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BCAMT Fall Conference Friday, October 20, 2023 -
Learning Mathematics: It's Relational
Primary Tasks
100 Hungry Ants by Elinor J. Pinczes
I used digit cards to create a 2-digit number pattern. The wind blew the cards and mixed them up. How might you place the loose digit cards into the following to complete a pattern? How do you know? How might you extend the pattern?
source: Vector, Spring 2016
Allow students time to explore the attributes of various 3-D shapes. Have them identify the faces, edges and vertices of the 3-D shapes? Present various problems for them to solve:
Have the students create their own clues to create a problem.
(From Vector, March 2015)
Take your class outside and have students collect 5 of an object (leaves, rocks, etc..). The task is for students to work in groups and find different ways to make 5. How many ways can you make 5? How can you show all of your ways?
Extensions: How about 4? How about 6?
Materials
Task Instruction
Source: https://www.youcubed.org/task/snap-it/
Materials:
Task Instruction
Example: I have 10 cubes and I decide to hide 4 in my cup. My group can see that I only have 6 cubes. Students should be able to say that I’m hiding 4 cubes and that 6 and 4 make 10.
Source: https://www.youcubed.org/task/how-many-are-hiding/
Intermediate to Grade 12
These open non-curricular math tasks are intended for intermediate to grade 12 students. They are non-curricular so that all students should be able to get started and investigate by drawing pictures, making guesses, or asking questions. When possible, extensions will be provided so that you can keep your students in Flow during the activity. Although they may not fit under a specific topic for your course, the richness of the mathematics comes out when students explain their thinking or show creativity in their solution strategies. I hope you and your class have fun with these weekly mathematics tasks.
Chicken nuggets come in boxes of 6, 9 and 20. What is the largest number of nuggets that you cannot buy when combining various boxes?
Source: http://www.playwithyourmath.com/
Take the number 25, and break it up into as many pieces as you want.
25 = 10 + 10 + 5 25 = 2 + 23 25 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 9 + 9
What is the biggest product you can make if you multiply those pieces together?
Will your strategy work for any number?
Source: http://www.playwithyourmath.com/
There is a hotel with _ rooms in a row. Every room has a door to the corridor and doors connecting it to the adjacent rooms. A woman has rented all _ rooms, telling the hotel manager that if he needs to contact her, she will always be in a room adjacent to the room she was in the day before. The hotel manager needs to contact the woman regarding an issue with her credit card; however, he can only check one room each day.
Can you devise a scheme such that the hotel manager is guaranteed to find the woman?
Start this task with 3 rooms, and when groups have convincing well-reasoned solutions, move to 4 rooms, 5 rooms, and so on.
Aliens from Jupiter capture three men. The aliens give the men a single chance to escape uneaten.
The captives are lined up in order of height, and are tied to stakes. The man in the rear can see the backs of his two friends, the man in the middle can see the back of the man in front, and the man in front cannot see anyone. The aliens show the men five hats. Three of the hats are black and two of the hats are white.
Blindfolds are then placed over each man’s eyes and a hat is placed on each man’s head. The two hats left over are hidden. The blindfolds are then removed and it is said to the men that if one of them can guess what color hat he is wearing they can all leave unharmed.
The man in the rear who can see both of his friends’ hats but not his own says, “I don’t know”. The middle man who can see the hat of the man in front, but not his own says, “I don’t know”. The front man who cannot see ANYBODY’S hat says “I know!”
How did he know the color of his hat and what color was it?
25 coins are arranged in a 5 by 5 array. A fly lands on one, and tries to hop on to every coin exactly once, at each stage moving only to the adjacent coin in the same row or column. Is this possible?
Extensions: Can you explain why some starting locations are not possible? What about 3D? What about rectangles?
Source: Thinking Mathematically by John Mason
When I make pancakes, they all come out different sizes. I pile them up on a plate in the warming oven as they are cooked, but to serve them attractively, I would like to arrange them in order with the smallest on top. The only sensible move is to flip over the topmost ones. Can I repeat this sort of move and get them all in order?
Extensions: What is the most complicated arrangement for 3, 4 or 5 pancakes? What is the minimum number of flips for these arrangements?
Source: Thinking Mathematically by John Mason
Note: This is an open mathematics problem. See: https://en.wikipedia.org/wiki/Pancake_sorting.
Five persons are standing on one side of a bridge.
They want to cross the bridge.
Without a torch, they cannot proceed.
Only one torch is available.
The torch has a remaining battery life of only 30 seconds.
