Finding and Fixing Errors in Rational Number Work | Grade 7 Math | LevelKite

A rational number is a number you can write as a fraction of two integers (numbers such as $-2$ , $0$ and $7$ ) with a denominator, or bottom number, that is not $0$ . Examples are $-\frac{3}{4}$ , $2.5$ and $-6$ . You already add, subtract, multiply and divide these numbers.

Now you take on a different job: reading someone else's finished work and judging it. This is error analysis. You find the first step that is not correct, explain why it is not correct, and fix the work from that step on.

A wrong final answer tells you only that something went wrong. Error analysis tells you what went wrong, so you can avoid the same slip in your own work.

Each line of someone's work makes a claim. Most lines are equations, which are statements with an equals sign. An equation claims that its left side and its right side have the same value. A line is valid when its claim is true and it starts from exactly what the problem and the earlier lines give. Otherwise the line is not valid.

A line can be valid even when it comes after a mistake. Suppose line 1 is not valid, and line 2 is worked out correctly from what line 1 says. Line 2 carries a wrong value, but the mistake did not happen there. So you look for the first line that is not valid, and you keep testing the later lines, because work can hold more than one error.

Key concept

Test each line against its rule

Read the work one line at a time, in order. For each line, name the operation it uses (addition, subtraction, multiplication or division) and test the line against the rule for that operation, including the condition that makes the rule true. The first line that fails is the first error. Say which rule it breaks, write a valid line in its place, and carry the correct value through the lines after it. A later line that is worked out correctly from a wrong value is not a new error. Keep testing to the end, because work can hold more than one error.

The rules you test against are ones you already use. Here are the words they need. Let $p$ and $q$ stand for any rational numbers.

The additive inverse, or opposite, of $q$ is $-q$ . It is the number that adds to $q$ to give $0$ . For example, the additive inverse of $-1.25$ is $1.25$ .
The absolute value $|q|$ is the distance of $q$ from $0$ , so it is never negative. For example, $|-6| = 6$ .
The direction of $q$ is positive, toward greater numbers, when $q$ is positive. It is negative, toward lesser numbers, when $q$ is negative. The number $0$ has no direction.
A factor is a number being multiplied. A sum is the result of adding, a product is the result of multiplying, and a quotient is the result of dividing.
A number is nonzero when it is not $0$ . The sign of a nonzero number is positive or negative.

The table gives each rule with the condition that makes it true. Using a rule outside its condition is a common cause of errors.

Rules to test a line against, where

p

and

q

are rational numbers

The line uses	Rule	Condition
Subtraction	$p - q = p + (-q)$ : subtracting $q$ adds the additive inverse of $q$	True for every $p$ and $q$
Addition	$p + q$ is $\lvert q \rvert$ away from $p$ , in the direction of $q$	True for every $p$ and $q$ ; if $q = 0$ , the sum is $p$
Adding opposites	$p + (-p) = 0$	True for every $p$
Sign of a product or quotient	Negative exactly when the count of negative numbers is odd; positive when that count is even	Only when every number multiplied or divided is nonzero
A factor of $0$	The product is $0$	Whenever any factor is $0$
$0$ divided by $q$	$0 \div q = 0$	Only when $q$ is not $0$
Division by $0$	Undefined: there is no value	For every number divided by $0$ , including $0$

Worked example

A subtraction slip

A student found $-3.5 - (-1.25)$ in two lines. Line 1: $-3.5 - (-1.25) = -3.5 - 1.25$ . Line 2: $-3.5 - 1.25 = -4.75$ . Find the first line that is not valid, explain why, and correct the work.

