5.2.2 Properties & Expressions
Apply the commutative, associative and distributive properties and order of operations to generate equivalent numerical expressions and to solve problems involving whole numbers.
For example: Purchase 5 pencils at 19 cents and 7 erasers at 19 cents. The numerical expression is 5 × 19 + 7 × 19 which is the same as (5 + 7) × 19.
Overview
Standard 5.2.2 Essential Understandings
Fifth graders do algebraic thinking well before formal courses in Algebra. The study of whole numbers and their operations is a perfect segue into algebraic thinking. Students show their understanding of equivalent expressions by expressing numbers in a variety of ways.
A multiplication problem can be rewritten in an expanded form using properties of arithmetic.
For example:
48 x 37 = (40 + 8) (37)
= (40 x 37) + (8 x 37)
= 40 x (30 + 7) + (8 x (30 + 7)
= (40 x 30) + (40 x 7) + (8 x 30) + (8 x 7)
Exactly the same process is used in algebra to multiply two binomials. (Thomas & Carpenter)
Fifth graders utilize the properties of arithmetic, which develops a deeper understanding of place value and the multiplication algorithm, and lays the foundation for future mathematics. They work with whole numbers and will transfer to solving and balancing equations with variables and verify why the properties work. Fifth graders' understanding of the base-ten number system is deepened as they come to understand its multiplicative structure. The focus on multiplicative reasoning in grades 3-5 provides foundational knowledge that can be built upon as fifth graders move to an emphasis on proportional reasoning in the middle grades.
Fifth graders develop an understanding of equivalence grades 3-5. Their ability to recognize, create, and use equivalent representations of numbers and geometric objects should expand. For example, 8 × 25 can be thought of as 8 × 5 × 5 or as 4 × 50; and three feet is the same as thirty-six inches, or one yard. They should extend their use of equivalent forms of numbers as they develop new strategies for computing and should recognize that different representations of numbers are helpful for different purposes.
Fifth graders should explore when and how shapes can be decomposed and reassembled and what features of the shapes remain unchanged. Equivalence also takes center stage as students study fractions and as they relate fractions, decimals, and percents. Examining equivalences provides a way to explore algebraic ideas, including properties such as commutativity and associativity.
All Standard Benchmarks
5.2.2.1 Apply the commutative, associative and distributive properties and order of operations to generate equivalent numerical expressions and to solve problems involving whole numbers.
Benchmark Group A
5.2.2.1 Apply the commutative, associative and distributive properties and order of operations to generate equivalent numerical expressions and to solve problems involving whole numbers.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Compose and decompose numbers.
- Use the commutative, distributive and order of operations to solve problems that are not in context.
- Apply the commutative, associative and distributive properties to generate equivalent expressions.
- Apply the order of operations to generate equivalent expressions.
- Generate multiple equivalent expressions for whole numbers.
- Recognize that the numerical expression 5 × 19 + 7 × 19 is the same as (5 + 7) × 19.
Work from previous grades that supports this new learning includes:
- Understand that the equal sign indicates balance and equality.
- Apply the properties of arithmetic: commutative, associative and distributive.
- Generalize patterns - a + 0 = a: identity property of addition.
- Understand that properties used with whole numbers can be simplified and proven with variables, which provides proof that these statements are true for all numbers.
- Verify these properties using the symbolic representation of the properties and generalize that these properties hold true for all numbers.
NCTM Standards
- Represent and analyze mathematical situations and structures using algebraic symbols. Grades 3-5 Expectations
- identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers;
- express mathematical relationships using equations.
Common Core State Standards
Write and interpret numerical expressions.
5.OA.1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
5.OA.2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "Add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
Misconceptions
Students may think...
- numbers on the right of the equation have the operation followed by the equals sign and the answer on the right For example, the answer comes next in a+b=c.
- if the answer is on the left followed by the equals sign and the operation is on the right, it is backwards and can't be solved (c = a+b).
- they can string a series of numbers and operation signs together with several equals signs (such as 8 + 4 = 12 + 5 = 17)
- there is no connection beween the arithmetic properties they already understand and the algebraic statements of the same properties (a+b = b+a is non-sensical).
