# 4.1.1A Mult/Div Facts

4
Subject:
Math
Strand:
Number & Operation
Standard 4.1.1

Demonstrate mastery of multiplication and division basic facts; multiply multi-digit numbers; solve real-world and mathematical problems using arithmetic.

Benchmark: 4.1.1.1 Multiplication & Division Facts

Demonstrate fluency with multiplication and division facts.

## Overview

Big Ideas and Essential Understandings

Fourth graders continue work with multiplication and division by demonstrating mastery of basic multiplication and division facts.  They use basic multiplication facts to support work with the multiplication of multi-digit numbers.  In addition, place value understanding supports multiplication by 10, 100 and 1000. Fourth graders solve multi-step problems involving multiplication, addition and subtraction.  They divide multi-digit numbers by one- and two-digit numbers in problem solving situations. They solve these division problems using a variety of strategies including

• repeated subtraction
• distributive, associative and commutative properties
• place value understanding
• mental strategies
• partial quotients

Fourth graders use rounding, benchmark numbers and place value to estimate and evaluate the reasonableness of results.

All Standard Benchmarks

4.1.1.1    Demonstrate fluency with multiplication and division facts.
4.1.1.2    Use an understanding of place value to multiply a number by 10, 100, and 1,000
4.1.1.3    Multiply multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms.
4.1.1.4    Estimate products and quotients of multi-digit whole numbers by using rounding, benchmarks and place value to assess the reasonableness of results. For example: 53 × 38 is between 50 × 30 and 60 × 40, or between 1500 and 2400, and 411/73 is between 5 and 6.
4.1.1.5    Solve multi-step real-world and mathematical problems requiring the use of addition, subtraction and multiplication of multi-digit whole numbers. Use various strategies, including the relationship between operations, the use of technology, and the context of the problem to assess the reasonableness of results.
4.1.1.6    Use strategies and algorithms based on knowledge of place value, equality and properties of operations to divide multi-digit whole numbers by one- or two-digit numbers.

Strategies may include mental strategies, partial quotients, the commutative, associative, and distributive properties and repeated subtraction.  For example: A group of 324 students is going to a museum in 6 buses. If each bus has the same number of students, how many students will be on each bus?

Benchmark Cluster

4.1.1.1    Demonstrate fluency with multiplication and division facts.

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

• readily derive the multiplication and division facts (factors including but not limited to 1-9).

Work from previous grades that supports this new learning includes:

• experiences with array models, skip counting, and relationships among facts (fact families)
• understanding the concepts of multiplication and division
• understand the relationship between multiplication and division
• relate multiplication to repeated addition
• relate division to repeated subtraction
Correlations

NCTM Standards

Understand numbers, ways of representing numbers, relationships among numbers, and number systems

• understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals;
• recognize equivalent representations for the same number and generate them by decomposing and composing numbers;
• develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers;
• use models, benchmarks, and equivalent forms to judge the size of fractions;
• recognize and generate equivalent forms of commonly used fractions, decimals, and percents;
• explore numbers less than 0 by extending the number line and through familiar applications;
• describe classes of numbers according to characteristics such as the nature of their factors.
##### Understand meanings of operations and how they relate to one another

• understand various meanings of multiplication and division;
• understand the effects of multiplying and dividing whole numbers;
• identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems;
• understand and use properties of operations, such as the distributivity of multiplication over addition.
##### Compute fluently and make reasonable estimates

• develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30 x 50;
• develop fluency in adding, subtracting, multiplying, and dividing whole numbers;
• develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results;
• develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students' experience;
• use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals;
• select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tool.

Common Core State Standards

• Use the four operations with whole numbers to solve problems.

4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

4.OA.3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

• Gain familiarity with factors and multiples.

4.OA.4. Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

• Generalize place value understanding for multi-digit whole numbers.

4.NBT.1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

4.NBT.2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4.NBT.3. Use place value understanding to round multi-digit whole numbers to any place.

• Use place value understanding and properties of operations to perform multi-digit arithmetic.

4.NBT.4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.

4.NBT.5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

4.NBT.6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

## Misconceptions

Student Misconceptions

Student Misconceptions and Common Errors

Students may think....

• multiplication functions the same way as addition. For example, multiplying by 0 means nothing changes and multiplying by 1 means getting 1 more.
• 6 x 7 and 7 x 6 are two separate facts and need to be learned independently. They are unable to apply the commutative property of multiplication when learning basic facts.
• they do not "know" the facts if they are not fast. In reality, students know their facts when they are able to derive an unknown fact from known facts. For example, 6 x 8 is the same as 2 x (3 x 8) or (3 x 8) + (3 x 8).

