Over the recent holiday weekend, I brought out my copy of Dice Games Properly Explained by Reiner Knizia (@ReinerKnizia). I enjoy playing dice games at home with my family as well as playing dice games in class with students. Dice games often have simple rules and typically don't require of lot of material other than dice. Dice are a great mathematics manipulative that can be used with wide range of ages. Even very young students can practice subitizing numbers from 1 to 6 by reading the dots on the face of the dice. Below are a few of my favourite mathy dice games. Shut the Box
,The object of the game is to cover as many of the 9 boxes as possible. Any numbers left uncovered at the end of your turn are added together. "Shutting the box", or covering all the numbers, leads to a perfect score of 0. The player with the lowest score is the winner. On the players turn, they roll the dice and add them together. You can then cover any boxes that are not already shut that sum to the total you rolled. For example, if I rolled a 2 and 4 my total is 6. I could cover the 6 box, the 5 and 1, the 4 and 2 or the 1, 2 and 3 boxes if they are uncovered. If I can't partition my number to cover any boxes then my turn is over and I add up any uncovered boxes to determine my score. This is a fun game with some strategy to it. It really focuses on early addition skills as well as partitioning numbers in a variety of way. It is a great game for a lower elementary classroom. NS Outcomes: Primary  N04 Students will be expected to represent and describe numbers 2 to 10 in two parts, concretely and pictorially. Mathematics 1  N04 Students will be expected to represent and partition numbers to 20. Mathematics 2  N10 Students will be expected to apply mental mathematics strategies to quickly recall basic addition facts to 18 and determine related subtraction facts. Even Minus OddIn this game, players take turns throwing six dice. You then total all the even dice together and all the odd dice together. Subtract the odd total from the even total to get your score. Take counters from the centre of the table equal to your score. If you have a negative total, pay that number of counters to the centre (don't worry, if you don't have any counters left, you're still in the game). When all the counters are gone from the centre, the game is over and the player with the most counters wins. Start with about 10 counters in the centre or more if you have a large group playing. NS Outcomes: Mathematics 2  N02 Students will be expected to demonstrate if a number (up to 100) is even or odd. Mathematics 2  N10 Students will be expected to apply mental mathematics strategies to quickly recall basic addition facts to 18 and determine related subtraction facts. Game of SixIn this category game, you need only one die and a score sheet. Players take turns rolling the die over six rounds. One your turn, roll the die and decide which category to score. Multiply the number on your die by the category value (1, 2, 3, 4, 5 or 6). Each category can be used only once each game. For example, if you roll a six on your first turn, you could score it in category 6 and earn 6x6 = 36 points. At the end of the six rounds, each player adds up their total points. The player with the most points wins. NS Outcomes: Mathematics 4  N05 Students will be expected to describe and apply mental mathematics strategies, to recall basic multiplication facts to 9 × 9, and to determine related division facts. Mathematics 5  N03 Students will be expected to describe and apply mental mathematics strategies and number properties to recall, with fluency, answers for basic multiplication facts to 81 and related division facts. NinetyNineThe focus of this game is on the order of operations for whole numbers. Five dice are rolled in this game and the player who rolls the dice calls out any number they wish between 33 and 99. This is the target number for the round. Once the dice are rolled, players create an expression using all five numbers on the dice and any operations (+,  , x, ÷). The goal is to create an expression whose value is as close as possible to the target number without going over. Divisions must work out without a remainder. Players secretly write down their expression. Once everyone has an expression (or a reasonable amount of time has passed), players reveal their expression. The player closest to the target scores a zero. All other players score the difference between their expression's value and the values that was closest (to a maximum of 5). Play as many rounds as their are players so each person can have a round setting the target. NS Outcomes: Mathematics 5  N03 Students will be expected to describe and apply mental mathematics strategies and number properties to recall, with fluency, answers for basic multiplication facts to 81 and related division facts. Mathematics 6  N09 Students will be expected to explain and apply the order of operations, excluding exponents, with and without technology (limited to whole numbers). A Focus on Mathematical ContentThere are lots of really fun dice games in Reiner's book although some are more suited to a mathematics classroom than others. I also really like the chapter on the theory of dice and probability (chapter 3). I think the games above are not only fun, but closely related to mathematics outcomes. A recent post I read from Hilary Kreisberg (@Dr_Kreisberg) discussed a protocol to assess good classroom tasks. One dimension of her protocol was to assess the mathematical content of a task and ask yourself if the task aligns well with specific gradelevel standards. I think this is an important aspect to remember and not just play games that are fun, but ones that also offer meaningful mathematical practice. EL
