Example: biology

Multiplication to Ratio, Proportion, and Fractions within ...

Multiplication to Ratio, Proportion, and Fractionswithin the Common CoreKaren C. Fuson1, Sybilla Beckmann21 Northwestern University, Professor Emerita2 Department of Mathematics, University of GeorgiaNCTM Annual Meeting, 2012 Karen C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC1 / 40 CCSS Grade 6 Critical Area 1 Students use reasoning about Multiplication and division to solve ratioand rate problems about quantities. By viewing equivalent ratios andrates as deriving from, and extending, pairs of rows (or columns) in themultiplication table , and by analyzing simple drawings that indicate therelative size of quantities, students connect their understanding ofmultiplication and division with ratios and rates.

multiplication table, and by analyzing simple drawings that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates.

Tags:

  Table, Ratios, Multiplication, Proportions, Multiplication table, Multiplication to ratio

Information

Domain:

Source:

Link to this page:

Please notify us if you found a problem with this document:

Other abuse

Transcription of Multiplication to Ratio, Proportion, and Fractions within ...

1 Multiplication to Ratio, Proportion, and Fractionswithin the Common CoreKaren C. Fuson1, Sybilla Beckmann21 Northwestern University, Professor Emerita2 Department of Mathematics, University of GeorgiaNCTM Annual Meeting, 2012 Karen C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC1 / 40 CCSS Grade 6 Critical Area 1 Students use reasoning about Multiplication and division to solve ratioand rate problems about quantities. By viewing equivalent ratios andrates as deriving from, and extending, pairs of rows (or columns) in themultiplication table , and by analyzing simple drawings that indicate therelative size of quantities, students connect their understanding ofmultiplication and division with ratios and rates.

2 Thus students expandthe scope of problems for which they can use Multiplication anddivision to solve problems, and they connect ratios and solve a wide variety of problems involving ratios and C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC2 / 40 Definitions of rate and ratioPeople do not agree about definitions of rate and CCSS learning path sought to support students to extend earlierunderstandings and avoid common errors and the R&P Progression for more C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC3 / 40 Notation ConfusionsBy Grade 6 what do students know about Fractions and the notation35?Karen C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC4 / 40 What does35mean?

3 3 parts of size15(15is 1 part when a whole is partitioned into 5 equal parts) 5=35(a fraction)The result of division can be expressed as a C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC5 / 40 Fractions versus ratiosFractions and ratios are different in their basic meanings: Fractions :are numbers telling how many parts of what sizeRatios:describe relationships between quantitiespart A to part B or part B to part A or part A (or B) to totalor total to part A (or B)It is too confusing to use the same notation for this new C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC6 / 40 Levels in learning ratioLevel 1:Grade 6 earlyUse 3:5 notation initially to build a new concept withwhole number 2:Grade 6 laterSee the quotient meaning35some people use for a ratioas a unit rate, the value of a ratio.

4 Relate Fractions andratios and all 3:Grade 7 ratios and proportions use Fractions such as34:25. Theconstant of proportionalityciny=cxis a unit this equation is actuallyBA, the unit rate forB:A,and is thereciprocalof the unit rate C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC7 / 40 Levels in learning ratioLevel 1:Grade 6 earlyUse 3:5 notation initially to build a new concept withwhole number 2:Grade 6 laterSee the quotient meaning35some people use for a ratioas a unit rate, the value of a ratio. Relate Fractions andratios and all 3:Grade 7 ratios and proportions use Fractions such as34:25. Theconstant of proportionalityciny=cxis a unit this equation is actuallyBA, the unit rate forB:A,and is thereciprocalof the unit rate C.

5 Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC7 / 40 Avoiding errorsMany proportion errors involve adding, not get into Multiplication -land first for ratio and C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC8 / 40 ResearchFuson, K. C. & Abrahamson, D. (2005). Understanding ratio andproportion as an example of the Apprehending Zone andConceptual-Phase Problem-Solving Models. In J. Campbell (Ed.),Handbook of Mathematical Cognition(pp. 213-234). New York:Psychology other articles you can get from Dor Abrahamsondor at our teaching experiments, Grade 5 students outperformedmiddleand high school students on proportion C.

6 Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC9 / 40 Karen C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC10 / 40 Karen C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC11 / 40 Karen C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC12 / 40 Extend a rate situation to be a class of rate situations with the sameunit rate and show them in a table . The unit rate involves started to save money. Every day she put three $1coins into her duck C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC13 / 40 RateDiscuss rate as an equal-groups hiding 1:$3 each day, $3 per day, $3 every day$3 each 1 day, $3 per 1 day, $3 every 1 dayThe unit rateis the amount in 1 group but we do not say the is how Multiplication with 3 numbers becomes aproportion with 4 numbers: it uses the 3=6 becomes 1:3=2:6 Karen C.

7 Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC14 / 40 Rate tablesStart with the term rate table as showing many situations with thesame show multiples of the unit rate starting with 1 in the first that these are just two columns of the ratio tables are introduced, we will notice that rate tables and ratiotables really are quite similar and behave alike (rows are multiples ofthe unit rate or basic ratio), so we consider rate tables as a specialcase of ratio tables and can call them ratio C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC15 / 40 What situations have a constant rate?Students discuss what situations have a constant rate and whichexample tables are rate and areas can be considered as equal groups (one row oronecolumn is the group), so rates can be used for such row is a multiple of the unit rate (later, of each other row, whenmultiplying by a fraction is included).

8 Karen C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC16 / 40 Finding unit ratesFind the unit rate given a product and the number of things:P n=unit ratePut this information in a scrambled rate table and fill in other scrambledrows of the C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC17 / 40 Relate table , equation, and graphKaren C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC18 / 40 From rate tables to ratio tablesRatios as the product columns from two linked rate s brother Tim saves $5 a day. Noreen and Tim startsaving on the same C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC19 / 40 Equivalent ratiosEquivalent ratios are two rows from a ratio can be written as6:10=21:35or6:10::21:35a)A basic ratio (Confrey s littlest recipe) is the least possible wholenumber ratio (from the 1s row of the MT).

9 Equivalent ratios are twomultiples of the basic )Equivalent ratios are multiples of each other (where one multiplecan be a fraction<1).Karen C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC20 / 40 ProportionsTwo equivalent ratios make a made applesauce using the same number of bagsof red and yellow apples. Her red apples cost $6, and heryellow apples cost $14. I used her recipe but made moreapplesauce. I paid $35 for my yellow apples. How much didmy red apples cost?Karen C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC21 / 40 Factor Puzzles, Ratio Tables, and Multiplication TablesThe Factor Puzzle and the Ratio table as columns from a MTimmediately makes a whole range of proportion problems it is important to explore the following three the with problems that have the information out of order:scrambled your assumption that makes the situation C.

10 Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC22 / 40 Factor Puzzles, Ratio Tables, and Multiplication TablesThe Factor Puzzle and the Ratio table as columns from a MTimmediately makes a whole range of proportion problems it is important to explore the following three the with problems that have the information out of order:scrambled your assumption that makes the situation C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC22 / 40 Additive structurecupsgrapecupspeach52104+5+2+5+2 156+5+2208+5+22510+5+2112233445678910111 213141556+5+2cups grapecups peach+5+2 Karen C. Fuson, Sybilla Beckmann (NU,UGA)Ratio, Proportion in CC23 / 40 Multiplicative structurecupsgrapecupspeach52104 2 3 20 2 3 2015620810040521122334456789101112131415 56156cups grapecups peach 3 3 Karen C.


Related search queries