# 5.5 - Calculations with Decimals

The student will

a) estimate and determine the product and quotient of two numbers involving decimals* and

b) create and solve single-step and multistep practical problems involving addition, subtraction, and multiplication of decimals, and create and solve single-step practical problems involving division of decimals.

*On the state assessment, items measuring this objective are assessed without the use of a calculator

### BIG IDEAS

• So that I can read and understand decimals in everyday situations (i.e. money, price tags, odometer reading, batting averages, swimmer/runner's time etc.)
• So that I can use decimals when measuring, cutting, and completing home projects
• So that pharmacists, seamstresses, engineers and others can make precise measurements
• So that when I am shopping, with a limited amount of money, I do not spend too much

### UNDERSTANDING THE STANDARD

• Addition and subtraction of decimals may be investigated using a variety of models (e.g., 10-by-10 grids, number lines, money).
• The base-ten relationships and procedures developed for whole number computation apply to decimal computation, giving careful attention to the placement of the decimal point in the solution.
• In cases where an exact product is not required, the product of decimals can be estimated using strategies for multiplying whole numbers, such as front-end and compatible numbers, or rounding. In each case, the student needs to determine where to place the decimal point to ensure that the product is reasonable.
• Estimation keeps the focus on the meaning of the numbers and operations, encourages reflective thinking, and helps build informal number sense with decimals. Students can reason with benchmarks to get an estimate without using an algorithm.
• Estimation can be used to determine a reasonable range for the answer to computation and to verify the reasonableness of sums, differences, products, and quotients of decimals.
• Division is the operation of making equal groups or shares.  When the original amount and the number of shares are known, divide to find the size of each share. When the original amount and the size of each share are known, divide to find the number of shares. Both situations may be modeled with Base-10 manipulatives.
• The fair-share concept of decimal division can be modeled, using manipulatives (e.g., Base-10 blocks).  Multiplication and division of decimals can be represented with arrays, paper folding, repeated addition, repeated subtraction, base-ten models, and area models.
• Students in grade four studied decimals through thousandths and solved practical problems that involved addition and subtraction of decimals. Consideration should be given to creating division problems with decimals that do not exceed quotients in the thousandths. Teachers may desire to work backwards in creating appropriate decimal division problems meeting the parameters for grade five students
• Division with decimals is performed the same way as division of whole numbers. The only difference is the placement of the decimal point in the quotient.
• When solving division problems, numbers may need to be expressed as equivalent decimals by annexing zeros. This occurs when a zero must be added in the dividend as a place holder.
• The quotient can be estimated, given a dividend expressed as a decimal through thousandths (and no adding of zeros to the dividend during the division process) and a single-digit divisor.
• Estimation can be used to check the reasonableness of a quotient.
• Division is the inverse of multiplication; therefore, multiplication and division are inverse operations.
• Terms used in division are dividend, divisor, and quotient.
dividend ÷ divisor = quotient

quotient
divisor ) dividend

dividend
divisor      =  quotient

• There are a variety of algorithms for division such as repeated multiplication and subtraction. Experience with these algorithms may enhance understanding of the traditional long division algorithm.

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

• Estimate and determine the product of two numbers in which:
• the factors do not exceed two digits by two digits (e.g., 2.3 × 4.5, 0.08 × 0.9, 0.85 × 2.3, 1.8 × 5); and
• the products do not exceed the thousandths place. (Leading zeroes will not be considered when counting digits.) (a)
• Estimate and determine the quotient of two numbers in which
• quotients do not exceed four digits with or without a decimal point;
• quotients may include whole numbers, tenths, hundredths, or thousandths;
• divisors are limited to a single digit whole number or a decimal expressed as tenths; and
• no more than one additional zero will need to be annexed. (a)
• Use multiple representations to model multiplication and division of decimals and whole numbers.
• Create and solve single-step and multistep practical problems involving addition, subtraction, and multiplication of decimals. (b)
• Create and solve single-step practical problems involving division of decimals. (b)

### KEY VOCABULARY

Updated: Aug 22, 2018