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1 Clock Model Decomposing or Splitting Denominator The clock model is used to develop the use of landmark fractions for addition and subtraction. With the hour as a common whole (60 minutes) fractions such as ½, /3, ¼, /6, /2 become landmarks. This lesson asks the student to build 0/6 of an hour. The student uses the clock model to combine unlike fractions (but the common unit of minutes) to make the whole and the part to find the number of minutes for the improper fraction. (Fosnot, 2002) The strategy of decomposing numbers up into friendlier pieces. In this case, 5/5 as a whole is decomposed to 4/5 + /5 = 5/5. Decomposing is also known as splitting. The number below the line in a common fraction that indicates the number of equal parts in which the unit is divided. 4 Doubles a Denominator to Halve a Fraction Doubles Numerator and Denominator to make Equivalent The strategy of doubling a denominator for the purpose of halving a fraction. The numerator of a unit fractions stays the same but each unit fraction is half as big. The value of the fraction is half what it was before doubling the denominator. This strategy is used to create equivalent fractions. In this example the ratio table clearly demonstrates the pattern. Tripling, quadrupling, etc. uses the same concept. x = Doubles Numerator to Multiply by Two Equivalence on a Double Number Line Equivalent The strategy of doubling the numerator of a fraction by multiplying the fraction by two. The value of the fraction is doubled when you double the numerator. The double number line is used to represent different fractional quantities that have a common whole. In this example the race course is the common whole with a set number of units as the distance. The snack tables, water stations, resting points, and food booths are concrete examples of the fractional quantities and the distance measured between each. Equivalent fractions are two different names for the same point on a number line and the samesized number. Every fraction is equal to an infinite number of other fractions. (Van de Walle, (206) 2 2 x = 4 4

2 For Equivalence the Ratio Must be Kept Constant Fraction Bar Chart are division (partitive and quotative) with a quotient less than one can be Thought of as Operators Express Relationships the Size or Amount of the Whole Matters To maintain equivalence, the ratio of the related numbers must be kept constant. This big idea is about thinking proportionally and understanding that 6/0 is not 3/5 doubled. (Fosnot, 2007) Model or tool that can be called rods, bars or blocks within DBL curriculum. The chart may be used as a reference or fraction bars may be used to build. This includes partitive and quotative division. Partitive division is dividing a total number of objects into groups. Example: How many groups of 5 fits into 20? Quotative division as a part-towhole relationship. Example: ¾ of submarine sandwich. The big idea that fractions are more than part of a whole, the whole matters and the fraction can be conceptualized as an operator, ½ can be understood as representing one-half of various wholes. (Fosnot, 2007) A fraction does not say anything about the size of the whole or the size of the parts. A fraction tells us only about the relationship between the part and the whole. (Van de Walle, 203) ½ of a 2 oz can of beans plus ½ of a 32 oz carton of chicken broth will yield: 6 oz of beans and6 oz of broth may Represent Division with a Quotient Less than One Generalized Use of a Repertoire of Strategies for Whole Number Operations The big idea that a fraction may represent division where the numerator is divided by the denominator. When the numerator is less than the denominator, the quotient is less than one and can be expressed as a decimal or percent. This is the strategy of generalizing the addition, subtraction, multiplication and division strategies for whole numbers to rational numbers. For example: you may decompose a decimal number and use the array model to solve 69 x 6.6. = 4 =.25 4

3 If Numerators are Common Only Denominators Matter when Comparing Improper Fraction Mixed Number This is the big idea that students grasp when understanding they understand the greater the denominator the smaller the piece. As a result, when numerators are common, they only need to compare denominators with the larger denominator representing the smaller fraction. For example, if a student understands this concept, they are able to list a set of fractions from least to greatest when the numerators are common and represent why this is the case visually. (Van der Walle, 2007) /3, /5, /8, /0 The term that denotes a fraction greater than one when not in mixed form. 6 5 The term that denotes a rational number greater than one in the form of a whole number and 5 fraction. For example, 5/4 = ¼ = 4 4 Money Model The money model is used to develop the use of landmark fractions for addition and subtraction. With the dollar as a whole (00 cents) fractions such as /0, /5, ¼, and ½ become landmarks. Multiplication is Connected to (e.g. ¾ = 3 x ¼) Number Line (length model) The idea that when you have multiple unit fractions, multiplying the unit fraction by a whole number provides a product. In this fair sharing example, 3 sub sandwiches divided among 4 students. Each sub is divided into ¼. Once the pieces are given to each student, the students realize that they have ¾ of a sub. 3 x ¼ = ¾ of a sub. The number line as a tool helps students recognize that fractions are numbers and they expand the number system beyond whole numbers. (Van de Walle, 203) Numerator The number above the line in a common fraction that indicates the number of parts represented out of a whole or set. 4

