3.2.2 Number Sentences
Understand how to interpret number sentences involving multiplication and division basic facts and unknowns. Create real-world situations to represent number sentences.
For example: The number sentence 8 × m = 24 could be represented by the question "How much did each ticket to a play cost if 8 tickets totaled $24?"
Use multiplication and division basic facts to represent a given problem situation using a number sentence. Use number sense and multiplication and division basic facts to find values for the unknowns that make the number sentences true.
For example: Find values of the unknowns that make each number sentence true
6 = p ÷ 9
24 = a × b
5 × 8 = 4 × t
Another example: How many math teams are competing if there is a total of 45 students with 5 students on each team? This situation can be represented by 5 × n = 45, or $\frac{45}{5}$=n, or $\frac{45}{n}$= 5.
Overview
Standard 3.2.2 Essential Understandings/Ideas
Third graders build on their previous work with operations as they write number sentences to represent a real world or mathematical situation involving multiplication and division. They write a real world problem to represent a given number sentence.
Work with number sentences continues as third graders determine if a number sentence is true or false. For example, is 9 x 7 = 73 true or false? They are also able to write their own true number sentences and false number sentences.
Third graders build on their understanding of the equal sign as they continue finding unknowns in number sentences. The unknowns in given number sentences are found in varying positions. For example, 5 x A = 20, 20 = B x 4, 20 ÷ K = 5. Third graders understand that when a letter representing an unknown is used more than once in a given number sentence it has the same value. They also understand that different letters in a given number sentence may or may not have the same value. For example, in the equation R x R = 16, "R" equals 4. In the equation A x B = 16, "A" and "B" could have same value (4) or different values (2 and 8 or 1 and 16).
Benchmark Group A
3.2.2.1 Understand how to interpret number sentences involving multiplication and division basic facts and unknowns. Create real-world situations to represent number sentences.
3.2.2.2 Use multiplication and division basic facts to represent a given situation using a number sentence. Use number sense and multiplication and division basic facts to find values for the unknowns that make the number sentences true.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Write a number sentence representing a real-world situation involving multiplication and division basic facts and unknowns.
- Write a real-world situation for a given number sentence involving multiplication, division basic facts and unknowns.
- Find values for unknowns that make number sentences true involving multiplication and division basic facts.
Work from previous grades that supports this new learning includes:
- Interpret number sentences that involve addition and subtraction as being true or false.
- Recognize 5 + m = 12 and be able to find the value of m in order to make the number sentence true.
- decompose and recompose numbers flexibly.
- interpret and find solutions for real world problems.
NCTM Standards
Represent and analyze mathematical situations and structures using algebraic symbols
- identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers
- represent the idea of a variable as an unknown quantity using a letter or a symbol
- express mathematical relationships using equations
Common Core State Standards
Represent and solve problems involving multiplication and division.
3.OA.4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?
Understand properties of multiplication and the relationship between multiplication and division.
3.OA.6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.8. Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Misconceptions
Students may think...
- there are rules that determine which number a letter stands for. For example, e = 5 because e is the fifth letter of the alphabet or y = 4 because y was 4 in the last number sentence.
- that a letter always has one specific value.
- different letters always represent different numbers.
- an equal sign means "and the answer is." In this way, when they see an equal sign, they want to carry out the operation preceding it. For example, when asked what the △ represents in the equation 4 x 3 = △ x 2 they say 12. They need to think of the equal sign as meaning "is the same as."
- the commutative property applies to division, e.g., 12 ÷ 3 = 4 is the same as 3 ÷ 12 = 4
Vignette
Vignette I Scroll down to see Vignette II.
The third graders in Mr. O's class have been working on solving multiplication and division story problems with a variety of contexts. They have been using models to represent problems, solving problems with the unknown in various positions and writing equations to match story problems.
Mr. O. knows that different structures in problems can provide differentiation for his students. Problems having the result of the operation as the unknown (24 ÷ 6 = n) are the easiest type of problems to solve, and in this case, write, because children have a great deal of experience with this problem type. Problems having an unknown divisor (24 ÷ n = 4) are harder for children to solve. Problems having an unknown dividend (n ÷ 6 = 4) can be the most difficult for children to solve.