- operationally and not relationally when they misunderstand the equals sign. For example, to determine if the number sentence 38 + 43 = 37 + 44 is true or false, students that are thinking relationally will perform the operations, one each side with no regard to the other side, until they have sums to compare. If they are thinking relationally they would look at the relationships between the numbers. Since 38 is one more than 37 and 43 is one less than 44, then this is a true sentence. Relational thinkers can balance equations without performing operations. Relational thinking is the foundation of algebra.
Vignette
Ms. Ardren begins her fifth grade math lesson by presenting the students with an open number sentence written on the whiteboard. The open number sentence is an equality sentence involving the distributive property. The students had been working on reviewing the commutative, associative, and distributive properties in a previous lesson. Ms. Ardren has the class work on the equality sentence together.
She reads the first equation aloud: "5 times 19 + 7 x 19 is the same as or equals the quantity blank plus blank times 19. What numbers do you think will go in the blanks to make both sides equal to each other? (5 × 19 + 7 × 19 = (__ + __) × 19)."
Ms. Ardren gave the students some silent think time and then asked a student to respond to her question
Student: "I think the numbers in the parentheses are the 5 and the 7."
Ms. Ardren asks the students for agreement with the response to raise their thumbs, disagreement thumbs down, and not sure thumbs sideways. Most thumbs went up, no thumbs were down, and a few put their thumbs sideways. Ms. Ardren then asked another student who had his thumb up why he agreed that 5 and 7 went into the parentheses. He responded that the 19 was already there so it had to be the 5 and the 7. Ms. Ardren asks the class to shake their heads yes if this response made sense to them. She asked if anyone could tell her more about why the 19 shouldn't be in the parentheses. She called on another student who had raised his hand to volunteer to say more about why the 19 wasn't in the parentheses.
Student: "The 19 can't be in the parentheses because look at the left side of the equation. It says 5 x 19 and 7 x 19. Now look at the right side of the equation. It says you add two numbers and then multiply the sum times 19. So the times 19 is already on both sides of the equals sign but the 5 and the 7 are missing on the right side. You could check to see if you are right by first multiplying 5 x 19 and 7x19 and then adding the two products together. Then add 5 + 7 and multiplying 12 x 19. You get the same answer 228 on both sides so 5 and 7 are the missing addends on the right side of the equation."
Ms. Ardren asks students if they have anything to add to this explanation. No student offers any further explanation and Ms. Ardren asks one of the students, who had indicated by a sideways thumb that they were unsure if 5 and 7 were the missing addends previously, to state the explanation already given in her own words, assessing student understanding and providing a summary of justification needed for the next part of the lesson.
Then Ms Ardren asks the students, "What number property is represented by this equation?" as she pointed to the example they had just worked on. The distributive property is the unanimous response. Ms. Ardren assigns four more number sentences which she writes on the whiteboard:
48 x 37 = ( ___ x 37) + ( ___ x 37)
34 + ____ + 17 = 28 + 17 + 34
4 x 13 = (10 x ___) + ( 4 x ___)
a + b = ___ + a
She asks the students to work independently and copy the equations, solve them, and identify and justify the number property they represent. The open number sentences are equality sentences involving the commutative, associative, and distributive properties.
The students begin copying the problems and working independently on the assignment. Ms. Ardren circulates the room while students are working. She is assessing understanding, noting misconceptions, and asking guiding questions when she sees students struggling. Then the teacher calls the group together and they go through the problems giving solutions, justifying and identifying the property represented. The students do very well solving the problems and identifying the properties and justifying their responses. The last open sentence draws some interesting comments and responses. When Ms. Ardren addresses the last number sentence, she said, "What is different about this equation?"
Student: "It doesn't have numbers, it has letters."
Ms. Ardren replied that the letters were called variables and variables can represent any number or value. She asks, "What variable belongs in the blank?"
Students identify that the blank should contain a "b" and the property represented is the commutative property. But when Ms. Ardren asks if there are any other comments, the following comments are expressed:
Student: "I learned about the commutative property a long time ago and I knew that you can change the order of the numbers in addition and multiplication and you get the same answer. But I never thought about using letters before."
Ms. Ardren asked if using letters was confusing. Then she asked, "What do the letters stand for?"
