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6th Grade · 6.NS.C.7

Absolute value in 6th grade

Absolute value is how far a number is from 0 on the number line, so ∣−7∣=7|-7| = 7 and ∣7∣=7|7| = 7. In 6th grade, your child also puts positive and negative numbers in order and explains what the order means in real life, such as which temperature is colder. Part of the work is knowing when to compare the numbers themselves and when to compare their absolute values.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.NS.C.7 Understand ordering and absolute value of rational numbers.

  • 6.NS.C.7.a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.
  • 6.NS.C.7.b Write, interpret, and explain statements of order for rational numbers in real-world contexts.
  • 6.NS.C.7.c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
  • 6.NS.C.7.d Distinguish comparisons of absolute value from statements about order.

What it means, in plain words

On a number line, a number to the right is always greater than a number to the left. An inequality is a statement such as −8<−2-8 < -2, read "negative 8 is less than negative 2". It is true because −8-8 sits to the left of −2-2. Comparing integers this way works for any pair, even two negative numbers. Ordering rational numbers (fractions and decimals, positive or negative) uses the same rule: place them on the line and read from left to right.

Order has a meaning in real life. A temperature of −10-10 degrees is colder than −4-4 degrees, because −10<−4-10 < -4. A diver at −30-30 feet is deeper than a diver at −5-5 feet. Your child writes and explains statements like these in words.

Absolute value is the distance from a number to 0, written with two bars: ∣−12∣=12|-12| = 12. A distance is never negative. In real life, absolute value gives the size of an amount without its direction. A bank balance of −25-25 dollars means a debt of 25 dollars, and ∣−25∣=25|-25| = 25 is the size of that debt. Absolute value in 6th grade also means telling the two kinds of comparison apart. A balance below −25-25 dollars is a debt of more than 25 dollars.

Your child should be able to:

  • Read an inequality such as −8<−2-8 < -2 as a statement about where two numbers sit on a number line
  • Order integers, fractions and decimals, including negative ones, from least to greatest
  • Find the absolute value of a number and explain it as a distance from 0
  • Tell when a real-life question asks about order and when it asks about absolute value

Example

Put these numbers in order from least to greatest: 33, −5-5, −34-\frac{3}{4}, 1.251.25 and −2.4-2.4.

  1. The negative numbers are the least. The one farthest from 0 comes first: −5-5, then −2.4-2.4, then −34-\frac{3}{4}.
  2. Then come the positive numbers, from smallest to largest: 1.251.25, then 33.
  3. From least to greatest, the order is −5-5, −2.4-2.4, −34-\frac{3}{4}, 1.251.25, 33.

Example

One morning it is −8-8 degrees Fahrenheit in one town and −3-3 degrees Fahrenheit in another town. Which town is colder, and which temperature has the greater absolute value?

  1. −8-8 is to the left of −3-3 on the number line, so −8<−3-8 < -3. The first town is colder.
  2. ∣−8∣=8|-8| = 8 and ∣−3∣=3|-3| = 3, so −8-8 has the greater absolute value.
  3. The first town is colder, and its temperature is also farther from 0.

Where children usually get stuck

  • Your child writes −9>−2-9 > -2, because 9 is bigger than 2.

    With positive numbers, the bigger number wins, and the child carries that rule over. For negative numbers, the one farther from 0 is the smaller one.

    What helps. Ask which is colder, 9 degrees below zero or 2 degrees below zero. The colder temperature is the smaller number.

  • Your child writes ∣6∣=−6|6| = -6.

    Absolute value was learned as "change the sign", so the child flips every number, positive ones too.

    What helps. Ask how many steps it takes to walk from 0 to 6, and then from 0 to −6-6. Both answers are 6, and a number of steps is never negative.

  • Asked who owes more money, a person with a balance of −40-40 dollars or a person with −15-15 dollars, your child picks −15-15 because −15>−40-15 > -40.

    The order is right, but the question is about the size of the debt, which is the absolute value. The child used the greater-than rule where absolute value was needed.

    What helps. Ask how many dollars each person owes. Owing 40 dollars is more than owing 15 dollars, even though −40-40 is the smaller number.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

  1. 1Warm-up
    Look at the inequality below.
    −2<3\displaystyle -2 < 3
    Select the number line that matches the inequality.
    A
    -505-23
    B
    -505-23
    C
    -505-23
    D
    -505-23
    Show the answer and how to solve it

    The correct diagram places -2 on the left side of 0 and 3 on the right side of 0.

