6th Grade · 6.NS.C.5

Positive and negative numbers in 6th grade

Positive and negative numbers describe amounts on opposite sides of 0, such as 10 degrees above zero and 10 degrees below zero. In 6th grade your child uses them to describe real situations and explains what 0 means in each one.

What the standard says

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

6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

What it means, in plain words

A positive number is greater than 0. A negative number is less than 0 and has a minus sign in front, like $-5$ , which is read "negative 5". For many children, 6th grade is the first year that negative numbers are part of math class.

The year starts with real situations that go in two opposite directions: temperatures above and below zero, heights above and below sea level, money earned and money spent, yards gained and lost in football. In each situation, 0 is the starting point, and its meaning depends on the situation. For heights, 0 is sea level. On a football play, 0 means the ball did not move forward or back. Your child should be able to say what 0 means each time.

Integers are the whole numbers, their opposites and 0: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ Opposites are two numbers the same distance from 0 on different sides, such as 7 and $-7$ . Children place positive and negative numbers on a number line that runs both ways from 0. This prepares them for adding and subtracting negative numbers in 7th grade.

Your child should be able to:

Use a positive or negative number to describe a real amount, such as $-10$ for 10 degrees below zero
Explain what 0 means in a situation, such as sea level for heights
Name the opposite of a number or a situation, such as a gain of 8 yards and a loss of 8 yards
Place integers on a number line on both sides of 0

Example

In a football game, a team gains 12 yards on one play and loses 6 yards on the next. A third play gains nothing. Which numbers describe the three plays, and what does 0 mean?

A gain moves the ball forward, so it is positive: 12.
A loss moves the ball back, so it is negative: $-6$ .
The third play is 0. Here 0 means the ball ended where the play started.
The three plays are 12, $-6$ and 0.

Example

The lowest point in Death Valley is 282 feet below sea level. What number describes its elevation?

Sea level is the starting point for elevation, so sea level is 0.
Below sea level is negative, so this point is at $-282$ feet.
A hilltop 282 feet above sea level is at the opposite number, 282 feet. Both are 282 feet from 0.
The elevation of the lowest point in Death Valley is $-282$ feet.

Where children usually get stuck

Your child writes 282 for a place 282 feet below sea level.
The word "below" already says where the place is, so the minus sign feels like saying it twice. Until now, numbers have only described amounts, never directions.
What helps. Ask what number a hilltop 282 feet above sea level gets. If both places get 282, the numbers cannot tell them apart. The minus sign does that job.
Your child sees a temperature of $-4$ degrees and asks what to take 4 away from.
For years the minus sign has meant subtraction. Now the same sign also marks a number below zero, and the child reads it as an instruction to do something.
What helps. Read $-4$ out loud as "negative 4", not "minus 4". For temperatures, also say "4 below zero", so the sign becomes a place, not a step.
Your child draws a number line from 0 to 10 and has nowhere to put $-3$ .
Every number line from earlier grades started at 0. The child has not seen that the line keeps going to the left.
What helps. Draw a thermometer standing up, with 0 in the middle and marks above and below it. Have your child mark $-3$ three steps below 0, then turn the drawing on its side.

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.

1Warm-up
Which situations need both positive and negative numbers? Select all that apply.
A
Temperature above or below 0 degrees
B
Number of books on a shelf
C
Miles walked in a day
D
Pages left in a book
Show the answer and how to solve it
The correct choice is Temperature above or below 0 degrees because it involves opposite values around the reference point of 0.
1. 1Positive and negative numbers are needed only when a situation has opposite values or directions around a clear reference point.
2. 2Temperature above or below 0 degrees fits this rule. The reference point is 0 degrees, and temperatures can be on either side of it — above (positive) or below (negative).
3. 3The other choices are one-sided amounts. Books on a shelf, miles walked, and pages left are all 0 or greater and never go below a reference point.
Common mistake. A common mistake is choosing Pages left in a book because the count is going down. But counting down to 0 is still a one-sided amount; it does not require negative numbers.
The idea underneath. Negative numbers are not for every situation that gets smaller; they are for situations that can go in opposite directions or lie on opposite sides of a reference point.

