This lesson is a nice follow-up to the Origami Frogs 3-Act Task. Since students have already made the origami frogs, they can now use them to investigate how the size of the frogs impacts how far they hop (on average).

The download contains a written lesson plan for Part 1 – the Origami Frogs 3-Act Task and Part 2 -the Hopping Frogs Investigation.

To begin the investigation, show the video below.

This lesson offers students an opportunity to collect and analyze data, use proportional reasoning, and use a variety of strategies to determine how far an origami frog might “hop” given its size.

Engage students in a notice & wonder discussion using Turn and Talk before sharing with the group.

Focus students on the main question from their wonders: How far will the big frog hop?

The Investigation

Since students don’t know how big the frog actually is, using proportional reasoning is difficult. So, students need to collect data on different size frogs and how far they “hop.”

Before students begin making origami frogs of different sizes, and measuring “hops,” engage students in a conversation about collecting “clean data.” In other words, in order to create a mathematical model to predict how far the big frog might “hop,” we all need to be measuring these “hops” by following the same parameters.

• Some questions to help students develop some measuring parameters:
• Should there be a starting line for the hop?
• Where should the frog be in relation to the starting line? (Front feet on the line or back feet on the line?)
• Where do you measure to after the hop? (To the point on the frog closest to the line after the hop or farthest from the line after the hop?)
• What if the frog lands and slides a little? Do we count the slide, or just mark where it lands? (Mark where it lands – the only frogs that will slide are the smaller ones. The larger ones tend to “stick” their landing). Tip: Cover the table with a table cloth, towel, or butcher paper to prevent sliding.
• Should be measure distance perpendicular to the line or at an angle? (Perpendicular)
• Should frogs get one hop, or should we get a mean distance for the hops?
• If we use the mean for the distance, how many hops should we use for each size frog?

The second question above may seem inconsequential, but since frogs hop primarily with their hind legs, it makes sense to have the back feet on the line. This will make it easier to measure some of the larger frogs’ hops as well.

Using the mean is optional, but it’s nice to include some measures of central tendency when appropriate and it certainly is here. Another option is to find the sum of all the distances hopped and use that. This will mean graphing larger numbers, but it should still work fine in a model to predict.

Once students have helped to set the parameters for measuring and collecting data for the different size frogs, pairs (or groups of three) of students can get to work folding origami frogs of different sizes.

How many different sized frogs should students make and test? This is a great question. The best predictions rely on clean data and a good mathematical model. Two frogs will not provide enough to make an accurate prediction, but ten would be time consuming, but four or so for each group with several hops for each frog is doable.

Printable directions to fold an origami frog can be found here.

Questions to Ask

• What do you think the relationship between frog length and hop length might be?
• What happens to the frogs’ hops as the size of the frog increases?
• Now that you have some data for several size frogs, how might it be useful to predict how far a frog might hop if we knew its length?
• What tools might you use to help visualize this relationship between frog length and hopping distance?

The last questions are meant to guide students to the realization that graphing the raw data might provide some insight to answer the question.

After students have graphed their data (Desmos would be a great tool here) and have created a regression line, they can then make some predictions about the hopping distance of different sized frogs.

Students may have concerns about their own data and the process which often act as a springboard to deeper discussions.

Potential concerns:

There are a few measurements that are really far off from the others (much larger or smaller than most of the other measurements). These measurements may have a negative impact on our models and our prediction. What should we do? There are several ways to “deal” with data points like this. You could remove the highest and lowest measurements from your data or you could remove only those data points that are outliers. This is a great opportunity to have a discussion with students about these two options. Alternatively, all data points from all groups could be combined to create an average for the class. This might be a good idea after groups share their predictions and compare with the actual data. The increased number of data points could address the problem of outliers, though students may still wish to remove them.

How do we know if our model is right? Great question. What students need to know is what statistician George Box stated in 1976 when referring to models: “All models are wrong, but some are useful.” When dealing with the real world, where variables may be difficult to account for, our models will likely be off. The purpose for investigating things like the hopping frogs is to reflect on all of the models presented to see what we can learn and where we might might have made an error in measurement, an incorrect assumption, or an estimation that was off.

Once students have created and shared their models, and created a regression, provide them with the length of the large frog and ask them to use their models to make a prediction. The large frog will “hop” the same number of times as the other frogs and the mean will be computed for these measurements. Students can then see how accurate their models were.

Engage students in a discussion about why their models may have been off.

After, students compare the actual with their models, they may ask how they might change their model to make it more accurate for frogs of any size. This is a great exercise in perseverance. Providing time to make models more accurate is incredibly valuable, making it well worth the time.

Whether their model was close or not, it may be a good idea to test their models against other origami frog sizes from larger squares of paper. Does the model work to make accurate predictions for these frogs? What about testing models for the hopping distances of smaller origami frogs? What’s interesting here, is that the models might work better (or worse) based on the frog sizes. This indicates a flaw in the model which should be addressed so that the model works for all frog sizes.