4.1.2B Decimals
Overview
Essential Understandings
Fourth graders expand their work with fractions to include representation of equivalent fractions. They use models to compare and order whole numbers and fractions, including improper fractions and mixed numbers. They are able to locate fractions on a number line. Fourth graders add and subtract fractions with like denominators and develop a rule for this action.
Fourth graders read, write and represent decimals in terms of thousands, hundreds, tens, ones, tenths, hundredths and thousandths. They use place value understanding, models, and number lines to compare and order decimals. They expand their understanding of rounding to include rounding of decimals to the nearest tenth.
Fourth graders use their knowledge of both fractions and decimals to read and write tenths and hundredths using both decimal and fraction notation. They know the decimal and fraction equivalents for halves and fourths.
All Standard Benchmarks
4.1.2.1
Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions.
4.1.2.2
Locate fractions on a number line. Use models to order and compare whole numbers and fractions, including mixed numbers and improper fractions. For example: Locate on a number line and give a comparison statement about these two fractions, such as "... is less than ..."
4.1.2.3
Use fraction models to add and subtract fractions with like denominators in real-world and mathematical situations. Develop a rule for addition and subtraction of fractions with like denominators.
4.1.2.4
Read and write decimals with words and symbols; use place value to describe decimals in terms of thousands, hundreds, tens, ones, tenths, hundredths and thousandths. For example: Writing 362.45 is a shorter way of writing the sum: 3 hundreds + 6 tens + 2 ones + 4 tenths + 5 hundredths, which can also be written as: three hundred sixty-two and forty-five hundredths.
4.1.2.5
Compare and order decimals and whole numbers using place value, a number line and models such as grids and base 10 blocks.
4.1.2.6
Read and write tenths and hundredths in decimal and fraction notations using words and symbols; know the fraction and decimal equivalents for halves and fourths. For example: = 0.5 = 0.50 and = = 1.75, which can also be written as one and three-fourths or one and seventy-five hundredths.
4.1.2.7
Round decimals to the nearest tenth. For example: The number 0.36 rounded to the nearest tenth is 0.4.
Benchmark Group B
4.1.2.4
Read and write decimals with words and symbols; use place value to describe decimals in terms of thousands, hundreds, tens, ones, tenths, hundredths and thousandths.
4.1.2.5
Compare and order decimals and whole numbers using place value, a number line and models such as grids and base 10 blocks.
4.1.2.6
Read and write tenths and hundredths in decimal and fraction notations using words and symbols; know the fraction and decimal equivalents for halves and fourths.
4.1.2.7
Round decimals to the nearest tenth.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Students will be able to:
- read and write decimals using words and symbolic notation
- describe decimals in terms of place value (thousands, hundreds, tens, ones, tenths, hundredths and thousandths)
- compare and order decimals
- place decimals on a number line
- represent tenths and hundredths in both fraction and decimal form
- know fraction and decimal equivalents for halves and fourths
- round decimals to the nearest tenth.
Work from previous grades that supports this new learning includes:
- read, write and represent whole numbers up to 100,000
- compare and order whole numbers
- used place value to represent numbers up to 100,000
- round whole numbers to the nearest ten thousand, thousand, ten and one.
NCTM Standards
Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Grade 3 - 5 Expectations:
- understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals;
- recognize equivalent representations for the same number and generate them by decomposing and composing numbers;
- develop understanding of fractions as parts of unit wholes, as parts of collection, as locations on number lines, and as divisions of whole numbers;
- use models, benchmarks, and equivalent forms to judge the size of fractions;
- recognize and generate equivalent forms of commonly used fractions, decimals, and percents;
- explore numbers less than 0 by extending the number line and through familiar applications;
- describe classes of numbers according to characteristics such as the nature of their factors.
Compute fluently and make reasonable estimates
Grade 3 - 5 Expectations
- develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30
Misconceptions
Student Misconceptions and Common Errors
Students may think...
- the word and can be used anywhere in a number and has no significance. They do not realize the use of the word and implies the decimal point.
