Development of Algebra throughout CMP3 Grade 6

The CCSS[1] for Grade six identify four critical areas for focused attention.  The third of those areas is algebra, “writing, interpreting, and using expressions and equations,” which is expanded to read as follows:

Following this paragraph, a description of the grade 6 algebra domain is elaborated as a list of nine specific standards in three clusters. This paper provides examples of how these nine specific standards are addressed in the 6th grade units follow each standard.

This standard calls for students to use exponential notation to write products like 5 × 5 × 5 × 5 in equivalent form as 54 and to write expressions like 73 in equivalent form as 343. Exponential notation is introduced in Problem 3.2 of Prime Time and place value thinking required by Investigation 3 of Comparing Bits and Pieces and throughout Decimal Ops. Of course, exponents also appear in key formulas of the Covering and Surrounding unit.

Writing numeric and symbolic expressions and sentences to represent the operations required to solve problems is one of the most important skills in applied mathematics. Each unit of CMP3 includes many activities that develop understanding and proficiency in work on such modeling tasks. For example, Problem 4.4 of Prime Time asks students to analyze five different word problem situations, to decide which operations are needed to solve the problems, and to write one or more expressions that represent those operations. Similar modeling tasks are presented in Problem 4.3 of Let’s Be Rational and Problems 1.1 and 4.4 of Decimal Ops.

The exercises that involve writing expressions to record problem-solving operations in Prime Time, Let’s Be Rational, and Decimal Ops result in number sentences like:

35 – 3(5) = 20   [Prime Time 4.4A];
3 ¹/₂ - ³/₄ = 2 ³/₄ [Let’s Be Rational 4.3C];
and 9.8(4.10 – 0.30) = 37.24   [Decimal Ops 1.1D].

Using letter named variables to write expressions and equations representing quantitative operations and relationships is a primary focus throughout the Variables and Patterns unit. To get some head start on this important algebraic skill, the earlier exercises in writing numeric expressions and equations could be extended to suggest variables.

For instance, a question that asks, “What is the cost per gallon of gasoline if 7.5 gallons cost $28.49” [Decimal Ops 1.1B] could be extended to ask, “What formula tells the cost per gallon if 7.5 gallons cost d dollars?” A question that asks, “What is the original price of an item if a 25% discount reduces that price by $24.75 [Decimal Ops 4.3C] could be extended to ask, “What is the original price of an item if a 25% discount equals d dollars?”

Understandings and skills used in writing numeric expressions and equations are extended to use of letter names for variables in several other problems of Let’s Be Rational and Decimal Ops. For example, in Problem 4.3H of Let’s Be Rational, students are given a problem context, and several equations with variables that might apply. Students then determine that the problem can be answered by equations 3(1 5/8+N)=10 and 10-4 7/8=3N. Note that these problems can be solved with numeric reasoning. Once students have solved the problem using various strategies, then their reasoning could be captured using variables.

Expressions with letter names for variable quantities also appear often in formulas derived in the Covering and Surrounding unit.

The mathematical terms sum, product, and factor are introduced early and used throughout the first sixth grade unit Prime Time and then also in Let’s Be Rational and Decimal Ops.. The term quotient is defined and used in development of both fraction and decimal division algorithms in  Let’s Be Rational and Decimal Ops

The mathematical term coefficient is not defined or used until problem 3.1 of Variables and Patterns. However, it is easily possible to introduce the term in Covering and Surrounding where formulas like A = πr2, P = 4s, and P = 2L + 2W involve coefficients.

Use of the word term to indicate any part of an algebraic or numeric expression consisting of a number, a letter, or the product of a number and a letter is not made explicit in any unit of the grade six CMP3 curriculum, except in the glossary of Variables and Patterns. We have avoided any attempt to define term because of the inherent ambiguity in the idea itself.  For example, does the expression 5(n + 7) involve two terms, three terms, or only one term?  