Only two people can go over the bridge at one time.
The torch needs to be returned to the remaining persons.
The five people take different times to cross the bridge.
One takes 1 second to cross the bridge.
The others take 3 seconds, 6 seconds, 8 seconds and 12 seconds.
Everyone must cross the bridge within 30 seconds.
How do they proceed?
Extensions: What if the time taken for two to cross is the average of their two times?
A 600-pound pumpkin was entered in a contest. When it arrived, it was 99% water. The pumpkin sat for days in the hot sun, lost some weight (water only), and is now 98.5% water. How much does it now weigh?
Extensions: At 98.5%, the pumpkin had lost 0.5% water. What if the pumpkin loses 1%, 2%, n%?
A magic square is a square grid with n rows and n columns, filled with distinct numbers from 1 to n2, such that the sum of the numbers in each row, column, and both long diagonals is the same.
In a box, you have 13 white marbles and 15 black marbles. You also have 28 black marbles outside the box. Remove two marbles, randomly, from the box. If they are of different colours, put the white one back in the box. If they are of the same colour, take them out and put a black marble back in the box. Continue this until only one marble remains in the box. What colour is the last marble?
Source: Vector 55(1) – p. 49: https://tinyurl.com/zaksq6b
If the pattern of arrows continues for ever, which point will the route visit immediately after (18, 17)? Explain how you know.
Extensions: How many points will be visited before the route reaches the point (9, 4)? Which point will be the 1000th to be visited.
Source: http://nrich.maths.org/5469
Consider the string 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Cross out any two numbers in this list and add the difference to the end of the list. This new number is now part of the list. Continue the process of crossing out two number on the list and adding the difference until there remains only one number. What can you say about the last number? Explore.
Source: Richard Hoshino, Quest University, Squamish
A marble can be placed in each of the 27 cubes that make a larger 3x3x3 cube. Imagine a 3D tic-tac-toe game where players take turns placing their own marbles in any of the 27 cubes. A player wins when they have 3 marbles in any straight line. How many different ways can someone win in this game?
Extensions: Investigate 4x4x4, 5x5x5, and nxnxn. What about a hypercube?
Source: Adapted from http://nrich.maths.org/895
You are a chef at a summer camp and you are frying 30 identical strips of bacon for this morning’s breakfast. A counselor comes in to inform you that there are only 18 campers coming in for breakfast and they all love bacon. What is the minimum number of cuts necessary? What is the minimum number of pieces?
Extensions: How do you know it is the minimum? What about sharing amongst 17 campers? 16 campers? n campers?
Adapted from Thinking Mathematically – John Mason
What fits better: A square peg in a round hole or a round peg in a square hole?
Source: http://www.peterliljedahl.com/teachers/good-problem
Picture to yourself a length of rope, lying on a table in front of you. The cross section of the rope is a regular N sided polygon. Slide the ends of the rope towards you so that it almost forms a circle.
Now, mentally grasp the ends of the rope in your hands. You are going to glue the ends of the rope together but before you do, twist your right wrist so that the polygonal end rotates through one nth of a full revolution. Repeat the twisting a total of T times, so that your mental wrist has rotated through T nths of a full revolution. NOW, glue the ends together, so that the polygonal ends match with edge glued to edges.
When the mental glue has dried, start painting one facet (flat surface) of the rope and keep going until you find yourself painting over an already painted part. Begin again on another facet not yet painted, and use a different colour.
How many colours do you need?
Source: Thinking Mathematically – John Mason
A band of 10 pirates are going to disband. They have divided up all of their gold, but there remains one GIANT diamond that cannot be divided. To decide who gets it the captain puts all of the pirates (including himself) in a circle. Then he points at one person to begin. This person steps out of the circle, takes his gold, and leaves. The person on his left stays in the circle, but the next person steps out. This continues with every second pirate leaving until there is only one left. Who should the captain point at if he wants to make sure he gets to keep the diamond for himself? What if there were 11 pirates? What if there were 12 pirates? What if there were 27 pirates? Etc.
Extensions: What if two pirates are skipped over? What if The Captain wants himself and his first lieutenant to be the two winners?
Source: http://www.peterliljedahl.com/teachers/good-problem
Students in a marching band want to line up for their performance. The problem is that when they line up in 2’s there is 1 left over. When they line up in 3’s there are 2 left over. When they line up in 4’s there are 3 left over. When they line up in 5’s there are 4 left over. When they line up in 6’s there are 5 left over. When they line up in 7’s there are no students left over. How many students are there?