- the words tens and tenths, hundreds and hundredths, thousand and thousandths are interchangeable.
- .52 is greater than .8 because 52 is greater than 8.
- decimals get smaller as the number of digits to the right of the decimal point increases. For example, students think .9 is less than .233 because two hundred thirty-three thousandths has more digits to the right of the decimal point...
Vignette
In the Classroom
Students in this fourth grade classroom are working on an activity that involves ordering decimals. The game they are playing incorporates a three by three grid and decimal cards. The object of the game is to place the cards in order from smallest to largest in each row and column. Students play in pairs, dealing five cards to each player and setting the extra cards to the side. Decimal cards in the "basic deck" are numbers in the tenths and hundredths. The numbers in the hundredths include but are not limited to "landmark" numbers, such as .25 and .75, and numbers that are often confused. For example, .01 and .1, and .5 and .05. Students wanting more of a challenge can add the "advanced cards" which include numbers in the thousandths. The object of the game is to be the first person to play all five cards.
The player with the smallest decimal starts the game in the spot indicated. The next player places a card with a greater decimal value to the right of the starting card or directly below it on the grid. In turn, players place cards. The card placed in the last spot of a row does not impact which number can be placed in the first spot in the next row. Only the column above or the card placed prior in that row affect the order of the cards.
Teacher stops at Group 1:
Student A: I will go first since I have the smallest number.
Teacher: That sounds like a fair way to decide who goes first. How do you know you have the smallest number?
Student A: The cards we are using only have numbers in the tenths and hundredths, and I have one-hundredth (card reads .01). Nothing else we are using is smaller.
Teacher nods and the student places the card in the top left corner of the board.
Student B: I am going to place one-tenth next to that card because I know a dime is more than a penny-which is one-hundredth.
Student A: Okay, I am going to the row below your one-tenth and putting two-tenths.
Student B: Why did you do that? Why didn't you finish the row?
Student A: It is part of my strategy. You will see.
Teacher: What strategy does this game require?
Student A: I know what numbers are in the deck and I know how many there are bigger than one-tenth and smaller than two-tenths. We can't play a smaller card behind or under a bigger one, so that is part of my strategy.
Student B: Hmmmm...I better start thinking about this.
Gameboard from Group 1 up to this point
Teacher moves to Group 2 (game already in progress).
Teacher: Wow, you are really making quick progress on the game board.
Group 2 game board at this point
Student C: Yeah, we are playing only with the thousandths cards. We should have mixed the deck.
Teacher: Why? Is it too challenging?
Student D: No, it is actually kind of boring. You know that all the cards will have to be in the thousandths, so there isn't much chance to surprise your opponent, and when we finish this game we are mixing all the cards together. Maybe we could add some whole number cards for next time we play. Can you or can we make some?
Teacher: We could consider that. Would you want just whole numbers or whole numbers and decimals mixed?
Student D: Maybe both- I'm not sure, but this game is about comparing numbers and making sure you lay your numbers in the right place at the right time. The more numbers and different types of numbers, the more options and more challenging.
Student C: It makes reading and thinking about the numbers a bigger deal when you have an opponent- this is a pretty good game.
Teacher: I'm glad you are enjoying it. I also like your ideas for improving it.
Teacher eventually makes her way back to Group 1:
Student B: I'm stuck. I can't go. I don't have a number smaller than six-tenths and that is what I need.
Student A: (to teacher) Can we add a rule?
Teacher: Well, we did come up with rules for this game as a class. What rule were you thinking of adding?
Student A: What if you had a one time "Go Fish" rule where you could have one chance to pull a card from the remaining deck?
Student B: Oh yeah, I like that rule. Plus, I could really use it right now!
Teacher: I like it too. What number would help you right now?
Student B: I know that there is a five-tenths card in the deck, and I could use that or the forty-five hundredths. Those are both less than six tenths but bigger than four tenths.
Teacher: Why don't you try the "Go Fish" rule. If you like it once you have tried it, add it to our list of rules on the board.
