Show Solution
Step 1. the problem. | |
Step 2. what you are looking for. | the number |
Step 3. Choose a variable to represent the number. | Let n=the number |
Step 4. Restate the problem as one sentence. Translate into an equation. | 2n⇒ The sum of twice a number +⇒ and 7⇒ seven =⇒ is 15⇒ fifteen |
Step 5. the equation. | 2n+7=15 |
Subtract 7 from each side and simplify. | 2n=8 |
Divide each side by 2 and simplify. | n=4 |
Step 6. is the sum of twice 4 and 7 equal to 15? 2⋅4+7=15 8+7=15 15=15✓ | |
Step 7. the question. | The number is 4. |
Step 1. the problem. | ||
Step 2. what you are looking for. | You are looking for two numbers. | |
Step 3. Choose a variable to represent the first number. What do you know about the second number? Translate. | Let n=1st number One number is five more than another. x+5=2ndnumber | |
Step 4. Restate the problem as one sentence with all the important information. Translate into an equation. Substitute the variable expressions. | The sum of the numbers is 21. The sum of the 1st number and the 2nd number is 21. n⇒ First number +⇒ + n+5⇒ Second number =⇒ = 21⇒ 21 | |
Step 5. the equation. | n+n+5=21 | |
Combine like terms. | 2n+5=21 | |
Subtract five from both sides and simplify. | 2n=16 | |
Divide by two and simplify. | n=8 1st number | |
Find the second number too. | n+5 2nd number | |
Substitute n=8 | 8+5 | |
13 | ||
Step 6. | ||
Do these numbers check in the problem? Is one number 5 more than the other? Is thirteen, 5 more than 8? Yes. Is the sum of the two numbers 21? | 13=?8+5 13=13✓ 8+13=?21 21=21✓ | |
Step 7. the question. | The numbers are 8 and 13. |
Step 1. the problem. | ||
Step 2. what you are looking for. | two numbers | |
Step 3. Choose a variable. What do you know about the second number? Translate. | Let n=1st number One number is 4 less than the other. n−4=2ndnumber | |
Step 4. Write as one sentence. Translate into an equation. Substitute the variable expressions. | The sum of two numbers is negative fourteen. n⇒ First number +⇒ + n−4⇒ Second number =⇒ = −14⇒ -14 | |
Step 5. the equation. | n+n−4=−14 | |
Combine like terms. | 2n−4=−14 | |
Add 4 to each side and simplify. | 2n=−10 | |
Divide by 2. | n=−5 1st number | |
Substitute n=−5 to find the 2 number. | n−4 2nd number | |
−5−4 | ||
−9 | ||
Step 6. | ||
Is −9 four less than −5? Is their sum −14? | −5−4=?−9 −9=−9✓ −5+(−9)=?−14 −14=−14✓ | |
Step 7. the question. | The numbers are −5 and −9. |
Step 1. the problem. | ||
Step 2. what you are looking for. | two numbers | |
Step 3. Choose a variable. One number is ten more than twice another. | Let x=1st number 2x+10=2ndnumber | |
Step 4. Restate as one sentence. | Their sum is one. | |
Translate into an equation | x+(2x+10)⇒ The sum of the two numbers =⇒ is 1⇒ 1 | |
Step 5. the equation. | x+2x+10=1 | |
Combine like terms. | 3x+10=1 | |
Subtract 10 from each side. | 3x=−9 | |
Divide each side by 3 to get the first number. | x=−3 | |
Substitute to get the second number. | 2x+10 | |
2(−3)+10 | ||
4 | ||
Step 6. | ||
Is 4 ten more than twice −3? Is their sum 1? | 2(−3)+10=?4 −6+10=4 4=4✓ −3+4=?1 1=1✓ | |
Step 7. the question. | The numbers are −3 and 4. |
Step 1. the problem. | ||
Step 2. what you are looking for. | two consecutive integers | |
Step 3. | Let n=1st integer n+1=next consecutive integer | |
Step 4. Restate as one sentence. Translate into an equation. | n+n+1⇒ The sum of the integers =⇒ is 47⇒ 47 | |
Step 5. the equation. | n+n+1=47 | |
Combine like terms. | 2n+1=47 | |
Subtract 1 from each side. | 2n=46 | |
Divide each side by 2. | n=23 1st integer | |
Substitute to get the second number. | n+1 2nd integer | |
23+1 | ||
24 | ||
Step 6. | 23+24=?47 47=47✓ | |
Step 7. the question. | The two consecutive integers are 23 and 24. |
Step 1. the problem. | ||
Step 2. what you are looking for. | three consecutive integers | |
Step 3. | Let n=1st integer n+1=2nd consecutive integer n+2=3rd consecutive integer | |
Step 4. Restate as one sentence. Translate into an equation. | n+n+1+n+2⇒ The sum of the three integers =⇒ is 42⇒ 42 | |
Step 5. the equation. | n+n+1+n+2=42 | |
Combine like terms. | 3n+3=42 | |
Subtract 3 from each side. | 3n=39 | |
Divide each side by 3. | n=13 1st integer | |
Substitute to get the second number. | n+1 2nd integer | |
13+1 | ||
24 | ||
Substitute to get the third number. | n+2 3rd integer | |
13+2 | ||
15 | ||
Step 6. | 13+14+15=?42 42=42✓ | |
Step 7. the question. | The three consecutive integers are 13, 14, and 15. |
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Standard 1.2.2.
