Changes

The number line is a geometric “model” of all numbers -- including 0 1, 2, 25, 374 trillion, and -5, Unlike counters, which model only counters, the number line models measurement, which is why it must start with zero. (When we count, the first object we touch is called "one." When we measur using a ruler, we line one end of the object we’re measuring against the zero mark on the ruler.  

The number line is a geometric “model” of all numbers -- including 0 1, 2, 25, 374 trillion, and -5, Unlike counters, which model only counters, the number line models measurement, which is why it must start with zero. (When we count, the first object we touch is called "one." When we measur using a ruler, we line one end of the object we’re measuring against the zero mark on the ruler.  

 

<br>Part of the power of addition and subtraction is that these operations work with both counting and measuring. Therefore, to understand basic operations like addition and subtraction, we need a number line model as well as counters.  

 

Part of the power of addition and subtraction is that these operations work with both counting and measuring. Therefore, to understand basic operations like addition and subtraction, we need a number line model as well as counters.  

The number line in teaching mathematics

The number line in teaching mathematics

One reason to use this mathematical object with students is that they need to see arithmetic in both contexts: counting and measuring. At the beginning, children may sometimes use the number line to find answers to arithmetic problems (e.g., figuring out what 3 + 10 is, before that becomes automatic to them) but that is never its purpose. We don’t rely on the number line for getting answers -- for that, we want the children to know basic facts and methods and use their heads -- but we do use the number line to understand things about the operation (addition and subtraction) and to understand what the answers mean.  

One reason to use this mathematical object with students is that they need to see arithmetic in both contexts: counting and measuring. At the beginning, children may sometimes use the number line to find answers to arithmetic problems (e.g., figuring out what 3 + 10 is, before that becomes automatic to them) but that is never its purpose. We don’t rely on the number line for getting answers -- for that, we want the children to know basic facts and methods and use their heads -- but we do use the number line to understand things about the operation (addition and subtraction) and to understand what the answers mean.  

For example, the answer to the subtraction problem 92 – 49 is the distance between those two numbers on the number line. That image can greatly help mental computation: 49 to 50 is one step, 50 to 90 is another forty steps, and 90 to 92 is another two steps, so altogether 43 steps. (See number line addition and subtraction below.) It also makes arithmetic with negetive numbers which has been already discussed. And the number line is essential for full understanding of ffractions and decimals. In fact, a ruler (a number line!) is one of the important places students encounter and need to use fractions.  

For example, the answer to the subtraction problem 92 – 49 is the distance between those two numbers on the number line. That image can greatly help mental computation: 49 to 50 is one step, 50 to 90 is another forty steps, and 90 to 92 is another two steps, so altogether 43 steps. (See number line addition and subtraction below.) It also makes arithmetic with negetive numbers which has been already discussed. And the number line is essential for full understanding of fractions and decimals. In fact, a ruler (a number line!) is one of the important places students encounter and need to use fractions. <br>

In counting

'''In counting'''

Number lines are first used just to show sequence—numbers standing on line in order! At this stage, neither the straightness of the line nor the distance between numbers is mathematically important,  

Number lines are first used just to show sequence—numbers standing on line in order! At this stage, neither the straightness of the line nor the distance between numbers is mathematically important, though our images are always standard anyway.  Children will look at chunks of the line, not always starting at 1, and will work forwards or backwards from some number that is placed on the line. They are learning about sequence and order, and that develops somewhat independently from counting.  <br>

though our images are always standard anyway.  Children will look at chunks of the line, not always starting at 1, and will work forwards or backwards from some number that is placed on the line. They are learning about sequence and order, and that develops somewhat independently from counting.   

'''In measurement'''<br>

In measurement

Students also learn about intervals on the number line, but just begin that process.  Kids used to gain the interval idea (in a slightly different form) from their experience playing board games. They knew that when they rolled a 5, they had to count their five spaces beginning with the next space. That is, they were counting moves, not positions.  Measurement depends on it. Addition and subtraction with counters does not depend on it, but those operations on the number line do depend on it. <br>

 

'''In addition and subtraction'''<br>

 

If we can add and subtract with counters, why use the number line? To connect these operations with measurement, and also because the counters no longer suffice when we get to fractions, decimals, and negative numbers. Over time, kids will connect number line images with thermometers, clocks, rulers (with fractional inches)… Coordinate graphs are based on perpendicular number lines; even bar graphs require the measurement idea more than the count idea, although they can begin with count. Addition and subtraction, or comparison, of distance is also why we use number lines.  Adding distance is further developed on open number lines. Students develop many ways to subtract (in second grade, they learn to subtract 8 from anything by subtracting 10 and then compensating, and then they extend that idea to other additions and subtractions).

Students also learn about intervals on the number line, but just begin that process.  Kids used to gain the interval idea (in a slightly different form) from their experience playing board games. They knew that when they rolled a 5, they had to count their five spaces beginning with the next space. That is, they were counting moves, not positions.  Measurement depends on it. Addition and subtraction with counters does not depend on it, but those operations on the number line do depend on it.  

In addition and subtraction

If we can add and subtract with counters, why use the number line? To connect these operations with measurement, and also because the counters no longer suffice when we get to fractions, decimals, and negative numbers. Over time, kids will connect number line images with thermometers, clocks, rulers (with fractional inches)… Coordinate graphs are based on perpendicular number lines; even bar graphs require the measurement idea more than the count idea, although they can begin with count.  

Addition and subtraction, or comparison, of distance is also why we use .  Adding distance is further developed on open number lines. Students develop many ways to subtract (in second grade, they learn to subtract 8 from anything by subtracting 10 and then compensating, and then they extend that idea to other additions and subtractions).

 

 

Hopping on the Number Line

=== Hopping on the Number Line ===

In this lesson, students generate sums using the number line model. This model highlights the measurement aspect of addition and is a distinctly different representation of the operation from the model presented in the previous discussion. The order (commutative) property is also introduced. At the end of the lesson, students are encouraged to predict sums and to answer puzzles involving addition.  

In this lesson, students generate sums using the number line model. This model highlights the measurement aspect of addition and is a distinctly different representation of the operation from the model presented in the previous discussion. The order (commutative) property is also introduced. At the end of the lesson, students are encouraged to predict sums and to answer puzzles involving addition.  

Activity 1 to introduce Number line

Activity 1 to introduce Number line