What a number line is, and how children use it from counting to negatives

A strip of tape across the kitchen floor, blocks at even steps, and a child hopping from one to the next: that is a number line, and it runs through homework from kindergarten to grade 6. The question "what is a number line?" usually comes up when a worksheet says "use the number line to solve". A number line is a straight line with numbers in order at equal spaces, and children use it to count, jump, and place fractions and negative numbers.

What is a number line?

A number line puts numbers in order at equal spaces. Wikipedia describes it as "a graphical representation of a straight line that serves as spatial representation of numbers". One point is chosen as 0, the marks on both sides of it are evenly spaced, and an arrow shows the direction in which the numbers get bigger.

On a school number line, bigger numbers sit to the right. OpenStax Prealgebra 2e states the rule plainly: "positive numbers are to the right of zero. Negative numbers are to the left of zero." A number to the right is always greater than a number to its left, which is why the number line is the picture behind greater than and less than signs.

How to draw a number line that works

A number line that works has four parts: a 0, equal spaces, tick marks and arrows. Draw a long straight line with a ruler, put an arrowhead on each end to show that the numbers keep going, and mark 0 near the left end (or in the middle once negatives arrive).

Make the tick marks the same distance apart, for example two squares on grid paper, and write one number under each: 0, 1, 2, 3. Every space must be the same size, because a child reads distance as amount.

How to use a number line to add and subtract, step by step

A number line turns addition into jumps to the right and subtraction into jumps to the left, shown here for 3 + 4 and 9 − 5.

1. Find the start. Put a finger or a dot on the first number: 3 for 3 + 4, 9 for 9 − 5.
2. Choose the direction. Plus means jump right, toward bigger numbers. Minus means jump left, toward smaller numbers.
3. Make the jumps. Draw one curved jump for each 1 you add or take away: four jumps right from 3, five jumps left from 9.
4. Count the jumps out loud. Say "one, two, three, four" as each jump lands, never as you leave a number.
5. Read where you land. Four jumps from 3 land on 7, so 3 + 4 = 7. Five jumps back from 9 land on 4, so 9 − 5 = 4.
6. Check the other way. Start at 7 and jump 4 back; you should be on 3 again. This ties addition and subtraction together, the same link that number bonds teach.

Count the jumps, not the tick marks

The jump rule is the part children get wrong most often. A child solving 3 + 4 who counts "3, 4, 5, 6" while touching tick marks lands on 6, one short, because the start was counted. The answer is the number of spaces crossed.

Kindergarten already practices this move. Common Core standard K.CC.A.2 asks children to "count forward beginning from a given number within the known sequence (instead of having to begin at 1)", and 1.OA.C.5 asks them to "relate counting to addition and subtraction (e.g., by counting on 2 to add 2)". Counting on from 3 to 7 is that standard on a number line.

The number line from kindergarten to grade 6

The number line grows with your child, from a counting track to the line that holds fractions, decimals and negative numbers.

Grade 2 is where the number line becomes a tool for bigger sums. Standard 2.MD.B.6 asks children to "represent whole numbers as lengths from 0 on a number line diagram" and to "represent whole-number sums and differences within 100 on a number line diagram". Here children meet the open number line: a blank line with only the numbers they need.

For 38 + 25, a child writes 38, jumps +10 to 48, +10 to 58, +2 to 60 and +3 to 63. For 52 − 17, the child can count up from 17: +3 to 20, +30 to 50, +2 to 52, and 3 + 30 + 2 = 35. Jumping by tens from any number is skip counting on a line, and jumping to the nearest ten is the picture behind rounding numbers.

Fractions and decimals on a number line

Fractions go on a number line by cutting the space from 0 to 1 into equal parts. Standard 3.NF.A.2 asks grade 3 students to "understand a fraction as a number on the number line" by "defining the interval from 0 to 1 as the whole" and "partitioning it into b equal parts". For fourths, cut the space into 4 equal parts: each part is ¼, and three of them end at ¾.

Two jumps of ¼ land on the same point as ½, so 2/4 and ½ are equal. Four jumps land on 1, and a fifth lands on 1¼, past 1. The guide to fractions for parents covers the fraction words that go with this.

Decimals follow in grade 4. Standard 4.NF.C.6 asks students to "use decimal notation for fractions with denominators 10 or 100", and its example places 0.62 on a number line: 10 parts from 0 to 1 make tenths, and 10 parts inside one tenth make hundredths.