4 Open Array Model for Multiplication This model for multiplication, also known as the array model, may be used to solve multiplication problems with fractions. The product illustrates the concept that when fractions are multiplied, a unit is divided into smaller parts. /3 x /4 = /2 Ratio Table Tool used to support multiple strategies and concepts such as fair-sharing relationships and finding equivalent fractions. Ratio Table for Division Standard Algorithms This model is used to generate equivalent ratios with fractions and use ratios to divide fractions. For example, students are given the time needed, as a fraction of an hour, to paint a fraction of a room. They use a ratio table to generate equivalent ratios to determine how much time it takes to paint one room. This is the same as dividing the time by the fraction of the room painted. This is the conventional strategy for solving computational problems for addition, subtraction, multiplication and division. x = The Properties that Hold for Whole Number Operations also Hold for Rational Numbers To Compare, Add or Subtract a Common Whole is Needed The understanding that you may use the same properties to compute whole numbers as you do for rational numbers. For example: you may use the distributive property to solve this problem by decomposing the decimal number to whole and decimal and solving. Common wholes make it possible to compare, add and subtract fractions. Example: If we want to calculate 3/4 of a dollar, thinking about a dollar as the whole is helpful. In this example, the student used a 50 cent piece, 2 dimes, and a nickel until he found 3/4 which was 75/00 of a dollar or 75 cents. Another type of common whole uses the clock model with one hour as the whole. (Fosnot, 2007) 69 x ( ) (69 x 6) + (69 x.6)

5 Unit Fraction A fraction in which the numerator is one when a whole has been divided into equal parts. ½, /3, ¼, /5, /6, /2 Uses a Common Whole to Compare Uses Money as Landmark Numbers This is using the strategy of finding a common whole (or context model) to compare fractions. For example, if a student is asked to compare 5/2 and 2/4, the student may use a clock as a model and convert the fractions to minutes of an hour. The whole becomes 60 minutes and the student is able to convert the fractions to 25/60 and 30/60 which make comparing the two fractions much easier. When students are able to associate the coin value with the fraction of the whole, these become landmark numbers for fractions and decimals. Coin Fraction Reduced Decimal of Dollar Fraction Nickel 5/00 / Dime 0/00 /0 0.0 Quarter 25/00 ¼ 0.25 ½ Dollar 50/00 ½ 0.50 Uses Multiplication and Division to make Equivalent Using a Common Whole to Add and Subtract When given a fraction, the student must use multiplication or division to create an equivalent fraction. In this example, the student uses a ration table as a tool and multiplies to determine if the fractions are equivalent. Students have trouble adding and subtracting fractions so the strategy of using a common whole enables them to use the context to add and subtract fractions. Models that use coins and clocks are helpful. When students are able to use this strategy independently, they demonstrate understanding of the related Big Idea. Reference To Compare, Add or Subtract a Common Whole is Needed as a Big Idea.

6 Using a Ratio Table as a Tool to Make Equivalent The ratio table enables students to find patterns and develop strategies to make equivalent fractions such as doubling and tripling. Using Landmark (e.g. ½, ¼ ) Using Place Value Understanding to Multiply and Divide by Powers of Ten Using Proportional Reasoning The strategy of using landmark numbers to compare and order fractions as well as solve. For example, if a student needs to know where ¼ or /8 is on the number line, dividing the line in half is a strategy for finding ¼ and /8. Once students use multiplying and dividing by fractions and decimals such as tenths and hundredths, they may extend this strategy to multiplication and division by powers of ten and be able to shift decimal points. (Fosnot, 2007) The ability to compare multiplicative relationships between quantities as a strategy to solve. For example: Calculating the advertised cost of $5 for 2 cans of soup to determine how much it will cost for 60 cans of soup. (Fosnot, 2007) = Cost $5 $5 $75 # Units With Decimal Equivalences the Numbers in a Different Place- Value Positions are Related by Powers of Ten With Unit, the Greater the Denominator, the Smaller the Piece is Extending the understanding of whole number place value to decimal numbers and their fraction equivalents. For example, students who understand this big idea are able to notice the relationship of the number of revolutions a hundredths dial makes when increasing or decreasing the dial in the ones place by a single digit. (Fosnot, 2007) An initial misconception is that students generalize their knowledge of whole numbers to fractions. The opposite is the case. For example, when 8 people share a pizza, each piece is smaller than when 7 people share it. The fraction bar chart illustrates this well.