# 3.1.2A Addition & Subtraction

3
Subject:
Math
Strand:
Number & Operation
Standard 3.1.2

Add and subtract multi-digit whole numbers; represent multiplication and division in various ways; solve real-world and mathematical problems using arithmetic.

Benchmark: 3.1.2.1 Addition & Subtraction: Procedures

Add and subtract multi-digit numbers, using efficient and generalizable procedures based on knowledge of place value, including standard algorithms.

Benchmark: 3.1.2.2 Real-World & Mathematical Problems

Use addition and subtraction to solve real-world and mathematical problems involving whole numbers. Use various strategies, including the relationship between addition and subtraction, the use of technology, and the context of the problem to assess the reasonableness of results.

For example: The calculation 117 - 83 = 34 can be checked by adding 83 and 34.

## Overview

Big Ideas and Essential Understandings

Standard 3.1.2 Essential Understandings

Students build on previous work with addition and subtraction of 2-digit numbers to include multi-digit whole number addition and subtraction.  They use various strategies to solve real-world and mathematical problems involving addition and subtraction.

Third grade students begin their formal work with multiplication and division by representing basic facts in a variety of ways.

They represent multiplication facts in a variety of ways including

• equal groups
• arrays
• equal jumps on a number line (skip counting)

They represent division facts in a variety of ways including

• repeated subtraction
• equal sharing
• forming equal groups

Students will develop an understanding of the relationship between multiplication and division which will lead to the development of a variety of strategies for multiplying and dividing.  They will solve real-world and mathematical problems involving multiplication and division.  Problem solving involving division will include both "how many in each group" and "how many groups" problems.

Multiplication extends to multiplying a two- or three-digit number by a one-digit number. Strategies may include mental strategies, partial products, the standard algorithm, and the commutative, associative and distributive properties.  Strategic thinking when solving problems is the focus not procedures.

Benchmark Cluster

Benchmark Group A

3.1.2.1:  Add and subtract multi-digit numbers, using efficient and generalizable procedures based on knowledge of place value, including standard algorithms.

3.1.2.2:  Use addition and subtraction to solve real-world and mathematical problems involving whole numbers. Assess the reasonableness of results based on the context. Use various strategies, including the use of a calculator and the relationship between addition and subtraction, to check for accuracy.

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

• Apply place value understanding when adding and subtracting multi-digit numbers. Students develop meaningful strategies to subtract across zero in three- and four-digit numbers.
• Solve real-world and mathematical problems involving multi-digit addition and subtraction using a variety of strategies. These strategies include assessing reasonableness of results, use of technology and the relationship between addition and subtraction, place value understanding when using the standard algorithm.
• Use multiple strategies to compose and decompose numbers flexibly in problem solving situations.
• Demonstrate an understanding of the relationship between addition and subtraction and using one of these operations to check for reasonableness of the answer.
• Compose and decompose numbers. For example, adding 184 and 37 could include 180 and 30, then adding 4 and 7 or 184 + 37 = (180 + 4) + (20 + 10 +7) = 180 + 20 + 10 + 4 +7.
• Expand their knowledge of "making 10" to include making multiples of 10 or 100 or 1,000 to add and subtract more efficiently.

Work from previous grades that supports this new learning includes:

• Composing and decomposing numbers. For example, adding 84 and 37 could include 80 and 20 and 10, then adding 4 and 7 or 84 + 37 = (80 + 4) + (20 + 10 +7) = 80 + 20 + 10 + 4 +7.
• Knowledge of "making a 10" to include making multiples of 10 to add and subtract more efficiently.
• Understand place value up to the thousands place.
• Place value understanding when adding and subtracting with two-digit numbers.
• Use multiple strategies to solve problems involving addition and subtraction.
• Round numbers to the nearest 10, 100.
• Write numbers in expanded form.
• Estimate sums and differences up to 100.
• Knowledge of basic addition and subtraction facts.
Correlations

NCTM Standards

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

Grades 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

## Misconceptions

Student Misconceptions

Students may think . . .

• there is only one way to solve a given addition or subtraction problem.
• the standard algorithm is just a set of steps to follow to get an answer. The steps are applied without understanding the underlying place value structure.
• all answers they get by following a "set of steps" are reasonable even though the answers do not make sense.
• there is only one way to compose or decompose a number. For example, they are unable to see 347 as 34 tens and 7 ones and only see it as 3 hundreds, 4 tens and 7 ones.
• "key words" in a problem dictate solution strategies.

