The sign reads "No Vehicles Over 3200 kg." I realized that I have no idea what the mass of my car is nor what type of vehicles would have masses greater than 3200 kg. I have personal referents for smaller masses such as 1 gram (such as a jellybean or a paperclip) and 1 kilogram (like a bag of sugar). I also know that my son weighs about 20 kg but I have no similar reference for 1000 kg. 1000 kg is also known as 1 megagram (Mg) and 1 metric ton (t). I know that a cubic metre of water has a mass of 1000 kg but that doesn't really seem to be helpful to me because I have no experience with an actual cubic metre of water.
This made me realize the importance of personal referents when teaching students about SI units of measurement. Students are introduced to grams and kilograms in Grade 3. The curriculum guides states that, "as with all measurement units, it is important that students have a personal referent for a gram and a kilogram. Students should recognize which mass unit (gram or kilogram) is appropriate for measuring the mass of a specific item." The curriculum guide also states that as they begin to estimate and measure masses using the gram (g) and kilogram (kg), they should develop a sense of what a kilogram feels like, "by lifting and holding a variety of objects that have a mass of 1 kg." One activity that the curriculum guide suggests is for students to make a kilogram mass of their own. "Provide students with materials such as sand, flour, sugar, and small cubes from base-ten materials to fill a container until it exactly balances with a 1 kg mass on a balance scale. Using this kilogram container they can now compare its mass to items in the classroom to help them find a personal referent for 1 kg."
Unfortunately, it is not feasible for me to fill a container with 3 200 000 jellybeans in order to compare its mass to a car ( ...but maybe you could use Skittles). I'm going to have to find my own reference for this mass. I found that a subcompact car, such as a Toyota Yaris hatchback has a curb weight of approximately 1000 kg. On the opposite end of the passenger vehicle size spectrum, a full size luxury SUV, such as a Cadillac Escalade, has a curb weight of about 2600 kg (I had to read up on the difference between curb weight and gross vehicle weight). Once students have an general sense of vehicle weight, it might be help solidify this understanding (and be a bit of fun) to challenge them to estimate the mass of a number of different vehicles... from motorcycles to truck cranes (similar to Dan Meyer's "How Old is Tiger Woods?" activity but with mass instead of age). I made a public Google Sides document with some photos and weights of various vehicles (still a work in progress).
Now, when I'm crossing the bridge, I'm constantly estimating the size of the different vehicles around me. I keep my distance from really big vehicles on the bridge... just in case.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 3 - M04 Students will be expected to demonstrate an understanding of measuring mass (g, kg) by: selecting and justifying referents for the units gram and kilogram (g, kg); modelling and describing the relationship between the units gram and kilogram (g, kg); estimating mass using referents; and measuring and recording mass.
Mathematics at Work 10 - M01 Students will be expected to demonstrate an understanding of the International System of Units (SI) by: describing the relationships of the units for length, area, volume, capacity, mass, and temperature; and applying strategies to convert SI units to imperial units.
Here are a few questions that I thought about:
Below are some additional photos:
Given the list of lego bricks, you could ask students a number of additional questions:
The Discovery Centre is currently working on a project to build Canada's largest Lego mosaic wall. The wall will be installed at the Discovery Centre's new location when it moves.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 7 - N07 Students will be expected to compare, order, and position positive fractions, positive decimals (to thousandths), and whole numbers by using benchmarks, place value, and equivalent fractions and/or decimals.
Mathematics 8 - N03 Students will be expected to demonstrate an understanding of and solve problems involving percents greater than or equal to 0%.
Mathematics 8 - N04 Students will be expected to demonstrate an understanding of ratio and rate.
Mathematics 9 - N03 Students will be expected to demonstrate an understanding of rational numbers by comparing and ordering rational numbers and solving problems that involve arithmetic operations on rational numbers.
Mathematics at Work 11 - G02 Students will be expected to solve problems that involve scale.
I recently read an article on Wired about the Solar Voyager. A pair of engineers, Isaac Penny and Christopher Sam Soon, designed and built an autonomous, solar powered vessel. On June 1st, 2016 the 18 foot vessel, named Solar Voyager set off on its trans-Atlantic adventure from Gloucester, Massachusetts to Portugal, a journey of more than 4800 kilometres. They are predicting that this trip will take 4 months, assuming that there are no catastrophic events mid-Atlantic. One cool thing about this trip is that the Solar Voyager reports it position and other data online every 15 minutes at http://www.solar-voyager.com/trackatlantic.html. Currently, about two-weeks into its journey, Solar Voyager is just South of Halifax, Nova Scotia where I live.
The image below shows how far the Solar Voyager has traveled during its first two weeks. That is 1/8 of the time estimated for the crossing. Based on the information below, do you think that it will reach its destination in 4 months? What factors did you consider when making your estimation?
Some factors you might consider are currents, weather, equipment malfunction, obstacles/collisions, wear and tear, etc. There are so many variables at play that it must be very hard to make an accurate estimation.
Some Questions/Estimates for Students:
One of the coolest things about this project is that these young engineers "built Solar Voyager in their free time, undertaking this voyage simply for the challenge." How can I commandeer this type of intrinsic motivation for students in math class? What about this project made them want to work so hard "just for the challenge" and not for some extrinsic reward. Was it because they were the ones who selected and designed the task? Did they have just the right skills so that they felt confident that they would be successful? What is something that was relevant to their lives? How did this project captivate their curiosity?
