GRADE 6: RATIOS AND PROPORTIONS BY: AMANDA ALVERSON, AMANDA GENTRY, AND DANIEL ORTIZ

PROGRESSION (THINGS CHANGE IN 6 TH GRADE)

6.RP.1 – Understand the concept of a ratio and use ratio language to describe a ratio between two quantities 6.RP.2 – Understand the concept of a unit rate (a/b) associated with a ration a:b with b ≠ 0 and use rate language in the context of a ratio relationship 6.RP.3 – Use ratio and rate reasoning to solve real- world and mathematical problems, e.g. by reasoning about tables of equivalent rates, tape diagrams, double number line diagrams, or equations STANDARDS RELATING FRACTIONS TO RATIOS Important Ideas: Associating fractions with ratios and rates Ratio situations always involve a comparison of 2 or more quantities, ie part to whole or part to part The notion of equivalence Finding equivalent rates The distinction between fractions, ratios, and rates Fraction is a comparison between numerator/denominator (a single number with one point on number line) while a ratio/rate may not be a single number

6.NS.1 – Interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem Important Ideas: Meaning of division involving two fractions Students need to make sense of what is being divided STANDARDS DIVIDING FRACTIONS BY FRACTIONS

Beginning with the concrete Use pattern blocks or fraction strips to model division calculations Using common denominators as a division strategy Find common denominators as a way to use the same units, e.g., 1/3 ÷ 1/10 = 10/30 ÷ 3/30 = 10/3 or 3 1/3 Multiplying the reciprocal as a division strategy Invert and multiply Estimating a quotient for a fraction IMPORTANT IDEAS (CONTINUED) How many triangles fit on the trapezoid? 1/2 ÷ 1/6 = 3

6.NS.5 – Understand that positive and negative numbers are used together to describe quantities having opposite direction or values. 6.NS.6 – Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axis familiar from previous grades to represent points on the line and in the plane with negative number coordinate a. Recognize opposite signs means opposite sides of zero c. Horizontal and vertical number lines 6.NS.7 – Understand ordering and absolute value of rational numbers a. Interpret statements of inequality as statements as position on number line b. Use rational numbers in real world context Important Ideas: Meaning of negative rational numbers -2/3 is the opposite of 2/3; they are the same distance from zero, 2/3 of a unit Renaming negative rational numbers -(2/3) = -2/3 = 2/-3 Comparing and ordering rational numbers Numbers farther to the left are less than numbers farther to the right STANDARDS INTERPRETING RATIONAL NUMBERS

MISCONCEPTIONS Relating Fractions to Ratios Students assume that a ratio is always a fraction, e.g., a ratio is not always out of 1 like a fraction Dividing a Fraction by a Fraction The answer is always smaller like when dividing whole numbers Not having an understanding of number lines, tape diagrams, and models – using them incorrectly Still not having a good idea of what a fraction is or what it represents Interpreting Rational Numbers Students having no knowledge of negative numbers Students not realizing that rational numbers can fit on the number line The value of a negative number is a positive; only seeing a negative as a negative Thinking a number that is farther from zero is always bigger than those numbers closer to zero

SUMMARY “The new concepts related to fractions that students must master in grade 6 involve division of fractions, the relationship between ratios and fractions, and the use of ‘negative fractions’ or rational numbers. Many students, however, are still likely to require more experience with the fraction operation foundation that are a focus in grade 5. The information learned in grade 6 is expanded in grade 7” (Small, pgs. 103, 121).

ACTIVITY “WHAT’S YOUR RATE?” This activity assesses students’ prior knowledge, asks students to think about the ways in which they have used ratios, and the type of information they compared with ratios. It allows teachers to determine students understanding of ratios. To begin the lesson, arrange students in pairs. Tell them to choose one of the following activities to do for one minute: Say the alphabet repeatedly Hop on one foot Do jumping jacks Have one student perform the selected activity while her partner counts and records the number of times the activity was completed in one minute. Students can use tally marks or another efficient way of recording the data. Have the partners switch roles and repeat the process. On the board, record the data that each pair has collected. Use only whole numbers. Disregard any half letters, hops, or jumping jacks. Ask students: Can you use your data to predict how many times you can complete the activity in one hour? Point out that this is the unit rate (per unit). The students should begin to see that with this data, they can estimate how many times the activity could be done in other time spans, such as one hour. Next, discuss the concept of proportion. Define proportion as two equal ratios. Explain to students that when they write proportions, they should use a variable in place of the unknown data in the equation. Example: 30 hops = X hops 1 min. 60 min. Use examples from the data students gathered to set up proportions and estimate results for different times. The students should use the data to practice solving proportions (they can use the number of times they completed the activity as their unknown). Check students’ work to make sure they are setting up proportions consistently, placing the time in the denominators of both sides of the proportion.

REFERENCES Ball, D. L., Lubienski, Mewborn, D. S. (2001). Research on Teaching Mathematics: The Unsolved Problem of Teachers’ Mathematical Knowledge. Handbook of Research on Teaching. New York. Virginia. Carborne, K. (2015). Whats your rate? Illuminations Math. Illuminations.nctm.org Cramer, K., Post, T., & Currier, S. (1993). Learning and Teaching Ratio and Proportion: Research Implications. InD. Owens (Ed.), Research Ideas For the Classroom (pp ). NY: Macmillan Publishing Company. Nunokawa, K. (2012). Multi-Relation Strategy in Students’ Use of a Representation for Proportional Reasoning. Ratios and Proportions. Small, M. (2014). Grade 6. Uncomplicating Fractions to Meet Common Core Standards in Math K-7. Columbia University: Teachers' College. Van de Walle, J. (2010). Elementary and Middle School Mathematics: Teaching Developmentally. Chapter 18 Ratios and Proportions. MA: Pearson Education D/Feldman%20Final%20Draft%20Common%20Core%20Fractions.cmap ?rid=1LK6PTWJ7-4GYTZK-104D&partName=htmljpeg