Represent real-world situations involving addition and subtraction basic facts, using objects and number sentences.
For example: One way to represent the number of toys that a child has left after giving away 4 of 6 toys is to begin with a stack of 6 connecting cubes and then break off 4 cubes.
Determine if equations involving addition and subtraction are true.
For example : Determine if the following number sentences are true or false
5 + 2 = 2 + 5
4 + 1 = 5 + 2.
Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as:
2 + 4 = $\square$
3 + $\square$ = 7
5 = $\square$ - 3.
Use addition or subtraction basic facts to represent a given problem situation using a number sentence.
For example : 5 + 3 = 8 could be used to represent a situation in which 5 red balloons are combined with 3 blue balloons to make 8 total balloons.
Standard 1.2.2 Essential Understandings
First graders build on their previous work with the composition and decomposition of numbers by writing number sentences to represent a real world or mathematical situation involving addition and subtraction. In addition, they will write a real world problem to represent a given number sentence.
Work with number sentences continues as first graders determine if a number sentence is true or false. For example, is 5 + 3 = 8 true or false? They are also able to write their own true number sentences and false number sentences.
First graders begin their work with variables by determining an unknown in a number sentence. These unknowns are found in varying positions in the number sentences.
For example, 5 + $\square$ = 8, 5 = $\square$ - 3, 3 + 5 = ∆ .
All Standard Benchmarks - with codes
1.2.2.1 Represent real-world situations involving addition and subtraction basic facts, using objects and number sentences.
1.2.2.2 Determine if equations involving addition and subtraction are true.
1.2.2.3 Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as: 2 + 4 = ∆ 3 + ∆ = 7 5 = ∆ - 3. 1.2.2.4 Use addition or subtraction basic facts to represent a given problem situation using a number sentence.
1.2.2.1 Represent real-world situations involving addition and subtraction basic facts, using objects and number sentences. 1.2.2.2 Determine if equations involving addition and subtraction are true. 1.2.2.3 Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as: 2 + 4 = ∆ 3 + ∆ = 7 5 = ∆ - 3. 1.2.2.4 Use addition or subtraction basic facts to represent a given problem situation using a number sentence.
What students should know and be able to do [at a mastery level] related to these benchmarks
3 + 6 = ∆ and ∆ = 3 + 6 3 + ∆ = 9 and 9 = 3 + ∆ ∆ + 6 = 9 and 9 = ∆ + 6 9 - 3 = ∆ and ∆ = 9 - 3 9 - ∆ = 6 and 6 = 9 - ∆ ∆ - 3 = 6 and 6 = ∆ - 3
Work from previous grades that supports this new learning includes:
NCTM Standards
Use mathematical models to represent and understand quantitative relationships.
Pre-K-2 Expectations
Common Core State Standards
Represent and solve problems involving addition and subtraction.
1.OA.1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions; e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
Work with addition and subtraction equations.
1.OA.7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
1.OA.8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 +? = 11, 5 = _ - 3, 6 + 6 = _.
Student Misconceptions and Common Errors
Students may think...
In the Classroom
First graders in Mr Xiong's class are determining if number sentences are true.
Mr. Xiong writes 3 + 5 = 8 on the board.