Mr. O.: Today we're going to build on activities we've been doing with division. I'm goingto give you some equations and you're going to write number stories to match them. What is an "equation?"
Marion: It's a number sentence with an equal sign in it.
Mr. O.: Thanks, Marion. That's a good definition.
Here are three equations we will be using today.
24 ÷ 6 = n
24 ÷ n = 4
n ÷ 6 = 4
Mr. O.: What do you notice about the equations on the board?
Julian: All of them are division...divided by.
Sureah: Some of them equal 4.
Peyton: Two of them start with 24.
Bryce: They all have the letter "n" in them but not in the same place.
Mr. O.: What does "n" represent in these number sentences?
Tyler: I think it stands for some number. We don't know what it is - that's what we have to figure out.
Paul: Well actually, all those equations are like a fact family.
Mr. O.: Say more about that.
Paul: I already know what the "n" stands for because the answer is right there in the three equations. See, the first two problems have the total number. Then the next problem has the other two numbers, 4 and 6. So, I know all the numbers I need - 24, 6, and 4. I already know that 4 times 6 is 24 and 6 times 4 is 24.
Mr. O: Nice connection, Paul. The point of this lesson is to write number stories that match the equations and then show your work as you answer the question in your number story.
The class gets to work.The children are used to working collaboratively and talking while they work. They know how to help each other by asking questions and not telling each other what to do. The children have solved many story problems during the year and they have experience choosing the equations that challenge them but still have an entry point so they can be successful. Mr. O. interacts with students as they work.
Mr. O. begins working with Ahlam because he has struggled with number stories in the past.
Mr. O.: Ahlam, what unit would you like to use in your number story?
Ahlam: Pencils.
Mr. O.: Okay. Read the equation you chose.
Ahlam: 24 divided by 6 equals n
Mr. O.: What can we say for about this symbol (points to equal sign)?
Ahlam: Same as. It means is the same as. 24 ÷ 6 is the same as n.
Mr. O.: Nice. So, how is your story problem going to start?
Ahlam: There are 24 pencils.
Mr. O.: That's a good start. Since this is a division problem, what kind of action are you going to use in your story?
Ahlam: Well, they could be given to people.
Mr. O.: Tell me more about giving 24 pencils to people. Pointing to the equation.
Ahlam: There are 24 pencils that 6 people are going to share them.
Mr. O.: OK. What are you trying to find out?
Ahlam: The answer.
Mr. O.: In this problem with people and pencils, what does the answer mean? What does it tell us?
Ahlam: It means how many pencils.
Mr. O.: I thought we already knew there were 24 pencils.
Ahlam.: No.....the answer tells how many pencils each person gets. The 24 tells how many pencils there are before sharing.
Mr. O.: (as he rewrites the equation) What I hear you saying is there are twenty four pencils that will be shared by sixchildren. You are going to find out how many pencils each child will get.
24 pencils ÷ 6 children = number of pencils each person gets.
Ahlam: Yes.
Mr. O.: Ahrem, are you finding the number of groups or the number in each group?
Ahlam: The number in each group. There are 6 children getting pencils and I will be finding how many pencil each person gets.
Mr. O.: Ahrem, how are these pencils being shared? Is everyone getting a different number of pencils?
Ahlam: No, they are sharing that means everyone gets the same amount.
Mr. O.: When writing a number story you need the words "shared equally" if you mean everyone gets the same amount.
Ahlam began writing, "There were 24 pencils. They are going to be shared equally with 6 children."
Mr. O. moves to another table.
Lisa: Mr. O., how do you spell "dolphins?
Mr. O.: Write down the sounds that you hear. In the word dolphins the "f" sound is made by the letters "ph."
Lisa: Okay.
Mr. O.: Which equation are you using?
Lisa: n ÷ 6 = 4
Mr. O.: What does the "n" stand for?
