Practice is important. Whether it is playing the piano, shooting free-throws, popping an ollie or solving a quadratic equation you need to practice to improve. Some practice routines are more effective than others at helping students solidify their understanding. Practice can often seem tedious and it can be difficult to maintain the motivation to practice.
In mathematics, students practice skills in a variety of ways. One style of practice that can help students stay motivated and engaged is purposeful practice. Instead of a page full of repetitive problems, students focus on an activity that has a mathematical goal to achieve. Dan Meyer wrote a blog post a few years ago titled "Purposeful Practice & Dandy Candies" that started me thinking about how to make activities in my classroom more purposeful.
One of my favourite sources of problems with purposeful practice is Open Middle. There is a large selection of questions organized by topic and grade level. Each question has an "open middle" meaning there are many ways to explore and solve the problem. Below is a question submitted to the Open Middle site by Robert Kaplinsky.
In this question, students try to find the arrangement of digits that yields the product closest to 50. Students will try numerous different arrangements of digits and get lots of practice multiplying decimal numbers without it seeming tedious. The question can also be quickly modified to give additional practice. For example, just add a hundredths place onto one of the factors and use 5 different digits.
Games can be a great way to encourage students to practice. There are lots of examples of but I'm going to mention just two. The first goes by several names. Joe Schwartz wrote a great post about Factor Captor. A similar game is described on the NCTM Illuminations site called the Factor Game. Students alternate turns playing on board filled with numbers. The first player selects a number to cover and adds that number to their score. The second player finds all the factors of that number, covers them and totals those number to add to their score. The roles are then reversed and play continues until there are no uncovered numbers remaining. There is a lot of math in this game and it is fun to play.
The second game is Horseshoes from Math4Love. This game is played with a deck of cards numbered 1-9. Two cards are drawn to form a two-digit target number. Then four more cards are drawn. Players use these four digits to create an equation using addition and subtraction that is as close as possible to the target number. For example, let's say that the target number is 25 and the four digits given are 1, 3, 6 and 9. A student might create the equation 39-16 = 23. Another student might make the equation 13+6+9 = 28. There are lots of way to tweak this game for different levels of complexity.
Both of these games allow for lots of numerical practice in a format that engages students. There are many excellent sources of ideas for mathematical games. My current 'go-to' resource is a book titled Well Played, 6-8.
There are several mathematical puzzles that include lots of practice with numerical computations in pursuit of a solution. KenKen puzzles and Maze 100 from NRICH are two such puzzles that I've used. I also think that Yohaku puzzles are great. They are numerical puzzles where you need to determine the number that is in each square in order to make the column and row sum/products.
For classrooms with the available technology, activities from Desmos.com are another way to practice with purpose. An activity that incorporates a lot of meaningful practice is Marbleslides. In this activity, student try to capture stars on a Cartesian grid by creating a path using functions that marbles roll down. Students work to refine their functions to capture as many marbles as possible. Another activity that generates lots of practice is Transformation Golf. Students use a series of rigid transformations to move a shape to specified location. They have to find an efficient path around several obstacles. Each successive challenge increases in complexity.
There are also activities that have a non-math goal for students to achieve. Lots of online math games have incentives for students such as badges to earn, experience points to accumulate or virtual prizes to win. There are also worksheets and activities with non-math goals. Worksheets such as "Algebra with Pizzazz" and "Punchline Algebra" have a riddle to be solved once all of the questions are completed.
In my classroom practice, I used a number of activities with these types of incentives and I think that many students find them exciting and fun. If students are excited to do math, I consider that a win. However, I think that these types of activities should be used with caution. We don't want to inadvertently send a message to our students that math isn't fun by itself so we have to disguise it (like sneaking vegetables into their favourite foods so kids will eat them).
If you have a favourite math activity, game or puzzle that gets students practicing math with a purpose, I'd love to hear about it. Please leave me a comment.
My son and I recently spent a lovely fall afternoon exploring the carnival games and amusement rides at a local fair. My son is quite adventurous when it comes to amusement park rides and is eager to try just about any ride that he meets the height requirement for. While we were walking through the midway, I spotted a carnival game called "Roll Down" that appeared to have a bit of mathematics involved.
The object of this "game of skill" is to roll six balls down and inclined ramp to land in one of six numbered bins. If the sum of the six rolls is under 10 or over 31, you win. Is this game worth the $5 price to play? What are my chances of winning? Should I go for under 10 or over 31?
The bins are just wide enough for a ball to fit so it is very difficult to aim a ball with accuracy. You also have to question if the balls roll straight and if the board is smooth and level. Lets just assume that the balls fall into a random bin (you could then play an equivalent game at home by rolling 6, six-sided dice). With six balls, the smallest sum possible is 6 (all 1's) and the largest is 36 (all 6's). How many ways are there to get each possible value?
