Decimal Ops Problem 2.1 implicit reasoning, explicit with extension

Evaluating numerical expressions involving exponents occurs explicitly in the Prime Time unit. The place value thinking throughout Decimal Ops incorporates the idea of exponents as powers of 10 as introduced in Comparing Bits and Pieces. For instance, in Problem 2.1 students use what they know about fractions to compare and add decimals. An extension that would make this powers of 10 use and reasoning explicit would be to rewrite some of the expanded forms in Problem 2.1 to include exponential notation. For example, a third question could be added here such as, “What is the correct answer in fraction form using powers of 10 in the denominator?” This type of thinking may help students later on when they study scientific notation.

Decimal Ops Problems 1.1 and 4.3 plus extensions including variables.

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. Problems 1.1 and 4.4 of Decimal Ops incorporate modeling tasks.

For instance, Problem 1.1 in Decimal Ops involves writing expressions to record problem-solving operations resulting in number sentences like:

9.8(4.10 – 0.30) = 37.24 [See problem 1.1 D]

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, “Pedro buys 7.5 gallons of What is the cost per gallon of gasoline for $28.49. How much does each gallon of gasoline cost?” [Decimal Ops 1.1 B above] 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 4.3 C below] 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 also extended to use of letter names for variables in several other problems of Decimal Ops.

Decimal Ops Investigation 3

he mathematical terms sum, product, and factor are introduced early and used throughout Decimal Ops. The term quotient is defined and used in development of the decimal division algorithm in Decimal Ops Investigation 3, p. 48, and is used consistently throughout.

Each number in a division problem has a name to help you explain your thinking. The number you are dividing by is the divisor. The number into which you are dividing is the dividend. The answer is the quotient.

The calculation 18 divided by 3 equals 6 can be written like this:

18 ÷3 = 6

dividend ÷ divisor = quotient

You can also write the above calculation in two other ways:

Decimal Ops Problem 4.1 and 4.3 extensions

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 which students are asked a variety of questions about sales tax 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. Refer to the examples below for ideas on how to extend these problems.

Problem 4.1 A extension

“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]

Problem 4.3 C (see p. 3 above) extension

“What formula shows how to calculate a 25% discount d on an item with original costc dollars?” [Ans: d = 0.25c]

Decimal Ops Problem 4.1 and 4.3 extensions

In a similar manner to standard CCSS.Math.Content.6.EE.A.2.c above, some numeric calculation problems can be extended to generalize equivalent expressions. In this case, students learn that the total price can be represented by the equivalent expressions of c + 0.06c = 1.06c.

Problem 4.1 A extension

“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]

Decimal Ops Problems 2.3 and 3.3

Students are engaged in solving equations and inequalities many times in 6^th grade CMP3 units. For example, 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.” To emphasize these algebraic ideas, assign the associated ACE problems as well.

Problem 2.3 and ACE #17, 18 and 19

ACE Problems

• 17. Find the value of n that make each sentence true. Then write the addition and subtraction fact family for the sentence.
  1. 22.3 + 31.65 = n
  2. 18.7 - 4.24 = n
• 18. Find the value of n that makes each sentence true. Use fact families to help.
  1. 2.3 + n = 3.24
  2. n - 11.6 = 3.75
• 19. Find the missing numbers.
  1. 36.03 + n = 45.218
  2. n + 0.488 = 13.762

Problem 3.3 and ACE #30

ACE Problem

• 30. Find the value of N that makes each equation true.
  1. 3.2 × N = 0.96
  2. 0.7 × N = 0.042
  3. N × 3.21 = 9.63
  4. N ÷ 0.8 = 3.5
  5. 2.75 ÷ N = 5.5
  6. 5.3N + 7.25 = 70.85

Explicit CMP3 attention to the meaning of solving an inequality does not occur until Variables and Patterns.

Decimal Ops Problems 1.1 and 4.3 and 4.4 extensions

As indicated in the comments on Standard 2a, writing expressions and equations to represent conditions of real or purely mathematical questions occurs throughout the units of the CMP3 grade six course. Use of letter variables in such expressions occurs explicitly in the formulas of Covering and Surrounding and then throughout Variables and Patterns.

But there are also opportunities in other earlier units to extend numeric calculations and expressions by asking students for general expressions involving letter names for variables. For example, as noted in the remarks about Standards 6.EE.A.2a, 6.EE.A.3, and 6.EE.A.4, one can ask students to give general formulas for calculating sales tax or discounts when working with percents in Decimal Ops.

Decimal Ops Problems 2.3 and 3.3 (See 6.EE.B.5 above)

Several CMP3 units develop student understanding and skill in work with these equations. The key idea in those earlier encounters with equation solving is fact families.

This use of fact families to solve one-step linear equations is developed and practiced in the 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.

Decimal Ops Problems 1.3 Introduces Unit Rate

This standard is essentially asking students to learn how to think about variables as quantities and equations as representations of functions or relationships.

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.