## Vignette

This third grade class is considering a combining problem that can be solved using a variety of strategies involving either addition or subtraction or a combination of the two.

 Jared has collected 1286 baseball cards, Kyle has collected 1527 baseball cards.  Jared would like to have as many cards as Kyle. How many more baseball cards does Jared need to collect so that he will have the same amount as Kyle?

Teacher: Please read the problem and discuss what is happening in the problem with your math partner.

The teacher asks the class to read the problem aloud before beginning a whole class discussion.

Teacher: Tell me what you know about the problem.

The teacher records information for all to see as students unpack the story problem.

Marquis: Jared and Kyle have baseball cards.

Jerry: Jared has 1286 cards and Kyle has 1527 cards.

Mia: Kyle has more cards than Jared.

Ian: Jared has less cards than Kyle. Kyle has the most.

Paul: Jared wants to have as many cards as Kyle has.

The teacher has recorded the following

Jared and Kyle have baseball cards.

Jared has 1286 cards.

Kyle has 1527 cards.

Kyle has more cards than Jared.

Jared has less cards than Kyle.

Kyle has the most.

Teacher: How could we use the bar model to represent what we know?

Note:  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.

Tina: Start with a bar to show all of Jared's cars.

Teacher: Tina, would you make that bar for us?

Tina makes a bar.

Teacher: What labels does the bar need?

Dante: It needs Jared's name and the number of cards he has.

Leo: He has one thousand two hundred eighty-six cards.

Tina adds the labels.

Jared

 1286

Teacher: Have we represented all the information with this bar?

Simone: We only showed a bar for Jared. We need a bar for Kyle, too.

Teacher: Will the bar for Kyle's baseball cards be longer or shorter than the bar Tina made to represent Jared's cards?

Mario: Kyle has more cards than Jared so his bar will be longer than the bar for Jared's cards.

Teacher: Mario, will you make the bar representing Kyle's cards?  Remember to add the labels for the bar.

Mario draws another bar and writes the labels. Note that the bars are aligned in order to make the comparison explicit.

Jared

 1286

Kyle

 1527

Teacher: Have we represented all the information in the bar models?

Students nod, indicating yes. The teacher decides to probe student understanding of the bar model and the relationships being represented.

Teacher:   Using the bar models, how do you know Jared has fewer cards than Kyle?

Isaac: The bar for Jared is shorter than the bar for Kyle. That means Jared has less than Kyle.

Teacher: How else do we know Jared has fewer baseball cards than Kyle?

Mia: Use the numbers. Kyle has 1527 cards and Jared has 1286 cards. We already know that 1257 is less than 1586.

Teacher: I see that Isaac used the bars and Mia used the labels. Both of them were able to justify why Jared had fewer cards than Kyle. Each of them used the bar model. The bar model may help you in deciding which strategy to use when you are solving a problem.

Teacher: Let's go back to the problem. We have listed all the things we know after reading the problem and we have represented that information using a bar model. Whatquestion are you trying to answer in order to solve this problem?

Donovan: How many more baseball cards does Jared need to collect so that he will have the same amount as Kyle?

Teacher: Can someone say it a different way?

Simone: Jared has less cards than Kyle but he wants to have the same amount.We have to find out how many more cards Jared needs to have the same number of cards as Kyle.

Teacher: Can someone say it another way?

Dante So ...Jared has 1286 cards but he wants to have 1527 just like Kyle. How many cards does Jared need to go from 1286 cards to 1527 cards?

The teacher adds to the previous recording.

Jared and Kyle have baseball cards.

Jared has 1286 cards.

Kyle has 1527 cards.

Kyle has more cards than Jared.

Jared has less cards than Kyle.

Kyle has the most.

Question:

Jared has 1286 cards but he wants to have 1527 just like Kyle. How many cards does Jared need to go from 1286 cards to 1527 cards?

Teacher: Work with your math partner to find the number of cards Jared will need in order to have the same number of cards Kyle has. You need to find the number of cardsJared needs in two different ways. Remember you can use any strategy thatmakes sense to you and any materials we have in our classroom. Be ready to share your solution strategies and your solution.

Note:  The teacher has the following extension problem choices for students if/when they finish the original baseball card problem. He will let some students choose a problem and will assign a particular problem for other students. This allows the teacher to differentiate to meet the needs of students who finish early.

