Diagnosing Error Patterns for Elementary Students

Session II – Multiplication, Division, Fractions, Expanded Notation

Multiplication

Determine the error pattern in the following:

Multiplication Student records the products sequentially with no regard

for partial products.

This type of error occurs early in the partial products multiplication process. Student needs both place value andknowledge of algorithmic structure.

Multiplication

Determine the error pattern in the following:

Multiplication

Student uses partial products

but does not account for the

place value when

recording and summing.

Work with the distributive property is recommended to recognize the correct value of the partial products: 54 x 6 = (50 x 6) + (4 x 6). Grid paper is always a good idea to reinforce this concept.

Multiplication

Determine the error pattern in the following:

Multiplication

Student has partial understanding

of the place value determined

by the multiplier but not all

partial products are recorded.

Again work with the distributive property (and grid paper) is recommended:

38 x 74 = (8 x4) + (30 x 4) + (8 x 70) + (30 x 70)

Division

Determine the error in the following:

Division

Problems with zero always have the potential for the student to skip that value place when computing.

Recommended would be an alternate division algorithm using repeated subtraction.

DivisionDetermine the error in the following:

Division

Remainder is recorded as part of the quotient.

Recommended is a clearer understanding that the remainder is a fractional representation of a portion of the divisor. 5 4) 23 5 ¾ or 5 with a remainder of 3 20 3

Fractions - Comparative Value

Determine the error pattern in the following:

Fractions – Comparative Value

Student does not understand fractions in terms of how the numerator and denominator function. Larger is assumed to be better. Note the on the last example the student also ignores the improper fraction.

Recommended is lots of practice with area fraction models and fraction bars.

Fractions - Computation

Determine the error pattern in the following:

Fractions - Computation

Student is using an additive or subtractive process instead of converting to like denominator.

The first suggestion would be to more firmly establish common fractions with area models. From there use of fraction bars to establish to trade for common denominators, then add or subtract.

Fractions - Multiplication

Determine the error pattern in the following:

Fractions - Multiplication

Student is incorrectly converting the second factor as that number over itself (1) instead of the number over 1.

1/2 x 3 = 3/6 is 1/2 x 3/1, not 3/6 x 3/3

Recommended estimating the answer using a repeated addition process.

Measurement – Elapsed Time

Determine the error pattern in the following:

George completed his “Iron Chef” competition in 3 hours, 18 minutes, 56 seconds. Roger completed the same competition in 2 hours, 32 minutes and 12 seconds. How much quicker was Roger than George?

Measurement –Elapsed TimeThis type of problem is related to the subtraction error

pattern where students take the smaller value from the larger regardless value.

Incorrect: Correct:

Problems like this – time, capacity, distance - require practice converting the regrouping according to the measure. It is more complicated because it is not always a base ten regrouping.

Multiplication with Decimals

Determine the error in the following:

Multiplication - Decimals

Student is correctly multiplying the factors but is not correctly using decimal place.

Many students learn to multiply with decimals using counting formulas. They add the decimal pales top and bottom to determine location in the product. While this can be accurate in an algorithmic sense, it can cause errors when are not careful. Better is to stress estimation strategies.

Expanded Notation

Determine the error in the following:

Expanded NotationStudent is not processing the zero placeholder correctly and using digits instead of

values for number interpretation.

407 = 40+7 Student is seeing the 40 as forty instead of 4 hundreds and 0 tens.

3,206 = 3,000+20+6 Student understands that the number begins in the thousands place, but processes the 20 as twenty instead of 2 hundreds and 0 tens.

44,056 = 4,000+400+50+6 Student does not process the zero instead using an incorrect left to right strategy beginning in the thousands place.

Practice is needed with place value understanding number what the digits represent in a numerical sequence containing a zero. Initial work with base ten blocks with mats to build number sense is recommended with number containing zero. Using a number strategy with zero as a placeholder is a good scaffolding technique. Example: 3,206 = 3 thousands, 2 hundreds, 0 tens, and 6 ones

= 3,000+200+0+6

= 3,000+200+6

Expanded Notation

Determine the error in the following:

Expanded NotationStudent does not have a good understanding of place value with decimal

values.

3.14  = 3 + .10 + .4

.207 = .200 + .7

1.618 = 1 + .6 + .18

In all three examples the student is trying to process the digits to the right of the decimal point either additively or sequentially or both. This is a difficult error pattern to break since students want to apply what they have learned about addition without full understanding of place value. Base ten blocks are recommended, but it requires patience since you are altering previously established values for this manipulative.

Example using 1.618 = 1 thousands cube (ones place), 6 hundreds flats (tenths place), 1 tens rod (hundredths place) and 8 unit cubes (thousandths place)

Resources

Error Patterns in Computation Ashlock, Robert B.

http://mathforum.org http://www.floridatechnet.org Special Thanks to the Teachers from the

Pinellas County School District