## Knots in a Rope

**Overview:**This lesson should be used in the middle of a scatter plots unit. Students are given a task similar (but opposite) to the classing math camp problem where a rope has knots tied into it resulting in a shorter rope. The difference between this lesson and other similar lessons is that instead of asking how long the rope will be after "n" knots are tied, a rope with the most number of knots possible is shown and students are asked to guess the length of the rope before the knots were tied. In this lesson students will be encouraged to identify their own questions and variables formulate their own models, and perform calculations before validating their conclusions. Before you begin: The rope is 151 inches long and has 121 knots tied into it.

**Set-up and Design:**I like to cut the ropes to uniform length OR make a marking at uniform lengths to keep the y-intercepts of the trend lines uniform. One of the most efficient ways to find the length of the rope is to find the rate of change in length based on knots and then use the point (121,151) and the rate to find the y-intercept of the whole rope function (Good point for discussion here: what does the y-intercept of the trend line represent?).

As for recording information, I like to break up the page into fourths. One fourth for notice/wonder, one fourth for goldilocks guesses, one fourth for important information, and the last fourth to start planning the model. On the back of the paper I include a blank graph in case the student wants to make a scatterplot or graph a line. I also have a separate handout that encourages students to use the rate and create a new y-intercept, useful for the unmotivated student but not necessary.

**Materials Needed:**- 3/16 in cotton rope (You can find it for cheap at Wal-Mart)
- Slide deck and/or image and video files
- Paper

**Procedures:**- Show the image of the rope with knots in it. Ask students what they notice and wonder about the image. You may also want to show the close up of the rope so that students can see the knots in the rope.
- Use these images to create the essential question: how long is the rope?
- Have students take their goldilocks guesses (too high, too low, just right) then ask the class what information could be useful? Use this discussion to have students come up with the variables and brainstorm their models. It is important in this step to reveal that the rope in the picture is 151 inches long and has 121 knots tied into it.
- Pass out a length of rope and a measuring tape to each ruler and have them solve their problem using whatever method they devised.
- Once all of the calculations have made, do another round of goldilocks guesses. In many cases the correct answer will be more than the largest guess (I did not tie my knots very tight for this purpose).
- Show the video to reveal the answer.
- Validate conclusions with the class. How accurate was your model? Where or why do you think your model failed?

**Resources:**