1.3.2C Money
Measure the length of an object in terms of multiple copies of another object.
For example: Measure a table by placing paper clips end-to-end and counting.
Tell time to the hour and half-hour.
Identify pennies, nickels and dimes; find the value of a group of these coins, up to one dollar.
Overview
Standard 1.3.2 Essential Understandings/Ideas
First graders move beyond direct comparison as they develop an understanding of linear measurement. They measure length by laying multiple copies of a unit end to end and counting the units. For example, the pencil is eight blocks long. First graders come to understand that using a larger unit when measuring means you will use fewer units, and using a smaller unit when measuring means you will use more units. When objects are being compared, first graders know that the units used to measure each object must be the same size.
When first graders use calendars or sequence events in stories, they are using measures of time in a real context. First graders learn to tell time to the hour and half-hour using analog and digital clocks. They relate these times to events during their day. For example, we eat lunch at 12:30pm.
First graders identify and know the value of dimes, nickels and pennies. They are able to count groups of dimes, nickels and pennies up a dollar.
All Standard Benchmarks - with codes
1.3.2.1 Measure the length of an object in terms of multiple copies of another object.
1.3.2.2 Tell time to the hour and half-hour.
1.3.2.3 Identify pennies, nickels and dimes; find the value of a group of these coins, up to one dollar.
Benchmark Group C
1.3.2.3 Identify pennies, nickels and dimes; find the value of a group of these coins, up to one dollar.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Identify pennies, nickels, and dimes.
- Know the value of a penny, a nickel and a dime.
- Count groups of pennies.
- Count groups of nickels.
- Count groups of dimes.
- Count a group of pennies, nickels and dimes up to a dollar.
- Use the cents symbol-¢.
Work from previous grades that supports this new learning includes:
- Count by ones to 120.
- Count by fives to at least 100.
- Count by tens to at least 100.
- Students may have real-life experiences with coins.
NCTM Standards
Understand measurable attributes of objects and the units, systems, and processes of measurement
Pre-K-2 Expectations
- Recognize the attributes of length, volume, weight, area, and time.
- Compare and order objects according to these attributes.
- Understand how to measure using nonstandard and standard units.
- Select an appropriate unit and tool for the attribute being measured.
- Apply appropriate techniques, tools, and formulas to determine measurements.
- Measure with multiple copies of units of the same size, such as paper clips laid end to end.
-
Use repetition of a single unit to measure something larger than the unit, for instance, measuring the length of a room with a single meter stick.
Common Core State Standards:
Measure lengths indirectly and by iterating length units.
1.MD.1. Order three objects by length; compare the lengths of two objects indirectly by using a third object.
1.MD.2. Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.
Tell and write time.
1.MD.3. Tell and write time in hours and half-hours using analog and digital clocks.
Misconceptions
Students may think...
- the size of coins indicates value. A nickel is worth more than a dime because it is larger.
- the value of coins depends on the "number " of coins. For example, one nickel is worth one and four dimes are worth four. It may be difficult for students to understand that the silver coins are composites representing many pennies.
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- Exploring coins with a magnifying glass helps students identify the characteristics of each type of coin.
- Using models that represent the composite coins with pennies gives support to students who are still counting by 1s or need support in exchanging pennies for nickels or dimes.
- Using manipulatives such as realistic plastic coins is essential for identifying and counting coins. Pictures are often misunderstood. It is easier for students to count coins when they can be moved.
- When children have difficulty finding the value of a group of coins, encourage them to sort the coins into separate piles first and then begin by counting the coins with the greatest value.
- As students count groups of coins, give a signal to indicate a new skip counting amount. For example, 10 (D), 20 (D) clap, 25 (N), clap, 26 (P), 27 (P), 28 (P).
- 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 U,s California Mathematics Council, 1995)
- Primary Economics
In this lesson, students play the role of a consumer as they learn how to use different combinations of coins to make money amounts up to 25 cents. Students will earn money and save it in their piggy banks until they have the exact amount to purchase an item of their choice.
Additional Instructional Resources
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.
New Vocabulary
"Vocabulary literally is the key tool for thinking."
Ruby Payne
cent sign:
penny:
nickel:
dime:
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 identifying and counting a group of coins (pennies, nickels, and dimes) at the first grade level? How do student misconceptions interfere with mastery of these ideas?
What experiences do students need in order to identify and count a group of coins (pennies, nickels, and dimes) successfully?
What common errors do first graders make when counting groups of coins?
When checking for student understanding of coin counting at the first grade level, 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 counting a group of coins (pennies, nickels, and dimes). 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 pre-k-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. (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.
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 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, M. (Ed.). (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.
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.
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.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
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.
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.
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.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Assessment
Performance Assessment
- Using real or plastic coins, ask students to find the value of the following combinations:
a. four dimes, three pennies
b. two dimes, two nickels
c. three dimes, four nickels, eleven pennies
d. six dimes, three nickels, seven pennies
Solution:
Benchmark: 1.3.2.3
- Using dimes, nickels and pennies, show the following in more than one way.
a. 38¢
b. 72¢
c. 19¢
d. 54¢
Solution: Correctly show each amount in at least two ways.
Benchmark: 1.3.2.3
Differentiation
Emergent learners benefit from working with pennies first and counting by ones. Placing pennies on a five or ten-frame and making connections to nickels and dimes will help them understand the value of coins. Emergent learners need to use actual coins when counting money amounts.
Students will need many experiences exchanging five pennies for a nickel and ten pennies for a dime.
Skip counting by 5's and 10's should be practiced before counting groups of nickels and groups of dimes.
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:
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.
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.
Students may recognize the difference between coins but not be able to use correct vocabulary in naming them. Labeled pictures provide a reference.
Understanding the value of a coin can be challenging since the value of one cannot be "built" from another even though they are related in value. Again, a chart that indicates the relationship between coins will provide immediate access for students.
- 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 the 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:
|
This is a _________________________. It is worth _____________________________. |
|
I count _________________ by ____________. |
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 Resource
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.
- After cutting coupons from the newspaper, students show each coupon amount in at least three ways.
- Students extend their learning by adding quarters and going beyond $1.00 when counting collections of coins.
- Using only dimes, nickels and pennies, how many ways can you make 25¢?
- How much money does each student have?
Read the clues and match the right amount to the correct child.
Clues:
Anna has 3 dimes and 3 pennies.
Carla has 1 coin.
Bob has more money than Carla, but less money than Anna.
Dave has 2 coins.
|
Student |
Amount |
|
Anna |
25¢ |
|
Carla |
35¢ |
|
Bob |
33¢ |
|
Dave |
28¢ |
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.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Parents/Admin
|
Students are: |
Teachers are: |
|
counting groups of coins dimes dimes and pennies dimes and nickels nickels and pennies dimes, nickels and pennies |
connecting the value of a collection of like coins to skip counting by the value of the coin. |
|
exchanging five pennies for a nickel, ten pennies for a dime, and two nickels for a dime. |
keeping the relationship between coins visible in the classroom. |
|
showing a given money amount using dimes, nickels and pennies. |
asking students to show an amount. |
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
Read Aloud Books
Alexander, Who Used to Be Rich Last Sunday, by Judith Viorst
The Coin Counting Book, by Rozanne Lanczak Williams
The Go-Around Dollar, by Barbara Johnston
Pigs Go to Market: Fun With Math and Shopping, by Amy Axelrod
Pigs Will Be Pigs, by Amy Axelrod