Objectives
The objectives of my curriculum unit is to develop the students' conceptual knowledge of the "unit", unit fractions, general fractions, comparing fractions, equivalent fractions, mixed numbers, improper fractions, and adding and subtracting fractions with like, related, and unlike denominators.
The "Unit"
The "unit" is essential in understanding a fraction. Often in an elementary mathematics textbook, a pie or a cookie is used to represent the unit. Students need to understand the unit and that the unit does not have to be one object. In dealing with fractions, it is always important to establish and clearly state what represents one. For example, if the smallest cube in base ten blocks is the unit then a "flat" is one hundred cubes. However, if the flat, or 100 cubes, is the unit, then one cube represents one hundredth.
Another important idea is that a unit changes in each new context. For example, if there are 24 cans of soda in a case and the unit is one can, then four units represents one sixth of the case. However, if there are 24 cans and the unit is two cans then two 2-packs represents one sixth. The above scenario models the need for students to be able to identify the unit in each context.
A real world example of identifying the "unit" occurs all the time in the grocery store. Which is a better deal, 18 ounces of cereal for $3.79 or 12 ounces of the same cereal for $2.88. By calculating the unit price, or the price per ounce, you can determine the best buy. The same problem could be solved by recognizing that 18 ounces is three 6 ounce units and 12 ounces is two 6 ounce units. Then the price could be determined by dividing $3.79 by three and $2.88 by two. This example demonstrates the benefit of thinking flexibly about units. This problem also shows the reasoning skills that will develop if students are provided opportunities to foster their unitizing ability. Providing students with many opportunities to develop their unitizing (chunking) skills will be an essential part of this unit. The curriculum unit will provide opportunities to unitize verbally and visually. By teaching my students to unitize and partition, finding common denominators will become a more innate and natural process for them.
Unit Fractions
The unit fraction is any fraction that has a numerator of one. For example, one third, one half, one fourth, and one eighteenth are examples of unit fractions. Mathematically if a whole is broken into d equal parts, then the unit fraction is 1/d. Once unit fractions are introduced, I would provide the students with many different types of models of the unit fractions and many opportunities to work with unit fractions of the same unit. For example, I would want my students to contrast 1/2, 1/3, 1/4, 1/5, 1/6 and so on of the same unit so they could determine that as the denominator gets larger the unit is divided into more pieces and the size of the piece gets smaller. The converse is also true. As the denominator gets smaller, the size of the pieces gets larger assuming that the unit fractions are related to the same size unit.
Another important understanding of the unit fraction 1/d is that multiplication by 1/d equals division by d. Multiplication by a fraction results in a smaller product. This concept tends to be confusing to students because their limited experience with whole numbers leads them to believe that multiplication always makes numbers larger. However, if students understand that multiplication by 1/d is really division by d, then the fact that the product gets smaller actually makes sense. In my fourth grade math curriculum I am not required to teach multiplication or division of fractions, however after learning about Singapore math it is very clear that these concepts are an integral part of working with fractions at any level. Multiplication of fractions is a part of naming general fractions. Two thirds is two copies, or two times one third. Five eighths represents five times one eighth. When students convert an improper fraction to a mixed number, they are really dividing by a fraction. If I have two and one third and I want to change it to a mixed number I am really dividing two and one third into as many one third pieces as I can. The answer is 7/3. This is an example of the concept of division being embedded in the work we will do with fractions.
General Fractions
General fractions are related to unit fractions through multiplication. If you have multiple copies (say, n) of a unit fraction (1/d), you have a general fraction n/d. For example, if I have three copies of 1/d, and d is four, then I would have three fourths, or 3/4. Simply put, a general fraction is multiple copies of a unit fraction. I could also say that I have c copies of 1/d or c/d. In this example c is called the numerator and d is called the denominator. The numerator tells the number of copies and the denominator tells the name of the pieces. You could also think of this in terms of adjectives and nouns. If you had three dogs, the three would be the adjective and the dogs would be the nouns. The adjective–noun model points to the principal that you can add like terms. The adjective-noun model removes some of the mystery and complication students have when dealing with fractions. Students often see a fraction and view it as two separate numbers instead of as one. By applying the adjective-noun technique to the fraction three fourths, the adjective would be three and the noun would be fourths. Thus, making the fraction three fourths seem less abstract. It also helps students to extend the concept of like terms to like denominators.
Equivalence
The next important idea is equivalence. There is more than one way to name a fraction. In fact, the same fraction can be named an infinite number of ways. It is important to understand equivalent fractions because just as with unitizing, sometimes it is helpful to "change the name" of a fraction in order to perform a calculation or to simplify a fraction. This is basically the notion of unitizing in different ways. If I had the fraction, one half, I could also call the fraction two fourths, three sixths, fifty hundredths, and an infinite number of names.
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