Extensions: What if there are over 200 students in the band? What if there are 6 left over when lined up in 7’s?
Source: John Grant McLoughlin
Consider a two-digit number: for example, 84. 84 is not a palindrome, so reverse the digits and add it to the original number: 84 + 48 = 132. This is still not a palindrome, so try it again: 132 + 231 = 363. 363 is a palindrome, so 84 can be called a depth 2 palindrome. Find the depth of all two-digit numbers.
Extensions: What about 3-digit numbers? What about the depth for the second time of becoming a palindrome? What happens when you shade a Hundreds Chart according to the number’s depth?
Source: http://www.magic-squares.net/palindromes.htm
A regular six-sided die has opposite sides that add to seven. If you removed this restriction, how many different six-sided die can you make?
Extensions: What about three-sided and eight-sided die? What if a six-sided die has two of the same numbers (1, 1, 2, 3, 4, 5) or 3 of the same numbers (1, 1, 1, 2, 3, 4)?
A square cake is to be divided amongst 5 people so that each person has equal portions of cake and icing. How should you cut the cake?
Extensions: What about a triangle cake? Hexagon? What about 3 people?
A secret number is assigned to each vertex of a triangle and pairs of numbers are added together to equal the number given along each edge. The task is not to solve each Arithmagon (although this is a good first step); rather, the task is to find structure, relationships and strategies for solving all Arithmagons.
Extensions: Investigate other polygons
Source: NRICH and Thinking Mathematically: https://nrich.maths.org/2670
Nine counters marked with the digits 1 to 9 are placed on the table. Two players alternately take one counter from the table. The winner is the first player to obtain, amongst his or her counters, three with the sum of exactly 15.
Extensions: What are some strategies for winning? What childhood game does this connect with?
Source: Thinking Mathematically – John Mason
Over the holiday season, I found two crazy soda flavours: Peppermint and Prune.
I poured a tall glass of each and then decided to mix them. I carefully measured 30 mL of the Peppermint soda and poured it into the Prune soda glass. After stirring the Prune soda mix, I measured 30 mL of this mix and poured it back into the Peppermint soda glass, creating a Peppermint soda mix. After all of this, which glass is purer? In other words, is the Peppermint mix more pepperminty or is the Prune mix more Pruney?
Extensions: What if we introduce a third soda: Cabbage flavour?
A goat is tethered by a 6 metre rope to the outside corner of a shed measuring 4 m by 5 m in a grassy field. What area of grass can the goat graze?
Extensions: What if the rope was fastened to the middle of one wall? What if the rope was 20 m long? What if the shed was circular?
Source: Thinking Mathematically – John Mason
Mountain bike Race. There are 25 racers and the track can only fit five racers at any time. Devise a strategy to determine gold, silver, and bronze. How many races are necessary?
There is a party for mothers and daughters. All of the mothers shake hands among themselves. Every daughter shakes hands with only the mothers. How many handshakes are there if the party involved 17 mother-daughter pairs?
Extensions: What if the mothers do not shake hands with their own daughters? What if grandmothers were brought into the mix?
A certain milkcrate can hold 36 bottles of milk. Can you arrange 14 bottles in the crate so that each row and column has an even number of bottles? (Thinking Mathematically – John Mason)
Extensions: What is the smallest array that can fit 14 bottles under this rule? What about 15 bottles?
How many diagonals are there in an n-gon (a polygon with n sides)?
Extensions: What about concave n-gons? What about n-hedrons?
It is 6:00. With a partner, take turns adding one of 15 min, 30 min, 45 min, or 60 min to the clock. The first player to reach 12:00 wins. (adapted from: http://wild.maths.org/approaching-midnight)
Extensions: Is there a winning strategy? What if the first to 12:00 is the loser?
What is it that makes them surprising?
Source: The Journal of Economic Perspectives, Vol. 19, No. 4 (Autumn, 2005), pp. 25-42
Prime numbers have exactly two factors – 1 and itself. Which numbers have exactly 3 factors? Exactly 4 factors? And so on. Given any positive integer, n, how can you tell exactly how many factors it has?
A circular road is 27 km long. On this road are six cottages, owned by 6 friends. The friends visit each other a lot, and they have noticed that every whole number from 1 to 26 (inclusive) is accounted for at least once when they calculate the distances from one cottage to another. Of course the friends can walk in either direction as required. Your task is to place these cottages at distances that will fulfill these conditions.
Extensions: Can you find more than one solution?
