# 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 $$57 \div 4$$.

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

• percentage

A percentage is a rate per 100.

For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.

• 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 . . .

• tape diagram

A tape diagram is a group of rectangles put together to represent a relationship between quantities.

For example, this tape diagram shows a ratio of 30 gallons of yellow paint to 50 gallons of blue paint.

If each rectangle were labeled 5, instead of 10, then the same picture could represent the equivalent ratio of 15 gallons of yellow paint to 25 gallons of blue paint.

• unit rate

A unit rate is a rate per 1.

For example, 12 people share 2 pies equally. One unit rate is 6 people per pie, because $$12 \div 2 = 6$$. The other unit rate is $$\frac16$$ of a pie per person, because $$2 \div 12 = \frac16$$.