2On level

Each row shows a situation and a claimed opposite number. Select Correct, Wrong sign only, Wrong magnitude only, or Wrong sign and magnitude for each row.

Situation	Claimed opposite	Classify the claim
A submarine is 9 m below sea level (-9).	+9
A hill is 7 m above sea level (+7).	+7
The temperature is 5° below 0 (-5).	+8

Show the answer and how to solve it

Row 1 is Correct, row 2 is Wrong sign only, and row 3 is Wrong magnitude only.

1To test an opposite-number claim, check 2 things: the sign must switch, and the distance from 0 must stay the same.
2Row 1 shows -9. Its opposite is +9. The claim is +9, so both the sign switched and the magnitude matches. Row 1 is Correct.
3Row 2 shows +7. Its opposite is -7. The claim is +7, which is the original number repeated — the sign did not switch, but the magnitude is correct. Row 2 is Wrong sign only.
4Row 3 shows -5. Its opposite is +5. The claim is +8: the sign switched correctly (positive matches), but the magnitude is 8 instead of 5. Row 3 is Wrong magnitude only.

Common mistake. A common mistake is accepting +7 in row 2 because the number part looks right, or accepting +8 in row 3 because the sign changed. Both sign and magnitude must be checked separately.

The idea underneath. Opposite numbers are the same distance from 0 on opposite sides of the number line.

3On level
Match each situation on the left to the situation on the right that has the same amount but the opposite direction or value.
Connect each item on the left to its match on the right.
Situations
Possible opposites
A balance increased by 7 dollars
A balance decreased by 11 dollars
The temperature rose 4 degrees
The elevator went down 9 floors
A balance decreased by 7 dollars
The temperature fell 4 degrees
A balance increased by 11 dollars
The elevator went up 9 floors
Use Tab to navigate between items. Click or tap a card to select it, then click or tap a card on the opposite side to create a match. You can also drag a card or its connector dot to a card on the opposite side.
Show the answer and how to solve it
The correct matches are: A balance increased by 7 dollars matches A balance decreased by 7 dollars; A balance decreased by 11 dollars matches A balance increased by 11 dollars; The temperature rose 4 degrees matches The temperature fell 4 degrees; and The elevator went down 9 floors matches The elevator went up 9 floors.
1. 1The rule for opposites in context: the amount must stay exactly the same, but the direction or type of change must reverse.
2. 2For the two balance situations, increased by 7 dollars matches decreased by 7 dollars, and decreased by 11 dollars matches increased by 11 dollars. Both pairs stay within the balance family and keep the original amount.
3. 3For the remaining pair, the temperature rose 4 degrees matches the temperature fell 4 degrees, and the elevator went down 9 floors matches the elevator went up 9 floors.
4. 4Double-check: every left item pairs with a right item that has the same number but the opposite action. No amount is reused in the wrong pair.
Common mistake. A common mistake is matching a balance change with another balance change just because both involve money, without checking the amount. A balance decreased by 11 dollars is the opposite of a balance increased by 11 dollars, not increased by 7 dollars.
The idea underneath. Opposite quantities in context keep the same size; only the direction or value change reverses.
4Stretch
A submarine is 47 feet under sea level. Enter the number that represents the submarine's elevation, using only the number.
Elevation:
Show the answer and how to solve it
Answer. -47
The answer is -47 because the submarine is 47 feet under sea level, so the elevation is 47 less than 0.
1. 1Sea level is the reference point 0.
2. 2The phrase 'under sea level' means the same as below sea level — the elevation is less than 0, so the number must be negative.
3. 3The submarine is 47 feet from sea level, so write the signed number as -47.
Common mistake. A common mistake is typing 47 instead of -47. The phrase 'under sea level' signals a negative elevation, just like 'below sea level' does. Also remember to type only the number with no unit.
The idea underneath. The reference point tells where 0 is, and words like above, below, under, debt, and credit tell which side of 0 the number belongs on.