4.1.1B Multi-Digit Multiplication
Overview
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 Group B
4.1.1.2 Use an understanding of place value to multiply a number by 10, 100 and 1000.
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 both 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.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- multiply one- and two-digit numbers by 10, 100 and 1000;
- use rounding, knowledge of benchmarks and place value to make a reasonable estimate of the product or quotient of a multi-digit problem;
- assess the reasonableness of answers and apply more than one strategy to check their work;
- understand and explain why the solution strategies work based on place value and properties of operations, and use them to solve problems;
- solve multi-step addition, subtraction and multiplication problems with solutions less than 100,000;
- accurately solve multi-digit multiplication problem: a one- or two-digit by a two- or three-digit number.
Work from previous grades that supports this new learning includes:
- understanding equal groupings and shared model for multiplication;
- understanding the relationship between multiplication and division;
- strategies to accurately find the product of multiplication of a 2 or 3 digit number by a single digit.
NCTM Standards
Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Grades 3 - 5 Expectations
- 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.
Grades 3 - 5 Expectations
- 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
Grade 3 - 5 Expectations
- 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
Generalize place value understanding for multi-digit whole numbers.
4.OA.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.OA.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.OA.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.OA.4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.
4.OA.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.OA.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.
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 and Common Errors
Students may think:
- when you multiply by a power of ten you just "add zeros to the end of the number." This "trick" leads to several errors in student thinking. The first is that placing a zero behind a digit is called "adding," and they confuse this with addition. The second is not understanding how they have impacted the magnitude of the new number. They will also extend this thinking to decimals and not recognize that this "trick" will not work. For example, 4.15 x 10 does not equal 4.150.
- it is not important to find an estimate before doing the actual problem. Students often think teachers are just asking them to do more work. In reality, knowing approximately how large your final answers are helps students know if answers are reasonable.
- place value is just used for naming numbers. Place value plays an enormous role in students' understanding and accuracy of solving problems involving large numbers. Understanding how multiplication impacts the size of the numbers being operated on and accurately representing that number depends largely on a student's number sense.
Vignette
In the Classroom
Students in this fourth grade classroom have been working on multi-digit multiplication. Teachers often assume that students who accurately apply the traditional algorithm, understand how to multiply; this is not always true. In addition to being able to apply the "traditional algorithm," students must explain how and why the algorithm works. In other words, "Can you show me the mathematics involved?" In this vignette, the traditional algorithm has been used to solve the problem. Fourth graders are sharing thinking strategies that can help them understand how the traditional algorithm works. Place value understanding is essential for understanding traditional algorithms.
Students are sharing different ways to show that 325 x 29 = 9,425
* The strategy Student A is about to share goes back to the classroom work and understanding about how large numbers are formed (expanded notation). This way of looking at numbers helps students to comprehend that while the digits have a face value, their location has a place value. In the traditional algorithm (as well as the lattice method) children use basic facts and follow a procedure. Other strategies, like the one below, maintain place value.
Student A: Well, the way I did it looks different, but it makes sense to me. First, I pull the number apart so it looks like this:
300 20 5 because that is the same as 325
x 20 9 because that is the same as 29
Student B: Is that like expanded notation?
Student A: Yes, but I left out the addition signs because I don't really need them.
Student A: So the next thing I know is that I need to multiply each of the numbers in the top number, (300, 20, and 5) by the numbers in the bottom number (20 and 9).
Note: This student had a huge realization when expanded notation was used as a way to demonstrate what was happening when solving 3-digit by single digit multiplication and has applied and used this strategy accurately and efficiently as the class moved onto multi-digit multiplication.
Student C: That's the same thing with my strategy!
This sparks a conversation among the fourth graders. Many of them have discovered the same thing about the strategy they are using. "You must multiply all digits in the first number by all the digits in the second number."
Student A: The most important part for me is to make sure I keep the numbers lined up as I solve the problems. I also like to draw lassos around the numbers as I multiply them; this helps me keep track.
Another Student: What do you mean a "lasso"?
Student A: Well, I draw a line from the 9 and a circle around the 5 and write 45 under my line.
(Student demonstrates this on the board).
Class snaps, gives thumbs up and waves hands in approval. They like that thinking!
Student A continues to explain the problem, narrating while doing so:
300 20 5
X 20 9
45 ⇒ The 45 is the product of the 9 x 5. Next, I lasso the 20 and solve 9 x 20
180 ⇒ This is like 9 x 2, which is 18, and then I multiply 18 by 10, so 180.
2700 ⇒ Next is 9 x 300. If you know your facts, this is easy. 9 x 3 is 27, and 27 x 100 is 2700!
Another Student: You could actually put the numbers you have on the board into columns like a place value chart, except your answers will be in the wrong spot. Student comes to board and adds lines around the numbers, making a mock-up place value chart.