    1. 1Read the inequality. It says -2 is less than 3.
    2. 2On a number line, greater numbers are farther right. -2 is negative, so it sits left of 0. 3 is positive, so it sits right of 0.
    3. 3The matching diagram must show -2 to the left of 0 and 3 to the right of 0.

    Common mistake. A common mistake is swapping the two points or placing both on the same side of 0. -2 is negative and belongs left of 0; 3 is positive and belongs right of 0.

    The idea underneath. On a number line, greater always means farther right, even when both numbers are negative.

  2. 2On level
    Lily's claim: "The inequality and number line below match each other."
    −3<−1\displaystyle -3 < -1
    -4-3-2-1012-3-1
    Select the best description of Lily's mistake.
    A
    The two negative numbers are reversed on the number line.
    B
    One point is one subdivision away from where it should be.
    C
    A negative number is on the wrong side of 0.
    D
    Both values should sit between integer ticks, not on whole-number marks.
    Show the answer and how to solve it

    The answer is that the two negative numbers are reversed on the number line because -3 should be to the left of -1, not to the right.

    1. 1Read the inequality first. It says -3 is less than -1.
    2. 2On a number line, numbers get greater as you move to the right. For two negative numbers, the one farther left is smaller, so -3 should appear left of -1.
    3. 3In Lily's diagram, the label -3 is placed at the position of -1, and the label -1 is placed at the position of -3. The two labels are fully swapped.
    4. 4So the best description is that the two negative numbers are reversed on the number line.

    Common mistake. A common mistake is to assume the error must involve the side of 0 because both values are negative. Both -3 and -1 are already on the left of 0; the only problem is their left-right order.

    The idea underneath. For negative numbers, farther left means smaller, even when the digits look larger.

  3. 3On level
    Look at the inequality below.
    −3>−7\displaystyle -3 > -7
    Drag the two number tiles to show the numbers from left to right on a number line.
    Drag a card to move it, or tap a card, then tap where it should go.
    −3\displaystyle -3
    −7\displaystyle -7
    Show the answer and how to solve it

    From left to right, place -7 first and -3 second.

    1. 1The inequality says -3 is greater than -7.
    2. 2Greater numbers are farther right on a number line. Since both numbers are negative, the number closer to 0 is greater. -3 is closer to 0 than -7, so -3 is greater.
    3. 3So -7 goes on the left and -3 goes on the right.

    Common mistake. A common mistake is putting -3 on the left because 3 is smaller than 7. With negative numbers, the value closer to 0 is greater, so -3 belongs to the right of -7.

    The idea underneath. Left to right on a number line always means least to greatest, so use the inequality to decide which number is smaller and which is greater.

  4. 4Stretch
    Look at the marked values on the number line below.
    −3.5\displaystyle -3.5
    −3\displaystyle -3
    −12\displaystyle -\frac{1}{2}
    0\displaystyle 0
    1\displaystyle 1
    2.5\displaystyle 2.5
    Drag the cards to show the values from least to greatest.
    Drag a card to move it, or tap a card, then tap where it should go.
    1
    2.5\displaystyle 2.5
    2
    −3\displaystyle -3
    3
    1\displaystyle 1
    4
    −3.5\displaystyle -3.5
    5
    −12\displaystyle -\frac{1}{2}
    Show the answer and how to solve it

    The correct order is −3.5-3.5, −3-3, −12-\frac{1}{2}, 11, 2.52.5 because those marked values go from left to right in that order on the number line.

    1. 1On a number line, values farther left are smaller.
    2. 2The two leftmost marked values are −3.5-3.5 and −3-3. These form a close pair on the negative side. Although 3.5>33.5 > 3 as positive numbers, as negatives −3.5<−3-3.5 < -3 because −3.5-3.5 lies farther to the left.
    3. 3Continuing right, the next marked value is −12-\frac{1}{2}, which equals −0.5-0.5. Then the line crosses zero and reaches 11 and 2.52.5.
    4. 4So the order from least to greatest is:

    Common mistake. A common mistake is placing −3-3 before −3.5-3.5 because 3<3.53 < 3.5. For negative numbers, the greater the absolute value, the farther left the point—so −3.5-3.5 is less than −3-3.

    The idea underneath. A number line shows order visually: moving left means smaller values, even when the numbers are negative.