Test line 1 against the subtraction rule: subtracting a number adds its additive inverse. The additive inverse of negative 1.25 is 1.25, because the two add to 0. So a valid line 1 changes the subtraction into adding 1.25.
$-3.5 - (-1.25) = -3.5 + 1.25$
Compare this with the student's line 1. The student subtracted 1.25, and subtracting 1.25 adds negative 1.25. The rule gives adding 1.25, not adding negative 1.25, so the two sides of line 1 have different values. Line 1 is not valid, and it is the first error.
$-3.5 - 1.25 = -3.5 + (-1.25) \neq -3.5 + 1.25$
Test line 2 by itself. Subtracting 1.25 adds its additive inverse, negative 1.25. The sum is 1.25 away from negative 3.5 in the negative direction, because negative 1.25 is negative. That gives negative 4.75, so line 2 is valid. It carries a wrong value only because it starts from line 1.
$-3.5 - 1.25 = -3.5 + (-1.25) = -4.75$
Correct the work from line 1. The sum is now 1.25 away from negative 3.5 in the positive direction, because the number added, 1.25, is positive.
$-3.5 + 1.25 = -2.25$
Check the new answer by adding. The answer plus the number you subtracted must give the number you started with. Negative 2.25 plus negative 1.25 is negative 3.5, the starting number, so the answer fits.
$-2.25 + (-1.25) = -3.5$

Answer

Line 1 is the first line that is not valid, because subtracting $-1.25$ means adding $1.25$ . The correct value is $-3.5 - (-1.25) = -2.25$ .

Worked example

A sign lost in a division

A student found $\left(-\frac{2}{3}\right) \times \left(-\frac{3}{4}\right) \div \left(-\frac{1}{6}\right)$ in three lines. Line 1: $\left(-\frac{2}{3}\right) \times \left(-\frac{3}{4}\right) = \frac{1}{2}$ . Line 2: $\frac{1}{2} \div \left(-\frac{1}{6}\right) = \frac{1}{2} \times 6$ . Line 3: $\frac{1}{2} \times 6 = 3$ . Find the first line that is not valid and correct the work.

Start with a quick sign check: decide whether the answer must be positive or negative. The calculation only multiplies and divides, and none of its 3 numbers is 0, so the result is negative exactly when the count of negative numbers is odd. All 3 numbers are negative and 3 is odd, so the result must be negative. The student's answer, 3, is positive, so at least one line is not valid.
Test line 1. Both factors are nonzero and 2 of them are negative. Since 2 is even, the product is positive. Multiply the top numbers and multiply the bottom numbers. Then divide the top and bottom by the same nonzero number, 6, which keeps the value. Line 1 is valid.
$\left(-\frac{2}{3}\right)\times\left(-\frac{3}{4}\right) = \frac{2\times 3}{3\times 4} = \frac{6}{12} = \frac{1}{2}$
Test line 2. Dividing by a nonzero number is the same as multiplying by its reciprocal, the number that multiplies with it to give 1. The reciprocal of negative one-sixth is negative 6, because their product is 1. The student multiplied by 6, but negative one-sixth times 6 is negative 1, not 1. Line 2 is not valid, and it is the first error.
$\left(-\frac{1}{6}\right)\times(-6) = 1 \qquad \left(-\frac{1}{6}\right)\times 6 = -1$
Write a valid line 2. The number you divide by is not 0, so you can change the division into multiplication by its reciprocal, negative 6.
$\frac{1}{2}\div\left(-\frac{1}{6}\right) = \frac{1}{2}\times(-6)$
Line 3 was correct arithmetic for the value it started from, so redo it with the corrected value. Both factors are nonzero and exactly 1 is negative. Since 1 is odd, the product is negative. Half of 6 is 3, so the product is negative 3.
$\frac{1}{2}\times(-6) = -\frac{6}{2} = -3$
Check in two ways. The answer is negative, which agrees with the sign check. Also, the answer times the number you divided by must give back the number you divided, one-half. Both factors are nonzero and 2 are negative, an even count, so that product is positive.
$-3\times\left(-\frac{1}{6}\right) = \frac{3}{6} = \frac{1}{2}$

Answer

Line 2 is the first line that is not valid, because the reciprocal of $-\frac{1}{6}$ is $-6$ , not $6$ . The correct value is $-3$ .

The sign check has conditions, and they matter. It works only for multiplication and division, and only when every number is nonzero.

