The first ever method of counting was counting on fingers. Tallies came later and were used for quite a long time. This gave way to Roman numerals. There used to exist many counting systems like clay tokens. After years of evolution a well-accepted and user-friendly decimal system with a base of ten that uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 came to existence. Place value plays an important role in the structure of the decimal system.
To have a strong base for the topic, it is recommended to include preschool math activities and learning games for kids while introducing the topic place values. To reinforce the concepts learned, there are many educational games for kids available on the internet.
In grade 1, they start with the place value definitions and deal with only tens and ones there using math games and place value activities. Gradually increasing the number of digits, by grade 4 they are comfortable with multi-digit whole numbers as well as decimals. By the end of the fourth grade, kids have a thorough knowledge about what is place value and its application in various fields of mathematics.
To revise the place value properties of whole numbers, consider the whole number 1269.
The number at the right most, 9 is at one's place, the number 6 at ten’s place, 2 at the hundred’s place and the number 1 at the thousand’s place.
The number can be written in expanded form as 
and can be decomposed as 
. That is, as you go from right to left, the place value gets multiplied by 10. In other words, as you go from left to right, the place value gets divided by 10.
A whole number ends at one’s place. But decimal numbers extend continuously. What happens if we go further right to the one's place? As per the logic so far, the place value should get divided by another 10. Let us use this logic to extend the place value system to decimal numbers.
For example, consider the decimal number 2341.568.
Here, the digit 5 that comes right after the decimal point has a place value of tenth. That is, the place value is 
. The digit right to it has a place value of hundredth or 
. Similarly, the right most digit, 8 has a place value thousandth or 
.
Thus, the expanded form of the number is 
.
It is clear from the expanded form that the place value of any digit gets multiplied by 10 if you move towards the left. It gets divided by 10, in other words, gets multiplied by if you move towards the right.
In the number 342.563, the place value of the 3 on the left most, 342.563 is hundreds making the value of the digit to be 300. But, the three at the right most 342.563 has a place value of thousandths, that gives the value 
.
Thus, the value of a digit depends on its position in a number.
SplashLearn is one of the math websites that provides well-structured math worksheets for various grades math skills. It's worksheets for practicing the place value strategies for decimals covers the required 5th grade math decimal topics well.
Children know that 10 ones form a single unit called “ten”, 10 tens forms a single “hundred” and so on. So, when a number is multiplied by 10, each of the digits gets multiplied by 10. That is, each “ones” gets multiplied by 10 to form that many “tens”. In a similar way, the number of tens in the number becomes the number of hundreds and so on.
The expanded form of a number 456.325 is
.
To multiply the number by 10, each product in the expanded form has to be multiplied by 10.
That is, 
.
So, when a decimal number is multiplied by 10, the place value of each digit gets multiplied by 10 and the decimal point shifts one unit to the right.
In a similar way, when a decimal number is divided by 10, the place value of each digit gets divided by 10 and the decimal point shifts one unit to the left.
If the multiplication by 10 is performed once again, that is equivalent to multiplying the original number by 10 times 10 or 100. This results in the decimal point shifting further one place to the right. Thus, the decimal point moves as many times to the right as the number of zeros in the multiple of 10. For example, if a decimal number is multiplied by 100,000, the decimal point shifts 5 places to the right. The similar result holds true for the division, with the shift of the decimal point to the left as well.
The multiples of 10 can mathematically be written using base and exponent. The base represents the number which is multiplied, 10 in this case. The exponent shows the number of times the base is multiplied.
Thus, when a decimal number is multiplied by a power of 10, the number of places that the decimal point moves are equal to the exponent.
Kids are already familiar with the decimals, fractions and their equivalence. Concrete models like base ten block, place value charts, hundred’s grid, and math manipulatives are used to reinforce the basic. They use fractional language to read decimals up to thousandth. Kids can write decimals in fractional form as well as in expanded form. Using the fractional equivalence, kids learn that 6 tenths are equivalent to 60 hundredths and 600 thousandths.
Kids use the equivalence between a decimal and its fractional form to compare two decimals.
For example, the decimal number 0.85 can be written in many forms:
The comparison can be done using the fact that all the above forms are equivalent.
For example, consider the decimals 0.089 and 0.123.
Writing the two in the fractional forms, 
and 
, it is clear that 0.089 < 0.123 as
89 < 123 .
Kids use equivalent fractions when the decimals in fractional forms do not have the same denominator.
Kids are familiar with rounding whole numbers in grade 3 itself. In grade 5, once they are comfortable with decimals and place value definitions, they use the same principle to round off the decimal numbers. A number line is the main math manipulative used for rounding.