Lisa: That's how many dolphins there were in the beginning.
Mr. O.: So, how are you going to start your story problem?
Lisa: There were "some" dolphins by the boat.
Mr. O.: What kind of action are you going to use in your problem situation.
Lisa: The dolphins are going to go to different aquariums so third graders can watch them and learn more about them. I really love dolphins. I saw some at the Minnesota Zoo.
Mr. O.: Will your number story involve the number of dolphins in each group or the number of groups of dolphins?
Lisa: Both. In this equation there will be 6 dolphins going to 4 different aquariums. Each aquarium will be getting 6 dolphins. So I will be using the number of groups and the number in each group. I have to figure out the number of dolphins in all.
Moving on to another student.....
Miko: I'm going to write about limos.
Mr. O.: Fun - Which equation are you going to use?
Miko: 24 ÷ n = 4
Mr. O.: What does the "n" stand for?
Miko: It's the number of limos.
Mr. O.: Tell me more about the limos in your story..
Miko: They're big and shiny each one is going to take 4 people to a concert.
Mr. O.: That makes sense. Are there going to be more or less than 24 limos?
Miko: Less because there are only 24 people and there will be 4 people in each one.
Mr. O.: Are you finding the number in a group or the number of groups?
Miko: I'm going to find how many limos I need so that each limo has 4 people. That is the number of groups because I already know there will be 4 in each limo.
Mr. O. went back to Ahlam to check on her progress. Ahlam wrote the following problem and was in the process of recording her answer.
24 ÷ 6 = n
There were 24 new pencils.
6 kids are going to share them equally.
How many pencils will each kid get if they get the same amount?
Mr. O.: In your problem, were you finding the number of groups or the number in each group?
Ahlam: I needed to find the number of pencils for each person. I already know the number of people that is the number of groups.
Mr. O.: How did you represent the problem?
Ahlam: I started by making 6 shapes for the six people. And then I made a line in each shape going across and starting again until I had 24 lines.
Mr. O.: What do the shapes mean? What do the lines mean?
Ahlam: The shapes are the people and the lines are the pencils.
Mr. O.: I see the six shapes representing six people in your problem and I see the lines for the pencils. Tell me about the numbers under the boxes.
Ahlam: That shows the number of pencils. See there were only 24 pencils so I had to make sure I really had 24. I just counted and wrote numbers when I counted.
Mr. O.: How many pencils will each of the six people get?
Ahlam: Four.
Mr. O. checks in with Lisa.
Lisa: Can I solve another problem now?
Mr. O.: Show me what you've done so far.
Lisa: Remember I was doing dolphins and aquariums and the last equation on the board. This shows the four aquariums and the six dolphins that were going to each aquarium.
Mr. O.: I see the four ovals for the aquariums and six x's for six dolphins in each aquarium. Where did the "24" come from in the equation 24 ÷ 4 = 6?
Lisa: The 24 is how many x's there are altogether in the ovals. See the ovals are the aquariums and the x's are the dolphins. I had to figure out how many dolphins there would be if each aquarium had six dolphins. Since each aquarium had six I counted by six to find how many in all ..... 6, 12, 18, 24.
Mr. O.: You have told me your number story. Now, write your number story for others to read.
Lisa wrote: There are some dolphins in aquariums at the zoo.
If there are 4 aquariums and each aquarium has 6 dolphins.
How many dolphins are at the zoo?
n ÷ 4 = 6
Mr. O.: Lisa, what if we used aquariums and dolphins but this was the equation?
n ÷ 6 = 4.
Would you write number story for this equation?
Lisa.: So you want me to figure it out for six aquariums when each aquarium gets four dolphins. It is almost the same as this one (points to her paper).
Mr. O.: It has the same numbers and the same things-dolphins and aquariums.
Lisa begins drawing. Mr. O. wants to see how Lisa responds to the same situation but using the same numbers to represent a "different" number of groups and a "different" number in each group.