Source: 536 Puzzles and Curious Problems by Henry Ernest Dudeney
You are backpacking through Europe. You would like to stay in the South of France for as long as possible, but you have run out of money. However, you have a 30 link gold chain and you have found a hotel that is willing to accept one link per night for payment of room and board. The manager wants payment every day and he is willing to help you out by cutting links for you. The problem is that he wants one gold link payment for every link he cuts. What is the most number of nights that you can stay in the hotel?
Extensions: What if the manager will accept payment after every second day?
Source: http://www.peterliljedahl.com/teachers/good-problem
On a 6 by 6 square array, dots can be placed at any intersection. If a line is drawn between any two dots, then this line can only pass through exactly two dots. Can 12 dots be placed on this array? This picture shows a mistake, because the green dot is a third dot on two lines.
Extensions: What about a 5×5 or 7×7? What if each line must contain exactly three dots?
Adapted from: http://galileo.org/classroom-examples/math/math-fair-problems/puzzles/minaret-with-a-view/
Imagine a long strip of paper folded in the same direction once, twice, and then a third time. When the strip is unfolded, how many creases will be on the paper? What directions will the creases be pointing? What about n folds?
Extensions: When the paper is unfolded, and the creases made to equal 90°, what do you notice in the shapes?
You have 7 pen caps on a table with the same side up. These caps all have to be turned over, but you can only turn over exactly three at a time. What is the smallest number of moves you can do this in? How do you know it is the smallest?
Extensions: What about 8, 9, or n pen caps? What if you turn over 4 at a time? What if the pen caps had a middle position: up, middle, down?
Source: https://www.youcubed.org/task/seven-flipped
How many matches are required to make n2 unit squares in a square array as in the following sequence?
Source: Thinking Mathematically by John Mason
This is a game for two players on a rectangular grid with a fixed number of rows and columns. Play begins in the bottom-left-hand square where the first player puts his mark. On his turn a player may put his mark into a square directly above or directly to the right of or diagonally above and to the right of the last mark made by his opponent. Play continues in this fashion, and the winner is the player who gets their mark in the upper-right-hand corner first.
Find a way of winning which your great aunt Maud could understand and use.
Extensions: What if you cannot move diagonally? What if the top right square means that the player loses?
Source: Thinking Mathematically by John Mason
You have 10 silver coins in your pocket (silver means that the coins could be any of nickels, dimes or quarters). How many different amounts of money could you have?
Consider the following pattern of 5 whole numbers, where each number is the sum of the previous two numbers:
3, 12, 15, 27, 42
I want the 5th number to be 100. Find all the whole seed numbers that will make this so (3 and 12 are the seed numbers in the above sequence).
Source: Peter Liljedahl
A number of pins are placed around a circle. A thread is tied to one pin, and then looped tightly around a second pin. The thread is then looped tightly round a third pin so that the clockwise gap between the first and second pin is the same as the clockwise gap between the second and third pin as shown:
The process is continued, always preserving the same clockwise gap until the first pin is reached. If some pin has not yet been used, the process starts again. Five pins with a gap of two use just one thread, while six pins with a gap of three use three threads. How many pieces of thread will be needed in general?
Extension: What if the gap alternates between two numbers?
Source: Thinking Mathematically by John Mason
How many squares are there on a chess board? And no, the answer is not 64.
Extensions: How many rectangles? How many triangles in this:
Three slices of bread are to be toasted under a grill. The grill can hold two slices at once but only one side is toasted at a time. It takes 30 seconds to toast one side of a piece of bread, 5 seconds to put a piece in or take a piece out and 3 seconds to turn a piece over. What is the shortest time in which the three slices can be toasted?
Source: Thinking Mathematically by John Mason
Extensions: The answer is not 151 sec. What about 4 slices? 5 slices?
A rope is wrapped tight around the Earth along the equator. The rope is cut, and 1 m of rope is added to the length and then stitched back together. A big green super hero lifts the rope and throws it so hard that it enters into circular orbit the Earth (bonus for the physicists: how fast does the rope need to spin to be in orbit?). How high is the rope hovering over planet Earth? (Earth radius is 6371 km)
Extensions: What about Jupiter, the Sun, or a basketball?
Instead of spinning the rope, the super hero lifts the rope straight up until it is tight again. How high can she lift the rope?
Imagine that you are at one end of a hallway with 1000 open lockers. The first student goes along and closes every single locker. A second student then goes and opens every second locker. A third student then changes the state of every third locker (meaning if the locker is open, the student closes it, and if the locker is closed, the student opens it). A fourth student changes the state of every fourth locker. This continues until 1000 students have gone through. When finished, which lockers are closed?
