Investigations where students discover the relationship between variables can help students build a deeper understanding of functions. Often these explorations are hands on and engaging lessons. They typically start with some sort of interesting video or question prompt such as "What makes for an exciting bungee jump?" or "Which cup will keep my coffee warm the longest?". I was prompted to think about my favourite investigations after seeing a post from Nat Banting on Twitter. Below are a collection of links and descriptions of my favourite secondary mathematics investigations. ## Linear Relationships- Vroom Vroom from Fawn Nguyen and Super Racers - Students use a toy pull back car to investigate how far it travels depending on how far it is pulled back.
- Knot Again! from Jon Orr and Ropes of Different Thickness from Alex Overwijk - Students investigate the relationship between the length that a rope can be stretched and how many knots are tied in the rope.
- Paper Clip Chain - Students investigate the relationship between the number of paper clips and the time it take to make a chain from them.
- Spaghetti Bridges from Mary Bourassa and Spaghetti Bridges from Andrew Busch - Students build a "bridge" using more and more strands of spaghetti and see how many coins/washers it can hold before breaking.
- Barbie Bungee from Fawn Nguyen and Barbie Bungee from Matt Vaudrey - Students investigate the number of rubber bands used in a bungee cord and how far the bungee stretches when attached to a toy doll.
## Quadratic Relationships
## Exponential Relationships- Bouncing a Ball from Andrew Busch - Students investigate how high a ball rebounds after each successive bounce. David Wees has a great suggestion of using audio to take more accurate measurements.
- Coin Flipping and Pennies and Dice from Andrew Busch - Students use coins, dice, candy (e.g. Skittles) or two coloured counters to investigate an exponential relationship based on the probability of landing on a specified side when thrown. I've seen variations of this activity in numerous textbooks.
- Too Hot to Handle from Andrew Busch - Students investigate how quickly a cup of how liquid cools using a thermometer.
I like the investigation above because they share several common features. - They use inexpensive or easy to find materials
- They are hands-on and engaging.
- They don't require extensive preparations or difficult cleanup (Spaghetti Bridges probably requires the most cleanup as it will leave bits of broken dried spaghetti all over the floor, plan accordingly)
- The instructions are easy to explain. The majority of class time is spent carefully collecting and analyzing data.
Do you have a favourite activity or resource for activities? Please let me know what it is. EL
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I was recently invited by a class to work with them on collecting and analysing data. After brainstorming some ideas with the classroom teacher, we settled on collecting data from pull back cars. I check out Fawn Nguyen's Vroom Vroom lesson and Simon Job's Car Racing lesson to get some ideas on how to organize this lesson. We started the lesson by sharing the first half of Simon's video of cars racing across the floor. We had the students do some notice and wonder about the action taking place in the video and then introduced activity. We showed students the recording sheet that we would be using and how we would be taking measurements (A link to the record sheet Google Doc is here). Then we brainstormed some ways to make sure that we all collected good data and avoided errors: we would all use the same units (centimeters), all measure our distances the same way (from the front bumper), not use data if the car bumped into a wall or a desk, etc. We split up into racing teams of three students each. Each group got a measuring tape, a pull back car and a recording sheet on a clip board. The classroom teacher and I circulated the room (and a bit of the hallway) to help students and answer questions. After student finished collecting their data and plotting their values we came back together as a class. We asked several groups to plot their data on the whiteboard at the front of the room. We then had a discussion about general trends as well as why each car had a slightly different graph. Cars might have different wind up springs, different tire grip, dusty floors, aerodynamics, etc. We finished the class with a bit of excitement... the 150 Challenge. Each team had to use the data for their car to predict how much they would need to pull back to make the car travel as close to 150 cm as possible. Teams huddled to interpret their data and select a pull back distance. Each team brought their car to the front of the class to give it their best shot. There was lots of cheering and excitement as some teams got very close. The winning distance was only 2.5 cm. Much more close than I had expected. This activity could be easily extended for higher grade levels by incorporating linear relationships, linear equations and linear regression. Nova Scotia Mathematics Curriculum Outcomes Mathematics 6 SP01 - Students will be expected to create, label, and interpret line graphs to draw conclusions. Mathematics 6 SP02 - Students will be expected to select, justify, and use appropriate methods of collecting data, including questionnaires, experiments, databases, and electronic media.Mathematics 6 SP03 - Students will be expected to graph collected data and analyze the graph to solve problems. Mathematics 9 PR02 - Students will be expected to graph a linear relation, analyze the graph, and interpolate or extrapolate to solve problems.Mathematics 10 RF07 - Determine the equation of a linear relation, given: a graph, a point and the slope, two points and a point and the equation of a parallel or perpendicular line to solve problems. (including RF07.06 Determine the equation of the line of best fit from a scatterplot using technology and determine the correlation)Mathematics Extended 11 S01 - Analyze, interpret, and draw conclusions from one-variable data using numerical and graphical summaries.EL
I recently did an activity with students to answer a question by collecting and analysing data. I was inspired by similar activities from Bruno Reddy, Mean Paper Aeroplanes, and Julie Reulbach, Paper Airplanes for Measures of Central Tendencies. We started class by watching a video of the Paper Airplane World Championship - Red Bull Paper Wings 2015. This short video (about 3 minutes) shows the highlights of three paper airplane competitions; Distance, Airtime, and Aerobatics. After watching the video I let students know that we would be making paper airplanes for a distance competition. |

