Embedded Files
Decimals
Decimals
As Parts of a Whole
As Parts of a Whole
Concrete
Concrete
Base ten blocks are an amazing (and underused) tool to help students understand decimals. To get started, your students will have to develop some new understanding. Until now, they have probably understood base tens as ones, tens, and hundreds.
But now, since they are older and learning more complex skills, we will use them differently. For the purpose of decimals, we will see these blocks as wholes, tenths, and hundredths. As shown by the picture below, the hundreds block will now represent one whole. The tens block represents one-tenth (because it is one-tenth of the whole or ten times smaller than the whole). The ones block represents one-hundredth because it is one-hundredth of the whole or one-hundred times smaller than the whole.
Your students will need time to develop this new understanding, so be sure to give them lots of time to work with base tens in this new way.
The big idea here is that we have to explicitly teach decimals (and fractions for that matter) in a hands-on, relevant way that students can really understand. To do this, we need to give opportunities to practice with concrete materials before moving on to abstract representations of decimals. All too often, students begin their decimal experience by simply shading in pictures on a worksheet, and get no concrete or real-life, relevant opportunities. If we want to prevent gaps in their understanding then we must give them these opportunities. Once they have had concrete experiences, and you are confident that they understand what decimals mean, here are some activities you may want to try.
Always remember to define the Whole (Which block type equals 1), from there you can build the decimal.
**Is it possible to build a value into the thousandths?Representational
Representational
Once students understand the concrete models of decimals, then you can have them visually represent the models using a Hundreds Chart grid. This chart represents the base ten blocks that the students have been using.
This model is great for demonstrating how to add and subtract decimals. It also is a great tool for building the connection between decimals and fractions. It is vital that they know that 0.01 = 1/100 and 0.1 = 1/10; that way when given a decimal such as 0.43 they can say that the decimal has 4 tenths and 3 hundredths.
Children may be asked to write 9/10 as a decimal. Again, the hundred number square can be useful for this: ask them to color in the 9/10 and they'll realize that this is the same as 90/100 which is 0.9 as a decimal.
Abstract
Abstract
Once students have experience with these models, then they can begin to apply the standard algorithms for decimals.
When the standard algorithm for adding and subtracting decimals is taught, students are told "Don't forget to line up the decimals!" Having experience with the other models first enables students to explain WHY they should line up the decimals when applying the algorithm.
Decimals on a Number Line
Decimals on a Number Line
A good way to represent positive decimals is as lengths; this way of representing decimals leads naturally to placing decimals on number lines. The meter, which is the main unit of length used in the metric system, is a natural unit to use when representing decimals as lengths because the metric system was designed to be compatible with the base-ten system.
To connect lengths with number lines, imagine representing a positive decimal as a length by using strips of paper. Now imagine placing the left end of the length of paper at 0. Then the right end of the length of paper lands on the point on the number line that the length represents. In this way, we can view number lines as related to lengths, and we can view points on number lines in terms of their distances from 0.
One way to think about decimals is as “filling in” the locations on the number line between the whole numbers. You can think of plotting decimals as points on the number line in successive stages according to the structure of the base-ten system.
At the first stage, the whole numbers are placed on a number line so that consecutive whole numbers are one unit apart.
At the second stage, the decimals that have entries in the tenths place, but no smaller place, are spaced equally between the whole numbers, breaking each interval between consecutive whole numbers into 10 smaller intervals each one-tenth unit long. Notice that, although the interval between consecutive whole numbers is broken into 10 intervals, there are only 9 tick marks for decimal numbers in the interval, one for each of the 9 nonzero entries, 1 through 9, that go in the tenths places.
We can think of the stages as continuing indefinitely. At each stage in the process of filling in the number line, we plot new decimals. The tick marks for these new decimals should be shorter than the tick marks of the decimal numbers plotted at the previous stage. We use shorter tick marks to distinguish among the stages and to show the structure of the base-ten system.
Stage 1: Whole numbers are represented on a number line.
Stage 2: Each unit is partitioned into 10 tenths.
Stage 3: Each tenth is partitioned into 10 hundredths.
Stage 4: Each hundredth is partitioned into 10 thousandths.
Stage 5: Each thousandth is partitioned into 10 ten-thousandths. Decimal numbers “fill in” number lines.
By “zooming in” on narrower and narrower portions of the number line, we can see in greater detail where a decimal is located.
Beckmann, Sybilla. Mathematics for Elementary Teachers with Activities (p. 14-16). Pearson Education. Kindle Edition.
Zero
Zero
In elementary school students start by learning to count from 1-5, and the concept of zero is introduced later. Zero is often a surprisingly difficult concept for students to grasp, as it is not a tangible value.
The numeral 0 serves two purposes in mathematics, it denotes nothing. How many apples do I have, 0.