## Vignette

In the Classroom

In this vignette, students are using rectangular array models, associative and distributive properties and known facts  to derive an unknown fact.  Students previously made their own array models using grid paper.  They recorded the number of rows and columns as a multiplication expression on the front of each model and the product on the back.

Both students selected the fact 6 x 7 as one they needed to work on.

Juan and Jessie were working together to figure out the answer to 6 x 7.

Juan A:  I have 6 x 7 and the array is almost a square.

Jessie:  But it isn't because it has to have the same number on both sides (rows and columns)

Juan: I know, I said almost -that might be helpful.

Jessie: Let's find some smaller arrays to cover it.

Juan and Jessie begin to search through the array models as the teacher approaches.

Mr. C: I see you have 6 x 7 as your challenging fact, what array are you looking for?

Jessie: Something that will cover part of it.

Juan: I am looking for something 7 long but smaller, maybe that would help us.

Jessie: But maybe not because I only know my 2s, 3s, and 5s really fast.

Juan: I know 2s, 3s, and 4s- because they are double 2s, 5s and 10s.

Jessie: Oh- I know my 10s too.

Mr. C: It sounds like you are on the right track- using what you know to discover what you need to learn.  Which rectangle are you placing on top now?

Juan:   I found the 5 x 6 which I know is 30---5s are easy for me because if I forget I can count by 5s really fast.

Jessie: Look there is a really small rectangle still uncovered- it's a 2 by 6!

Knowing Juan and Jessie are on the right track, the teacher moves to another group of students.

Jessie: Two times six is twelve- so...(Juan interrupts)

Juan: 30 and 12 is 42!  6 times 7 is 42- oh yeah!

Jessie:  Let's record it on our sheet.

Students write:

5 x 6 = 30

2 x 6 = 12

30 + 12 = 42

7 x 6 = 42

Jessie: I think we should look for another solution- if we don't he (the teacher) will ask us to anyway.

Juan: Okay, let's try to cover half of it. I really like doing doubles.

Jessie: I don't get that. How is a double going to help us with this?

Juan: If we find something that covers half of it..... it is going to be a pretty small array.  We just have to find the answer to it and then double it..... like times two-ing it.

Jessie: Alright, let's look for an array that covers half.

At this point, Juan and Jessie look for an array that covers half of the rectangle so they can double that amount.

Juan:  Got it!   I found  3 x 7  and it covers half.

Jessie:  So we need another  3 x 7 to cover the whole thing. I get it...3 x 7 is 21 and we need to double it.

Juan:  I can write this one.

Student writes:

3 x 7 = 21
21 + 21 = 42
7 x 6 = 42

Note

While the words distributive property and associative property were not used during this activity, students were applying these properties as they worked to derive an unknown basic fact.  In the first example,  Juan and Jessie used the distributive property of multiplication over addition.

Using the Distributive Property of Multiplication over Addition,

6 x 7 = 6 x (5 + 2) = (6 x 5) + (6 x 2) = 30 + 12 = 42

The second example could also be represented by using the distributive property of multiplication over addition, but in this case, the students described it using the associative property of multiplication.

Using the Associative Property of Multiplication

6 x 7 = (2 x 3) x 7 = 2 x (3 x 7) = 2 x 21 = 42

What other known facts could these students have used to derive

6 x 7?  What properties would students be using with these other facts?

## Resources

Instructional Notes
• Students may need support in further development of previously studied concepts and skills.
• Timed tests do not support the learning of basic facts. Timed tests may help students achieve faster recall of facts they already know but do not help students learn unknown facts.
• Students need strategies for deriving multiplication facts. Many of these strategies use the associative and distributive properties. Although facts do not need to be learned in any particular order, here are some strategies students use when deriving facts.

Relate multiplying by 2 to addition doubles.

Relate multiplying by 3 to doubling and adding one more set.
Ex.  3 x 8 is the same as (2 x 8) + 8.

Relate multiplying by 4 to doubling and doubling again.

Relate multiplying by 5 to skip counting by five.

Relate multiplying by 6 to multiplying by 3 and doubling.
Ex.  6 x 9  is the same as (3 x 9) x 2.

Relate multiplying by 7 to using known facts.
Ex.  7 x 8  is the same as (5 x 8) + (2 x 8).

Relate multiplying by 8 to multiplying by 4 and then doubling.

Relate multiplying by 9 to multiplying by 10 and subtracting one set
Ex.  9 x 8 is the same as (10 x 8) - 8.

Relate multiplying by 10 to place value knowledge of groups of ten.