Mr. Xiong: I want you to think about this number sentence. Is this number sentence true or false? Skyler, what do you think?
Skyler: I think it's true.
Mr. Xiong: Why do you think it's true?
Skyler: Well, look. I have five here (flashes five fingers) and three here (flashes three fingers). That's eight! So it's true.
Mr. Xiong: Brie, I noticed that you were agreeing with Skyler. Why do you think it's true?
Brie: I started at the five and counted three more, 5 - 6..7..8..so I know it's true.
Mr. Xiong writes 8 = 3 + 5 on the board.
Mr. Xiong:. What about this number sentence? True or false? He looks around the room and sees some puzzled faces. 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 8 + 3 isn't 5, it's 11. Look at the other one, that's how you write it.
Mr. Xiong: Let's look at the first equation. Alan, can you read it for us?
Mr. Xiong points to 3 + 5 = 8.
Alan: Three plus five equals eight.
Mr. Xiong: Alan, can you read the problem another way?
Alan: Pauses and thinks for a moment. Three and five make eight.
Mr. Xiong: Okay...good. How about another way? Could anybody read it another way?
Beth: Three plus five is the same as eight.
Mr. Xiong: Good. Now, let's take a look at this problem. Tia, will you read this number sentence.
Mr Xiong points to 8 = 3 + 5.
Tia: Eight equals three plus five.
Mr. Xiong asks the class to read the number sentence with Tia as he points to each part of the number sentence.
Brie: I can read it another way. Eight is the same as three plus five.
Mr. Xiong: Points to 3 + 5 = 8 . Three plus five is the same as eight.
Points to 8 = 3 + 5. Eight is the same as three plus five.
Does it matter where we put the equal sign, at the beginning or at the end?
Skyler: It doesn't matter. It's just like turning things around, but it means the same thing.
Mr. Xiong used Alan's response as a springboard for the conversation about the meaning of the equal sign.
Teacher Notes
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
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
Asking first graders to determine if a number sentence is true or false is important as first graders develop a sense of equality. It is important to use number sentences with "easy numbers." Number sentences such as 5 = 5 and number sentences which are false (3 + 5 = 10) challenge student thinking. It is also important for first graders to consider the following types of number sentences: 6 = 4 + 2, 3 = 10 - 7, 4 + 2 = 2 + 4, 5 + 3 = 5 + 3.
The following number sentences might be used the first time first graders are asked to determine if a number sentence is true or false.
True or False?
3 + 2 = 5
2 + 3 = 5
5 = 3 + 2
2 + 3 = 4
5 = 5
5 = 3 + 3
2 + 3 = 2 + 3
2 + 3 = 2 - 3
Additional Instructional Resources
Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001). Navigating through algebra in prekindergarten-grade 2 . Reston, VA: National Council of Teachers of Mathematics.
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.
Von Rotz, L., & Burns, M. (2002). Grades k-2 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.
number sentence : mathematical sentence written in numerals and mathematical symbols
addition: to join two or more numbers to get one number (called the sum or total)
subtraction: an operation that gives the difference between two numbers. Subtraction can be used to compare two numbers, or to find out how much is left after some is taken away
equal: having the same amount or value, the symbol is =
" 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 Use: Multiple and varied opportunities to use the words in context. These 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.
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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 first grade level? How do student misconceptions interfere with mastery of these ideas?
What kind of number sentences should first graders see related to equality in an instructional setting?
Write a set of number sentences you could use with first graders in exploring their understanding of equality. Which are the most challenging for first graders?
When checking for student understanding, what should teachers
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 first grade level? How do student misconceptions interfere with mastery of these ideas?
What kind of equations should first graders experience when identifying an unknown in an equation?
Write a set of equations you could use with first graders in exploring their understanding of identifying unknowns in equation. Which are the most challenging for first graders?
Examine student work related to a task involving identifying unknowns. 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 first 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.
Carpenter, T., Franke, M., & Levi, L. ( 2003). Thinking mathematically integrating arithmetic & algebra in elementary school , Portsmouth, NH: Heinemann.
Chapin, S., & 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. (2010). Young mathematicians at work constructing algebra. 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.
Sammons, L. (2011). Building mathematical comprehension-using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions pre-k 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, M. (Edt).(1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
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.
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.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course . Sausalito, CA: Math Solutions.
