Rate, Ratio, and Proportional Reasoning Using Equivalent Fractions
TABLE OF CONTENTS
Concepts & Skills to Maintain
Selected Terms and Symbols
In this unit students will:
• gain a deeper understanding of proportional reasoning through instruction and
• will develop and use multiplicative thinking
• develop a sense of proportional reasoning
• develop the understanding that ratio is a comparison of two numbers or quantities
• find percents using the same processes for solving rates and proportions
• solve real-life problems involving measurement units that need to be converted
Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the eight practice standards should be addressed constantly as well. To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement this unit. This unit provides much needed content information, but excellent learning activities as well. The tasks in this unit illustrate the types of learning activities that should be utilized from a variety of sources.
STANDARDS ADDRESSED IN THIS UNIT
Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics. Teaching of the CCGPS should embed the eight mathematical practices. The Standards for Mathematical Practice (MP) describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.
Understand ratio concepts and use ratio reasoning to solve problems.
MCC6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
MCC6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 (b not equal to zero), and use rate language in the context of a ratio relationship.
MCC6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g. by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
MCC6.RP.3a Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
MCC6.RP.3b Solve unit rate problems including those involving unit pricing and constant speed.
MCC6.RP.3c Find a percent of quantity as a rate per 100, (e.g. 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole given a part and the percent.
MCC6.RP.3d Use ratio reasoning to convert measurement units, manipulate and transform units appropriately when multiplying or dividing quantities.
Standards for Mathematical Practice:
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning , strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately) and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
1. Make sense of problems and persevere in solving them.
In grade 6, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”
2. Reason abstractly and quantitatively.
In grade 6, students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations.
3. Construct viable arguments and critique the reasoning of others.
In grade 6, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like “How did you get that?”, “Why is that true?” “Does that always work?” They explain their thinking to others and respond to others’ thinking.
4. Model with mathematics.
In grade 6, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students begin to explore covariance and represent two quantities simultaneously. Students use number lines to compare numbers and represent inequalities. They use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences about and make comparisons between data sets. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context.
5. Use appropriate tools strategically.
Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 6 may decide to represent similar data sets using dot plots with the same scale to visually compare the center and variability of the data. Additionally, students might use physical objects or applets to construct nets and calculate the surface area of three dimensional figures.
6. Attend to precision.
In grade 6, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to rates, ratios, geometric figures, data displays, and components of expressions, equations or inequalities.
7. Look for and make use of structure.
Students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist in ratio tables recognizing both the additive and multiplicative properties. Students apply properties to generate equivalent expressions (i.e. 6 + 2x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality), c=6 by division property of equality). Students compose and decompose two‐ and three dimensional figures to solve real world problems involving area and volume.
8. Look for and express regularity in repeated reasoning.
In grade 6, students use repeated reasoning to understand algorithms and make generalizations about patterns.
During multiple opportunities to solve and model problems, they may notice that a/b ÷ c/d = ad/bc and construct other examples and models that confirm their generalization. Students connect place value and their prior work with operations to understand algorithms to fluently divide multi‐digit numbers and perform all operations with multi‐digit decimals. Students informally begin to make connections between covariance, rates, and representations showing the relationships between quantities.
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
MCC6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fraction, e.g. by using visual fraction models and equations to represent the problem.
Compute fluently with multi-digit numbers and find common factors and multiples.
MCC6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.
MCC6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
MCC6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100. And the least common multiple of two whole numbers less than or equal to 1. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no remainder.
•A ratio is a number that relates two quantities or measures within a given situation in a multiplicative relationship (in contrast to a difference or additive relationship).The relationships and rules that govern whole numbers, govern all rational numbers.
•Making explicit the type of relationships that exist between two values will minimize confusion between multiplicative and additive situations.
•Ratios can be express comparisons of a part to whole, (a/b with b ≠0), for example, the ratio of the number of boys in a class to the number of students in the class.
CONCEPTS & SKILLS TO MAINTAIN
It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.
• Multiples and Factors
• Divisibility Rules
• Relationships and rules for multiplication and division of whole numbers as they apply to decimal fractions
• Understanding of common fractions
SELECTED TERMS AND SYMBOLS
The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.
The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.
The websites below are interactive and include a math glossary suitable for middle school children.
Note – At the middle school level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks.
This web site has activities to help students more fully understand and retain new vocabulary
Definitions and activities for these and other terms can be found on the Intermath website. Intermath is geared towards middle and high school students.
• Percent: A fraction or ration in which the denominator is 100
• Proportion: An equation which states that two ratios are equal.
• Rate: A comparison of two quantities that have different units of measure
• Ratio: compares quantities that share a fixed, multiplicative relationship.
• Rational number: A number that can be written as a/b where a and b are integers, but b is not equal to 0.
• Unit Ratio: are ratios written as some number to 1.
• Quantity: is an amount that can be counted or measured.