# Lesson 5

A New Way to Interpret $a$ over $b$

### Lesson Narrative

In this lesson, students apply the general procedure they just learned for solving $$px=q$$ in order to define what $$\frac{a}{b}$$ means when $$a$$ and $$b$$ are not whole numbers. Up until now, students have likely only seen a fraction bar separating two whole numbers. This is because before grade 6, they couldn't divide arbitrary rational numbers. Now an expression like $$\frac{2.5}{8.9}$$ or $$\frac{\frac12}{\frac35}$$ can be well-defined. But the definition is not the same as what they learned for, for example, $$\frac25$$ in grade 3, where they learned that $$\frac25$$ is the number you get by partitioning the interval from 0 to 1 into 5 equal parts and then marking off 2 of the parts.  That definition only works for whole numbers. However, in grade 5, students learned that $$2 \div 3 = \frac23$$, so in grade 6 it makes sense to define $$\frac{2.5}{8.9}$$ as $$2.5 \div 8.9$$

### Learning Goals

Teacher Facing

• Comprehend that the notation $\frac{a}{b}$ can be used to represent division generally, and the numerator and denominator can include fractions, decimals, or variables.
• Describe (orally) a situation that could be represented by a given equation of the form $x+p=q$ or $px=q$.
• Express division as a fraction (in writing) when solving equations of the form $px=q$.

### Student Facing

Let's investigate what a fraction means when the numerator and denominator are not whole numbers.

### Student Facing

• I understand the meaning of a fraction made up of fractions or decimals, like $\frac{2.1}{0.07}$ or $\frac{\frac45}{\frac32}$.
• When I see an equation, I can make up a story that the equation might represent, explain what the variable represents in the story, and solve the equation.

Building On

### Glossary Entries

• coefficient

A coefficient is a number that is multiplied by a variable.

For example, in the expression $$3x+5$$, the coefficient of $$x$$ is 3. In the expression $$y+5$$, the coefficient of $$y$$ is 1, because $$y=1 \boldcdot y$$.

• solution to an equation

A solution to an equation is a number that can be used in place of the variable to make the equation true.

For example, 7 is the solution to the equation $$m+1=8$$, because it is true that $$7+1=8$$. The solution to $$m+1=8$$ is not 9, because $$9+1 \ne 8$$

• variable

A variable is a letter that represents a number. You can choose different numbers for the value of the variable.

For example, in the expression $$10-x$$, the variable is $$x$$. If the value of $$x$$ is 3, then $$10-x=7$$, because $$10-3=7$$. If the value of $$x$$ is 6, then $$10-x=4$$, because $$10-6=4$$.