volume of the box. Can you determine a rule for finding the volume of a box if you know its width, depth, and height?  This exploration helps students develop the concept of volume for rectangular prisms.

Dynamic Paper teachers can create nets for polygons, graph paper, number lines, number grids, tessellations, and spinners. Need a pentagonal pyramid that's six inches tall? Or a number line that goes from ‑18 to 32 by 5's? Or a set of pattern blocks where all shapes have one-inch sides? You can create all those things and more with the Dynamic Paper tool. Place the images you want, then export it as a PDF activity sheet for your students or as a JPEG image for use in other applications or on the web.

Cube Nets is an interactive activity where students need to identify the nets for cubes. A net is a two-dimensional figure that can be folded into a three-dimensional object.

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 

rectangular prism - a solid figure that has only rectangles for faces. Rectangular prisms are sometimes referred to as "boxes."

surface area - the sum of the areas of all faces of a three dimensional figure.

volume - the amount of space a three-dimensional figure takes up, measured in cubic units. The amount a three-dimensional object will hold or the number of units needed to fill the object with no gaps.

"Vocabulary literally is the

 key tool for thinking."

Ruby Payne

Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions.  Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.

Learning vocabulary in the mathematics classroom is contingent upon the following:

Integration:   Connecting new vocabulary to prior knowledge and previously learned vocabulary.  The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.

Repetition:    Using the word or concept many times during the learning process and connecting the word or concept with its meaning.  The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.

Meaningful Use:  Multiple and varied opportunities to use the words in context.  These 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.

Frayer Model

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.

example /non -example chart

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 developing and then using formulas to determine the area of triangles, parallelograms and other figures that can be decomposed into triangles at the fifth grade level?  How do student misconceptions interfere with mastery of these ideas?

What experiences do fifth graders need in order to develop the formula for finding the area of a triangle, parallelogram or a figure that can be decomposed into triangles?

When checking for student understanding related to finding the area of triangles, parallelograms and figures that can be decomposed into triangles, what should teachers

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

Examine student work related to finding the area of triangles, parallelograms and figures that can be decomposed into triangles. 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 tools and strategies might fifth graders use to measure the volume and surface area of objects that are shaped like rectangular prisms?

What are the key ideas related to developing and using the formulas V = ℓwh and V = Bh to determine the volume of a rectangular prism? What misconceptions do fifth graders have related to the volume of rectangular prisms?

What experiences do students need in order to successfully develop the formulas V = ℓwh and V = Bh?

Examine student work related to finding the volume of a rectangular prism. 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., & 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.

Fuson, K., SanGiovanni, J., Lott Adams, T., & Beckmann. S. (2009). Focus in grade 5, 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.

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.

References 

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.

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.

Reeves, D. (2007). Ahead of the curve: The power of assessment to transform teaching and learning. Indiana: Solution Tree Press.

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

  • A triangle has a height of 25 feet. The length of its base is 12 feet.

What is the area of the triangle?

Solution:  150 sq. ft
Benchmark 5.3.2.2 
MCA III Item Sampler

  • A trapezoid is shown.

trapezoid

What is the area of the trapezoid?
A. 48 sq. cm
B. 138 sq. cm
C. 168 sq. cm
D. 173 sq. cm

Solution: B. 138 sq.cm.
Benchmark 5.3.2.1 
MCA III Item Sampler

  • Amy covers the box shown with paper.


Amy's Box

What is the surface area of the box?
A. 180 sq. in.
B. 296 sq. in.
C. 592 sq. in.
D. 960 sq. in.

Solution: C. 592 sq.
Benchmark 5.3.2.2
MCA III Item Sampler

  • A rectangular prism has a height of h cm. The area of its base is B cm 2.

How much does the volume of the prism increase when the height is
increased by 1 cm?

A. 1 cu. cm
B. h + 1 cu. cm
C. B cu. cm
D. B + 1 cu. cm

Solution: C. B cu.cm.
Benchmark 5.3.2.4
MCA III Item Sampler

Differentiation

Struggling Learners 

Students need more work with nets of rectangular prisms. They need many opportunities to cut boxes apart into the rectangular faces and find the area of each face.

Students should fill transparent containers with 1 cm cubes to find the volume. NOTE: Emphasis needs to be placed on students understanding that the key to finding the volume is determining a base layer and the number of layers needed to fill the space. Multiplication can be used to count the units.

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 
  • Students may need more work with nets of rectangular prisms. The language of surface area needs to be developed as students find the area of each section of the net.
  • Students should fill transparent containers with 1 cm cubes to illustrate the concept of volume. When finding the volume, the emphasis needs to be placed on students understanding that the key to finding the volume is determining a base layer and the number of layers needed to fill the space. Multiplication can be used to count the units.
  • Word banks need to be part of the student learning environment in every mathematics unit of study. 
  • Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.

Frayer Model

  • 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:

The surface area is ________________.  That means _______________________.

The surface area is ___________________.  I found it by _______________________.

The volume is ________________.  That means _______________________.

The volume is__________________.  I found it by ________________________ .

  • 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 might investigate finding the areas of various triangles and look for patterns. Next they can investigate finding areas of quadrilaterals. After these investigations students can sort and compare and contrast their methods for triangles and quadrilaterals. Then the students could hypothesize and test how they can use this information to find the area of any regular polygon.  The goal is to generalize how knowing how to find areas of triangles can be used to decompose any shape into triangles as one way to find the area of any given polygon.

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 ...

developing formulas for finding the area of a triangle, parallelogram, trapezoid and figures that can be decomposed into triangles.

modeling the decomposition of two-dimensional shapes into triangles and using this information to help students develop a formula for finding area.

finding the surface area of rectangular prisms by decomposing them into their nets.

modeling the decomposition of rectangular prisms into nets and asking students to explain their thinking as they decompose a prism into a net.

finding the volume of rectangular prisms by counting the number of layers of cubes and the number of cubes in each layer and using multiplication to find the total number of cubes.

checking for student understanding of volume as the number of cubes needed to fill the rectangular prism.

developing a formula for finding the volume of a rectangular prism.

checking for student understanding that multiplication can be used to help count the number of cubes needed to describe the volume of a rectangular prism. 

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 
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