 Choose your own numbers. Jared has collected ________ baseball cards, Kyle has collected ________.  Jared would like to have as many as Kyle. How many more baseball cards does Jared need to collect so that he will have the same amount as Kyle?
 Write your own combining problem using five-digit numbers. Solve the problem in more than way.  Explain your thinking.

The teacher checks in with pairs of students as they work in order to check for understanding using the following prompts:  What are you finding?  What strategy are you using?  What does this number represent?  Why are you adding/subtracting?  Does that make sense? How do you know?  How can you check to see if your answer is reasonable?

The teacher brings the class together to share solution strategies and solutions.

Teacher:   Who would like to begin sharing one of you strategies?

Gina/Sami: We used the Empty Number Line. We started at 1286 and figured out how many it takes to get to 1527. It looks like this.

Teacher: How many cards does Jared need in order to have the same number of cards as Kyle?

Sami/Gina: 241

Teacher: Can you put that into a complete sentence?

Sami/Gina: Jared needs 241 more baseball cards.

Teacher: Can someone tell us why they started with 1286 and ended with 1527.

Tamara: The 1286 shows the number of cards Jared has. The 1527 is how many Jared wants to have. That is the number that Kyle has.

Teacher: What about the other numbers they used on their number line?

Benjamin: I think they tried to think of numbers that are easier to use. We know that if we get to 1300 we could get to 1500 really fast and then to 1527. So they started with 1286 and went 4 to get to 1290 and then 10 more to get to 1300 and then the really big jump to 1500 and another jump to 1527.

Teacher: Gina and Sam used the Empty Number Line to keep track as they counted up to find the number of cards needed to go from 1286 to 1527.

Teacher: Did anyone else use the Empty Number Line?  Several hands went up.

Teacher: Did anyone have an Empty Number Line that used different jumps than Gina and Sam used as they went from 1286 to 1527.

Trent/Davis: We used the same starting and ending numbers but different numbers in the middle.

The teacher leads a discussion comparing and contrasting the number line models and the numbers the students used in each.

Teacher: Did someone use a different strategy in finding the number cards Jared needs in order to have the same number as Kyle?

Dante/Ryan: First we made the bar model. We drew the bars and lined them up. Thenwe made a line on the bar for Kyle that matched the end of Jared's bar. We knew that was 1286 because it matched the bar for Jared and he has 1286 cards.

Jared     1286

Kyle      1527

 1286

Dante/Ryan: So then we did the number line like Trent and Davis. We counted up from 1286 to 1527 to find the missing part.

Teacher: Who used a different strategy?

Sarah/Aiza: We knew we needed to find the difference between 1286 and 1527 so we subtracted 1286 from 1527.

Teacher: Can someone explain how Sarah and Aiza got 300?

Abe: There are not any thousands because 1,000 minus 1,000 is zero. Then you get to the hundreds and 500 minus 200 is 300.

Teacher: Sarah and Aiza, did that explain your thinking?

Both students nod in agreement.

Teacher: I see you have used a negative number. Tell us about -60. How did you get that number?

Sarah/Aiza: It is like the empty Number Line going backwards. We started at 20 and had to go back 80 in all. So....starting at 20 and going back 20 gets us to 0 and then we still had to go back 60 more and that was -60.

Teacher:   Can someone tell us why they went back 20 and then went back 60?

Doreen: Because they had to go back 80 and 0 is a good stopping point. So they got to 0 by going back 20 but had to go back 80 so that means they had 60 more to go.

Teacher: Can someone show us these moves on an Empty Number Line?

Eric: They were doing 20 minus 80. It is like this. It is going backwards. They started at 20 and went backwards to 0 which was a jump of 20. Then they still had to go back 60 more because they were subtracting 80. That means they got to -60.

Teacher: Sarah and Aiza, can you tell us how you got 241 as the number of cards Jared needs so he would have the same number of cards as Kyle?

Sarah/Aiza: We started at 300 and went back 60 to 240 and then added 1 more to get to 241.

Teacher: What questions do you have for Sarah and Aiza?

There were no further questions for Sarah and Aiza.

Teacher: Who has a different strategy to find how many more cards Jered needs in order to have the same number of cards as Kyle?

Abduhl/Joe: We changed the numbers at the beginning and then we had to change at the end.

Teacher:  Can you show us the changes you made?

Abduhl/Joe:  This is what we did.