There are only 31 possible sums (6-36) that you can score. To roll a sum of under 10, you can score 6, 7, 8 or 9. To roll a sum of over 31, you can score 32, 33, 34, 35, or 36. At first glance, it looks like you have a 9/31 chance of winning but this is not correct.
This reminds me of a bet in the casino game craps that looks good, but on further inspection is really bad. The field bet is a bet on the sum of the next roll of two six-sided dice. If the sum of the two dice is 2, 3, 4, 9, 10, 11 or 12 you win. If the sum is 5, 6, 7 or 8 then you lose. It has the illusion that there are more ways to win than lose, but you are much more likely to roll one of the losing numbers.
With practice at Roll Down, you might be able to achieve better than the random results that I detailed above. Instead of this practice, I decided to spend my $5 at the concession stand to buy a hand-battered, deep fried corn dog. A midway concession stand can also be considered a bit of a gamble, but in this case it was a delicious win!
My son and I were looking through an issue of Fun to Learn Friends magazine recently and we ran across a game called "Spring Bingo". He was quite interested and wanted to give it a try. He is in grade primary and can subitize the pips on a six sided die as well as confidently sum the values on two dice. Just the skills we need to play this game.
We played several times. At the end of most of the games, my son got frustrated trying to roll either a 2 or a 12. He doesn't quite understand why it takes so long to roll one of these numbers. George's bingo card has both of these numbers out of the eight numbers on the card. This seems unnecessarily cruel. Ted's card has the 2 but not the 12. This made me wonder if this is a fair game or if one of the cards has a better probability of winning. Time for some math...
It appears that Card B is slightly better than Card A. I'm not sure if this would make a significant difference in the outcome of the game (i.e. rolling all the numbers on your card before your opponent does). I wonder what the average number of rolls it takes to complete each bingo card is? How much of an advantage does going first in this game give? If the player with Bingo Card A goes first does this equalize the advantage of the better Bingo Card B? These would be great questions for the Mathematics 12 research project outcome (MRPO1).
To try to answer some of these questions, I thought that writing some code would be helpful. I know a grade 8 student that completed the Introduction and Intermediate Programming with Python courses from Art of Problem Solving. I contacted him and he graciously created a very nice Python program to simulate this game for me. I modified the code a bit so that it just plays one bingo card and counts how many rolls it takes to complete the card. The average number of rolls in 100,000 games for "George's Card" was 58. The average number of rolls in 100,000 games for "Ted's Card" was 48. I was a bit surprised that there was this much difference. Playing with Ted's bingo card appears to be an advantage.
Next I modified the code again so that it plays the game with George going first to count how often George wins. In 100,000 games, when George went first, he won 50703 of the games. I again modified the code so that Ted plays first. In 100,000 games, when Ted went first, he won 50806 of the games. It seems that going first is an even greater advantage than having the better bingo card.
What I really like about this game is that there are mathematical outcomes that can be addressed with this activity across a wide range of grade levels. At younger ages, students are practicing subitizing and adding numbers. One variation of this game is to play it solo. This might be a nice option for a math station. I found several examples of "Roll and Cover" games where students have a sheet of paper filled with numbers (from 1 to 6 or from 2 to 12) and students roll the die or dice and cover the number (with a token or a bingo dauber) when they roll it. Just do a Google search for "roll and cover math game" and you'll find lots of examples posted online.
For students looking for an opportunity for enrichment, they can make variations of this game. They could also write computer code to simulate this game (using Scratch or Python or some other language). They could also do some statistical analysis of the game to see how fair it is. There are so many options with this simple game.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 1 N02 - Students will be expected to recognize, at a glance, and name the quantity represented by familiar arrangements of 1 to 10 objects or dots.
Mathematics 1 N09 - Students will be expected to demonstrate an understanding of the addition of two single-digit numbers and the corresponding subtraction, concretely, pictorially, and symbolically in join, separate, equalize/compare, and part-part-whole situations.
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.
Mathematics 5 SP04 - Students will be expected to compare the likelihood of two possible outcomes occurring, using words such as less likely, equally likely, or more likely.
Mathematics 6 SP04 - Students will be expected to demonstrate an understanding of probability by: identifying all possible outcomes of a probability experiment; differentiating between experimental and theoretical probability; determining the theoretical probability of outcomes in a probability experiment; determining the experimental probability of outcomes in a probability experiment; comparing experimental results with the theoretical probability for an experiment.
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.
Mathematics 12 MRP01 - Research and give a presentation on topic that involves the application of mathematics.
Scale is a concept that is found at numerous grade levels in the Nova Scotia Mathematics curriculum. Scale drawings and models, similar polygons, and proportions are all found in mathematics outcomes. In math, scale is the ratio of the length in an image (or model) to the length of the actual object.