Zero is also a place holder in place value. For example, we can say we have 3 tens, with an absence of ones. We can't just write the value 3 with nothing in the ones place. It is confusing, what does the 3 mean? 3 ones, 3 tens, 3 hundreds? We could always write 3 and the word "tens", but that's a lot of work-- 4 whole letters! So the numeral 0 is used to denote that there is an absence of items in the indicated place. 30, 3 tens and 0 ones.
Negative Numbers
Negative Numbers
On a number line, we display the negative numbers in the same way as we display 0 and the numbers greater than 0. To the right of 0 on the number line are the positive numbers. To the left of 0 on the number line are the negative numbers. The number 0 is considered neither positive nor negative. We can think of a negative number, -N, as the “opposite” of N. Negative numbers are commonly used to denote amounts owed, temperatures below zero, and even for locations below ground or below sea level.
Positive and negative numbers are placed symmetrically with respect to 0 on the number line. Note that this symmetry on the number line should not be confused with the symmetry of place values with respect to the ones place.
Note that positive numbers get larger in value as you move to the right on the number line. This means that the farther from zero a positive value is, the larger that value. This is the opposite for negative numbers. Values for negatives get smaller the farther left you move. This means that the farther from zero a negative value is, the smaller that value.
Beckmann, Sybilla. Mathematics for Elementary Teachers with Activities (p. 18). Pearson Education. Kindle Edition.
Exposing a New Concept
I’ve learned that with negative numbers you need to expose your child to the concept in the elementary grades. A good time to begin is in the second half of fourth grade. At this point in their math journey, children are ready to explore negative values but not quite ready to perform mathematical operations with those numbers.
Interacting with Negative Numbers
So how do we introduce children to negative numbers? As with all math concepts, we need to make observations with our children as we do hands-on activities—but how? What makes negatives difficult for children initially is that you can’t look at or touch negative three apples. The best way to interact with negative numbers is with a number line.
Activity
Activity
Give your child the number line and the markers.
Ask your child to place the blue marker above the number 5. This will be a point of reference.
Ask your child to place the red marker above the number 7. Ask, “Which of the two numbers is larger?” and “Is it farther to the left or farther to the right?” Point out that the larger number is to the right and the smaller is to the left.
Direct your child to take the red marker from 7 and put it above the number 1.
Ask, “Now which number is the larger?” and “Is it farther to the left or to the right?”
Have your child move the red marker above number negative 1.
Ask, “What do we call that first number to the left of zero?” Your child might say, “Minus 1.” Explain that it is called negative one and that any time that negative symbol appears before a stand-alone number, it shows the number is less than zero.
What is Difficult About Decimal Words?
What is Difficult About Decimal Words?
The names for the values of the places to the right of the ones place are symmetrically related to the names of the values of the places to the left of the ones place.
There are several common errors associated with the place value names for decimals. One error is not distinguishing clearly between the values of places to the left and right of the decimal point. For example, students sometimes confuse tens with tenths or hundreds with hundredths or thousands with thousandths. The pronunciation is similar, so it’s easy to see how this confusion can occur! Teachers must take special care to pronounce the place value names clearly and to make sure students understand the difference. Another error occurs because students expect the symmetry in the place value names to be around the decimal point, not around the ones place. Some students expect there to be a “oneths place” immediately to the right of the decimal point, and they may mistakenly call the hundredths place the tenths place because of this misunderstanding.
A cultural convention is to (1) say decimals according to the value of the right-most nonzero decimal place and (2) say “and” for the decimal point. For example, we usually say 3.84 as “three and eighty-four hundredths” because the right-most digit is in the hundredths place. Similarly, we say 1.592 as “one and five-hundred ninety-two thousandths” because the right-most digit is in the thousandths place. From a mathematical perspective, however, it is perfectly acceptable to say 3.84 as “3 and 8 tenths and 4 hundredths” or “three point eight four.” In fact, we can’t use the usual cultural conventions when saying decimals that have infinitely many digits to the right of the decimal point. For example, the number pi, which is 3.1415... must be read as “three point one four one five.. .” because there is no right-most nonzero digit in this number! Furthermore, the conventional way of saying decimals is logical, but the reason for this will not be immediately obvious to students who are just learning about decimals and place value.
Beckmann, Sybilla. Mathematics for Elementary Teachers with Activities (p. 17-18). Pearson Education. Kindle Edition.
Information in this section corresponds to: 1.2 Decimals and Negative Numbers: Beckmann, Sybilla. Mathematics for Elementary Teachers with Activities (p. 13-20). Pearson Education.
Information in this section corresponds to: 1.2 Decimals and Negative Numbers: Beckmann, Sybilla. Mathematics for Elementary Teachers with Activities (p. 13-20). Pearson Education.
Page updated
Report abuse