In this unit, students solidify their understanding of the base-ten number system, extend their use of the standard algorithms to add, subtract, and multiply decimals beyond tenths and hundredths, and learn to use algorithms to calculate quotients. The work here builds on what students learned in earlier grades about operations on whole numbers and decimals.

Students begin by exploring the use of decimals in a shopping context and by revisiting addition and subtraction of decimals, using both concrete representations and numerical calculations. The activities in the first section reinforce ideas about place value, properties of operations, the algorithms for adding and subtracting, and the relationship between addition and subtraction. 

Next, students investigate various ways to find the product of two decimals: by using decimal fractions, writing equivalent expressions with whole numbers and unit decimals (such as 0.1 and 0.01), using diagrams and partial products, and reasoning about the relationship between a decimal and a related whole number. Students notice that the different methods of reasoning are governed by the same structure based on place value, which also underlies the standard algorithm for multiplication.

The next section focuses on division. Students have an opportunity to use base-ten blocks or diagrams to represent division of multi-digit numbers before exploring other numerical methods, such as using partial quotients and long division. Students progress through calculations of increasing complexity. They first divide whole numbers that give a whole-number quotient, and then divide whole numbers with a (terminating) decimal quotient. Next, they divide a decimal by a whole number, and finally a decimal by a decimal.

Mai’s diagram for

 

Lin’s calculation for

In the last section, students apply the mathematics from the unit to solve problems in applied situations. These require students to interpret quantities and results in context, and to consider appropriate levels of precision in their work. 

A note about materials:

Base-ten blocks and paper versions of them will be useful throughout the unit. Consider preparing commercially produced base-ten blocks, if available, or printing representations of base-ten units on card stock, cutting them out, and organizing them for easy reuse.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as explaining, interpreting, and comparing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Explain

• Processes of estimating and finding costs (Lesson 1).
• Approaches to adding and subtracting decimals (Lesson 4).
• Reasoning about products and quotients involving powers of 10 (Lesson 5).
• Methods for multiplying decimals (Lesson 8).
• Reasoning about relationships among measurements (Lesson 15).

Interpret

• Representations of decimals (Lesson 2).
• Base-ten diagrams showing addition or subtraction of decimals (Lesson 3).
• Area diagrams showing products of decimals (Lesson 7).
• Base-ten diagrams representing division of a whole number or a decimal by a whole number (Lessons 9, 12).
• Calculations showing partial quotients or steps in long division (Lessons 10, 11, 12).

Compare

• Base-ten diagrams with numerical calculations (Lesson 4).
• Methods for multiplying decimals (Lesson 6).
• Methods for finding quotients (Lessons 10, 11, 12).
• Measurements of two- and three-dimensional objects (Lesson 15).

In addition, students are expected to describe decimal values to hundredths, generalize about multiplication by powers of 10 and about decimal measurements, critique approaches to operations on decimals, and justify strategies for finding sums, differences, products, and quotients.

The table shows lessons where new terminology is first introduced in this course, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.