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I recently had the opportunity to work with a calculus class on curve sketching and how derivatives affect the shape of a graph. The classroom teacher and I brainstormed some ideas about how we might infuse some hands on activity into the lesson. We decided to try an activity called Functions on the Floor. I originally saw an outline of this activity in a presentation from Liana Dawson called HandsOn Calculus Activities. Warm up  Functions and Their DerivativesWe started the class out with a Demos activity called Functions and Their Derivatives. We had students work in pairs on this activity. In the first part of the activity, students are presented with the graphs of three functions and they have to decide which is the original function, which is the derivative and which is the second derivative. In the second part of the activity, students create their own challenge and then get an opportunity to try out the challenges created by other groups. I really like the collaboration and discussions creating by working on these challenges. Functions on the Floor InstructionsAfter the warm up, we transitioned to the Functions on the Floor activity. We used masking tape to create several coordinate systems on the floor with the x and yaxis labeled from 3 to 3. At each of these stations was a list of information about a continuous function. Students used a small rope to create a function on the axis that satisfied all of the conditions listed. They then drew their function into a Desmos activity I had prepared. Using the drawings in the Desmos activity we could monitor students activity and plan for our review of the functions at the end of class. We heard some really constructive conversations taking place. The Google slides for the stations can be found here. Sharing our WorkI thought that this activity was a nice way to incorporate both handson physical problem solving while still leveraging some of the power of online tools like Desmos. I learned a few lessons doing this activity and the classroom teacher and I had a productive discussion after the lesson. We talked about how the lesson went and where we saw areas for improvement. We both agreed that eight stations was more than needed as it took some time for the discussions in the student groups to come up with a reasonable graph. We thought that some of the stations could have fewer constraints to consider as well. Also, I think next time I would use something besides rope. I had pretty inexpensive rope and it wasn't as pliable as I would have liked. I think a thick piece of yarn might have worked just a well. If you give this activity a try, let me know how it goes. Nova Scotia Mathematics Curriculum Outcomes Calculus 12 B15  Demonstrate an understanding of critical points and absolute extreme values of a function Calculus 12 B16  Find the intervals on which a function is increasing or decreasing Calculus 12 C5  Apply the First and Second Derivative Tests to determine the local extreme values of a function Calculus 12 C6  Determine the concavity of a function and locate the points of inflection by analyzing the second derivative EL
We started the lesson by reviewing strategies for decimal division. To create a need for division we had an activity where students found an average. We used the Estimating Time activity from NRICH. We challenged several student volunteers to estimate 5 seconds, giving each student three tries. We then asked the class to find the average of each student by dividing the total of their three attempts by three (just a note that students don't have a formal introduction to mean, median and mode until grade 7 SP01). After students finished, we highlighted several strategies that we saw student using and had the students explain their division strategies. We saw students making equal groups with base 10 blocks, partial quotients and the standard long division algorithm. Check out Graham Fletcher's progression of division video for some background on division strategies. Designing a Spinning Top
Testing DesignsGroups used a stopwatch (on their phone app or on https://stopwatch.onlineclock.net/ using a Chromebook) to see how long each top would spin. The tested each design three times and recorded their times. They then used division to find the average spin time for each design. They used this data to decide which of their designs was the best. Sharing DesignsAt then end of the class we led a class discussion about their design process. We asked students what characteristics made a top spin longest: size of the top (number of cubes used), shape of the top (symmetry). How it was spun seemed to be a big factor. One student spun the top from the sides instead of the top and was able to get a long spin (average of about 25 seconds). Further PracticeNova Scotia Mathematics Curriculum Outcomes Mathematics 6 N08  Students will be expected to demonstrate an understanding of multiplication and division of decimals (onedigit whole number multipliers and onedigit natural number divisors). 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 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 onedigit divisors or more than twodigit multipliers, the use of technology is expected). Mathematics 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. EL
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
Quadratic Relationships
Exponential Relationships
I like the investigation above because they share several common features.
