Proportional Relationships

This week we talked about proportional relationships; specifically ratios and rates. Rates and ratios are concepts that are introduced initially in grade 4, but are developed later on in grade 7.

There are several different types of rates and ratios (rates, part-part ratios, part-whole ratios, equivalent ratios, etc), and I'm not going to go through them all. Instead I'd like to focus first on some engaging ratio problems, and a potential strategy for teaching them.

The engaging math word problem is as follows:

Based on the diagram below, which of the lots is fuller?
MacCuish (2015). Parking lot ratio question. 

The above question is based on one I found at EDUGains in one of their 'Big Ideas' resources (page 3).  If the question is written out it is as follows: in parking lot A there are 24 of 40 spots filled. In Lot B there are 56 of 80 spots filled. Which parking lot is fuller?

Students may get caught up with the 'absolute' numbers and think that lot B is fuller since there are more spots filled (i.e. 56 vs 24 cars), and yes lot B is fuller, but it has to do with the ratio of filled spots to empty ones.

When teaching rates and ratios one can follow a 3-part lesson plan. The three parts are as follows:

  1. Minds On
    • students re-activate previous learning 
    • teachers can gather the students in a group to read aloud the question and discuss (think-pair-share)
    • teachers can also focus on different ways of solving the problem by asking multiple students to explain how they found the answer
  2. Action
    • students prepare to work in small groups to solve a new question by going through with the teacher previously established success criteria for working in groups
    • then students collaborate to solve a new math problem
    • once done, students have to write an explanation for how they solved the problem as well as present it to the class
    • while students are working, the teacher can assess in an ongoing way students progress, as well as ask triggering questions if they get stumped
  3. Reflection
    • once each student has presented their work then the teacher highlights key aspects of each group's method
    • again, students are asked to think-pair-share
    • as an 'exit ticket' students must complete 2-3 questions independently
This specific set of activities for a 3 part lesson plan was adapted from Learn, Teach, Lead which is a wonderful website with plenty of resources for teachers looking to amp-up their math teaching game. 




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