Many "ooh's and oh's" from the class.
Student A: Yeah, I kind of think like that when I write the numbers this way. It keeps it all the right size, the 3 is a 300 like it is supposed to be.
Student: When I do the other way (points to the traditional algorithm), I say in my head "9 x 300" but I am solving 9 x 3.
Student A: Well, I only solved 9 x 3 but then I say in my head, "times the power of 100" so I remember the place value and the meaning in the zeroes. Remember when Mrs. X showed us those hundred's blocks? Well, I need 9 piles of those in 3 stacks- right? If I just did 9 x 3 I would have 9 piles of 3 little cubes!
One student scrambles to the shelf and pulls off the blocks being discussed. The students nod and make related comments.
Students A: So once I have used the 9 to multiply all three top numbers, I move onto the 20.
300 20 5
X 20 9
45
180
2700
100 ⇒ 20 x 5 is 100, that one is easy, next 20 x 20 (draws the "lasso" on each step)
400 ⇒ I just know this one! The powers of ten help you know how big it will be!
6000 ⇒ For 20 x 300 I thought "2 x 300" then 600 times 10! I'm done multiplying.
9425 ⇒ I add the products together and wa-la! The same final product as the other way!
At this point, the teacher has students look for similarities between Student A's strategy and the traditional algorithm.
Student sharing: This is the traditional algorithm. When you multiply the 9 and the 5 together, you get 45 in both methods. Where you record the numbers is different though. In A's way, you keep it together and write it below. You could do that same thing using this method if you wanted, but I learned to put it above the 2, which is in the tens place in 325.
Student sharing: Here is how I set up the problem using the place value chart. It helps me keep track. I wrote the answers and problems off to the side as a way to double check. You can see how it looks like Student A's strategy.
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills, such as review of the commutative, associative, and distributive properties.
- Understanding "expanded notation" or "expanded form" at this level is very helpful to students. When students are able to rewrite a number in expanded notation, for example the number 42,456 as 40,000 + 2,000 + 400 + 50 + 6, it re-enforces their understanding of place value and helps them to see how operating (multiplying) on the number will impact the number.
- The relationship among powers of ten is a difficult concept. Fourth graders need to use physical models in connection with numbers. A helpful visual of the relationship among powers of ten is the place value chart. As you move to the left on the chart, the number increases by a factor of ten:
1,000x10 100x10 10x10 1x10
↶ ↶ ↶ ↶
ten-thousand | thousand | hundred | tens | ones |
10,000 | 1,000 | 100 | 10 | 1 |
80,000 | 8,000 | 800 | 80 | 8 |
- Understanding what it means to multiply by 10 (or a power of 10) is more than following a rule to "add a 0." This latter "trick" leads to several errors in student thinking, the first being that placing a zero behind a digit is called "adding," and they confuse this with addition. The second "trick" is not understanding how they have impacted the magnitude of the new number. They will also extend this thinking to decimals, not recognizing that the "trick" does not work. For example, 4.15 x 10 does not equal 4.150. Consider having students use base-ten blocks to show that when multiplying a whole number by 10, the digits move to the next bigger place value and that there will never be any ones left in the one's place, which is why there will be a zero in the one's place. For example, for 10 x 34, we can begin with base-ten blocks to represent 34.
Students can then show what it means to multiply by ten by taking each piece and showing that they have 10 of the first "long" (rod, or group of 10), 10 of the second long, 10 of the third long, 10 of the first unit, 10 of the second unit, 10 of the third unit, and 10 of the fourth unit as shown in the picture below.
Students need to connect this representation to the product when written as 340. Students should be given ample opportunities to make this connection and eventually generate a rule for multiplying by 10.
- Student understanding of partial products and other alternate strategies for recording multiplication will support the understanding of the steps in the standard algorithm. The standard algorithm stores partial products, too!
- The array model represents multiplication of multi-digit numbers while retaining place value structure. Place value blocks can be used to build an array model or it can be drawn on paper. When drawn on paper, the array becomes a graphic organizer and may not be proportional.
- While the idea of teaching "key words" might be appealing, it does not support student comprehension of word problems and can lead to incorrect solutions.
- Number Talks are a great way to keep strategies visible in the classroom. During a Number Talk, students are given three to five problems related to the problem to be solved. They are asked to think about how knowing these related problems can help them in their strategy to find the final product. Fourth graders, for example, might look at these sets of problems:
- Related problems are the focus of this set.
75 x 3 74 x 3 74 x 30 74 x 10 74 x 2 74 x 38 |
Students use 25 as a landmark number when multiplying, like using quarters. Because of this, multiples of 25 are easier for them to multiply as well. They can easily picture 3 stacks of 3 quarters, thus, 3 x 75.
