This lesson deals with teaching not only proportions but proportional reasoning that will have connections from 6th grade through 12th grade. This is a very important foundational skill for not only future math courses but science courses and elective courses as well. Let’s start with connecting to fractions. Students will have seen fractions and their equivalents on a basic level so let’s connect with that prior learning. Place the slide below on the board and ask the students:

- Why is each one true?
- What math is being done to the first fraction to get the second fraction?

Now ask if they can share a false example. Let them use a CAS device and they can investigate before answering which gives them more confidence to answer.

Now let’s look at these fractions in context. Context is always an important part of a lesson so students can relate to the numbers. For my example I am using boxes and crates. This can be modeled easily with blocks and cups, M&M’s on a certain size paper, even real boxes if you can get the different sizes. The key is nothing fits perfectly.

Discussion Question:

If 6 boxes fit into 4 ½ crates, how many crates will it take for 1 box? for 100 boxes?

Using the CAS students can easily use the words “boxes” and “crate” as variables to do the investigating and keep it contextual. Students can look at the different representations and make connections between the numbers and the application. For example the graph below is created from the spreadsheet above without retyping anything. Create a graph page and SELECT your variables created by the spreadsheet. It reinforces the relationship between the data, coordinates, and graphing. All graphs come from data not randomly generated numbers.

After this introduction some good activities to use throughout the unit are TI’s Recipe: Unit Rate, Proportionality in Tables, Graphs, and Equations, and/or Proportions in Stories activities. I really like the **Recipe: Unit Rate** because it ties the previous unit rate knowledge into proportional reasoning and it is contextual. The other two activities move into the more advanced graphing and equations part of proportionality but still shows the students that slope is just using proportional reasoning for more complex problems.

I feel that we as educators sometimes tend to treat graphing as a separate entity, so the students see it as something completely new. Actually it really is an extension of proportions and proportional reasoning. “Slope” is not a new concept it is just a different use for proportions. If we build on student’s knowledge of equivalent fractions and unit rates to progress into proportional reasoning then proportion calculations and graphing, students will see a natural flow instead of a new beginning.