Do you have a favourite activity or resource for activities? Please let me know what it is. EL
I was recently looking for an activity to explore a linear relationship, preferable one that included some practice with decimals. I found a few examples but none of them really seemed to satisfy what I was looking for. Here are a few of my criteria for good experiments to explore function relationships:
Notice and WonderThere were some great questions about volume and surface area, weight, and size of the paper clips (what is a #4 sized paper clip?). The questions the we went on to investigate was how long would it take to make a paper clip chain from all 100 paper clips. I was inspired by Dan Meyer's Guinness World Record for the longest paperclip chain in 24 hours. Dan blogged about breaking the record as well as asking student to see how many paperclips they could chain in one minute. EstimationI asked students to estimate how long they thought it would take to create a chain of 100 paper clips. I also asked them to think about an estimate that they know was too low (that creating a chain this fast was not possible) and too high (that they would have no problem creating a chain in this time even going slowly). Most students thought that a time between 5 and 6 minutes was a good "just right" estimate. Gathering Data
Revising EstimatesAfter collecting and analyzing some data, I ask students if they'd like to revise their estimate for 100 paper clips. Then we test their revised estimate using a plot of the values they collected and extrapolating. Below is one student's data plotted in Desmos. They estimated 300 seconds (5 minutes) to chain all 100 paperclips. This lesson could be modified to include outcomes from a number of different grade levels. I closed the lesson by showing students the record for the most paper clips linked together in one minute and asked students how they would compare. 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 7 PR02  Students will be expected to create a table of values from a linear relation, graph the table of values, and analyze the graph to draw conclusions and solve problems. Mathematics 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 onedigit divisors or more than twodigit multipliers, the use of technology is expected). Mathematics 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. Mathematics 7 SP02  Students will be expected to determine the effect on the mean, median, and mode when an outlier is included in a data set. EL
Multilink cubes are an incredibly versatile manipulative for mathematics class. You may see these 2 cm interlocking cubes referred to by several different names including: linking cubes, multilink cubes, Snap cubes, CubeaLinks and HexaLink. It's a manipulative that I've seen used at nearly every grade level from Primary to 12. Recently, John Rowe (@MrJohnRowe) stated a conversation on Twitter about math manipulatives. The conversation prompted me to reflect on how I've used manipulatives, and especially multilink cubes, as part of instruction and inquiry. I thought it would be nice to list a few of my favourite examples here: Speedy Squares from Mary Bourassa (@MaryBourassa) Speedy Squares is an activity that asks students to predict how long it would take them to build a 26 x 26 square out of linking cubes. Students start by building smaller squares and recording their times. They can then use this time to extrapolate an answer. Students could use quadratic regression to make a more accurate prediction. Jon Orr (@MrOrr_geek) also blogged about this activity and how he introduced it to his class. NS Outcomes: Math Extended 11 RF02, PCAL 11 RF04 and Math 12 RF01
Skyscraper Puzzles from Brainbashers This is a logical reasoning puzzle that you can play with just pencil and paper. The game become more focused on spatial reasoning when you actually build the towers using multilink cubes. Lots of educators have written blog posts about how they use this puzzle in their math classrooms including Mary Bourassa, Sarah Carter, and Amie Albrecht. Mark Chubb (@MarkChubb3) has blogged about this puzzle and shared some great templates for using with multilink cubes. NS Outcomes: Math at Work 10 G01, Math 11 LR02 Orthographic Projections from Jocelle Skov (@mrs_skov) Each student creates a 3D object using multilink cubes. Next they draw the top, front and side views of their object. Once every student has finished the three views of their object, they trade drawings with another student. That student then tries to build the 3D object in the drawing. They check their work with the original object on the teachers desk. NS Outcomes: Math 8 G01, Math at Work 11 G03 3D Linear Relations inspired by Alicia Potvin (@AliciaPotvin1) Each group of students builds three terms of a linear pattern using multilink cubes. Ask students to use one colour for the part of the pattern that stays the same and another colour for the part of the pattern that changes. Groups then rotate through the room and for each pattern, record a table of values, a graph and the equation. You could also ask students to determine how many cubes would be in the 43rd term as suggested at http://www.visualpatterns.org/. NS Outcomes: Math 9 PR01, Math 10 RF04