The many CMP3 exercises involving the distributive property, like reasoning that perimeter of a rectangle can be expressed in two equivalent ways 2(L + W) and 2L + 2W, highlight the need to view parts of complex expressions as single entities.  In the case of 2(L + W), order of operations conventions imply that one has to calculate the sum of length and width before multiplying that single number by 2.

The obvious place where students develop skill in evaluating expressions for specific values of their variables is the measurement unit Covering and Surrounding.  However, there are also opportunities in other CMP3 units.  For example, Investigation 4 of Decimal Ops is all about applications of percents.  In Problem 4.1 of that unit students are asked a variety of questions about sales taxes and discounts on purchases.  In answering those questions they are implicitly writing general formulas for calculating such taxes and discounts and then using the formulas in a number of specific cases.  To more explicitly satisfy CCSS 6.EE.2a and 2c, you could ask students to write general formulas using letter names for variables and then ask them to show how to use those formulas in specific cases.  For example, 

“What formula shows how to calculate a 20% discount d on an item with original cost c dollars?” [Ans: d = 0.25c]

or

“What formula shows how to find the total price p of an item costing c dollars when a tax of 6% must be added to the original cost.” [Ans: p = c + 0.06c or p = 1.06c]

The conventional rules for order of operations in evaluating expressions are stated and practiced in Problem 4.3 of Prime Time and used consistently thereafter.

The fundamental concept of equivalent expressions is introduced in Investigation 4 of Prime Time where students focus on application of the distributive property of multiplication over addition to generate equivalent numeric expressions and use those expressions to reason about odd and even numbers.

The idea of equivalence comes up again in considering equivalence of fractions and ratios in the Comparing Bits and Pieces and Let’s Be Rational units. Equivalence of algebraic expressions is considered in Covering and Surrounding where the distributive property is again used to explain why perimeter of a rectangle can be calculated following the directions of 2(L + W) or (2L + 2W).

Of course, the Variables and Patterns unit gives much more extensive and explicit attention to equivalence of algebraic expressions.

The idea that equivalence means two expressions give the same outputs is first highlighted in Investigation 4 of Prime Time. Featuring applications of the distributive property, the most common principle for generating equivalent expressions, problem 4.3 of that investigation states the property in generality with letter names for variables a(b + c) = a(b) + a(c). As noted just above, this property is used in a powerful way to explain equivalence of the two common formulas for perimeter of rectangles P = 2(L + W) and P = 2L + 2W.

More explicit consideration of equivalence for algebraic expressions occurs in Investigation 4 of Variables and Patterns.

Explicit and thorough attention to this standard in CMP3 occurs in the Variables and Patterns unit. Students are also engaged in solving equations and inequalities many times in earlier units. For example, application and connection exercises in Investigation 4 of Prime Time ask students to “replace m with a whole number to make 7(4 + m) = 49 true” and “find whole number values of n so that 3(n + 2) is a multiple of 5.” Exercises in Investigations 2 and 3 of Decimal Ops ask students to “Find the value of n that makes n – 11.6 = 3.75 true” and “Find the value of N that makes N ÷ 08 = 3.5 true.”

Explicit CMP3 attention to the meaning of solving an inequality does not occur until Variables and Patterns. However, inequalities do occur in earlier units, offering opportunities to give attention to solving inequalities there. For instance, Problem 3.2 of Comparing Bits and Pieces asks students to insert < or > symbols to complete number sentences like 0 __ -3. An easy extension of such exercises could ask students, “What values of x will make the sentence x < -3 true?”

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

But there are also opportunities in other earlier units to ask students for general expressions involving letter names for variables. For example, as noted in the remarks about Standard 2a, one can ask students to give general formulas for calculating sales tax or discounts when working with percents in Decimal Ops.

