# Number Line

A number line is a straight line for representing the position of numbers relative the origin (the number 0). Intervals are usually marked with ticks and labels. Number lines come in so many different formats and varieties. Classroom number lines are hung on the wall of a class while individual number lines can fit on a desk and sometimes and can collapse or break into segments for more compact storage. Number lines can come in integer, decimal, or fraction formats. Number lines are incredibly flexible and versatile manipulatives that can be used for locating numbers, comparing numbers, fractions and mixed numbers, decimals, integers, absolute value, addition, subtraction, inequalities, muiltiples, common denominators, and more.

Virtual Number Line Manipulative

Virtual manipulatives, such as Brainingcamp's Number Line Manipulative, allow users to do things that are difficult or impossible with physical number line manipulatives. The virtual number line can toggle between decimal and fraction format. Users can easily zoom in and out, so¸ the manipulative supports a very large range of numbers. Colored markers and resizable arrows can be added and moved around with ease. There are also special modes for inequalities, multiples and for working with a double number line.

Using a Number Line

Locating numbers on the number line

Hide some or all of the labels on the number line. Guess the location of a number on the number line by placing a marker where you think it should be. Reveal the actual location of the number. Show that a value can be thought of both as a location on a number line, or a distance from zero.

Fractions and mixed numbers

Using a fraction number line, focus on fractions with a denominator of 5. The denominator represents the number of equal size steps between zero and one. The numerator represents the number of actual steps. So the fraction 2/5 represents 2 steps out of 5 equal steps between zero and one.

Equivalent Fractions

Use a double number line, with both number lines displaying fraction intervals. If possible, display intervals of 1/3 on one number line and 1/9 on the other. Draw a line or a bar from 0 to 2/3 on the first number line. Drag the line or bar to the origin of the second number line. Notice that bar extends from the origin to 6/9, so 2/3 and 6/9 are equivalent fractions. Fractions are equivalent when one fraction can be made from the other by multiplying and dividing the numerator and denominator by the same whole number. Multiplying both then numerator and denominator of the fraction 2/3 by 3 makes the equivalent fraction 6/9.

Comparing and Ordering

Notice than when two fractions have the same denominator, the fraction with the larger numerator is greater. Then notice than when two fractions have the same numerator but different denominators, the fraction with the smaller denominator is greater. To compare fractions with different denominators express the fractions as equivalent fractions with a common denominator.

To model the addition 5 + 3, draw a positive-facing arrow of length 5 units from the origin and another positive-facing arrow of length 3 units from the end of the first arrow. The end of the second arrow is at location 8 on the number line, so 5 + 3 = 8. To model the subtraction 5 – 3, draw a positive-facing arrow of length 5 units from the origin and a negative-facing arrow of length 3 units from the end of the first arrow. Notice that we used a negative-facing arrow for the second arrow to indicate subtraction. The end of the second arrow is at location 2, so 5 – 3 = 2. Subtracting a number is the same as adding the additive inverse, so 5 – 3 = 5 + (–3). As a last exercise, draw a positive-facing arrow of length 5 units from the origin. Think about what number can be subtracted from 5 to create -4. Try it.

Using a fractions number line focus on intervals of 1/5. To model the addition 1/5 + 2/5, draw a positive-facing arrow of length 1/5 from the origin and another positive-facing arrow of length 2/5 from the end of the first arrow. The end of the second arrow is at location 3/5 on the number line, so 1/5 + 2/5 = 3/5. Fractions with different denominators can be added by expressing the numbers as equivalent fractions with a common denominator and then summing the numerators.

Absolute Value

The absolute value of a number is its distance from 0. Notice that the absolute value of both +3 and -3 is 3. The absolute value of opposite numbers is the same. Because absolute value is a measure of distance from the origin, it is always positive.

Graphing Inequalities

To graph the inequality x > 3, draw a hollow circle at the point +3 on the number line and then draw a right pointing arrow from the dot until the end of the number line. The inequality represents all the numbers that are greater than 3. Since the circle is empty, the number 3 is not included in the inequality x > 3. Shading the center of the circle would represent the inequality x ≥ 3, which includes the number 3. A left pointing arrow would represent the inequality x ≤ 3, all the numbers less than or equal to 3.

Commutative Property

Model 4 + 2 by placing a right facing arrow of length 4 followed by a right facing arrow of length 2. Next model 2 + 4 by placing a right facing arrow of length 2 followed by a right facing arrow of length 4. Both sums have equal value. The Commutative Property says that changing the order of the terms does not change the result.

Multiples

To find multiples of 3, skip count by 3's and draw the hops as you go. Do the same for multiple of 5. The common multiples of 3 and 5 are all the points where the hops meet each other on the number line (15, 30, 45, …). The least common multiple is the smallest of these points (15).

Common Denominator

To find common denominators for the fractions 1/3 and 1/4, use a fractions number line. Find the multiples of 3 and 4 by skip counting and drawing hops. Notice that the multiples intersect on the number line for the fist time at location 12, so 12 is the least common denominator of 1/3 and 1/4. On the number line, mark the locations 1/3 and 1/4. Notice that 1/3 and 3/4 are equivalent to the fractions 4/12 and 3/12.

Multiplying and Dividing

Represent the multiplication 2 × 3 by starting at the origin and placing 2 arrows of length 3. The result is equal to a length of 6, so 2 × 3 = 6. Represent the division 8 ÷ 4 by starting at the origin and seeing how many arrows of length 4 it takes to make a length 8. It takes 2 arrows of length 4 to equal an arrow of length 8, so 8 ÷ 4 = 2.

Opposites

Place a marker at location +3. Next place a marker with the opposite value, -3. Notice that opposite values are the same distance from the origin, but on different sides of the origin.

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