Student: "When you use a letter in math, it can stand for any number. When the letters are different, each letter stands for a different number. So in this equation, if a stands for a number and b stands for a different number, it means you can change the order you add the numbers and you will still get the same sum."
When Ms. Ardren asked if anyone wanted to add information, another student raised her hand.
Student: "I just thought of something; using letters to show how a property works is kind of a shortcut to show how a property works."
When Ms. Ardren asked her to tell them more about that, the student said, "Well if a letter means any number, then you don't have to keep trying different numbers to prove that the property works, you can just use a letter to prove it works."
Ms. Ardren wrote the following equation on the board,
20 + 12 + 30 = 20 + 30 + 12. She asked, "Is this a true sentence?"
Student: Yes, because both sides have a sum of 62.
Ms. Ardren asked what property was represented with this equation. The students responded the commutative property because the order of the addends was changed.
Ms. Ardren asked the class if they know the statement is true without adding and finding both sums. She continued by turning to the last student who had spoken and asked if she could replace the numbers with letters to represent the equation.
Student: a + b + c = a + c + b.
Ms. Ardren: Is this a true statement? We can't check in by adding the letters like some students did above. Is there another way to prove that each side has the same value?
Student: The letters are the same only in a different order. In the problem with numbers - the numbers are the same only in a different order. It's not necessary to add to check sums. You can look at both sides and see each number on both sides. Since it doesn't matter what order you add - the answer is the same. So adding is an extra step you don't have to do to show that both sides are the same.
The class then proceeded to supply letters for the other open sentences they had just solved.
Ms. Ardren summarized their work by telling them that properties of arithmetic such as commutative, associative, and distributive properties simplify solving equations. She poses the following problem written on chart paper: "Mark has $12. He spends $6 and then gets his allowance which is $5. How much does he have left?"
Before she sent the students back to their seats to work with a partner she posted the following questions on chart paper: "Does it matter if the problem is solved $12 - $6 + $5; $12 - ($6 + $5); or ($12 + $5) -$6? Why or why not? Do you get the same solution for each? What do you think is the correct solution?"
Ms. Ardren asks the students to work with a partner to solve the problems and discuss the questions. She continues to circulate the room noting students who are having difficulty and students who are using understanding of order of operations.
The teacher pulls the whole group together going through the problem and questions. The teacher focuses on students' comments that express some understanding of order of operations. The teacher discusses order of operation rules and their relationship with the commutative, associative, and distributive properties.
Complete the operations on parentheses first.
Perform multiplication and division from left to right
Perform addition and subtraction from left to right
Then the teacher gives students a set of 2 number sentences without parentheses and asks them to find the solutions:
5 + 6 x 7 (solution 47)
7 + 24 / 3 x 2 - 5 (solution 18)
She then adds the problem 6 x 8 x 7 and tells them there are many ways to multiply the numbers in this problem. The teacher asks them to find as many ways as they can think of to get the correct product. The students are to work with a partner. Ms. Ardren circulates the room observing understanding of order of operations and in the last problem observing students thinking that the factors can be composed and decomposed in numerous ways to get the correct solution, writing down examples to bring to the whole class.
Working with the whole class, Ms Ardren and students work through the problems discussing correct solutions and order of operations rules utilized and properties applied. For the last problem, the teacher takes a few examples from students who want to share and then says there were some interesting examples that the teacher saw and would like the whole class to discuss. This is where the teacher can bring up examples like 2 x 3 x 7 x 2 2 x 2 or
(6 x4) x (2 x 7) where students agree that numbers can be combined or decomposed to get the same product.
Ms. Ardren follows this up asking which properties make this possible and does order matter for the examples given? She then assigns some follow up problems centered on order of operations and combining and decomposing numbers to represent numerous ways to solve number sentences for homework.
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- Teachers need to consistently engage students in discussions and activities about the equals sign - appropriate use and meaning and solving equations- such as using True- False statements and open-ended statements.
- Teacher notation should be mathematically correct and not include 'run on' sentences such as stringing the equals sign when continuing to operate on a quantity; e.g., 2+5 = 7 + 9 = 16 (7 does not equal 16!).
- Teachers need to help develop students relational thinking vs.operational thinking; e.g., 35 + 47 = _ + 46. Students would solve relationally by adjusting numbers without using any computation. The missing number must be 36 since 46 is one less than 47, then 35 must be increased to account for the adjustment on the other side of the equals sign.