Using related facts is the focus of this set of problems.
32 x 1 16 x 2
8 x 4 4 x 8
2 x 16 1 x 32 |
"Doubling one factor while halving the other" is a great strategy for students to see how multiplication facts are related. There is an activity that can accompany this called "Splitting Rectangles" in which students first build the arrays on graph paper, then literally cut them in half and reattach them to make the next rectangle. This could also be done with blocks or tiles.
Implementing Number Talks: Helpful Hints
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.
- Modeling word problems is critical as students develop an understanding of operations and the relationships that exist between and among those operations. The bar model is an effective tool that can be used to represent the relationships in a variety of problem solving situations.
- For more information on the bar model, read Modeling Word Problems in the Mathematics Best Practice tab in the Resources section of the Minnesota Mathematics Framework.
- 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)
Burns, M., & Wickett, M. (2001). Teaching arithmetic: Lessons for extending multiplication, grades 4-5. Sausolito, CA: Math Solutions.
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.
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.
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.
expanded form. A multi-digit number is expressed in expanded form when it is written as a sum of single-digit multiples of powers of ten. For example: 643 = 600 + 40 + 3.
"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.
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 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?
- ask during classroom discussions?
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.
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 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., & 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, & 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
- (From MCA III Item Sampler)
A truck has 50 boxes of jump ropes.
Each box contains 100 jump ropes.
How many jump ropes are on the truck?
A. 50 B. 500 C. 5,000 D. 50,000
Solution: C. 5,000
Benchmark: 4.1.1.2
- The fourth grade students are having a bake sale to raise money for their new playground.
32 children each volunteer to bring in a dozen (12) baked goods.
Part a: How many items will they have to sell?
Part b: How much money will they earn if each item sells for 25 cents?
Solution: Part a: 384 baked goods Part b: $96
Benchmark: 4.1.1.5
- (From MCA III Item Sampler )
Two numbers are multiplied together.
\begin{align*}724\\\underline{\times 8\square}\\62,264\end{align*}
Which digit goes in the box?
A. 0 B. 1 C. 4 D. 6
Solution: D. 6
Benchmark: 4.1.1.3
- Using rounding, which of the following is the most reasonable estimate for 489 x 72?
A. 3,500 B. 35,000 C. 28,000 D. 32,000
Solution: B. 35,000
Benchmark: 4.1.1.4
Differentiation
- Multi-step problems can be problematic for students. While the idea of teaching "key words" might be appealing, it does not support student comprehension of word problems and can lead to incorrect solutions. Another approach might be "story problem theater." When students listen to a story problem and then try to "act out" what is happening in the problem, it is easier to see what mathematics might be involved.
- The lattice method is another option for students having difficulty with multi-digit multiplication. Further information about the lattice method can be found here: http://mathworld.wolfram.com/LatticeMethod.html
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.
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.
- Multi-step problems can be problematic for students. While the idea of teaching "key words" might be appealing it does not support student comprehension of word problems and can lead to incorrect solutions. Another approach might be "story problem theater." When students listen to a story problem and then try to "act out" what is happening in the problem, it is easier to see what mathematics might be involved.
- 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:
When I multiply ____ by ten I get _______________________. |
When I multiply ______ by one hundred I get _______________________. |
This is how I solved the problem. First, ____________________________________________________________. Second, __________________________________________________________. . . . Last, _____________________________________________________________. |
I rounded______________ to ______________ because ______________________. |
- When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding.
Additional ELL Resources
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.
- Students should examine the similarities and differences between several solution strategies.
- Students should compare solution strategies to the standard algorithm.
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: |
using strategies to estimate products prior to solving multiplication problems. | encouraging students to round numbers or use landmarks to make estimations. |
accurately multiplying numbers by 10, 100 and 1,000. | providing models with base ten blocks and grid paper to illustrate the magnitude of numbers multiplied by powers of ten. |
translating story problems into number sentences or pictures to help solve them accurately. | offering ways of thinking about multi-step problems so students have access to solving them (acting it out, using manipulatives, drawing pictures). |
solving multiplication problems (using more than one strategy as a way of checking for accuracy). | recognizing student thinking and strategies, asking questions to check for understanding and accuracy. |
unpacking story problems to demonstrate understanding of the problem and what is being asked. | asking questions about story problems that will help students understand the structure of the problem and what is being asked. |
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 edition. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC.: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
For Administrators
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Parent Resources:
Mathematics handbooks to be used as home references:
Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Helping your child learn mathematics
Provides activities for children in preschool through grade 5
What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN
Help Your Children Make Sense of Math
Ask the right questions
In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.
Good questions, and good listening, will help children make sense of the mathematics, build 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
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