Extensions: Why are these lockers closed? What if they are not lockers, but they are dials with three positions: A, B or C?
Write each number from 1 – 10 using exactly 4 fours and any mathematical sign or symbol.
Extensions: 1 – 20 or 1 – 100. Can you do this with 4 fives or 4 threes etc…
Three green frogs are trying to change position with3 orange frogs. Green frogs and orange frogs can only move forward onto an empty lily pad or leap frog a single frog onto an empty lily pad.
How many moves are required to solve this puzzle?
Extension: What if there were 4 frogs on each side (or 5 frogs or n frogs…)? What if the number of frogs on each side is not equal? How can you communicate the solution to a friend over the phone?
Source: https://britton.disted.camosun.bc.ca/frog_puzzle.htm
You have a 3 L jug and a 5 L jug and an unlimited supply of water. How can you measure exactly 4 L of water? https://youtu.be/BVtQNK_ZUJg
Extensions: What if not 4 L? What if not 3 L and 5 L? What if the jugs were 3 L and 6 L? or 3 L and 7 L?
Start with the cards ace, two, three, four, and five. Arrange the cards in such a way so that they come out in increasing sequence when you deal the cards out like this:
What is the pattern? What is your strategy?
Extensions: Add more cards to the deck. Can you do 13 cards? 52 cards? 104 cards?
Source: Peter Liljedahl. https://youtu.be/FOcqqV0IdQ8
Players take turns picking any number from 1 through to 6. Each time a number is picked, it is added to the total score. The player who makes the total score add to 31 wins.
Extensions: What if 31 loses? What if you can choose from 2 through 6?
On a table, there are 1001 loonies lined up in a row. I then come along and replace every second coin with a nickel. After this, I replace every third coin with a dime. Finally, I replace every fourth coin with a quarter. After all this, how much money is on the table?
Extensions: Why is the repeating pattern 12? Design a task that has a repeating pattern of 15. How many starting loonies are needed to make a total of $100?
Start with a collection of paychecks, from $1 to $12. You can choose any paycheck to keep. Once you choose, the tax collector gets all paychecks remaining that are factors of the number you chose. The tax collector must receive payment after every move. If you have no moves that give the tax collector a paycheck, then the game is over and the tax collector gets all the remaining paychecks. The goal is to beat the tax collector.
Turn 1: Take $8. The tax collector gets $1, $2 and $4.
Turn 2: Take $12. The tax collector gets $3 and $6 (the other factors have already been taken).
Turn 3: Take $10. The tax collector gets $5.
You have no more legal moves, so the game is over, and the tax collector gets $7, $9 and $11, the remaining paychecks.
Total Scores:
You: $8 + $12 + $10 = $30.
Tax Collector: $1 + $2 + $3 + $4 + $5 + $6 + $7 + $9 + $11 = $48.
Source: http://wordplay.blogs.nytimes.com/2015/04/13/finkel-4/?_r=0
Extensions: What is the highest score you can achieve? What is the lowest score? What if you had 18 paychecks?
My friend Peter has built a new dart board for his son. The board has two regions: the centre circle, valued at 9 points, and the outside circle, valued at 4 points. What is the largest number that cannot be achieved as a score in this game?
Extensions: What if not 9? What if not 4? What if there was another scoring circle?
What’s a better fit, a round peg in a square hole, or a square peg in a round hole?
Source: Peter Liljedahl http://www.peterliljedahl.com/teachers/good-problem
Place 17 jelly beans into the two circles below so that each circle contains the same number of jelly beans. How many ways can you do this? What if you want the ratio of jelly beans for each circle to be 4:3? m:n?
Can this net be folded into an open-topped box? How do you know?
How many different nets can be folded into an open-topped box?
Extensions: What if not open-topped? How many different nets are there for other polyhedra?
Rock, Paper, Scissors is considered a fair 2-person game with three options: R, P, S. Why is it fair? Can you design a fair game with 4-options? 5-options?
Source: David Peterson, Fraser Lake Elementary School
Each side of a cube requires paint. You have only two colours of paint, white and red. How many different ways are there to paint the cube?
Extensions: What if not two colours? What if not a cube?
There are nine people spaced evenly in a 3×3 grid (try this in your class). The person in the top right corner is removed, leaving an empty space. How can you move the person in the bottom left corner into this empty space? People can only move horizontally or vertically into empty spaces. How many moves are required? Is this the minimum number of moves?
Extensions: What if not 3×3? What if not one empty space?
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