Mean, Median, Mode and Range with Linking Cubes from Jana Barnard and Cathy Talley Ask each student to reach into a large box of linking cubes to grab as many as they can with one hand. Students then build a tower with their linking cubes. As a class, students organize their towers in order from shortest to tallest. To get the class range, subtract the height of the shortest tower from the tallest tower. Is there a height that occurs more often than any other? That is the mode. To get the median, find the tower in the middle of the row (if an even amount of towers, average the two middle towers). To get the mean, even out all the towers until they are the same height saving any "left over". Suppose you had 12 towers, each with a height of 10 and 5 remainder cubes. This would give a mean of 10 and 5/12 cubes. NS Outcomes: Grade 7 SP01 Spinners from Shaun Mitchell and Mike Wiernicki (h/t Jen Carter) Students, in small groups, design a spinning top made of multilink cubes. The goal is to design a top that spins the longest. Once the group settles on their design they collect some data. They spin the top and record the time it spins in seconds to the nearest hundredth (or tenth). They do this three or four times and then average the time (hence they have to add three or four decimal numbers and then divide that decimal by 3 or 4). They could also model their decimal numbers using decimal squares. NS Outcomes: Grade 6 N08, Grade 6 SP02 A few links to some documents that provide some additional suggestions for using linking cubes:
The examples above are mostly from secondary math classes. Multilink cubes are also incredibly useful in elementary math classes (counting, measuring with nonstandard units, composing and decomposing numbers, etc). What are your favourite linking cube activities? Let me know and I'll add them to this post. EL
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 onevariable data using numerical and graphical summaries. EL
So a slight disaster struck our household recently. (and I mean slight; this is filtered through the lens of a child! My kids were recently at my husband's place of work (a large high school), waiting for him to finish up and drive them home. They kids are used to hanging around there and are often trying to find ways to amuse themselves while they wait for dad. My oldest son had his reaction ball with him and they decided to play catch with it in the open foyer of the school. In the usual brotherly fashion, they started to argue about who gets the ball. One of them (he asked to remain nameless!), threw the ball, hitting the trophy case in the foyer. The ball is deceptively heavy (about 272 g; a tennis ball is about 58 g); it ended up cracking the glass of the trophy case! The child who threw it says it hit the frame of the case three sections down from where the crack is; he’s floating a theory that the crack was already there. He’s also playing with the idea that the force of the hit on the frame sent vibrations throughout the whole case, causing the crack so far from the point of impact. #science #physics You can imagine how upset the kids were. They are really empathetic, kind, never any trouble at school and generally well behaved (A biased opinion, I know! But I’m their mom and #1 fan) When I arrived home and heard what went down, I encountered a very sheepish looking older brother and a very sad little brother who sent himself to his room. I went to have a chat with him and he was not in the mood to be cheered up. He told me that is was going to take more than 100 years to pay off the damage they had done. I tried to assure him it would not take that long and he said: "But Mom! I did the math" Here’s how the conversation went: Child: “My allowance is $10 every two weeks so that is only $120...wait $240 a year.” Me (in my head): Actually you are assuming that you get your allowance only 2 times a month, some months you get it 3 times. You are using a bimonthly calculation, not biweekly. Me (to my son): awww, honey :( Son (through tears): “and the glass is going to cost $30 000 dollars so that means like 100 years!!!! I’ll never be able to get that much money!!” Me (in my head): Well, you are assuming that your earning potential for the next 100 years is going to stay the same. As your mother, I am hoping that you will have a job at some point that pays more that $10 every two weeks. Also, how much do you think glass costs??? Hmmmm… 30 000 divided by 240 is 125. That’s a pretty good estimate using these assumptions. My child is a genius. Me (to my son): “How big was the glass? It can’t possibly cost $30 000!” (30 minutes of debating the cost of glass and listening to his various mental calculations) He went on to explain to me how he did his calculations and after a lengthy discussion, I finally convinced him that 1) the glass did not cost that much, 2) he was not going to be spending the next 100 years paying for this glass, and 3) he was most likely not the cause of the small crack in the glass. This conversation got me thinking about all of the little mathematical conversations parents have with their kids. I know my own kids are experts at negotiating timelines for bed, justifying how much screen time they should have and estimating how long it takes to get out the door for activities (factoring in travel time, and whether they are going to a game or a practice). As a math teacher, I notice and capitalize on these moments. My kids would argue I notice this too much. Sometimes when the kids ask a question that could be reasoned through mathematically, they preface the question with a “I just want the answer  don’t talk to me about the math!!!”. If you are looking to create these kinds of moments with your students or supporting their parents in having these kinds of conversations, check out and share the following websites:
As per my kid’s request, I’m working on not asking too many questions.
Do you know of any other great resources like these? Let me know! K.