Student B: I'll try it right now.
At the end of the session, the teacher leads a discussion about student strategies used during the game. Some students had advanced ideas, while others were more basic. The following strategies were shared during the discussion:
- I always put my cards in order in front of me so I can see the range of numbers I have. I try to place them in order on each of my turns. I lay my lowest number, then the next lowest and so on.
- I lay my smallest number on my first turn and my largest on my second turn. I try to put the smallest in the top left corner and the greatest number in the bottom right. I don't play to block my opponent, I play to lay all of my cards!
- My strategy is to let my partner go first, and then lay my next biggest card compared to the card he laid directly behind or below his card. I try not to leave gaps.
- I didn't really have a strategy, but I was really lucky! Now I have some ideas to try!
At the conclusion of sharing their strategies, many pairs of students were interested in expanding the playing board to a 4x4 and coming up with alternative rules, all related to making the mathematics more challenging. Some suggested rules were:
- The first card played can go anywhere on the board.
- Add the "Go Fish" rule: If you do not have a card to play, draw one.
- The entire board must go in order from square 1 (top left corner) to square 9 (bottom left corner).
- Add an "Old Maid" rule: Instead of drawing from the pile, you have a "one time" chance to pull a card from your opponent's hand and give away one of yours.
- The partners play with cards face up and work together to place all of their cards with the smallest possible difference from square 1 to square 9. Compare differences with other students playing by this rule.
Resources
Teacher Notes
- Students could play a game to practice comparing decimals.
Using cards, with decimal numbers on them less than zero, they could play a game similar to "Compare" or "War." Each person in a pair could have half of a deck and then they compare their top cards. Each child could read their number and they can decide whose decimal is larger (or you could player smaller wins). If they are having trouble deciding, they could build it with coins.
You could even make a more challenging set (including thousandths) as students show they are ready. This would mean they had gone beyond relying on coins, since we do not have a thousandths coin in our currency.
- Comparing decimals challenges students as they are accustomed to seeing the number with the most digits having the largest value.
Take the case of comparing .44, .456, and .45. Look at the first digit following the decimal, if they are the same go to the second digit. In this case they all have four in the tenths place. In the hundredths however, forty-four hundredths is less than forty-five hundredths. Continue on to the next place value. Only one has a digit in the thousands, so that is our largest decimal.
- Visuals of grids divided into tenths, hundredths and thousands help students see how decimal numbers are related. It also helps students see the relationship between fractions and decimals. Students do not always hear or see the connections between the two when working with decimals.
- The Rational Number Project (Fraction Operations and Initial Decimal Ideas) provides researched based strategies and lessons supporting conceptual understanding of fractions and decimals including connections to operations with fractions and decimals.
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: Fraction operations and initial decimal ideas.
- Students should locate decimals on number lines with varying intervals. For example, where would the decimal .75 be located on the following number lines?
- 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.
NCTM Illuminations:
- Lesson Name: Numbers and Language: Post Office Numbers
Lesson Objectives: Students will discuss, describe, read, and write about whole numbers to thousands, decimals to hundredths, and common fractions.
- Web Based Activity: This website offers links to a wide variety of games including those that allow practice at ordering and rounding.
Additional Instructional Resources
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: Fraction operations and initial decimal ideas.
Duncan, N., Geer, C., Huinker, D., Leutzinger, L., Rathmell, E., & Thompson, C. (2007). Navigating through number and operations in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
tenth: one part of one whole divided into ten parts; the first place value to the right of the decimal point.
hundredth: one part of one whole divided into a hundred parts; the place value to the right of the tenths place.
thousandth: one part of one whole divided into a thousand parts; the place value to the right of the hundredths place.
"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 a word bank as the need arises. Students refer to a word bank which leads to greater understanding and application of words in context when communicating mathematical ideas.
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 students in thinking about the meaning of words.
Vocabulary Strips: Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.
word | definition | illustration |
Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.