We changed 1527 into 1499 plus 28. Then we could subtract 1286 from1499 with no problems but we had to remember to add the 28 back in at the end. Instead ofadding 28 to 213 it was easier to add 30 and then go back 2 to end up at 241.

Teacher: Who can explain where the numbers 1499 and 28 came from?

Fran: 1499 and 28 make 1527 when you add them. They just broke 1527 apart into two other numbers to make the subtraction easier.

Teacher: It is an efficient strategy that we will be practicing tomorrow. You have already used this strategy to help you learn your basic facts. But, sometimes it feels different when we use a strategy with larger numbers.

These third graders used a variety of strategies in solving a single problem. Students strengthen their number sense when given the opportunity to use different algorithms in problem solving situations. Understanding that algorithms record thinking keeps the focus on mathematical reasoning rather than on a set of rules. This class will work with other algorithms including the standard algorithm in subsequent classes.

## Resources

Instructional Notes

Teacher Notes

• Students may need support in further development of previously studied concepts and skills.
• Students need to unpack real-world problems, clarifying the meaning of the problem--defining words, retelling the problem, acting out the problem, if appropriate.
• 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.
• Common student errors that need to be addressed during classroom instruction:
• Students may not understand what to do when the sum is ten or larger in column addition.

428

+349

7617

Students may not understand borrowing/trading when completing a subtraction problem.

605

-289

484

Students may not choose the right operation when solving a word problem (when they see numbers the tendency is to add).

Students may not reason about the problem solution (looking at an addition problem like 428+349, they don't estimate that the sum should be in the 700s).

Students may use addition to correctly solve a subtraction problem but incorrectly record the problem using subtraction.  For example,  when solving the problem:  There are 198 girls and 245 boys at Monroe Elementary School.  How many more boys than girls are at the school?  The student thinks--It takes 2 to get from 198 to 200 and then 45 more to get to 245 so  2 + 45 = 47, but records it as

198 - 247 = 47

• Use models, including base-ten blocks, when demonstrating addition and subtraction problems to record thinking when using the standard algorithm.
• Students are developing multiple strategies for solving multiplication and division problems. Work with multiplication and division, at the third grade level, is not about the standard algorithm.
• 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.
http://scimathmn.org/stemtc/resources/mathematics-best-practices/modeling-word-problems
• Some third graders may need further practice with 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 5, 6, and 11 with addition and subtraction.

5 + 6 = 11        11 - 5 = 6         6 + 5 = 11        11 - 6 = 5

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.

(insert Addition Fact triangles)

Questioning

Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started

What do you need to find out?

What do you know now? How can you get the information? Where can you begin?

What terms do you understand/not understand?

What similar problems have you solved that would help?

While Working

How can you organize the information?

Can you make a drawing (model) to explain your thinking? What are other possibilities?

What would happen if...?

Can you describe an approach (strategy) you can use to solve this?

What do you need to do next?

Do you see any patterns or relationships that will help you solve this?

How does this relate to...?

Why did you...?

What assumptions are you making?

Reflecting about the Solution

How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?

How can you convince me your answer makes sense?

What did you try that did not work? Has the question been answered?

Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)

Tell me more.

Can you explain it in a different way?

Is there another possibility or strategy that would work?

Is there a more efficient strategy?

Help me understand this part ...

(Adapted from They're Counting on Us, California Mathematics Council, 1995)

Instructional Resources

Instructional Resources

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. & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.

Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.

Yeatts, K., Battista, M., Mayberry, S., Thompson, D., & Zawojewski, J. (2004). Navigating through problem solving and reasoning in grade 3. Reston, VA: National Council of Teachers of Mathematics.

New Vocabulary

New Vocabulary

"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.

Professional Learning Communities

What are the key ideas related to multi-digit addition and subtraction at the third grade level?  How do student misconceptions interfere with mastery of these ideas?

What experiences do students need in order to develop an understanding of multi-digit addition and subtraction?

What strategies would a third grader use in solving a multi-digit addition problem or a multi-digit subtraction problem?

What mistakes do third graders make when solving a multi-digit addition or subtraction problem using the standard algorithm? How are these mistakes related to place value understanding?

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 multi-digit addition or multi-digit subtraction. 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 third grade level?

Professional Learning Community Resources

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.

Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades k-8, (2nd ed.). Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.

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.

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.

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 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.

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.

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.

