Presentation on theme: "Fraction Sense* How do we get there?"— Presentation transcript:
1Fraction Sense* How do we get there? Decimal DisconnectsAnd, what to do about them…*which includes decimals, and percent…
2Why Fractions?50% of 8th graders could not order three fractions from least to greatest (NAEP, in NCTM, 2007).Less than 30% of 17-year olds correctly represented as 29/1000 (Kloosterman, 2010).When asked which of two decimals, and 0.83 is greater, most 5th and 6th graders chose (Rittle-Johnson, Siegler, and Alibali, 2001).
3Grade 4.NFExtend understanding of fraction equivalence and ordering.Explain why a fraction a/b is equivalent to a fraction (n x a/n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.Compare two fractions with different numerators and denominatorsC. Understand decimal notation for fractions and compare decimals fractions.Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators of 10 and 100.Use decimal notation for fractions with denominators 10 or 100.Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of the comparisons with the symbols <, =, or >, and justify the conclusions (e.g. by using a visual model.SKIP
4Understand the place value system. Grade 5.NBTUnderstand the place value system.Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of ten.Read, write, and compare decimals to thousandths.Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., = 3 x x x 1 x 3 x (1/10) + 9 x (1/100) + 2 x (1/1000).Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results.4. Use place value understanding to round decimals to any place.SKIP
5What’s important here? Representations – models, drawings Equivalence – very importantComparing and Ordering Decimals – linked to equivalenceConnections – connecting math-wise (e.g. to fractions, ratio, proportion)Contexts – tasks in settings (e.g. measurement)Fennell, Kobett, and Wray, MTMS (2014)
6Think about and do…Represent the following using what’s in front of you:0.20.871.230.999 (good luck)What materials do you use for decimals?
70.7What happens to the value of the decimal if a 0 is inserted AFTER the 7?What happens to the value of the decimal if a 0 is inserted BEFORE the 7?How would you demonstrate that 0.15 > 0.149
8Which is closest to 1? 0.9 1.1 0.91 1.09 How do you know? Fennell, Kobett, and Wray, MTMS (2014)
9Comparing and BetweenThink of a number between the following – show how you know.1/4 and 1/3and16% and 19%Note: finding a fraction that falls between is more challenging (for most) than finding a decimal or percent.
10More on ComparingGive the students cards with the numbers 1, 4, 5, and 9 and one with a decimal point on them (or some other random digits Maybe different pairs get different sets?Then ask them to create the smallest number they can. The biggest number they canA number that is closest to 5. A number that is closest to 50.
11More on ComparingAlan tried to make a decimal number as close to 50 as he could (using 1, 4, 5, & 9). He arranged them in this order: Jerry thinks he can arrange the same digits to get a number that is even closer to 50. Do you agree or disagree? Explain your thinking.
13Mathematics Teaching Cases Carne Barnett-Clarke, et al Take a look…Mathematics Teaching CasesCarne Barnett-Clarke, et al
14Grade 5.NBT B. Perform operations with multi-digit whole numbers and with decimals to hundredths. 7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.SKIP
16What contexts do you use? Advocate for? Decimal ContextsGrade 4.MDSolve problems involving measurement and conversion of measurement from a larger unit to a smaller unit.Grade 5.MDA. Convert like measurement units within a given measurement system.What contexts do you use? Advocate for?SKIP
17What about percent? Common percents and you? 6.RP.A.3.c – Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent.7.RP.A.3 – Use proportional relationships to solve multistep ratio and percent problems (interest, taxes, etc.).Standards and curriculum!
18Coming next: middle school… Grade 6 – Ratios and Proportional RelationshipsUnderstand ratio concepts and use ratio reasoning to solve problems.Concept of ratioConcept of unit rateRatio and rate reasoningGrade 7 – Ratios and Proportional RelationshipsAnalyze proportional relationships and use them to solve real-world and mathematical problems.Compute unit rates associated with ratios of fractions…Recognize and represent proportional relationships between quantities.Use proportional relationships to solve multistep ratio and percent problems.SKIP
20Critical AreasGrade 5Extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations.Grade 6Connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems;