Misconceptions and Error Patterns with Decimal Operations
Table of Contents
Facilitator Information and instructions 1
Raymond: Activity Sheet 2
Facilitator Notes 3–4
Afsana: Activity Sheet 5
Facilitator Notes 6–7
Jamil: Activity Sheet 8
Facilitator Notes 9–11
Yolanda: Activity Sheet 12
Facilitator Notes 13–14
Additional Sheets for Teachers:
Treatments of remainders in division situations 15
Some common errors and misconceptions about decimals 16
Decimal Outcomes for grades 4-6 17
What we learned from the EMLA about students' mistakes 18–19
Misconceptions and Error Patterns with Decimal Operations
Facilitator Notes
Sometimes students learn erroneous concepts and procedures. It is important that we be alert to these erroneous learnings.
In this workshop computational student work is presented for the operations of addition, subtraction, multiplication and division of decimals. Each student has made errors in his/her computations. Looking for misconceptions and error patterns will allow the teacher to gain specific knowledge about a student’s strengths and be able to plan further instruction for that student.
For students to have computational fluency, they need “conceptual understanding” – comprehension of mathematical concepts, operations and relationships – and “procedural fluency”- skilled in carrying out procedures flexibly, accurately, efficiently, and appropriately.
Teaching students rote procedures before teaching them the related concepts and how these concepts are applied to those procedures actually interferes with meaningful learning. Understanding the concepts and reasoning involved in an algorithm leads to a more secure mastery of the procedure.
Directions for facilitators:
This activity can be done as a think- pair- share. The purpose of this activity is to have teachers examine student work looking for error patterns. They will then have a discussion about what they notice, what might be some of the causes of these errors and how they can plan their instruction to help students.
Time : 3 hrs. +.
Materials:
ü 4-6 curriculum guide
ü Base-10 blocks
ü Alge-tiles
ü Activity handouts as indicated below.
1. Print off a copy of file 01B-Error patterns in decimals (01B_student_work_error_patterns_dec.doc) for each teacher. Do not staple; keep each sheet separate.
2. Pass out the first sheet on Raymond to each teacher. His errors are in the addition of decimals. Have the teachers do the sheet individually, and then discuss with a partner or group. A whole group discussion should follow.
3. Materials needed for Raymond are base-10 blocks, number lines, hundredth chart
4. The same process can then be done with each successive sheet of student work.
ü Afsana (subtraction of fractions) – base 10-blocks
ü Jamil (multiplication of fractions) – base-10 blocks, Alge-tiles, hundredth chart
ü Yolanda (division of fractions) –base-10 blocks
5. The additional sheets provided (pages 15–19) can be run off and given to teachers. It is important to discuss pages 18 and 19 (04_what_learned_from_EMLA_decimals.doc) with teachers as part of this workshop.
6. Activity handouts to be prepared for this session:
File / Instructions01B_student_work_error_patterns_dec.doc / print and use each page individually
02A_number_lines.doc
03_hundredth_charts.doc
A. Here is an example of Raymond’s work where he was practicing addition of decimals. The procedure he used resulted in incorrect answers.
ü Can you find a pattern in his work? Can you determine the misconception?
ü Discuss your ideas with a partner.
ü Think about and discuss what you will need to do to help Raymond.
Raymond:1. What error pattern do you notice? ______
Use Raymond’s procedure to complete these examples.2. What does he understand about decimals?
______
3. What does he not yet understand?
______
4. What would you do to help Raymond?
______
Raymond-
ü adds decimals as we would add whole numbers.
ü does not understand where to place the decimal because in each case it is placed to the left of the sum.
ü could be asked to say the problems in E and F out loud. In E he might say three-tenths plus five-tenths is eight-tenths which is correct. Then he goes on to say for F, seven-tenths plus seven-tenths is fourteen-tenths. What do you do?
What can be done to help Raymond?
1. Use estimation
Students should become proficient at estimating the answer to a decimal computation before they use paper and pencil. Students need to realize that 10 tenths make one whole, 100 hundredths make one whole etc.
Discuss with the teachers the proper kinds of questions to ask so students can estimate the sum:
§ Is the sum more or less than one? Why?
§ Is the sum between one and two? Why?
§ The sum is between what two whole numbers? Why?
§ The sum is close to what whole number? Why?
Then discuss each of Raymond’s problems asking teachers what questions they would ask to get students to estimate the answer.
If students realize, through estimation, that the sum to 0.6 + 0.9 must be greater than one because 0.9 is close to one, then they can make judgments about the incorrectness of their answer.
When working with bigger numbers, students can make estimates by rounding the numbers to friendly whole numbers.
Have the teachers estimate and discuss their thinking for the following sum
6.907 + 122.01 + 56.1234
When estimating, thinking focuses on the meaning of the numbers and the operation and not counting decimal places.
2. Use base-10 blocks
Teachers may refer to the 4-6 curriculum guide for information on how to use base-10 blocks to represent decimals and model the addition of decimals.
In grade 4, outcomes
A2- interpret and model decimal tenths and hundredths
A7- compare and order decimals with and without models
B1 – add and subtract decimals involving tenths and hundredths, and whole numbers to five digits
B12 – solve and create word problems involving adding and subtracting decimals (to hundredths)
B13- estimate sums and differences of whole numbers and decimals
C1 – demonstrate an understanding of the relationship of adding decimals and adding whole numbers
and in grade 5, outcomes
A2- interpret and model decimal tenths, hundredths, and thousandths
A7- read and represent decimals to thousandths
B1-find sums and differences involving decimals to thousandths
B8-solve and create addition and subtraction problems involving whole numbers and/or decimals
B10-estimate sums and differences involving decimals to thousandths
are outcomes that give this information. Remember that with decimals, the base-10 block that represents one will be different than operations on whole numbers. The EMLA results indicate that students are not comfortable with flexibility of thinking around changing what represents one whole so this must be carefully developed. When working with tenths and hundredths, the flat can represent one; when working with thousandths, the large cube can represent one.
