Lesson objective: Understand that real-world situations can be represented by quotients with negative numbers.

From 6.RP.3, students bring prior knowledge of unit rates. From 6.NS.5, students bring prior knowledge that positive and negative numbers can be used to represent quantities in real-world contexts. This prior knowledge is extended as students represent real-world situations with quotients of negative numbers. A conceptual challenge students may encounter is that situations with different descriptions are represented with different but equivalent quotients.

The concept is developed through an extended example of temperature change, which leads students to write rates as fractions (or quotients) with negative numbers.

This work helps students deepen their understanding of equivalence because they discover that different placements of the negative sign in a fraction can result in equivalent quotients. Specifically, they learn that -p/q = p/-q = -(p/q).

Students engage in Mathematical Practice 2 (Reason abstractly and quantitatively) as they decontextualize and contextualize rational numbers by translating real-world temperature changes to numerical expressions and identifying equivalent representations.

**Key vocabulary:**

- ascended
- quotient
- rate

**Special materials needed:**

- none