We started by brainstorming with students the characteristics of a good paper airplane that will fly a long distance. Most students have some experience in making paper airplanes. Several features that we discussed were: the shape (a glider or dart shape, wing angles), design features (symmetry, vertical flaps or a heavy nose), construction methods (sharp creases and accurate folds), materials (type of paper) and how it is thrown (launch angle, strength and accuracy of the thrower). |

## Predictions

## The Experiment

## The Results

## Reflection

Students had a handout where they were asked to reflect on how they could make this experiment more accurate and reliable. We thought that a few practice throws would help before we started collecting data (most student's last throw was their farthest distance). Another option would be to have more than three trials for each plane to increase the reliability of the data collected. There were lots of other really good suggestions as well. |

**Nova Scotia Mathematics Curriculum Outcomes**

**Extended Mathematics 11**

**S01**- Analyze, interpret, and draw conclusions from one-variable data using numerical and graphical summaries.

**Mathematics 9 SP03**- Students will be expected to develop and implement a project plan for the collection, display, and analysis of data by: formulating a question for investigation; choosing a data collection method that includes social considerations; selecting a population or a sample; collecting the data; displaying the collected data in an appropriate manner; drawing conclusions to answer the question.

- The x-axis has an inconsistent scale (sometimes 3 months, sometimes 4 months) that was automatically generated by Excel.
- Is a y-axis necessary or would a simple timeline do? Perhaps with a count below the timeline for successful land/drone ship vs failed land/drone ship landings.
- The graph doesn't indicate the two periods of investigation following launch failures.
- The choice of a line graph is a controversial one since these are discrete data points and not continuous data.
- The label on the y-axis should say "Number of successful landings to date". Otherwise it it looks like there were 9 successful landing in March 17 when there were actually 9 total successful landings by March 17.
- Using red and green dots will make this graph meaningless to people who are colorblind. You could use different symbols as well as different colors to remedy this.
- The success rate is more important than the total number of successful launches. Also, some successes are more ambitious than others (e.g. ocean/land/drone ship or the first re-used booster).
- Why not differentiate between land and drone ship landing attempts?.
- Not all launch data is included in this graph. How do you decide which launches to include (e.g. ocean landings where there was no drone ship).

## In the Classroom

**Nova Scotia Mathematics Curriculum Outcomes**

**Grade 8 SP01**- Students will be expected to critique ways in which data is presented.

**Mathematics Essentials 11 F2**- Select an effective data display for a given set of data and explain the reasons for the choice.

**Mathematics at Work 11 S01**- Students will be expected to solve problems that involve creating and interpreting graphs, including bar graphs, histograms, line graphs, and circle graphs.

Last summer, my son and I took a bag of coins from our piggy bank to our local TD Bank to cash them in. My son is 5 years old and likes to feed the coins into the machine. No less exciting is taking the proceeds down the street to Woozles, our favourite bookstore. Unfortunately, when we got the the bank, the coin counting machine was gone. The teller told us we would have to roll our own coins from now on. |

## Questions and Estimations

How would you design an experiment to test the accuracy of TD's coin counting machines? Would you test lots of different machines or a few machines multiple times? How many trials would you run to be confident in your results? What factors might contribute to the errors discovered in these machines?

**Nova Scotia Mathematics Curriculum Outcomes**

**Mathematics 11 S02**- Interpret statistical data, using: confidence intervals, confidence levels and margin of error.

**- Critically analyze society’s use of statistics.**

Mathematics 11 S03

Mathematics 11 S03

**Grade 9 SP03**- Students will be expected to develop and implement a project plan for the collection, display, and analysis of data by: formulating a question for investigation; choosing a data collection method that includes social considerations; selecting a population or a sample; collecting the data; displaying the collected data in an appropriate manner; drawing conclusions to answer the question.

**Grade 7**

**SP06**- Students will be expected to conduct a probability experiment to compare the theoretical probability (determined using a tree diagram, table, or other graphic organizer) and experimental probability of two independent events.

## Noticing

## Wondering

I tried sorting the burgers in a variety of ways but there appears to be no pattern to the difference in price between chips and fries. This restaurant is in Oregon, which has no sales tax (one of only five such states), so the prices are not set so that when tax is applied, the total is a round number. I'm assuming that every burger gets the same amount of fries, but perhaps this isn't the case. I wonder if they get many questions about this?

**Nova Scotia Mathematics Curriculum Outcomes**

**Grade 7 SP01**- Students will be expected to demonstrate an understanding of central tendency and range by: determining the measures of central tendency (mean, median, mode) and range; determining the most appropriate measures of central tendency to report findings.

**Grade 7 N02**- Students will be expected to demonstrate an understanding of the addition, subtraction, multiplication and division of decimals to solve problems (for more than one-digit divisors or more than two-digit multipliers, the use of technology is expected).

**Mathematics 11 S01**- Demonstrate an understanding of normal distribution, including: standard deviation and z-scores.

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