Zero changes the test. Any factor of $0$ makes the product $0$ . Since $0$ is neither positive nor negative, counting negative numbers no longer tells you anything. Division needs two separate rules. Zero divided by a nonzero number is $0$ . Division by $0$ is undefined, which means it has no value at all.

Remember also what a quick check can and cannot do. When the sign check applies and the sign of the answer is wrong, at least one line is not valid. When the sign is right, the size can still be wrong, so you still test each line. The table tests four single lines against these rules.

Four single lines: valid or not valid?

Student's line	Valid?	Reason	Correct line
$0 \div \left(-\frac{3}{4}\right)$ is undefined	No	$-\frac{3}{4}$ is not $0$ , and $0$ divided by a nonzero number is $0$	$0 \div \left(-\frac{3}{4}\right) = 0$
$-2.5 \div 0 = 0$	No	Division by $0$ is undefined: no number times $0$ equals $-2.5$	$-2.5 \div 0$ is undefined
$(-7) \times 0 \times (-1.2) = 8.4$	No	One factor is $0$ , so the product is $0$ ; counting negative numbers does not apply	$(-7) \times 0 \times (-1.2) = 0$
$(-2) \times (-0.5) \times (-3) = -3$	Yes	No factor is $0$ ; $3$ factors are negative and $3$ is odd; $2 \times 0.5 \times 3 = 3$	No change needed

Worked example

A diver's elevation

Elevation is measured in meters from sea level: positive above it, negative below it. A diver starts at $-12.5$ meters. The diver's elevation changes by $-1.5$ meters each minute for $4$ minutes. Then the diver rises $3.75$ meters. A student wrote: Line 1: $4 \times (-1.5) = -6$ . Line 2: $-12.5 + (-6) = -6.5$ . Line 3: $-6.5 + 3.75 = -10.25$ . Find every line that is not valid and the correct final elevation.

Test line 1. The elevation changes by negative 1.5 meters in each of 4 minutes, so the total change is 4 times negative 1.5 meters. Both factors are nonzero and exactly 1 is negative. Since 1 is odd, the product is negative, and 4 times 1.5 is 6. Line 1 is valid: the change is negative 6 meters.
$4\times(-1.5) = -6$
Test line 2. The sum is 6 away from negative 12.5 in the negative direction, because the number added, negative 6, is negative. So the sum is less than negative 12.5. The student's negative 6.5 is greater than negative 12.5, so line 2 is not valid. It is the first error.
$-12.5 + (-6) = -18.5$
Keep testing. Line 3 starts from the value in line 2, negative 6.5, and adds 3.75. The sum is 3.75 away from negative 6.5 in the positive direction, because 3.75 is positive, so it is greater than negative 6.5. The student's negative 10.25 is less than negative 6.5. Line 3 is a second error.
$-6.5 + 3.75 = -2.75 \neq -10.25$
Correct the work. Line 3 must start from the corrected line 2, negative 18.5. The sum is 3.75 away from negative 18.5 in the positive direction.
$-18.5 + 3.75 = -14.75$
Check against the situation by adding the two changes first. Down 6 meters and up 3.75 meters make a total change of negative 2.25 meters, so the diver must finish below the start. Negative 14.75 is less than negative 12.5, so the answer fits. The student's negative 10.25 is greater than negative 12.5, so it fails this check.
$-12.5 + (-6 + 3.75) = -12.5 + (-2.25) = -14.75$

Answer

Lines 2 and 3 are not valid. Line 2 is the first error, because $-12.5 + (-6) = -18.5$ , and line 3 moves in the wrong direction. The diver's final elevation is $-14.75$ meters, which is $14.75$ meters below sea level.

Takeaway. Error analysis means testing each line of work, in order, against the rule for the operation in that line. Use each rule only under its condition. For example, count negative numbers only when you multiply or divide nonzero numbers, use distance and direction for a sum, and never give a value to division by $0$ . The first line that fails its test is the first error. Name the rule it breaks, write a valid line in its place, carry the correct value forward, and keep testing to the end.