- Teachers need to incorporate the properties of arithmetic when teaching facts or standard algorithms.
- Students need help to generalize and generate conjectures; e.g., what happens when
odd + odd odd + even even + even etc.
- Teachers need to vary equations so that all are not in the form a + b = cm.
- 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)
Students decompose 2-digit numbers, model area representations using the distributive property and partial product arrays, and align paper-and-pencil calculations with the arrays. The lessons provide conceptual understanding of what occurs in a 2-digit multiplication problem. Partial product models serve as transitions to understanding the standard multiplication algorithm.
Use this tool to strengthen understanding and computation of numerical expressions and equality. In understanding equality, one of the first things students must realize is that equality is a relationship, not an operation. Many students view "=" as "find the answer." For these students, it is difficult to understand equations such as 11 = 4 + 7 or 3 × 5 = 17 - 2.
This interactive pan balance allows numeric or algebraic expressions to be entered and compared. You can "weigh" the expressions you want to compare by entering them on either side of the balance. Using this interactive tool, you can practice arithmetic and algebraic skills, and investigate the important concept of equivalence.
Students are giving 5 numbers and asked to use standard arithmetice operations to create a target number. The rules of Krypto are amazingly simple - combine five numbers using the standard arithmetic operations to create a target number. Finding a solution to one of the more than 3 million possible combinations can be quite a challenge, but students love it. And you'll love that the game helps to develop number sense, computational skill, and an understanding of the order of operations.
Additional Instructional Resources
Cuevas, G., & Yeatts, K. (2001). Navigating through algebra in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
Small, M. (2009). Good questions:Ggreat 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.
Wickett, M., Kharas, K., & Burns, M. (2002). Grades 3-5 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.
New Vocabulary
expression: a symbolic statement and can be either an arithmetic expression, such as 2 + 3, or an algebraic expression, such as 2x + 3. Expressions may be constants (e.g., 3, 4, --2), variables (e.g., m, x, w), operations (e.g., addition, subtraction), and grouping symbols (e.g., parentheses).
"Vocabulary literally is the key tool for thinking."
Ruby Payne
Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions. Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.
Learning vocabulary in the mathematics classroom is contingent upon the following:
Integration: Connecting new vocabulary to prior knowledge and previously learned vocabulary. The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.
Repetition: Using the word or concept many times during the learning process and connecting the word or concept with its meaning. The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.
Meaningful Multiple and varied opportunities to use the words in context. These
Use: opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems. Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.
Strategies for vocabulary development
Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.
Mathematics Word Bank: Each unit of study should have word banks visible during instruction. Words and corresponding definitions are added to the word bank as the need arises. Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.
Labeled pictures and charts: Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.
Frayer Model: The Frayer Model connects words, definitions, examples and 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 generating equivalent numerical expressions at the fifth grade level? How do student misconceptions interfere with mastery of these ideas?
What skills do fifth graders need in order to successfully generate equivalent numerical expressions?
What experiences do students need in order to generate equivalent numerical expressions?
What are the key ideas related to the successful use of the commutative, associative and distributive properties? What misconceptions do fifth graders have related to the commutative, associative, and distributive properties?
When checking for student understanding related to the use properties or equivalent expressions, what should teachers
- listen for in student conversations?
- look for in student work?
- ask during classroom discussions?
Examine student work related to generating equivalent numerical expressions. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.
How can teachers assess student learning related to these benchmarks?
How are these benchmarks related to other benchmarks at the fifth grade level?
Professional Learning Community Resources
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Chapin, S., and Johnson, A. (2006). Math matters: Understanding the math you teach, grades K-8. (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.
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.
Burns, Marilyn. (2007). About teaching mathematics: A k-8 resource (3rd ed.). Sausalito, CA: Math Solutions Publications.
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Caldera, C. (2005). Houghton Mifflin math and English language learners. Boston, MA: Houghton Mifflin Company.
Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades K-8. (2nd ed.). Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.
Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 2 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., ... & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centered mathematics grades K-3. Boston, MA: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Assessment
An expression is shown.
4 + 3 (6 +10) ÷ 2
What is the value of the expression?