Additional Resources for Vocabulary Development
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Professional Learning Communities
Reflection - Critical Questions regarding the teaching and learning of these benchmarks.
What are the key ideas related to the understanding of decimals at the fourth grade level? How do student misconceptions interfere with mastery of these ideas?
How would you know a student understands the decimal system when using numbers from .001 to .1?
What representations should a student be able to make for the number 365.472 if they understand place value?
What experiences do students need in order to develop an understanding of rounding decimals to the nearest tenth?
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 place value task involving decimals. 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 fourth 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.
Barnett-Clarke, C., Fisher, W., Marks, R., Ross, S., Charles, R., & Zbiek, R. (2010). Developing essential understanding of rational numbers for teaching mathematics in grades 3 - 5. Reston, VA.: National Council of Teachers of Mathematics.
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.
Empson, S., & Levi, L. (2011). Extending children's mathematics: Fractions and decimals. Portsmouth, NH: Heinemann.
Fosnot, C., & Dolk, M. (2002). Young mathematicians at work: Constructing fractions, decimals, and percents. 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.
Schielack, J. (2009). Focus in grade 4: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
The Rational Number Project (Fraction Operations and Initial Decimal Ideas) provides research based strategies and lessons supporting conceptual understanding of fractions and decimals including connections to operations with fractions and decimals.
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: Fraction operations and initial decimal ideas.
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Barnett-Clarke, C., Fisher, W., Marks, R., Ross, S., Charles, R., & Zbiek, R. (2010). Developing essential understanding of rational numbers for teaching mathematics in grades 3 - 5. Reston, VA.: National Council of Teachers of Mathematics.
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades 3-5. Sausalito, CA: Math Solutions Publications.
Burns, M. (Eds). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Cavanagh, M. (2004). Math to know: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades k-8, 2nd Edition. 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.
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: Fraction operations and initial decimal ideas. http://www.cehd.umn.edu/rationalnumberproject/rnp2.html
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5.Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC.: National Academies Press.
Duncan, N., Geer, C., Huinker, D., Leutzinger, L., Rathmell, E., & Thompson, C. (2007). Navigating through number and operations in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
Empson, S., & Levi, L. (2011). Extending children's mathematics: Fractions and decimals. Portsmouth, NH: Heinemann.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Fosnot, C., & Dolk, M. (2002). Young mathematicians at work: Multiplication and division. Portsmouth, NH: Heinemann.
Hyde, Arthur. (2006). Comprehending math: Adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010).Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
McNamara, J., & Shaughnessy, M. (2010). Beyond pizzas & pies: 10 essential strategies for supporting fraction sense, grades 3-5. Sausalito, CA. Math Solutions Publications.
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.
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.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J. (2009). Focus in grade 4: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
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 3-5. Boston, MA: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Assessment
Performance based:
- Benchmark 4.1.2.4:
Build the largest number you can by putting the following numbers into the blanks provided.
6, 8, 2, 4
____. ____ ____ ____
- Benchmark 4.1.2.7:
Round .438 to the nearest tenth.
- Benchmark 4.1.2.5
Place these decimals on the line above the fraction they represent: .75, .25, .50
- Benchmark 4.1.2.4:
Shade in the grid to show .45 or forty-five hundredths
- What is 9.582 rounded to the nearest tenth?
A. 9.5
B. 9.58
C. 9.6
D. 10
Solution: C. 9.6
Benchmark 4.1.2.7
- In the number 200.358, which digit is in the hundredths place?
A. 2
B. 3
C. 5
D. 8
Solution: C. 5
Benchmark: 4.1.2.4
- A decimal is shown on a grid.
Solution: D: 0.275
Benchmark: 4.1.2.5
- Which fraction is equivalent to 0.23?
A. [math]\frac{1}{23}[/math]
B. [math]\frac{23}{10}[/math]
C. [math]\frac{23}{100}[/math]
D. [math]\frac{2}{3}[/math]
Solution: C. [math]\frac{23}{100}[/math]
Benchmark: 4.1.2.6
Differentiation
- Using grids that model decimals in comparison to a whole are very beneficial in building understandings of what decimals represent.