Uittenbogaard, W., & Fosnot, C. (2008). Mini-lessons 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. A., & Lovin, L. H. (2006). Teaching student-centered mathematics grades K-3. Boston, MA: Pearson Education.

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

Wickett, M., Ohanian, S., &  Burns, M. (2002). Teaching arithmetic: Lessons for introducing  division, Grades 3-4. Sausalito, CA: Math Solutions.

Yeatts, K., Battista, M., Mayberry, S., Thompson, D., & Zawojewski, J. (2004). Navigating through problem solving and reasoning in grade 3. Reston, VA: National Council of Teachers of Mathematics.

Zemelman, S., Daniels, H., & Hyde, A. (2005). Best practices: Today's standards for teaching and learning in America's schools. Portsmouth, NH: Heinemann.

## Assessment

• Subtract. 6,905 - 37

A. 3,205

B. 6,868

C. 6,932

D. 6,968

Solution: B 6868

Benchmark:    3.1.2.1, 3.1.2.2

MCA III Item Sampler

• Jeff had 1,350 glass beads and 695 clay beads.

He sold 138 glass beads and 47 clay beads.

How many beads did Jeff have left?

A. 470

B. 746

C. 1,860

D. 2,230

Solution: C 1860

Benchmark: 3.1.2.1, 3.1.2.2

MCA III Item Sampler

• Chase has 249 video games. Zachary has 1,589 games. What is the total number of games they have?

A. 2,128

B. 1,838

C. 1,249

D. 1,986

Solution: B. 1838

Benchmark: 3.1.2.1, 3.1.2.2

MCA III Item Sampler

• Jennifer's bird club spotted 527 birds. Bill's bird club spotted 297 birds. Did the two bird clubs spot more than 870 birds? Explain your answer.

A. Yes

B. No

Solution: B. No. Possible explanation: I added 527 + 297 and the answer was 824, which is less than 870.

Benchmark: 3.1.2.1, 3.1.2.2

MCA III Item Sampler

6,340

-3,451

A. 2,889

B. 2,499

C. 3,889

D. 3,111

Solution: A. 2,889

Benchmark: 3.1.2.1, 3.1.2.2

MCA III Item Sampler

• The school sold 1,588 tickets to the football game. There are 2,500 seats available in the stadium. How many more tickets could be sold for the football game?

A. 1,088

B. 912

C. 1,912

D. 88

Solution:  B. 912

Benchmark: 3.1.2.1, 3.1.2.2

MCA III Item Sampler

• Aaron's book has 326 pages. He read 26 pages on Monday, 49 pages on Tuesday, and 36 pages on Wednesday. How many pages does he have left to read?

A. 127

B. 300

C. 274

D. 215

Solution: D. 215

Benchmark: 3.1.2.1, 3.1.2.2

MCA III Item Sampler

## Differentiation

Struggling Learners

Struggling Learners

• Students need to unpack real-world problems, clarifying the meaning of the problem--defining words, retelling the problem, acting out the problem.
• Students will need appropriate place value models during real-world problem solving.
• Students may need support in finding the mathematics vocabulary to explain solution strategies.
• Some third graders may need further practice with 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 5, 6, and 11 with addition and subtraction.

5 + 6 = 11        11 - 5 = 6         6 + 5 = 11        11 - 6 = 5

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.

(insert: Addition Fact triangles)

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., & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.

Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.

English Language Learners
• Students need to unpack real-world problems, clarifying the meanings of words in the problem, retelling the problem.
• Students need appropriate place value models during real-world problem solving.
• Students may need support in finding the mathematics vocabulary to explain solution strategies.
• 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:

 I used this strategy to solve the problem. First, __________. Then ______.
 The answer makes sense 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.

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

Solve multi-step problems involving addition and subtraction.

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... Teachers are... using manipulatives such as base ten blocks, bundles of sticks, place value charts, or graph paper to add and subtract numbers. modeling the think aloud strategies used by students. explaining and justifying solution strategies. making connections between and among student solution strategies. using multiple strategies when solving addition and subtraction problems. using place value materials to demonstrate problem solving. connecting addition and subtraction and explaining the relationship between the two. listening to and asking clarifying questions as students explain and justify thinking.

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 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.

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

Parent Resources

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

What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN

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

• Third graders may need further practice with 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 5, 6, and 11 with addition and subtraction.

5 + 6 = 11     11 - 5 = 6    6 + 5 = 11    11 - 6 = 5

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.