After they have had time to finish their number stoires, students come together to share. As each student shares his/her number story, Mr. O. records each number story on a separate sheet of paper. In the coming weeks, the students will write equations to match each others number stories.
Vignette II
Third graders in Mr. Xiong's class are determining if number sentences involving the operations of multiplication or division are true. In previous grades, these students had determined if number sentences involving the operations of addition or subtraction were true.
Mr. Xiong writes 3 x 5 = 15 on the board.
3 x 5 = 15
Mr. Xiong: I want you to think about this number sentence. Is this number sentence true or false? What do you think?
Skyler: I think it's true.
Mr. Xiong: Why do you think it's true?
Skyler: Well, look... 3 x 5 is three groups of five. So I counted three groups of five- - 5,10, 15. Three times five is 15. This equation is true.
Mr. Xiong: Evie, I noticed that you were agreeing with Skyler. Why do you think it's true?
Evie: I used my hand for a group of five and I need three hands for three groups of five. then I skip counted just like Skyler.
Mr. Xiong: Do you all agree that 3 x 5 = 15?
Students nod in agreement.
Mr. Xiong writes 15 = 3 x 5 on the board.
Mr. Xiong: What about this number sentence? Is it true?
15 = 3 x 5
He looks around the room and sees many nods and some puzzled faces.
Mr. Xiong: Alan, what do you think?
Alan: It's false. But you wrote it wrong, Mr. Xiong. Are you trying to trick us? You can't write it that way. The equal sign goes at the end.....and so it's false because 15 x 3 isn't 5, it's way more because it is fifteen groups of 3. Now look at the other one, that's how you write it!
Mr. Xiong: Let's look at the first equation. (Mr. Xiong points to 3 x 5 = 15.)
Alan, can you read it for us?
Alan: Three times five equals fifteen.
Mr. Xiong: Alan, can you read the problem another way?
Alan: Pauses and thinks for a moment. Three groups of five make fifteen.
Mr. Xiong: Okay...good. How about another way? Could anybody read it another way?
Katie: Three plus five is the same as fifteen
Mr. Xiong: Nice thinking. Now, let's take a look at this problem. Tia, will you read this
number sentence. Mr Xiong points to 15 = 3 x 5.
Tia: Fifteen equals three times five.
Leon: I can read it another way. Fifteen is the same as three times five.
Mr. Xiong: Points to 3 x 5 = 15. Three times five is the same as fifteen.
Mr. Xiong: Points to 15 = 3 x 5. Fifteen is the same as three times five.
Does it matter where we put the equal sign, at the beginning or at the end?
Jillian: It doesn't matter. It's just like turning things around, but it means the same thing.
Mr. Xiong writes 3 x 5 = 3 x 5. He picked this equation because he wants to see if students are thinking operationally (solving each side of the equal sign and comparing the results) or if students are thinking relationally (comparing what is on each side of the equal sign, looking for likenesses and differences).
Mr. Xiong: Is this equation true?
Hal: True. True. True. It's true because three times five is fifteen and three times five is fifteen. It is like fifteen is the same as fifteen. Fifteen equals fifteen. Like I said true.
Mr. Xiong sees many heads nodding. He assumes these students were thinking operationally--solving both sides of the equal sign and then comparing the results. He wants to see if anyone thought about it in a different way.
Mr. Xiong: Yes, the equation is true. Who thought about it in a different way?
Jae: Well...I just looked at one side and then the other. Both of them were three times five. See....(pointing to the equation) three times five and three times five. Since they are both the same then it is true. I didn't have to do three times five is fifteen because they were the same.
Mr. Xiong: I see that you looked at the parts on each side of the equal sign and saw that they were exactly the same. Since the parts were exactly the same the equation was true. Mr. Xiong underlines 3 x 5 on either side of the equals sign in the equation.
Mr. Xiong wants to see what students will do with this new information. He wonders how well they understand what Jae did.
Mr. Xiong: How about this one? Is it true?
3 x 5 = 3 + 5
Mr. Xiong sees many students nodding indicating yes but Ira is shaking his head to indicate no, not true. Mr. Xiong decides to ask a student who thinks it is true to respond first.
