How to Teach Proportional Reasoning: Strategy & Activities

Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education

Proportional reasoning can be a difficult concept for students, and even for some adults! In this lesson, we will investigate strategies and activities that can assist the teacher when teaching proportional reasoning.

Traveling Abroad

If you have ever traveled to another country you know that you have to exchange your money. For example, if you have American dollars and you are traveling to Japan, you would exchange dollars to Japanese yen. The big question is, what is the exchange rate? In other words, if you exchanged 1 dollar, how many yen would you get in return? This is where ratios, and proportional reasoning comes into play and any mathematical process used to solve problems dealing with money must be important! Let's discuss how to teach these concepts.


Before proportional reasoning can be addressed, the students need to understand what a ratio is. A ratio is a way to compare two values. When it comes to math, it is imperative to give examples. Sometimes the examples can almost speak for themselves. Picking the right example will help keep the students engaged so it might be a good idea to use a sports example. For example, in basketball, there are 5 players on the court while in football there are 11 players on the field. This makes the ratio of basketball players to football players 5:11.

Explain to the students that ratios can be put into fractions and if there are 5 basketball players to 11 football players, we can write the fraction with either sport on top.

Ratios can be written either way

It is a good idea to make each student come up with a ratio of their own and turn it into a fraction, which is high on the level of Bloom's taxonomy of learning. If they can come up with their own ratio, they are definitely ready to move on to the next step, which consists is proportional reasoning.

Proportional Reasoning

Giving the students the formal definition of proportional reasoning may cause their eyes to roll towards the back of their head. As we discussed earlier, examples can be the perfect definition so leave the stuffy mathematical definitions to the textbooks.

Continue on with the same scenario you used to teach ratios to introduce proportional reasoning. Let's pretend the basketball and football rules regarding how many players are allowed have changed to allow more people to play, but the proportion of players for each sport has to be kept the same. Ask the students, ''if one more player is allowed to play on the basketball court, how many more football players are allowed to play if the proportion is to be kept the same?'' Let the students attempt to figure out how to solve this problem. Be prepared for them to say since one more basketball player was added, one more football player should be added. This is incorrect, so you should have the explanation ready for why it is incorrect. Here's a sample of an explanation you can provide.

Ask them if adding one player to each team would keep the fraction the same value. Show them it won't.

5/11 is about 0.45 and 6/12 is 0.5 so adding one to each team is the wrong process

Tell them since adding one to each team doesn't work they should try multiplying, and this begins with setting up a proportion (step 1).

Setting up the proportion between two fractions

Point out that both numerators have the same ''units'', which is basketball players, and both denominators have the same ''units'', which is football players.

Step 2 in the explanation is to show them how to cross-multiply. Draw arrows connecting the values to be multiplied diagonally or connect them with figure-eights. Using different colors can be beneficial.

For this step, tell them to ''Butterfly'' the proportion, which is a mnemonic device showing that the proportion looks like a butterfly after they draw their diagonal figure-eights.

Circling the values to be multiplied makes it look like a butterfly

Let them know to look for the fractions to disappear after they have completed this step. It may help to abbreviate basketball player (bp) and football player (fp) so the equation doesn't look so extended.

Equation after cross multiplying

It might help to tell the students this equation is ''mixed up'' since you have basketball players and football players on the same side of the equation. Basketball and football players don't play together in the same game in real life so we have to separate them now too. To ''straighten out'' this equation, you divide by the value next to the ''x''.

Straightening out the equation

Now tell them to look for the side of the equation with the ''x'' to have all of the numbers and units cancel, but only the units will cancel on the other side. When you show it on the board, erase everything that cancels illustrating that canceling means disappearing!

Simplified equation after canceling

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