Negative numbers on a number line

Negative numbers sit to the left of 0 on a number line, as the mirror image of the positive numbers. In grade 6, standard 6.NS.C.6 asks students to "extend number line diagrams and coordinate axes familiar from previous grades" to negative numbers and to "recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line".

The numbers 3 and −3 are opposites: each sits 3 spaces from 0, on different sides. That distance is the absolute value. OpenStax defines it in one line: "The absolute value of a number is its distance from 0 on the number line." Standard 6.NS.C.7 uses a debt as its example: an account balance of −30 dollars is written |−30| = 30, the size of the debt.

The number to the right is always bigger, so standard 6.NS.C.7 asks students to "interpret −3 > −7 as a statement that −3 is located to the right of −7", and −3 °C is warmer than −7 °C. The full set of sign rules is in the guide to negative numbers.

Play free in your browserComparing NumbersK–Grade 1

Open and closed circles on a number line

An open circle on a number line means the number at the circle is left out, and a closed (filled) circle means it is included. Circles appear when the number line shows a range of numbers, such as all numbers greater than 2. The graph of x > 2 has an open circle at 2 and an arrow to the right, because 2 is not greater than 2.

The graph of x ≥ 2 has a closed circle at 2, because "greater than or equal to" lets 2 in. Use open circles for < and >, and closed circles for ≤ and ≥.

Number line practice at home, and how to check progress

Number line practice at home works best with a line your child can walk on. Put masking tape on the floor, mark 0 to 10 with sticky notes one big step apart, and call out "start at 4, jump 3". Your child hops, counts each landing and says where they ended: 7.

Move the game to paper after a week or two: draw a line, write only the start number, and let your child mark the jumps. Later, put 0 in the middle of the tape with −1 to −5 on the left. A thermometer is a vertical number line, a bridge to the coordinate plane, where two number lines cross at 0.

Play free in your browserRounding NumbersGrade 3–4

A five-question check for the number line

Five short questions show whether your child understands the number line. Ask one or two a day and let your child draw the line.

• Counting on: solve 6 + 5 with jumps. A jump of 4 reaches 10, and 1 more lands on 11.
• Subtracting: solve 43 − 18 on an open number line. Jump back 10 to 33, then 8 to 25.
• Fractions: mark ¾ between 0 and 1. The line needs 4 equal parts, with ¾ at the third mark.
• Order: which is greater, −4 or −1? The answer is −1, because −1 sits to the right of −4.
• Distance: how far is it from −2 to 3? Two spaces to 0 plus 3 spaces to 3 make 5.

A child who gets four or five right in a week is ready for the next row of the grade table. A child who lands one off on the first two questions is counting tick marks, and needs another week of floor hopping.

Common number line mistakes

Four mistakes cause most wrong answers on number line homework, and each has a quick fix.

• Counting tick marks instead of jumps: for 3 + 4, counting "3, 4, 5, 6" lands on 6. Count the landings only, starting with the first jump.
• Uneven spacing: a hand-drawn line with squeezed ticks makes 7 look closer to 5 than to 9. Use grid paper or a ruler so every space is the same size.
• Starting at 1: a line without 0 hides the first space, so 2 + 3 or a length from 0 comes out wrong. Always draw and label the 0.
• Reading negatives backward: a child who thinks −6 is bigger than −2 because 6 is bigger than 2 is ignoring position. Ask which one sits farther right on the line.

Edge case: the double number line

A double number line is two number lines drawn one above the other with marks that line up, and it shows a ratio. Standard 6.RP.A.3 lists "double number line diagrams" among the tools grade 6 students use for ratio and rate problems.

Take 3 apples for $2: the top line counts apples 0, 3, 6, 9, 12, and the bottom line counts dollars 0, 2, 4, 6, 8 under the same marks. Reading straight down from 12 apples gives $8. The guide to ratios shows the same idea as a table.

Your number line checklist and next steps

• Zero: draw and label 0 first, with arrows on both ends.
• Spacing: make every space the same size, using grid paper or a ruler.
• Direction: plus jumps right, minus jumps left, and the number to the right is bigger.
• Jumps: count the landings, never the starting tick.
• Fractions: cut the space from 0 to 1 into equal parts before you label it.
• Circles: open for < and >, closed for ≤ and ≥.

The free comparing numbers and rounding numbers games play in the browser with no sign-up and practice the same "which side of the line" thinking. The free MathIt app for iOS and Android adds about 90 game types with adaptive difficulty for a few minutes of daily practice.