Emma: It is true because both sides are the same.
Mr. Xiong: Tell me more about both sides are the same.
Felix: See it is three and five on this side and three and five on the other side of the equal side. Both are the same so it is true just like the last one we did.
Mr. Xiong sees several students nod in agreement.
Ira: I disagree. It is not the same on both sides. Pointing to 3 x 5, this one is three TIMES five and, pointing to 3 + 5, this one is three PLUS 5. They are not the same. One is times and one is plus. So they are not the same even though they have the same numbers.
Several Students: It's false.
Mr. Xiong continues the conversation using the following number sentences.
3 x 5 = 5 + 5 + 5
3 x 5 = 5 x 3
3 x 5 = 3 x 4 + 4
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- Provide opportunities to find an unknown in number sentences with the unknown in various locations. For example: 36 ÷ 6 = m, 3 x ∆ = 21, 6 = m ÷ 3, t = 6 x 4.
- Third graders need opportunities writing number stories that match number sentences before they write number stories for number sentences involving unknowns.
- 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.
http://scimathmn.org/resources/mathematics-best-practices/modeling-word-problems
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)
Cuevas, G., & Yeatts, K. (2001). Navigating through algebra in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
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 k-3. Boston, MA: Pearson Education.
Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
Usiskin, Z. (1999). Doing algebra in grades k-4. Algebraic Thinking, Grades K-12: Readings from NCTM's School-Based Journals and Other Publications, 5-6. Reston, VA: National Council of Teachers of Mathematics.
New Vocabulary
"Vocabulary literally is the key tool for thinking."
Ruby Payne
number sentence - an equation or inequality expressed using numbers and common symbols.
equation - a mathematical statement that asserts the equality of two expressions.
value - relative worth, magnitude; numerical quantity that is assigned or determined by calculation or measurement; number represented by a figure, symbol.
represent - refers both to process and to product, the act of capturing a mathematical concept or relationship in some form and to the form itself.
Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions. Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.
Learning vocabulary in the mathematics classroom is contingent upon the following:
Integration: Connecting new vocabulary to prior knowledge and previously learned vocabulary. The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.
Repetition: Using the word or concept many times during the learning process and connecting the word or concept with its meaning. The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.
Meaningful Multiple and varied opportunities to use the words in context. These
Use: opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems. Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.
Strategies for vocabulary development
Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.
Mathematics Word Bank: Each unit of study should have word banks visible during instruction. Words and corresponding definitions are added to the word bank as the need arises. Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.
Labeled pictures and charts: Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.
Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.
Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words. Teachers should use these during the instructional process to engage student in thinking about the meaning of words.
Vocabulary Strips: Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.
word | definition | illustration |
Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.
Additional Resources for Vocabulary Development
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
What are the key ideas related to an understanding of equality at the third grade level? How do student misconceptions interfere with mastery of these ideas?
What kind of number sentences should third graders see related to equality in an instructional setting? Do third graders understand that the amounts are equal on each side of the equal sign? (working with number sentences such as 3 x K = 12 + 6)
Write a set of number sentences you could use with third graders in exploring their understanding of equality. Which are the most challenging for third graders?
When checking for student understanding, what should teachers
- listen for in student conversations?
- look for in student work?
- ask during classroom discussions?
Examine student work related to a task involving equality. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.
What are the key ideas related to identifying an unknown in an equation at the third grade level? How do student misconceptions interfere with mastery of these ideas?
What kind of equations should third graders experience when identifying an unknown in an equation?
Write a set of equations you could use with third graders in exploring their understanding of identifying unknowns in equation. Which are the most challenging for third graders?
When checking for student understanding, what should teachers
- listen for in student conversations?
- look for in student work?
- ask during classroom discussions?
Examine student work related to a task involving identifying unknowns in equations. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.
How can teachers assess student learning related to these benchmarks?
How are these benchmarks related to other benchmarks at the 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., and Johnson, A. (2006). Math matters: Understanding the math you teach, grades K-8. (2^{nd} ed.). Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.