Extensive and systematic attention to solving one-step linear equations occurs in the Variables and Patterns unit.  However, several other units develop student understanding and skill in work with these equations.  The key idea in those earlier encounters with equation solving is fact families.  For any three numbers a, b, and c, if a + b = c, then we can infer that b + a = c, a = c – b, and b = c – b.  Similarly, for any three numbers a, b, and c, if ab = c we can infer that ba = c, a = c ÷ b (b ≠ 0) and b = c ÷ a (a ≠ 0).  These relationships can be used to solve any one-step linear equation.  

For example, to solve n + 5 = 34 we reason as follows:

If n + 5 = 34, then n = 34 – 5 or n = 29
Checking, we see that 29 + 5 = 35.

Similarly, to solve 5n = 35 we reason as follows:

If 5n = 35, then n = 35 ÷ 5 or n = 7
Checking, we see that 5(7) = 35.

This use of fact families to solve one-step linear equations is developed and practiced in the Let’s Be Rational and Decimal Ops units. So students should have efficient and easy to recall strategies for solving x + p = q and px = q before they get to Variables and Patterns. Those early encounters with equation solving are generally embedded in the context of real world quantitative questions.

Note that the fact family way of thinking about and solving one-step linear equations is essentially an un-doing strategy.  If one finds the 25% discount on an item by multiplying the original cost by 0.25, then to retrieve the original cost when the discount is known one divides the discount by 0.25. The common equation solving strategy that relies on the idea of maintaining the equality of both sides of an equation (add, subtract, multiply, or divide equals by equals) is developed and applied later in Variables and Patterns.

Fact family reasoning does not apply, without considerable care, to solution of inequalities. So students will not learn a systematic strategy for solving inequalities prior to the Variables and Patterns unit. However, inequalities are introduced in Investigation 3 of Comparing Bits and Pieces and related to number lines in the same unit. The work that is given there could be extended relatively easily to develop the idea that the solution of an inequality like x > c or x < c is an infinite set of numbers, represented as half of a number line.

This standard is essentially asking students to learn how to think about variables as quantities and equations as representations of functions or relationships. These ideas are developed fully in Variables and Patterns, but there are a number of occasions in earlier units that provide opportunities to lay the groundwork for-full blown treatment of this standard.

For example, Covering and Surrounding Problems 1.2 and 1.3 examine the relationship between perimeter and length for rectangles of fixed area and the relationship between area and length for rectangles of fixed perimeter. In each case students are asked to express the relationship with a table and a graph of sample ordered pairs satisfying the relationship and then to describe the pattern relating values of the dependent variable to values of the independent variable (though the terms independent and dependent variable are not used at this point). A simple extension of each given problem could also ask students to express the relationship as an equation with letter variable names.

In that same unit other formulas, like those for perimeter and area of any square or surface area and volume of a cube give additional opportunities to graph dependence of one variable (most naturally perimeter, area, or volume) on another (most naturally side length).

Problem 2.3 of Comparing Bits and Pieces introduces the idea of rate table and asks students to generate sample pairs of values for related variables. Then, in problem 1.3 of Decimal Ops, students explore the concept of unit rate, in each example setting up a natural opportunity to study relationships between a dependent variable and an independent variable linked by an equation in the form d = rt, with the coefficient r as the unit rate. In those situations thinking about unit rates is a natural invitation to thinking about how one variable changes as another related variable changes.

Taken together, the examples from Prime Time, Comparing Bits and Pieces, Let’s Be Rational, Covering and Surrounding, and Decimal Ops, as well as the suggestions about how problems in those units could be extended, show that there is indeed considerable material addressing algebra standards in early units of the CMP3 grade six curriculum. Those prior encounters with CCSSM algebra standards give students significant preparation for the PARCC or SBAC testing, and they also provide valuable prior knowledge to help students succeed in the Variables and Patterns unit and its thorough and explicit treatment of all the algebra standards for grade six.

The examples of the preceding problems from Prime Time, Comparing Bits and Pieces, Let’s Be Rational, Covering and Surrounding, and Decimal Ops are occur in separate documents. To help teachers see more clearly how they might plan emphasize and extend the algebra Expressions and Equations CCSS content, the problems discussed above are outlined by unit.