Models, language, and symbols are all used to describe or communicate a mathematical concept. By supplying students with various models, students are able to see different representations of a concept. While students are used to hearing the concept verbally, and seeing it with mathematical symbols, models provide a real-world, hands-on connection. This, in turn, enables teachers and students to communicate a concept using alternate methods.

The models represented below will be utilized in the additional teaching strategies portion of the Decimals and Percents lesson.

In fifth grade, students begin working with rational numbers and should be familiar with their representation in equivalent forms; fractions, decimals, and percents. Students should be adept at applying the operations associated with decimal numbers. Students should also be adept at finding percentages, as well as converting between its different representations. They should understand conceptually that a percent is a part of 100. In sixth grade, students will apply their knowledge of percentages in real-world situations, such as finding sales tax, interest, discounts, tips, etc. By the end of grade 6, students should have a mastery of the models, language, and symbols associated with decimal numbers. A strong understanding of decimal number operations, as well as an understanding of its value in comparison with other numbers, allows students to make estimations and determine if their answers are reasonable. In seventh grade, the problems become more complex with finding percent increase and decrease, as well as estimating percentages to determine if answers are reasonable.

When working with decimal numbers, one is referring to a quantity that considers both the unit value as well as a portion of the unit value. If the unit value is a whole number, then the values after the decimal point are a portion of the whole number. For example, if the water temperature is 68.2 degrees, then a student should recognize that the 0.2 represents a portion of one degree.
Money is another model used when dealing with decimal numbers. Using money is a powerful tool that students can draw from their own knowledge to interpret the idea of a decimal number. In most cases the unit value is one dollar and the values after the decimal place represent a portion of the dollar. In the United States, the portion of a dollar is called cents.
However, the unit value does not always have to be a dollar, it could be a million. For instance, a famous actor may earn 2.3 million dollars. The 0.3 represents a portion of the million dollars, which is equivalent to three hundred thousand dollars.

The need for a decimal representation of a percent becomes apparent when students begin using percentages to solve problems. Similar to the values to the right of the decimal point, a percent represents a portion of a whole. Therefore, one could represent a percent as a decimal value.

Consider the case where a student drank 25% of a gallon of water. One could say she drank 0.25 gallons of water. (Note: The process that is used to convert percents to decimals is shown in the Symbols section.)
Another representation of $2.43 with money would be to show 2 dollar bills, 4 dimes, and 3 pennies.

Now consider the situation where the container had 5 gallons of water in it.  If the student drank 25% of the 5-gallon container, how much did she drink? In particular, we are asking, what is 25% of 5 gallons? In this case the percent must first be converted into a decimal value in order to perform the calculations.

Convert to a decimal value              Calculate the amount
25% → 0.25                                            0.25 x 5=1.25  or 1.25 gallons of water

The correct mathematical language becomes essential when communicating mathematical ideas since some common English words have mathematical implications. For example, the word “and” is sometimes used incorrectly as a pausing place in a large number (e.g., 2,345 is incorrectly read as two thousand and three hundred forty-five). However, when reading a decimal number in a math class the word “and” is used to represent the decimal point. The number 12.34 is written as twelve and thirty-four one-hundredths and should be expressed verbally in the same manner.

Outside the classroom, one may hear people talking about giving 110% effort.  Some will say this is impossible, however, mathematically, 110% is a valid representation of an amount. The mathematical interpretation is that the value in reference is itself plus ten percent. Therefore, the final value is greater than the original amount.

110% of 50 is:
                       50    plus     10% of 50
             or      50    plus     5
             or      55. 
                                   
Notice that 55 is more than the original amount of 50.     

Percentages can also carry a negative value. The negative sign usually represents a decrease. This is seen when calculating percent increases and percent decreases in seventh grade. 

The decimal expansion of a number is its representation in base 10. For example,

Symbolic notation becomes a necessity when standard Arabic numbers become impractical due to the length of the number. For example, the decimal expansion for one third is approximately,

To represent this exactly, one would have to write the number three forever. As accepted notation or convention, the decimal number can be represented with a vinculum or “bar” over the repeating part of the decimal. This informs the reader that the numbers under the bar will repeat. 


(Notice that the 1 is not repeated.)

 

If a number does not end or repeat yet continues on forever, then for most practical purposes it would be appropriate to round off the number. A common number that is neither a finite decimal nor a repeating decimal is π(pi). Most people think of π as 3.14 however, it is actually

.

These types of decimals are called transcendental numbers, which belong to the  set of irrational numbers. Other transcendental numbers include

When converting between decimals and percents, consider the meaning of a percent: per 100 or 1/100. The decimal representation of 1/100 is 0.01.  Therefore, when taking a percentage and expressing it as a decimal, one would take the value and multiply it by 0.01.


Many students will conclude at this point that to convert percents to decimals, one would move the decimal point two places to the left. Likewise, the opposite is true for converting decimals to percents. The above shows the mathematical process of converting a decimal to a percent.