# Lesson 5

Say It with Decimals

### Lesson Narrative

In this lesson students continue to study situations of fractional increase and decrease. They start to use decimal notation to express the situations. For example, they see that "one quarter less than $$x$$" can be expressed as $$\frac34 x$$ or as $$0.75x$$.

### Learning Goals

Teacher Facing

• Comprehend and use the term “repeating” (in spoken language) and the notation $\overline{\phantom{“ “}}$ (in written language) to refer to a decimal expansion that keeps having the same number over and over forever.
• Coordinate fraction and decimal representations of situations involving adding or subtracting a fraction of the initial value.
• Use long division to generate a decimal representation of a fraction, and describe (in writing) the decimal that results.

### Student Facing

Let’s use decimals to describe increases and decreases.

### Required Preparation

Print and cut up slips from the Representations of Proportional Relationships Card Sort blackline master.  Prepare 1 copy for every 2 students. These can be re-used if you have more than one class. Consider making a few extra copies that are not cut up to serve as an answer key.

### Student Facing

• I can use the distributive property to rewrite an equation like $x+0.5 x=1.5 x$.
• I can write fractions as decimals.
• I understand that “half as much again” and “multiply by 1.5” mean the same thing.

Building Towards

### Glossary Entries

• long division

Long division is a way to show the steps for dividing numbers in decimal form. It finds the quotient one digit at a time, from left to right.

For example, here is the long division for $$57 \div 4$$.

$$\displaystyle \require{enclose} \begin{array}{r} 14.25 \\[-3pt] 4 \enclose{longdiv}{57.00}\kern-.2ex \\[-3pt] \underline{-4\phantom {0}}\phantom{.00} \\[-3pt] 17\phantom {.00} \\[-3pt]\underline{-16}\phantom {.00}\\[-3pt]{10\phantom{.0}} \\[-3pt]\underline{-8}\phantom{.0}\\ \phantom{0}20 \\[-3pt] \underline{-20} \\[-3pt] \phantom{00}0 \end{array}$$

• repeating decimal

A repeating decimal has digits that keep going in the same pattern over and over. The repeating digits are marked with a line above them.

For example, the decimal representation for $$\frac13$$ is $$0.\overline{3}$$, which means 0.3333333 . . . The decimal representation for $$\frac{25}{22}$$ is $$1.1\overline{36}$$ which means 1.136363636 . . .