- Students need to connect manipulatives with mathematics concepts. Representations using decimal grids will help students compare and order decimals. Decimal grids can help students translate between decimals and their respective fraction equivalents.
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 based on three stages during the learning process that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.
The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts. At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level. Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task. Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.
The Representational Stage is the drawing stage. Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems. They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking. Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.
The Abstract Stage is the symbolic stage. Teachers model mathematical concepts using numbers and mathematical symbols. Operation symbols are used to represent addition, subtraction, multiplication and division. Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding. Moving to the abstract level too quickly causes many student errors. Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5.Sausalito, CA: Math Solutions.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
English Language Learners
- English Language Learners need to connect manipulatives with mathematics concepts and the language used to communicate mathematical ideas. Using grids that model decimals in comparison to a whole are very beneficial in building understandings of what decimals represent. Representations using decimal grids will help students compare and order decimals and translate between decimals and their fraction equivalents.
- Word banks need to be part of the student learning environment in every mathematics unit of study. Refer to these throughout instruction.
- Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words such as count, first, second, third, etc.
- Sentence Frames
This decimal has ________ tenths, _________ hundredths and _______ thousandths.
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:
This decimal has ________ tenths, _________ hundredths and _______ thousandths. |
The decimal ________ is larger than the decimal ________ 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 3-5. Sausalito, CA: Math Solutions Publications.
Students who understand reading, writing and comparing decimals are most likely ready to practice operating on them using real-life situations. Money is always a great start for operating on decimals. The stamp problem from Figure This! has students using a combination of stamps to add up to $1.77, which is great practice in adding decimals to get to a predetermined total. (http://www.figurethis.org/challenges/c08/challenge.htm) Not a live site.
Solve and write decimal riddles.
- I'm less than 0.9, greater than 0.7, and 0.05 from 0.87. I am _______.
- We're greater than 0.75, less than 1.2, and 0.04 from 0.92. We are ____.
- I'm less than 0.4, greater than 0.12, and 0.07 from 0.35. I am ______.
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: |
reading, writing and representing decimals. For example, they write .65, say sixty-five hundredths, and shade sixty-five hundredths on a blank hundredths grid. | listening for correct word usage and watching to see if what students write matches what they say, and vice-versa. |
describing decimals in terms of thousands, hundreds, tens, ones, tenths, hundredths and thousandths. | supporting the description of decimals using place value terms with physical models and other representations. |
comparing and ordering decimals with and without number lines and justifying the placement of decimals. | asking for and listening to mathematical justifications or reasons for decimal ordering. |
translating between equivalent decimal and fraction notations. | providing representations that facilitate the connection of decimals and their corresponding fraction representation. |
rounding decimals to the nearest tenth and explaining their reasoning. | checking student strategies for accurate rounding. |
What should I look for in the mathematics classroom?
(Adapted from SciMathMN,1997)
What are students doing?
- Working in groups to make conjectures and solve problems.
- Solving real-world problems, not just practicing a collection of isolated skills.
- Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
- Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
- Recognizing and connecting mathematical ideas.
- Justifying their thinking and explaining different ways to solve a problem.
What are teachers doing?
- Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
- Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
- Connecting new mathematical concepts to previously learned ideas.
- Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
- Selecting appropriate activities and materials to support the learning of every student.
- Working with other teachers to make connections between disciplines to show how math is related to other subjects.
- Using assessments to uncover student thinking in order to guide instruction and assess understanding.
Additional Resources
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k-8, 2nd edition. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). (2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
For Administrators
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Parent Resources
Mathematics handbooks to be used as home references
Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Helping your child learn mathematics
Provides activities for children in preschool through grade 5
What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN.
Help Your Children Make Sense of Math
Ask the right questions
In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.
Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if...?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Can you make a prediction?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work?
Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part...
Adapted from They're counting on us, California Mathematics Council, 1995
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