A. 16
B. 26
C. 28
D. 56
Solution: C. 28
Benchmark 5.2.2.1
MCA III Item Sampler
Which value makes the equation 5b + 15 = 30 true?
A. b=3
B. b=9
C. b=10
D. b=75
Solution: A. b=3
Benchmark: 5.2.2.1
Which value makes the equation 15x + 11 = 56 true?
A. x = 5
B. x = 6
C. x = 4
D. x = 3
Solution: D. x = 3
Benchmark: 5.2.2.1
MCA III Item Sampler
What is the value of 4k+6(j-2) when k=3 and j=5?
A. 26
B. 30
C. 40
D. 108
Solution: B. 30
Benchmark: 5.2.2.1
MCA III Item Sampler
Differentiation
Students need to begin with visual representations of equalities/inequalities using drawings and manipulatives. Students describe what they see happening in the equality/inequality, and the teacher models their description using numerical representation including an equal sign, greater than sign, or a less than sign. Students use concrete models for a short time and then progress to representing equalities and inequalities with words, stating "equal, greater than, and less than" with justification and finally expressing the equality/inequality with numbers, symbols, and variables.
Student Think Alouds are an effective tool for students with math difficulties as they help students make meaning of their mathematics. The benefit of the Think Aloud slows down their thinking and makes them focus on the relationship between expressing the equality/inequality in words and the numerical representation.
Concrete - Representational - Abstract Instructional Approach
The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.
The CRA approach is based on three stages during the learning process:
Concrete - Representational - Abstract
The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts. At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level. Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task. Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.
The Representational Stage is the drawing stage. Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems. They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking. Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.
The Abstract Stage is the symbolic stage. Teachers model mathematical concepts using numbers and mathematical symbols. Operation symbols are used to represent addition, subtraction, multiplication and division. Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding. Moving to the abstract level too quickly causes many student errors. Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5. Sausalito, CA: Math Solutions.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
Students can draw pictures to help solidify the commutative, associative, and distributive properties. Associate the names of the properties with their root words such as "commute" (go back and forth), "associate" (to connect or joining together), or "distribute" (to give out in shares or apportion).
- 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:
Give an example of the Associative Property of Addition. |
Give an example of the Distributive Property. |
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.
Provide students challenging equalities and inequalities and have them prove or disprove the equations using the properties of arithmetic.
Students write their own equalities and equalities. They can then prove or disprove each others' equalities and inequalities.
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
Administrative/Peer Classroom Observation
Students are ... |
Teachers are ... |
engaged in the activity participating in a manner specified by the teacher (whole group, independently, or working with a partner. |
beginning the lesson with an activity or investigation building on prior knowledge, or as an introduction to the day's lesson such as coordinate graphing, properties of numbers, solving algebraic equations. Teachers are working with the whole group or circulating around the room if students are working independently or with a partner. |
participating in the discussion, responding when asked questions, or asking clarifiying questions. |
pulling group together to summarize activity or investigation, asking guided questions so students have a base of prior knowledge or common background to begin the day's lesson. |
fully engaged in the lesson. They exchange ideas and they ask clarifying questions |
setting the background for the lesson: an extension to prior investigation where students work with a partner or small group, a teacher led presentation where students are actively involved, a hands on lesson where manipulatives or models are used, or an activity such as a game |
sharing their thinking and understanding, discussing and asking questions, and completing any assigned follow up work. |
providing closure to the lesson through some type of summary. This may be teacher led with teacher asking questions to assess student understanding, group reports from group activities, a final problem worked together, or assigned follow up work. |
What should I look for in the mathematics classroom?
(Adapted from SciMathMN, 1997)
What are students doing?
- Working in groups to make conjectures and solve problems.
- Solving real-world problems, not just practicing a collection of isolated skills.
- Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
- Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
- Recognizing and connecting mathematical ideas.
- Justifying their thinking and explaining different ways to solve a problem.
What are teachers doing?
- Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
- Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
- Connecting new mathematical concepts to previously learned ideas.
- Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
- Selecting appropriate activities and materials to support the learning of every student.
- Working with other teachers to make connections between disciplines to show how math is related to other subjects.
- Using assessments to uncover student thinking in order to guide instruction and assess understanding.
Additional Resources
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k-8. (2nd ed.). Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
For Administrators
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
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.