Update: Solar Voyager ran into some trouble south of Nova Scotia. It appears it got tangled in some fishing gear and the props and rudders were fouled. After drifting for over a week, the vessel was picked up by HMCS St. John's, a Canadian Navy offshore patrol vessel.
I was introduced to row games while reading Kate Nowak's blog several years ago. A row game is an activity for a pair of students to work on together. Problems are organized in two columns. The first student completes all of the problems in column A and the second student completes all of the problems in column B. The questions in each column are different but the answers are the same. Students collaborate to verify that their answers match. If they do, they move on to the next question. If the answers do not match, the students work together to find out where the error was made and how to fix it. This allows students to have immediate feedback on their work. It also generates great discussions between students as they check each other's work. Another benefit is that students can correct each other's computational errors and the teacher's time can be focused on helping students with more serious comprehension errors.
Row Game Links
There are a couple of great resources for row games online. Kate Nowak has a shared Google drive folder packed with mathematical row games for a variety of grade levels and topics. Another row game collection is available on John Scammell's Orchestrated Experiences for High School Math website.
Nova Scotia Mathematics 10 Cumulative Review Row Game
Below is a row game that I created as a cumulative review for Mathematics 10. I created about half of the questions myself and appropriated the rest from row games created by Kate Nowak, John Scammell, and David McGuinness.
Open middle questions make for great classroom discussions. These are problems that have a closed beginning and end (i.e. an uncomplicated initial question and a single final answer) but an open middle where students can explore different paths and strategies to get to the solutions. Openmiddle.com is a great resource to find these types of questions. Below are a pair of open middle style problems I created for some of the probability outcomes in the NS Mathematics curriculum.
Problem #1 - The Spinners Problem
Directions: Select three of the spinners below (you may pick more than one of each) such that the total number of sectors in all three spinners totals 10. Select spinners so that the probability of all three spinners landing in the shaded sector is the smallest (or largest).
Hint: There are only 4 different ways to select 3 spinners whose sectors add up to 10. In other words, how many ways can you a make a expression with 3 terms that add up to 10 using the numbers 2, 3, 4, 5 or 6.
Solution: The four different options for selecting spinners are below. Option 1 yields the greatest probability and option 4 yields the least probability.
1) The greatest probability: 1/2 * 1/2 * 1/6 = 1/24
2) 1/2 * 1/3 * 1/5 = 1/30
3) 1/4 * 1/4 * 1/2 = 1/32
4) The least probability: 1/3 * 1/3 * 1/4 = 1/36
Extension: How would this problem change if you removed the restriction of exactly three spinners? If you could create as many spinners as you wanted such that the sectors totaled to 10 (e.g. 1 spinner with 10 sectors or 5 spinners with 2 sectors, etc.) what would the greatest probability be?
1) The greatest probability: 1/10 = 1/10
2) 1/2 * 1/2 * 1/2 * 1/2 * 1/2 = 1/32
3) 1/5 * 1/5 = 1/25
Problem #2 - The Marbles Problem
Directions: "There are _____ red marbles and _____ blue marbles in Bag A. There are _____ red marbles and _____ green marbles in Bag B. " Place a different whole number from 1 to 9 in each blank to make the probability of drawing a red marble from either bag A or B the same. How many different ways can you find to do this?
Hint: How can you have a different number of red marbles in each bag but the same probability of selecting a red marble?
Solutions: The are 40 different solutions! I first found 5 solutions and then realized that if you just flip the number of red marbles with the other colour, you get another 5 solutions. Then I realized that if you swap the bags, you double the solutions again. I made a Python program to check my work and realized that I had still missed a few and added them in.
1) Bag A: 1 red and 2 blue; Bag B: 3 red and 6 green
For each solution below, the following 3 rearrangements are possible.
1B) Bag A: 2 red and 1 blue; Bag B: 6 red and 3 green
1C) Bag A: 3 red and 6 blue; Bag B: 1 red and 2 green
1D) Bag A: 6 red and 3 blue; Bag B: 2 red and 1 green
2) Bag A: 1 red and 2 blue; Bag B: 4 red and 8 green
3) Bag A: 1 red and 3 blue; Bag B: 2 red and 6 green
4) Bag A: 1 red and 4 blue; Bag B: 2 red and 8 green
5) Bag A: 2 red and 3 blue; Bag B: 4 red and 6 green
6) Bag A: 2 red and 3 blue; Bag B: 6 red and 9 green
7) Bag A: 2 red and 4 blue; Bag B: 3 red and 6 green
8) Bag A: 2 red and 6 blue; Bag B: 3 red and 9 green
9) Bag A: 3 red and 4 blue; Bag B: 6 red and 8 green
10) Bag A: 3 red and 6 blue; Bag B: 4 red and 8 green
If you want to make this problem a bit easier, just use the numbers from 1 to 6. This gives a total of 16 solutions. These would be 1, 3, 5, and 7 from above plus the 3 additional re-arrangements of each.
Extension: Make the number of green marbles a 2 digit number.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 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.
Mathematics 8 - SP02 Students will be expected to solve problems involving the probability of independent events.
Mathematics 10 Essentials - G1 Express probabilities of simple events as the number of favourable outcomes divided by the total number of outcomes
Mathematics 12 - P03 Solve problems that involve the probability of two events.