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. 

Learning Targets

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.

CCSS Standards

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 \(10-6=4\).

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