Fosnot, C., & Dolk, M. (2002). Young mathematicians at work: Multiplication and division. Portsmouth, NH: Heinemann.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Otto, A., Caldwell, J., Wallus Hancock, S., & Zbiek, R.(2011). Developing essential understanding of multiplication and division for teaching mathematics in grades 3 - 5. Reston, VA.: National Council of Teachers of Mathematics.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J. (2009). Focus in grade 3, teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.
Burns, Marilyn. (2007). About teaching mathematics: A k-8 resource (3rd ed.). Sausalito, CA: Math Solutions Publications.
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Caldera, C. (2005). Houghton Mifflin math and English language learners. Boston, MA: Houghton Mifflin Company.
Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades K-8. (2nd ed.). Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.
Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 2 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
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.
Willoughby, S. S. (1997). Functions from kindergarten through sixth grade. Teaching Children Mathematics, Feb., 314-18.
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
- What values for M and R will make both sentences true?
M x R = 12
R x R = 16
Solution: M represents 3 and R represents 4.
Benchmark: 3.2.2.1
- Mr. Lindstrom packed 48 books into 6 boxes. Each box has the same number of books. Which number sentence represents this situation?
(a) 48 - 6 = __ (b) 48 + 6 =__ (c) 48 ÷ 6 = __ (d) 48 × 6 = __
Solution: c
Benchmark: 3.2.2.2, 3.2.2.1
- What number makes this number sentence true 3 + 5 = __ × 2?
Solution: 4
Benchmark: 3.2.2.1
- Which story problem can be represented using the number sentence 2 × n = 18?
A. Tom had 18 pencils. He gave some pencils away and had 2 left over. How
many pencils did Tom give away?
B. Alice bought some books and spent $18. Each book cost $2. How many
books did Alice buy?
C. Maya had some rocks and 2 baskets. She put 18 rocks in each basket. How
many rocks did Maya have?
D. Pedro saw 2 kinds of birds. He saw 18 robins and some crows. How many
crows did Pedro see?
Solution: B
Benchmark: 3.2.2.1, 3.2.2.2
MCA III Item Sampler
- An equation is shown. 3×7= __+7
Which number makes the number sentence true?
A. 3
B. 14
C. 21
D. 28
Solution: B. 14
Benchmark: 3.2.2.1
MCA III Item Sampler
Differentiation
Model equations involving unknowns with manipulatives.
Students should work with number sentences involving unknowns using multiplication and division basic facts they have mastered. This provides an opportunity for them to develop an understanding of writing and interpreting open number sentences which will, in turn, help them learn the facts they do not know.
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 k-3. Boston, MA: Pearson Education.
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
- Model equations involving unknowns with manipulatives using language to develop student understanding.
- Word banks need to be part of the student learning environment in every mathematics unit of study. Refer to these throughout instruction.
- Word 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:
The equation is __________ because ___________________________. |
The unknown is ______________ 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 expand their understanding of equality by writing true or false number sentences, involving multiplication and division, on slips of paper and placing them in a container. After drawing a slip of paper from the container, students determine if the number sentence is true or false.
Students can by write open number sentences on slips of paper and place them in a container. After drawing a slip of paper from the container, students tell or write a number story to match the open number sentence and then find the unknown.
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... |
solving problems involving unknowns. Using concrete materials in solving problems involving unknowns. | providing multiple experiences for students to solve problems involving unknowns. |
creating real world situations for an equation involving unknowns. | presenting multiple representations of problems; e.g., 4 x n = 12, n x 4 = 12, as well as 12 = 4 x n, 4 x n = 3 x 8 and 12 = n x 4. |
writing equations using unknowns such as: 4 x n = 12, n x 4 = 12, as well as 12 = 4 x n , 4 x n = 3 x 8 and 12 = n x 4. | providing opportunities for students to talk and work cooperatively in small groups to solve problems. |
talking with a partner about equations using unknowns. | asking students to justify their thinking when determining the value of an unknown. |
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