The facilitator should model how to do one of Raymond’s 6 problems (where trading is necessary) and have the teachers work on the others. Teachers should be encouraged to model with base-10 blocks, record their model and put in the symbolic work beside the model.
3. Use a number line
Pass out file 02A-Number Lines. Work with the teachers, showing them how to record the addition using arrows as indicated below. Have them compare the sum with the symbolic work. Have teachers represent several of Raymond’s problems on a number line.
Explain to teachers that they should provide students with the number lines already constructed. If students draw them, the spacing for tenths may not be consistent.
4. Using a hundredth chart such as the one pictured below will help determine the reasonableness of his answers. Ask teachers to represent problem E on the hundredth chart. Pass out copies of file 03-Hundredth Charts. Have the teachers discuss with a partner how this can be done. Since the sum for E is less than one, representing the problem on a hundredth chart is easily done.
Then ask the teachers to discuss with a partner how the hundredth chart can help illustrate that the sum of fourteenths-hundredths in F is not a reasonable answer.
Summary: Students need to realize that when adding decimals they line up the place values. If students are told “to line up decimal points” that is a rote version that does not promote conceptual understanding.
B. Here is an example of Afsana’s work where she was practicing subtraction of decimals. The procedure she used resulted in incorrect answers.
ü Can you find a pattern in her work? Can you determine the misconception?
ü Discuss your ideas with a partner.
ü Think about and discuss what you will need to do to help Afsana.
Afsana:1. What error pattern do you notice? ______
Use Afsana’s procedure to complete these examples.2. What does she understand about decimals?
______
3. What does she not yet understand?
______
4. What would you do to help Afsana?
______
______
Afsana-
ü When the subtrahend has decimal parts (tenths, hundredths, etc.) but there are no corresponding decimal parts in the minuend, Afsana simply brings down these digits as part of the answer (difference).
ü When both numbers have digits in the same place value position, she has no problem subtracting correctly.
What can be done to help Afsana?
1. Use estimation
Students should become proficient at estimating the answer to a decimal computation before they use paper and pencil. Students need to realize that 10 tenths make one whole, 100 hundredths make one whole etc.
Discuss with the teachers the proper kinds of questions to ask so students can estimate the difference
§ Is the difference more or less than 10(or other appropriate number)? Why?
§ The difference is between what two whole numbers? Why?
§ The difference is close to what whole number? Why?
Then discuss each of Afsana’s problems asking teachers what questions they would ask to get students to estimate the answer or if estimating the answer first will help determine the error.
If students realize, through estimation, that the answer (difference) to 87 - .31 must be less than 87, then they can make judgments about the incorrectness of their answer.
When working with bigger numbers, students can make estimates by rounding the number to friendly whole numbers. Have the teachers estimate and discuss their thinking for the following subtraction
459.9 – 12.354
When estimating, thinking focuses on the meaning of the numbers and the operation and not counting decimal places.
2. Use base-10 blocks
Teachers may refer to the 4-6 curriculum guide for information on how to use base-10 blocks to represent decimals and model the subtraction of decimals decimals.
In grade 4, outcomes
A2- interpret and model decimal tenths and hundredths
B1 – add and subtract decimals involving tenths and hundredths, and whole numbers to five digits
B12 – solve and create word problems involving adding and subtracting decimals (to hundredths)
B13- estimate sums and differences of whole numbers and decimals
and in grade 5 outcome
A2- interpret and model decimal tenths, hundredths, and thousandths
A7- read and represent decimals to thousandths
B1-find sums and differences involving decimals to thousandths
B8-solve and create addition and subtraction problems involving whole numbers and/or decimals
B10-estimate sums and differences involving decimals to thousandths
are outcomes that give this information. Remember that with decimals, the base-10 block that represents one will be different than operations on whole numbers. The EMLA results indicate that students are not comfortable with the flexibility of thinking around changing what represents one whole so this must be carefully developed. When working with tenths and hundredths, the flat can represent one; when working with thousandths, the large cube can represent one.
The facilitator should model how to do one of Afsana’s 6 problems (where trading is necessary) and have the teachers work on the others. Teachers should be encouraged to model with base-10blocks, record their model and put in the symbolic work beside the model.
3. Teachers can also use the following with students: (it is not expected to work with these in this session)
ü a place value chart that has been relabeled for use with decimals.
ü a number line showing whole numbers and tenths
ü a hundredth grid
NOTE: More questions on the addition and subtraction of fractions can be found in the 4-6 curriculum guide on pages 4-19 and 5-27.
NOTE: Some sources indicate that money should also be used. Please read what is said about money and the conceptual development of decimals on page 4-4 in grade 4 of the 4-6 curriculum guide.
Summary: Usually the need to add or subtract decimals occurs when students are doing problems that involve the contexts of measurements or money. When working with measurements students need to be sure they are expressed in the same units and with the same precision before the numbers can be added or subtracted. Problems involving money often occur in the format that causes Afsana problems (e.g. $4 - $.35).
C. Here is an example of Jamil’s work where he practiced multiplication of decimals. The procedure used resulted in incorrect answers.
ü Can you find a pattern in Jamil’s work? Can you determine the misconception?
ü Discuss your ideas with a partner.
ü Think about and discuss what you will need to do to help Jamil.
Jamil:Add a decimal in front of the 5 in part b and the 4 in part c above
1. What error pattern do you notice? ______
Use Jamil’s procedure to complete these examples.2. What does he understand about decimals?