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.
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 .
Sammons, L. (2011). Building mathematical comprehension-Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., ... & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: Aquest 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.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann .
There are seven turtles in the pond. Three more turtles go into the pond. How many turtles are in the pond?
Solution: 7 + 3 = 10 Benchmark: 1.2.2.4
7 - 2 = 5 Solution: Contexts will vary but will represent seven minus two equals five. Benchmark: 1.2.2.4
5 - 1 = 6 9 = 4 + 5 6 + 4 = 6 + 4 6 = 6 4 + 1 = 5 + 2 5 + 4 = 4 + 5 Solution: F, T, T, T, F, T Benchmark: 1.2.2.2
3 + 4 = ∆ 10 - ∆ = 6 8 = ∆ + 5 Solution: 7, 4, 3 Benchmark: 1.2.2.3
Students may need to use materials to find the unknown in open number sentences involving addition and subtraction. For example, when presented with 3 + ∆ = 7, a teacher may lay out three counters and ask how many more are needed to make seven.
Students may need to use materials in order to determine if a number sentence is true or false.
Concrete - Representational - Abstract Instructional Approach
The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.
The CRA approach is based on three stages during the learning process:
Concrete - Representational - Abstract
The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts. At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level. Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task. Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.
The Representational Stage is the drawing stage. Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems. They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking. Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.
The Abstract Stage is the symbolic stage. Teachers model mathematical concepts using numbers and mathematical symbols. Operation symbols are used to represent addition, subtraction, multiplication and division. Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding. Moving to the abstract level too quickly causes many student errors. Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.
Additional Resources:
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k- 2. 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.
Using materials to model the action in a story problem will help students write an equation which matches the problem.
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 think this number sentence is _____________ because _______________. |
I used ___________________ to find the missing number. |
I know __________________________ is true/false because _____________________. |
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 skills by writing true or false number sentences 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 write number stories for equations involving three addends having a sum less than or equal to twelve.
Students write their own number sentences involving unknowns and find the unknown in each other's number sentences.
What Number Goes Where?
Students are given a set of digits 0-9 on small cards. Each playing board has 10 missing numbers involving addition, subtraction or counting. Using each digit only once on the playing card, first graders fill in the missing numbers to make the number sentences true or to complete the counting sequence.
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 k-2. Sausalito, CA: Math Solutions.
Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann .
Administrative/Peer Classroom Observation
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determining if number sentences involving addition and subtraction are true or false and explaining their thinking.
| providing a sequence of number sentences that will challenge first graders' thinking about equality. Asking students to explain their thinking.
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finding the unknowns in open number sentences involving addition and subtraction and explaining their thinking.
| providing open number sentences involving addition and subtraction with the unknown in different positions. Listening to students' ideas and strategies for finding an unknown. |
writing number stories to match given number sentences involving addition and subtraction. | providing students with number sentences for students to write number stories. Noting strategies used to solve number sentences. |
writing number sentences to match number stories | providing students with number stories for students to write number sentences. Providing materials to assist students. |
Students are trying to solve problems with the use of objects and number lines to identify the missing numbers. | Teachers are encouraging them to come up with ways to solve the problem. Using vocabulary that is needed for understanding. |
What should I look for in the mathematics classroom? ( Adapted from SciMathMN,1997)
What are students doing?
What are teachers doing?
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.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
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
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.
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?
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
Read Aloud Books:
Domino Addition by Lynette Long, Ph.D.
The Hershey's Kisses Addition Book by Jerry Pallotta
One More Bunny: Adding From One to Ten by Rick Walton
In this number sentence activity, students learn the steps to write a number sentence to solve mathematical equations. Students solve a number sentence and then determine if there is another way to solve the problem.
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Learn how to use number sentences to solve a word problem.We hope you are enjoying this video! For more in-depth learning, check out Miacademy.co (https://ww...
Example 2: Complete the math sentence so that it is true. 6 + 7 = 9 + ―. Solution: 6 + 7 = 13. So, to make the sentence true, 9 + ― must be equal to 13. Therefore, the missing number must be 13 - 9 or 4. Example 3: Substitute the value into the variable (x) and state whether the resulting sentence is true or false.
A number sentence is a combination of numbers and mathematical operations that children are often required to solve. Examples of number sentences include: 32 + 57 = ? 5 x 6 = 10 x ? 103 + ? = 350. They will usually be composed of addition, subtraction, multiplication or division - or a combination of all four!