• Student awareness and use of the commutative property reduces the number of multiplication facts to be mastered from 100 to 55.
• Understanding the relationship between multiplication and division aids in learning basic facts. Teachers need to make this relationship visible as students develop basic fact knowledge. For example, 12 divided by 4 equals 3 because 4 times 3 equals 12. The array for 4 times 3 is the same as the array for 12 objects being put into 4 rows.
• Students should work with various forms of the operation signs for multiplication and division including:

• Use the appropriate vocabulary as students work with basic facts - product, factors, quotient, divisor, and dividend.
• Games and activities can be effective ways for students to practice basic facts.
• Fact Triangles are tools used to help build mental arithmetic skills. Fact Triangles are effective for helping students memorize facts because of their emphasis on relationships in a fact family.

A fact family is a collection of four related facts linking two inverse operations.

For instance, the following four equations represent the fact family relating 3, 4, and 12                with multiplication and division.

4 x 3 = 12      12 ÷ 4 = 3      3 x 4 = 12     12 ÷ 3 = 4

To use the Fact Triangles:

ask students to write or say the four equations that represent the fact family for the numbers on the card.
cover one of the numbers and ask students to identify the missing number for the fact family. Once the missing number has been identified, students can write all four equations for the fact family.

• 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?

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

Lesson Name: It's in the Cards

Lesson objective: In this unit, students use the properties of multiplication to help them master the multiplication facts. The most effective and efficient way to help the students learn the number facts is to build an understanding of the operation, then encourage the students to use the helpful features of the number system, and finally to provide interesting activities for retention. This unit focuses on the second and third aspects of this learning process.

Duncan, N., Geer, C., Huinker, D., Leutzinger, L., Rathmell, E., & Thompson, C.(2007). Navigating through number and operations in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.

O'Connell, S., & SanGiovanni, J.. (2011). Mastering the basic math facts: Multiplication and division, strategies, activities & interventions to move students beyond memorization.  Portsmouth, NH:  Heinemann.

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

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

Uittenbogaard, W., Fosnot, C.. (2008). Minilessons for early multiplication and division, grades 3-4. Portsmouth, NH: Heinemann.

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

factor: the numbers multiplied together to get a product,
example: factor x factor = product   or   4 x 7 = 28,     4 and 7 are factors of 28

multiple: a multiple of a number is the product of that number and any other whole number. zero is a multiple of every number.

product: the result of two numbers being multiplied together,
example: factor x factor = product  or   4 x 7 = 28;  4 and 7 are factors of 28

divisor: the number that will divide the dividend, for example in 72 ÷ 9 , 9 is the divisor

dividend: the number that will be divided, for example in 72 ÷9, 72 is the dividend

quotient: the answer to a division problem.

"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 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.

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 multiplication and division basic fact fluency at the fourth grade level?  How do student misconceptions interfere with mastery of these ideas?

What experiences do students need in order to develop an understanding of basic multiplication facts and the related division facts?

How might fourth graders use the Distributive Property of Multiplication Over Addition and the Associative Property to derive basic facts?

Why do timed tests hinder fact acquisition?

When checking for student understanding, what should teachers:

• listen for in student conversations?
• look for in student work?

How can teachers assess student learning related to these benchmarks?

Examine student work related to a task involving basic multiplication and related division facts.  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 are these benchmarks related to other benchmarks at the fourth grade level?

Professional Learning Community Resources

Carpenter, T., Franke, M., & Levi, L. ( 2003). Thinking mathematically integrating arithmetic & algebra in elementary school, Portsmouth, NH: Heinemann.

Chapin, S. and Johnson, A. (2006).  Math matters: Understanding the math you teach: Grades k-8, 2nd edition.  Sausalito, CA: Math Solutions.

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: 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 4: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

References

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 3-5. Sausalito, CA: Math Solutions Publications.

Burns, M. (Ed) (1998).  Leading the way: Principals and superintendents look at math instruction. Sausalito, CA:  Math Solutions.

Cavanagh, M.. (2004). Math to know: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Chapin, S. and Johnson, A. (2006).  Math Matters: Understanding the math you teach, grades K-8, 2nd Edition.  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., & Lynch, J. (2007). Math for all: differentiating instruction grades 3-5.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.

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.

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.

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

O'Connell, S., & SanGiovanni, J. (2011). Mastering the basic math facts: Multiplication and division, strategies, activities & interventions to move students beyond memorization.  Portsmouth, NH:  Heinemann.

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., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., Trevino, E., & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence.  Reston, VA.: National Council of Teachers of Mathematics.

Schielack, J. (2009).  Focus in grade 4: Teaching with curriculum focal points. 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.

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

Uittenbogaard, W., Fosnot, C.. (2008). Minilessons for early multiplication and division, grades 3-4. Portsmouth, NH: Heinemann.

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.

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

## Assessment

1. Eight friends share 32 stickers. How many stickers did each friend get if the stickers  were shared equally?

A. 3                  B. 4                  C. 5                 D.  6

Solution:  B. 4

Benchmark:  4.1.1.1

2.   Mandy handed out four stickers to each of her friends at her party.  If she passed   out 32 stickers, how many friends were at her party?