The name of this card game can be traumatic for some students. For students who have lived in areas of armed conflict, "war" is not an appropriate name for a lighthearted card game. An alternative name you might consider is "Topit". The basic game is easy to learn. Here are the basic rules:
CardsMost of these versions can be played with a regular deck of playing cards. (You could remove the face cards, make all face cards equal 10 or use J=11, Q=12, K=13. Jokers could be removed or have a value of zero. Aces can be either 1 or 11.) If you want to use basic number cards, you can steal them from a UNO deck or any other numerical card game you might have laying about (e.g. SkipBo, Zero Down, Krypto, etc). Some might need special cards to be printed off such as (Logarithm War). Mathy VersionsRepresenting Numbers War  A special deck of cards is used showing a variety of ways to represent numbers from one to twenty. Representations might include numerals, tally marks, ten frames, dominoes, dot patterns, fingers, etc. Place Value War  Students flip two cards (A9) to make a two digit number (or three cards for a three digit number). They get to choose which represents the tens place and which is the units place. Ask students to read their number out loud, “five tens and three ones equals fiftythree.” Addition or Subtraction War  Each student flips two cards and either adds or subtracts them to get their value. Ask students to read aloud the number sentence created by their cards. For example, if the student draws a 4 and a 6, they should say, "four plus six equals ten" or "the sum of four and six is ten".
Decimal War  Similar to place value war, each student flips two or more cards to create a decimal number. You could create special deck to include fractions, decimals and percents as well as pictorial representations of rational numbers.
Evaluating Functions War  Students determine the value of a function at two different points to see which is greater. Students get practice evaluating functions from a table, equation, or graph. Sarah Carter has shared a set of cards she created for this activity. Radians and Degrees War  Practice mentally comparing angle measurements in radians and degrees using a special set of cards. Sarah Carter has a link to a free set and a description of this activity. Trig War  Play with cards with sine, cosine and tangent trig expressions and special angles on the unit circle (in both radians and degrees). Sam Shah has a link to a set of cards to download and a description of this activity.
Do you use any other versions of War in your classroom? Do you know a great deck of custom cards ready to download? Let me know. EL
I recently had the opportunity to work with a student to investigate parabolas and quadratic functions. We used one activity to investigate two different quadratic relationships. First we observed the shape of the stream of water coming out the side of a water bottle and then we observed the rate the water drains. The Water FountainI set up a cylindrical bottle of water on a crate. The bottle had a whole in it covered with a piece of tape. I asked students for some predictions. What will the shape of the water coming out of the side of the water bottle look like. What will happens to the stream of water as the water level goes down? I noticed that the student drew the water stream coming out of the bottle like it comes out of a water fountain (where we had just filled the bottle). We took the tape off the hole and then watched the water come out while making some observations and taking some photos. We selected a good photo (the black bulletin board in the background really helped) and loaded into Desmos. Then we used a table to record some points along the steam of water. After that we did a linear and then a quadratic regression on the point to see that the parabola was a much better fit than a line. We then had a chat about parabolas and projectile motion. Draining the TankWe set up the water bottle again but this time instead of looking at the shape of the stream of water, we focused on how fast the water level fell. I asked the student to predict what this might look like. You might ask students to predict what a graph of the water level might look like over time for the two situations below. How would the graph look when filling the tank compared to emptying the tank? The water flowing into a tank should rise at a linear rate. Students should expect that when the water drains from an open tank, the flow will be greatest at first and then gradually decrease as the water level decreases. (This is an application of Torricelli's Law). Next we taped a measuring tape to the side of the bottle and collected some data as the water flowed out of the bottle (A similar experiment is described in CanavanMcGrath, Foundations of Mathematics 12, 429). We used the stopwatch on my cell phone to record the time at each centimeter of height. This wasn't as accurate as I had hoped due to some distractions in the room. We set up the experiment again and the second time I recorded the water falling using a video (I used the CoachMyVideo app). We were able to get much more accurate values this way. We entered the data in a table on Desmos and then did a quadratic regression to fit a curve to our points. I was a bit surprised at how well the data from the video analysis on our second attempt fit to a quadratic curve (R^2 = 0.9999. I really liked how we could use the exact same setup to investigate two different quadratic relationships. Nova Scotia Mathematics Curriculum Outcomes Mathematics 11 RF02  Demonstrate an understanding of the characteristics of quadratic functions, including: vertex, intercepts, domain and range and the axis of symmetry. Precalculus 11 RF04  Students will be expected to analyze quadratic functions of the form y = ax^2 + bx + c to identify characteristics of the corresponding graph, including vertex, domain and range, direction of opening, axis of symmetry, xintercept and yintercept, and to solve problems. EL

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