A number sentence is an arrangement of numbers and mathematical operation symbols. They are a common way of formatting math questions throughout the elementary school years. Children are taught how to solve these problems as well as how to write these types of math statements in class. Number sentences can be written in a variety of ways to ...
This video explains the Number Sentence Strategy for solving word problems. This strategy is most commonly used. It helps us identify the mathematical operat...
A Third Space Learning online tuition lesson using number sentences to solve problems. ... These math sentences or statements are introduced as a maths skill in Year 1, where pupils read, write and interpret mathematical statements involving mathematical symbols including addition (+), subtraction (-) and equals (=) signs.
Write a number sentence representing a real-world situation involving multiplication and division and unknowns. ... Use the four operations with whole numbers to solve problems. 4.OA.1. Interpret a multiplication equation as a comparison; e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. ...
A number sentence is a "mathematical sentence" used to express various mathematical relationships, namely equality and inequality. Number sentences are made up of: Numerals. Operations (addition, subtraction, multiplication, division, etc.) Equality / inequality symbols. Below are some examples of number sentences.
Word form can be difficult for students to figure out, but understanding the language of math can help to be able to write out and solve the number sentence that is being used in the problem.
A number sentence is an arrangement of numbers and symbols. Also referred to as a "sum" or "problem," number sentences are a common way of formatting questions in K-5 math. It's crucial that children learn this early, as it is how the majority of the work in their math lessons will look. "Number sentence" is the term that's used in K-5-level ...
A number sentence is a mathematical statement made up of two expressions and a relational symbol (=, >, <, etc). An equation is a number sentence whose relational symbol is the equal sign. An inequality is a number sentence whose relational symbol is anything else. Emphasizing the connection between equations and inequalities supports sense making.
Grade 1 - Unit 1 - Lesson 6G1 - 1-6 Problem-Solving Strategy Write a Number SentenceWatch on YouTube: https://youtu.be/XttfT9io8t4 Download The Application: ...
The problems within this post are meant to help students solve problems by writing a number sentence. These problems are designed to be used with first, second or third grade math students. Within this collection you will find two variations of each problem. You will easily be able to create additional problems using the wording below as a base.
Writing and Solving Word Problems 1.3.1 Write and solve number sentences from problem situations involving addition and subtraction. Author: Phylicia Created Date:
You need to know that there are 18 pencils and they are divided into 3 equal groups 2) PLAN:How can you solve the problem? You can write a number sentence to solve the problem. Write a division sentence to divide the pencils in 3 equal groups. 3) SOLVE: 18 ÷ 3 = 6 There are 6 pencils in each group.
Write a number sentence representing a real-world situation involving multiplication and division basic facts and unknowns. ... 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 ...
Write a number sentence with a variable for the unknown. Answer the question, "How many more players are needed?" Solution: 6 + n = 11, 5 more players are needed (n = 5) Benchmark: 2.2.2.2. Write a story to go with this number sentence: 42 - n = 28; Solve the number sentence for n.
Talking about basic concepts, understanding what number sentences are and how to write them is particularly important. A number sentence uses mathematical operations from addition, subtraction, multiplication to division. Symbols used in any number sentence vary depending upon what they indicate. These are numerical expressions of a word problem.
This Problem-Solving Strategy: Write a Number Sentence: Problem Solving Worksheet is suitable for 4th - 5th Grade. In this number sentence instructional activity, students use the problem solving strategy of understand, plan, solve, and look back to help them write number sentences to solve the word problem.
In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don't usually arise on an everyday basis, but they provide a good introduction to practicing the Problem-Solving Strategy. Remember to look for clue words such as difference, of, and and. Find the number. Step 1.
For example: Determine if the following number sentences are true or false. 7 = 7. 7 = 8 - 1. 5 + 2 = 2 + 5. 4 + 1 = 5 + 2. Benchmark: 1.2.2.3 Missing Numbers. Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as: 2 + 4 = .
Usage Permissions. This Write a Number Sentence: problem Solving Worksheet is suitable for 1st - 2nd Grade. For this number sentence worksheet, students learn the steps to write a number sentence to solve mathematical equations. Students solve a number sentence and then determine if there is another way to solve the problem.
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