Why Learn Fractions, Decimals and Percents? Title I Directors’ Meeting – March 9, 2010 Waterfront Place Hotel – Morgantown
Why Fractions, Decimals and Percents? “A major goal for K-8 mathematics education should be proficiency with fractions (including decimals, percents and negative fractions), for such proficiency is foundational for algebra.” - National Math Panel
In one recent study…51% of adults could not calculate a 10% tip on a lunch bill.
Item from 8th Grade NAEP Assessment Ms. Thierry and 3 friends ate dinner at a restaurant. The bill was $67. In addition, they left a $13 tip. Approximately what percent of the total bill did they leave as a tip? A) 10% B) 13% C) 15% D) 20% E) 25%
Results from NAEP Ms. Thierry and 3 friends ate dinner at a restaurant. The bill was $67. In addition, they left a $13 tip. Approximately what percent of the total bill did they leave as a tip?
More from the National Math Panel • The Panel continues…“A conceptual understanding of fractions and decimals and the operational procedures for using them are mutually reinforcing. One key mechanism linking conceptual and procedural knowledge is the ability to represent fractions on a number line. Instruction focusing on conceptual knowledge of fractions is likely to have the broadest and largest impact on problem‐solving performance when it is directed toward the accurate solution of specific problems. “
WV CSO’s Addressing Fractions, Decimals and Percents WV CSOs addressing fractions (including decimals, percents, and negative fractions): M.O.K.1.7,M.O.1.1.9, M.O.2.1.7, M.O.3.1.2, M.O.3.1.6, M.O.3.1.7, M.O.4.1.1, M.O.4.1.4, M.O.4.1.5, M.O.4.1.6, M.O.4.1.7, M.O.5.1.1, M.O.5.1.3, M.O.5.1.5, M.O. 5.1.6, M.O.5.1.7, M.O. 5.1.11, M.O.6.1.4, M.O.6.1.6, M.O.6.1.8, M.O.7.1.1, M.O.7.1.5, and M.O.8.1.3.
M.O.K.1.7 identify and name halves and wholes using concrete models M.O.1.1.9 identify, name, and explain why a given part is a half, third or fourth of a whole or part of a group, using concrete models. M.O.2.1.7 identify and explain fractions as part of a whole and as part of a set/group using models. M.O.3.1.5 demonstrate an understanding of fractions as part of a whole/one and as part of a set/group using models and pictorial representations. M.O.3.1.6 create concrete models and pictorial representations to compare and order fractions with like and unlike denominators, add and subtract fractions with like denominators, and verify results. M.O.3.1.7 use concrete models and pictorial representations to demonstrate an understanding of equivalent fractions, proper and improper fractions, and mixed numbers. M.O.4.1.4 using concrete models, benchmark fractions, number line compare and order fractions with like and unlike denominators add and subtract fractions with like and unlike denominators model equivalent fractions model addition and subtraction of mixed numbers with and without regrouping.
M.O.4.1.5 analyze the relationship of fractions to decimals using concrete objects and pictorial representations. M.O.5.1.7 analyze and solve application problems and justify reasonableness of solution in problems involving addition and subtraction of: fractions and mixed numbers, decimals. M.O.6.1.4 analyze and solve real‐world problems involving addition, subtraction , multiplication and division of whole numbers, fractions, mixed numbers, decimals, integers, and justify the reasonableness by estimation. M.O.7.1.1 compare, order, and differentiate among integers, decimals, fractions, and irrational numbers using multiple representations (e.g., symbols, manipulatives, graphing on a number line). M.O.8.1.3 analyze and solve grade‐appropriate real‐world problems with whole numbers, decimals, fractions, percents, percent increase and decrease, integers, and including, but not limited to, rates, tips, discounts, sales tax and interest and verify solutions using estimation techniques.
Item from 4th Grade NAEP Assessment What fraction of the group of umbrellas is closed? A) 1/3 B) 3/7 C) 4/7 D) 3/4
Results from NAEP What fraction of the group of umbrellas is closed?
Grade 4, Benchmark 3 Acuity Item Dominic had a candy bar that was divided into 8 pieces. He gave 1/8 of the candy bar to Rose and 1/4 of the candy bar to Cathy. How much of his candy bar did Dominic have left? A. B. C. D. 2_ 12 5_ 8 1_ 2 3_ 8
Results from 8 counties(1 from each RESA) 2_ 12 5_ 8 1_ 2 3_ 8 n = 2852
Why Learn Fractions, Decimals and Percents? How should we teach
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What is a good story or model for the problem: 1 ¾ ÷ ½ ? What is the quotient 1 ¾ ÷ ½ ? from "Knowing and Teaching Elementary Mathematics" by Liping Ma
What is a good story or model for the problem: 1 ¾ ÷ ½ ? • 9 US teachers (43%) calculated a correct and complete quotient (6 teachers could not give any answer) • 72 Chinese teachers (100%) calculated a correct and complete quotient • 1 US teacher COULD create a model (6 could not come up with a story, 16 created stories with misconceptions) • 1 Chinese teacher COULD NOT create a story represented by the division problem. from "Knowing and Teaching Elementary Mathematics" by Liping Ma
How is it possible to learn about operations with fractions without learning algorithms to do the operations?
Representations of Rational Numbers • Part Whole (including unitizing) • Fractions as quotients • Fractions as measures • Fractions as operators • Fractions as ratios and rates
A Serving of Waffle Cake You have a paper disc on your table. Use this disc to represent a waffle cake. Joey is allowed to have 1/6 of the waffle cake. Show Joey’s share by folding the paper. Explain your answer.
More waffle cakes Jane has two waffle cakes (represented by 2 paper discs). Jane is allowed to have 1/6 of the total amount of waffle cake. Show Jane’s share by folding the paper Explain your answer.
Still More Waffle Cakes Chris has 1 ½ waffle cakes. Chris is allowed to have 1/6 of this amount of waffle cakes. Show Chris’s share by folding the paper. Explain your answer.
Working on a Number Line Locate ¾ on each of the number lines. Be prepared to explain your thinking. 0 1 0 5
Working on a Number Line Locate ¾ on each of the number lines. Be prepared to explain your thinking. 3 4 0 1 3 4 0 5
Fractions as measures Problem is from a 2006 study from the University of Arizona. This task was given to 56 7th-grade students Only 9 of the 56 students showed correct representations on both number lines 36 students were correct on the 0-1 number line but incorrect on the 0-5 number line.