Number, Operations, Rates and Ratio
Every branch of mathematics and nearly every application of mathematics uses numbers and operations in essential ways for reasoning and problem-solving tasks. The applications in Connected Mathematics (CMP) involve whole numbers, integers, fractions, decimals, percents, ratios, and irrational numbers. The overarching goal in the CMP3 Number and Operations strand is to extend student understanding and skill in the use of numbers and operations to represent and reason about quantitative information.
There are ten Units in the CMP3 Number, Operations, and Proportional Reasoning strand. These Units aim to:
- Extend knowledge of whole numbers, fractions, and decimals, develop the ability to know which representation and which operation to choose, and develop fluency in operations with those numbers;
- Develop concepts of ratio, rate, and proportion, particularly equivalence of ratios, and apply to real-world problems;
- Develop understanding of the meaning of percents and facility with applications;
- Introduce negative, irrational, and complex numbers and the structural properties of each number system.
The learning progressions that work toward each of those broad goals are described in the following sections.
Extending Whole Numbers, Fractions and Decimals
Proficiency in the use of common fractions depends on understanding the multiplicative structure of whole numbers. This includes understanding the concepts of factor, multiple, prime number, greatest common factor, and least common multiple. Prime Time begins the CMP3 curriculum by developing ideas of elementary number theory while engaging students in games and problem-solving tasks that set the tone for a classroom that operates as a community of learners in which all participants are active participants. Prime Time also begins the Algebra and Functions strand of the curriculum by developing student understanding of arithmetic expressions, the Distributive Property, and the Order of Operations. The review and extension of fraction and decimal number understanding and skill is focused in the three Grade 6 Units, Comparing Bits and Pieces, Let’s Be Rational and Decimal Ops. In theseUnits, students develop number sense and problem solving habits of mind by addressing four key questions:
- What kind of numbers will accurately represent the quantities involved?
- What operations on those numbers will provide answers to the core questions posed?
- What is a good estimate for the result of those operations?
- What exact result is produced by applying a standard computational algorithm for the operations?
The name change from Bits and Pieces II (CMP2) to Let’s be Rational, Bits and Pieces I to Comparing Bits and Pieces, and Bits and Pieces III to Decimal Ops signals an extension of students’ understandings about fractions and decimals. Negative numbers are introduced and students order these numbers on the number line, observing that, for example, - ¾ and ¾ are the same distance from zero but on opposite sides of zero. A consequence of the addition of by-hand computations with multidigit numbers in Decimal Operations is that students can see that any fraction a/b can be written as a decimal, which will either repeat or terminate, by computing a ÷ b.
Let's Be Rational and Decimal Ops Units focused on fraction and decimal operations also begin the work on basic ideas of algebra by exploring fact families that connect pairs of related operations, equivalent equations, and equivalent expressions.
The multiplication/division fact family relating 4, 6, and 24 includes:
4 x 6 = 24
6 x 4 = 24
4 = 24 ÷ 6
6 = 24 ÷ 4
Students work on computational skills and simultaneously develop skill in solving simple equations. Use of fact-family reasoning enables solution of one-step equations. For example, to solve 4x = 24, students reason that x = 24 ÷ 4, or to solve 6% of x = $4.80, students reason that x = $4.80 ÷ 0.06.
The Distributive Property of Multiplication Over Addition and Subtraction, introduced in Prime Time allows writing of arithmetic calculations in a variety of equivalent forms.
4(5 + 1) = 4(5) + 4(1).
The Distributive Property is used in the Algebra and Functions strand in support of the overarching goal of writing and interpreting equivalent expressions. As the integer, rational, real, and complex number systems are developed in succeeding Units, students are repeatedly asked whether such very useful structural properties of operations do or should continue to apply in the new contexts.
Ratio, Rates, Scale Factors and Proportions
Problem 1.1 in Comparing Bits and Pieces presents some fundraising goals and then asks students whether some claims about each fundraising goal are true. The aim of Problem 1.1 is to focus student attention on the common mathematical challenge of comparing quantities accurately and fairly. It also highlights the difference between comparison by subtraction (or addition) and comparison by division (or multiplication).
Problem 1.1 introduces the proportional reasoning concepts of ratio, rate, and percent that are then developed in subsequent Problems of the Unit. The Problem serves a second important function of helping teachers to determine the breadth and depth of students’ prior knowledge of ratio, rate, and percent. Problem 1.1 also illustrates our guiding principle that students will grasp new concepts most readily if they encounter them in familiar contexts to which they can first apply their informal sense-making and then abstract underlying common mathematical structures.
Subsequent problems in Comparing Bits and Pieces develop the concept of ratio and rate more explicitly, based on the idea that a ratio of a to b means every a of one quantity is related to b of a second quantity. Equivalence of ratios is developed by reasoning about what makes sense in meaningful contexts and then equivalence of ratios is connected to equivalence of fractions. The ways that working with ratios is different from working with fractions are also highlighted.
Suppose there are 10 boys and 15 girls in one class and 12 boys and 9 girls in another class. The ratio of boys to girls in the two classes combined is 22 to 24.
From this informal, sense-making, intuitive beginning, the concepts of ratio and rate develop in breadth and depth across the Grade 6 and Grade 7 CMP3 courses. Unit rates and rate tables are introduced in the second Investigation of Comparing Bits and Pieces and revisited in an algebraic context in Variables and Patterns near the end of Grade 6.