Below is a question relating to scale factors. A scale factor is the ratio of any two corresponding lengths in two similar geometric figures. Take a look at the three different versions of Connect Four. Estimate the scale factor between each pair of game boards from the given pictures. Estimate the radius of each of the coloured chips. Is the scale factor of the radius of each coloured chip the same as the scale factor of their volume?
You might ask students how scale is different from proportion. Try out this question: How big would a game board of Connect Four Hundred be (or even Connect Four Million) compared to Connect Four?
In visual arts, scale refers to the size ratio between objects within an image. Using a consistent scale will make a drawing look more realistic. Objects do not appear too large or too small when compared to each other. Sometimes however, an artist might intentionally change the scale of certain objects in an image. One such technique is called 'Hieratic scale' or sometimes 'Hierarchical proportion'. This technique can be seen in paintings and sculpture from the middle ages where powerful or holy people were sometimes painted larger than ordinary or less important people to show their relative importance. The larger a person was, the greater their importance.
It would be fun to show students examples of how visual artists play with scale in order to make an impact on the viewer. Students might even be given an opportunity to create a piece of art that has an exaggerated or inconsistent scale or plays with forced perspective.
Nova Scotia Mathematics Curriculum Outcomes
Grade 6 N05 - Students will be expected to demonstrate an understanding of ratio, concretely, pictorially, and symbolically.
Grade 8 N04 - Students will be expected to demonstrate an understanding of ratio and rate.
Grade 8 N05 - Students will be expected to solve problems that involve rates, ratios, and proportional reasoning.
Grade 9 G03 - Students will be expected to draw and interpret scale diagrams of 2-D shapes.
Math at Work 10 G03 - Students will be expected to demonstrate an understanding of similarity of convex polygons, including regular and irregular polygons.
Mathematics Essentials 11 D9 - calculate scale factors in 2-D scale diagrams and 3-D scale models understand the relationship among the scale factor and the related change in area or volume.Math at Work 11 G02 - Students will be expected to solve problems that involve scale.
Mathematics 11 M03 - Demonstrate an understanding of the relationships among scale factors, areas, surface areas and volumes of similar 2-D shapes and 3-D objects.
High school exams in Nova Scotia (grade 10-12) typically run in the second to last week in June. This is the time that grade 9 students are writing some exams as well. The week prior to exams is often set aside by teachers for a cumulative review. This is a time to revisit the curriculum outcomes, consolidate learning and make final connections. What teachers do with this review time can vary greatly from class to class. Many teachers create a booklet of review questions that cover the main topics from the course. This can be a helpful resource for some students but not all that engaging.
Here are some additional ideas for reviewing outcomes that might increase student engagement and allow for some differentiation.
No matter how you decide to review for the exam, by the time you get to exams, "the hay is in the barn". Here is a note from Alberta Education... "The best way to prepare students for writing the achievement test is to teach the curriculum well and to ensure that students know what is expected. Many of the skills and attitudes that support test writing are, in fact, good skills and strategies for approaching all kinds of learning tasks."
Coins in a Row is a simple game that I have used in lots of classrooms at different grade levels. It is a great way to tackle some mental math and logical reasoning. I first learned about this game in an article from Ivar Peterson's Math Trek. The game originally appeared in Peter Winkler's Mathematical Puzzles: A Connoisseur's Collection (2004).
The Rules: Arrange a bunch of coins of various denomination in a row (any even number of coins will do). The first player chooses a coin from one of the ends and slide it over to her pile. Your opponent then chooses a coin from one of the ends of the row of remaining coins. You and your opponent take turns removing a coin in this manner until they are gone. The player with the highest total value of coins wins.
The fun part of this game is that it is unfair. Unfair games are a great way to get out of doing some chores while you're visiting in-laws over the holiday break (e.g. "Lets play a game, the loser has to wash the dishes"). The player who goes first can always win if they know the correct strategy. After playing this game a few times, I let students know that it is an unfair game and ask them if they can find a winning strategy. Here is how the winning strategy works. Label the coins from 1 to n going from left to right. Add up the value of all the odd labeled coins then add up the value of all the even labeled coins. The first player can choose either an odd or even labeled coin. The second player only has two evens to choose from if the first player took the first coin from the row. The second player only has two odd coins to choose from if the first player to the coin from the end of the row. The first player then just takes a coin from whichever end the second player chose from and the first player can guarantee that they get either all the odd or even labeled coins.
This strategy guarantees at least a tie for the first player. It is not necessarily the "best" strategy however. According to Ivars Peterson, no one has yet worked out an optimal strategy that works for any number of coins.
Alternate versions are quite easy to create for this game. For example, Coins in a Circle. In this version, the first player can pick any coin, then play continues as in the previous version.
Another way to play would be to use a deck of cards (remove the face cards) and deal out 10 cards in a row.