Round 25.458 to the nearest tenth.
As in the case of whole numbers, the first step is to find the ‘tenths’ that comes before and after the number. On a number line, 25.458 lies between 25.4 and 25.5, closer to 25.5 when plotted.
Thus, 25.458 rounded to the nearest tenth is 25.5.
The math practice worksheets for plotting decimals on a number line, rounding decimals, and comparing decimals will help students to have a good command over the topic.
In grade 4, kids have learned how to multiply one-digit and two-digit numbers. In grade 5, kids apply various strategies for multiplying multi-digit whole numbers.
Breaking up into comfortable numbers: Consider the product 425 by 13.
Kids can break the two digit number 15 as 10 + 5 so that the multiplication is simpler. That is, 
.
Now the products are simpler and can even be done as a mental math. A multiplication by 10 results in an increase of place value of each digit by 10. Thus, 
. There are 4 hundreds, 2 tens, and 5 ones in 425. A multiplication by 3, would result in multiplying each component by 3.
Thus, 
.
Area Models: Here, the multiplicand, as well as the multiplier, are decomposed and the corresponding rectangular area models are drawn.
Now, the product of the two numbers is same as the area of the rectangle with sides equal the numbers. The bigger rectangle is divided further into smaller rectangles which make the computations easy.
Standard Algorithm: The standard algorithm of multiplying two numbers is also used. The method involves multiplying the multiplicand by each digit of the multiplier, aligning the place values.
In grade 4, kids performed division up to a two-digit number by a one digit whole number using the connections between division and multiplication. Also solving the logical question of how many divisors are “contained” in the dividend. In grade 5, when the dividend is a four-digit whole number and the divisor is a two digit number, the mathematics behind is the same with trickier calculations.
There are 2,852 soft drink cans in a warehouse. These have to be packed in cartons. Each carton can hold 21 cans. How many cartons would be required?
Here, the number of cartons is equal to 
.
Kids apply various strategies to find the quotient.
Using the connections between division and multiplication: The dividend 2852 can be written in expanded form as 
.
Consider some friendly multiples of 21.
Subtract 2100 from 2852 to get 752.
The next step is to find the number of 21s contained in 752.
Subtract 630 from 752 to get 122.
Now, find the number of 21s in 122. It is 5.
and 
which is a number less than the divisor. So, it is the remainder. This procedure can be tabulated as:
Expanded Form: The total number of 21s in 4852 would be the number of 21s in each of these addends in the expanded form.
Adding the quotients and remainders, Q: 135 and R:17.
Area Models: Similar to the area models for multiplication, here the area of the rectangle and one of the dimensions, say height are available. This can be used to find the length of the rectangle.
SplashLearn provides well-organized 5th grade worksheets from easy to difficult math problems for both multiplication and division.
Addition and Subtraction: The equivalence between decimals and fractions would give a visual image for addition and subtraction of decimals. For example, 0.3 is equivalent to 
and can be shown by shading 3 parts of a square divided equally into 10 parts. Similarly, another decimal in tenths, 0.5 can be represented by shading 5 parts of a similar unit square divided equally into 10 parts. The addition of the two would result in shading 8 parts out of 10 which is equal to 0.8. Subtraction follows a similar procedure.
A standard algorithm would include lining-up the decimals according to the place value and adding the digits in the corresponding place values.
Multiplication: Multiplication can be considered as a repeated addition. Here again, the equivalence between decimals and fractions can be used to find the products.
For example, 0.25 × 3 . The decimal 0.25 is equivalent to 
and multiplying by 3 denotes adding it three times, resulting in 
or 0.75.
When there are two decimals like 0.4 and 0.6, the fractional strips for multiplication can be used.
The decimal numbers that are greater than or equal to 1 can be converted to a mixed fraction to perform the multiplication
The standard algorithm for multiplying the decimals aligns the digits according to the place value and follow the computational algorithm for multiplication. The decimals can be multiplied without the decimal point and the decimal point can be marked in the product as the sum of the number of decimal places in the factors.
Division: The division of decimals is explained using the relation between the multiplication and division. When there is a division problem, 
, we are trying to find how many groups of 1.2 is contained in 3.6. That is, we are trying to find a number, say n such that 
. Since kids are more comfortable working with whole numbers than decimals, we can consider the whole number version of this division as 
. That is, how many 12s are there in 36. We have:
Similarly:
Thus, 
.
Plenty of place value worksheets for operations in decimal are available in SplashLearn.
