Presentation on theme: "Multiplication and Division of Fractions: Thinking More Deeply"— Presentation transcript:
1Multiplication and Division of Fractions: Thinking More Deeply Nadine Bezuk and Steve KlassSession CMC-S 2005
2Today’s Session Welcome and introductions Meanings for operations How do we help children model and reason about the operationsMultiplication and division with whole numbersMultiplication and division with fractionsModels for multiplication of fractionsSet, Array, Area, MeasurementModels for division of fractionsQuestions
3What Students Need to Know Well Before Operating With Fractions Meaning of the denominator (number of equal-sized pieces into which the whole has been cut);Meaning of the numerator (how many pieces are being considered);The more pieces a whole is divided into, the smaller the size of the pieces;Fractions aren’t just between zero and one, they live between all the numbers on the number line;A fraction can have many different names;Understand the meanings for operations for whole numbers.
4Types of Models for Considering Fractions Area/regionFraction circles, pattern blocks, paper folding, geoboards, fraction bars, fraction strips/kitsLength/linearNumber lines, rulers, (fraction bars, fraction strips/kits)Set/discreteChips,counters, painted beans
5What Does Elliot Know? What does Elliot understand? What concepts is he struggling with?How could we help him understand how to model and reason about the problem?
6What Do Children Need to Know in Order to Understand Division With Fractions?
7Reasoning About Division Whole number meanings for division6 ÷ 2 = 3Sharing / partitiveWhat does the 2 mean? What does the 3 mean?Repeated subtraction / measurementNow what does the 2 mean and what does the 3 mean?
8Reasoning About Division With Fractions Sharing meaning for division:1 ÷One shared by one-third of a group?How many in the whole group?How much does each group get?How does this work?
9Reasoning About Division With Fractions Repeated subtraction / measurement meaning1 ÷How many times can one-third be subtracted from one?How many one-thirds are contained in one?How does this work?How might you deal with anything that’s left?
10Dealing With Remainders Two or more units being measured?
11A Context For Division of Fractions You have 1 cups of sugar. It takes cup to make 1 batch of cookies. How many batches of cookies can you make?How many cups of sugar are left?How many batches of cookies could be made with the sugar that’s left?
12Models for Reasoning About Division Area/Measurement (batches of cookies)Linear/Measurement (ribbon - external units)How do you deal with the remainder?
13Materials for Modeling Division of Fractions How would you use these materials to model?Paper tapeFraction circlesYou could also use:Pattern blocksFraction Bars / Fraction Strips
14Multiplication of Fractions Consider:How do you think a child might solve each ofthese?What kinds of reasoning and/or models mightthey use to make sense of each of theseproblems?
15An Eighth Grade Problem? George has of a pie. He ate of that.How much pie did he eat?17% of 13-year-olds answered correctly, though 60% could correctly calculate the product(NAEP, 1983, p. 26)
16Another ContextAt one school 3/4 of all eighth graders went to one game. Two-thirds of those who went to the game traveled by car. What part of all the eighth graders traveled by car to the game?12% chose to multiply, while about 55% decided to subtract and about 8% to divide! (Sowder, 1988)
17Reasoning About Multiplication Whole number meanings - U.S. conventions4 x 2 = 8Set - Four groups of twoArray - Four rows of two columnsMeasurement - Four units by two units2 x 4 = 8Set - Two groups of fourArray - Two rows of four columnsMeasurement - Two units by four units
18Reasoning About Multiplication Fraction meanings - U.S. conventionsSet - Two-thirds of one group of three-fourthsArray - Two-thirds of a row of three-fourths of one columnMeasurement - Two-thirds of one unit by three-fourths of one unitSet - Three-fourths of one group of two-thirdsArray - Three-fourths rows of two-thirds of one columnMeasurement - Three-fourths of one unit by two-thirds of one unit
19Contexts for Multiplication Finding part of a part (a reason why multiplication doesn’t always make things “bigger”)Pizza (pepperoni on )Brownies ( is frosted, of the that part has pecans)Lawn ( is mowed, of that is raked)
20Models for Reasoning About Multiplication Area/measurement models (fraction circles)Linear/measurement (ribbon)Set models (eggs in cartons)
21Materials for Modeling Multiplication of Fractions How would you use these materials to model ?Paper tapeFraction circlesYou could also use:Pattern blocksFraction Bars / Fraction Strips
23ReferencesNational Assessment of Educational Progress. (1983) The third national mathematics assessment: Results, trends and issues. Report No. 13-MA-01. Denver: Education Commission of the States.Sowder, L. (1988) Concept-driven strategies for solving story problems in mathematics. Final Tech. Rep, Center for Research in Mathematics and Science Education, San Diego State University, San Diego, CA (ERIC Document Reproduction Service No. ED )
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