A. 7                  B. 5                  C.  8                D. 6

Solution:  B. 8

Benchmark:  4.1.1.1

3.      There are 35 students going on a class trip. The students ride in vans. There are 7 students riding in each van. How many vans are needed to take all the students?

A. 4                  B. 5                  C. 6                 D. 7

Solution:  B. 5

Benchmark:   4.1.1.1               MCA III  Gr  4 Sampler

4.      Tom has 72¢.   New pencils cost 9¢ each.  How many pencils can Tom buy if he spends all of his money?

A. 9                  B. 7                  C. 6                 D. 8

Solution:   D.  8

Benchmark 4.1.1.1

Performance Assessment

1.      Using array cards, show two different ways to find the product of  8 x 7 .  Explain your thinking.

Solution:   Solutions will vary.  Ex:  2 x 7  and  6 x 7

Benchmark;  4.1.1.1

2.      List all single digit factor pairs that give a product that is more than thirty and less than forty.

Solution:  4 x 8, 4 x 9, 5 x 7, 6 x 6

Benchmark:  4.1.1.1

## Differentiation

Struggling Learners
• Students who struggle with basic facts should have the opportunity to build models for multiplication and division. This can be done using connecting cubes, tiles, grid paper and other materials that show how multiplication can be seen as repeated addition and division can be seen as repeated subtraction.
• Relate skip counting on a hundreds chart to repeated addition and equal size groups. This understanding can then be connected to the symbols. For example,

6 + 6 + 6 + 6 = 24  is the same as  4 x 6 = 24.

• Use grid paper arrays to model the commutative property of multiplication.
• Fact Triangles are tools used to help build mental arithmetic skills. Fact Triangles are effective for helping students memorize facts because of their emphasis on relationships in a fact family.

• A fact family is a collection of four related facts linking two inverse operations.

For instance, the following four equations represent the fact family relating 3, 4, and 12 with multiplication and division.

4 x 3 = 12      12 ÷ 4 = 3      3 x 4 = 12     12 ÷ 3 = 4

To use the Fact Triangles,

ask students to write or say the four equations that represent the fact family for the numbers on the card

cover one of the numbers and ask students to identify the missing number for the fact family. Once the missing number has been identified, students can write all four equations for the fact family.

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.

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 who struggle with basic facts should have the opportunity to build models of multiplication and division. This can be done using connecting cubes, tiles, grid paper and other materials that show how multiplication can be seen as repeated addition and division can be seen as repeated subtraction.
• Relate skip counting on a hundreds chart to repeated addition and equal size groups. This understanding can then be connected to the symbols. For example,

6 + 6 + 6 + 6 = 24  is the same as  4 x 6 = 24.

• 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.

• 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:

 3 and 7 are factors of ________________________________________.
 The factors of  24 are _________________________________________.
 The product of ________  and   ________ is  _________.
 I used these array cards, ________________________ to solve __________________.

●     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.

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

Extending the Learning

Students can begin to explore all possible factors of a number and/or multiples of a number.

Students can learn to use parentheses to explain their array combinations, exploring the distributive and associative properties (see vignette).

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.

Classroom Observation

 Students Are:  (descriptive list) Teachers Are:  (descriptive list) building models of multiplication and division facts using tally marks, cubes, tiles, and grid paper and recording the symbolic representation of each model. asking questions to unpack student understanding of models used to represent multiplication and division. using the missing factor approach with "triangle" flash cards to relate multiplication to division. intentionally connecting the concepts of multiplication and division. using hundreds charts to discover pattern when skip counting by a given number. helping students to see connections between facts they struggle with and facts they know- " How can the 'known' fact 5 x 8 help you to learn 6 x 8?" representing both partitive and measurement division. insuring that students spend time developing basic fact knowledge for the most difficult facts.

What should I look for in the mathematics classroom?

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.

For Mathematics Coaches

Chapin, S. and Johnson, A. (2006).  Math matters: Understanding the math you teach: Grades k-8, 2nd edition.  Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC.: National Academies Press.

Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.

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

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

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.

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

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?

What did you try that did not work?
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

Multiplication/Division Fact Triangles

• Fact Triangles are tools used to help build mental arithmetic skills. Fact Triangles are effective for helping children memorize facts because of their emphasis on relationships in a fact family.

A fact family is a collection of four related facts linking two inverse operations.

For instance, the following four equations represent the fact family relating 3, 4, and 12 with multiplication and division.

4 x 3 = 12      12 ÷ 4 = 3      3 x 4 = 12     12 ÷ 3 = 4

To use the Fact Triangles:

• ask your child to write or say the four equations that represent the fact family for the numbers on the card.
• cover one of the numbers and ask your child to identify the missing number for the fact family. Once the missing number has been identified, your child can then write all four equations for the fact family.