In Grade 7, Stretching and Shrinking develops thinking about ratios in terms of geometric similarity and scale factors, and Comparing and Scaling brings ratios, rates, scale factors, and rate tables together to develop strategies for solving proportions. These fundamental proportional reasoning ideas are then extended and applied to work on linear functions, rates of change, and slope of graphs in the algebra Units Moving Straight Ahead in Grade 7 and Thinking with Mathematical Models at the start of Grade 8.
The Number strand and the Algebra and Functions strand both make use of the concepts of rate and proportionality.
Throughout this development of proportional reasoning concepts and skills, students are continually asked two key questions.
When does it make sense to compare quantities by ratio or rate?
How can ratios or rates be expressed in equivalent forms to answer questions that involve proportions?
Building on the concepts of equivalence and scaling, Comparing and Scaling enables students to develop reasonable algorithms for solving proportions, underpinned by sound conceptual understanding.
In almost every practical quantitative reasoning task, percents are the most common tool for framing and resolving questions that call for comparison of quantities. They offer standardized language and procedures for describing the relationship between two quantities from both additive and multiplicative perspectives.
Additive Perspective Examples
If an item originally priced at $15 is on sale for $9, the price reduction is $6. Also, $6 is 40% of the original price, or the sale price is 60% of the original price.
If a city’s population increases from 25,000 to 30,000, the increase is 5,000 which is 20% of the original population. The new population is 120% of the original population.
While each of these examples represent additive reasoning, the increase or decrease can be expressed as a percent.
Multiplicative Perspective Examples
What is 30% of 90?
0.30 x 90 = n
36 is what percent of 90?
p x 90 = 36
45 is 25% of what number?
45 = 0.25 x n
The prior work with fact families relating multiplication and division pays rich dividends in learning how to deal with percent problems. Of course, percents appear throughout the geometry, data, probability, and algebra Units of each grade.
Traditional introductions to negative numbers focus on integers-positive and negative whole numbers and 0. The development in CMP3 takes a somewhat different path. In compliance with CCSSM requirements, we begin in the sixth-grade Unit Comparing Bits and Pieces by extending the rational number line to include negative numbers. The development there is limited to location of positive numbers and their opposites and the concept of absolute value. Later in Grade 6, a problem in Variables and Patterns extends the coordinate plane to all four graphing quadrants, using negative numbers informally with the assumption that sixth-grade students will almost certainly have had some encounters with them.
The heart of the development of negative numbers is the Grade 7 Unit Accentuate the Negative. We promote sense-making through games with wins and losses and other story contexts, number line models, and chip models to build on student intuitions and derive standard rules for operating with negative numbers. In the spirit of connected mathematics, we mix tasks about negative rational numbers with those limited to integer values.
In addition to using informal reasoning and models for numbers and operations, we again connect with the fact-family idea in the derivations of algorithms.
After establishing rules for multiplication (product of two negative numbers is the subtle case), rules for the division follow:
(-30) ÷ 5 = -6 because -30 = 5(-6)
(-30) ÷ (-5) = 6 because -30 = (-5)(6)
The final Investigation of Accentuate the Negative takes a retrospective view of the family of number systems—whole numbers, integers, and rationals—and highlights the basic structural properties that are common to all, especially the distributive property. Since Accentuate the Negative occurs relatively early in Grade 7, negative numbers are available for applications in other Units, most notably in Moving Straight Ahead about linear functions.
Irrational and Complex Numbers
The development of standard number systems and operations in CMP3 is completed in two Units of the Grade 8 curriculum-Looking for Pythagoras and Function Junction. Looking for Pythagoras makes the standard connection between square roots and diagonals of squares or hypotenuses of right triangles. It deals with decimal representation of rational (repeating) and irrational (non repeating) numbers, and, per CCSSM specifications, introduces cube roots as well. The number π pops up earlier, in Grade 7, when circumference and area of circles is tackled in Filling and Wrapping.
Complex numbers have traditionally been a topic of high school Algebra II courses, but the CCSSM syllabus for Algebra I calls for introduction of complex numbers. Thus, in our effort to offer CMP3 materials that support a full introductory algebra course, we have included complex numbers in Function Junction. The development there is mathematically fairly standard, continuing the pattern of extending the number system to include meaningful numbers that provide solutions for equations not solvable in preceding simpler number systems.
3x = 2 is not solvable in whole numbers, but it is solvable in rational numbers;
3 + x = 2 is not solvable in positive numbers, but it is solvable in integers;
x2 = 2 is not solvable in rational numbers, but it is solvable in real numbers;
x2 = - 1 is not solvable in real numbers, but it is solvable in complex numbers.
Conceptual Knowledge and Procedural Skills
The specific Number and Operations Units of CMP3 develop all the concepts and procedural skills specified in the CCSSM with a consistent focus on meaningful derivations of ideas, techniques, and applications. When students complete all Units in the strand and the important connections in other content strands, they should be well prepared conceptually and technically:
- To represent quantities with appropriate numerical forms, to identify operations that will answer questions of interest;
- To estimate results of planned operations, to perform algorithms to produce exact computational results; and
- To interpret those results in the contexts from which questions arose.
- They should have sound understanding of the key structural properties of number systems that allow and guide the more general reasoning about quantity using algebra.