Central Idea
"Mathematicians can find patterns and record what they find in different ways to help them make predictions, share information and solve problems."
"Mathematicians can find patterns and record what they find in different ways to help them make predictions, share information and solve problems."
Chance
An inquiry into...
An inquiry into...
- If luck exists (PERSPECTIVE)
- The language of probability (FORM)
- How we can work out probability visually (FUNCTION)
- How probability relates to fractions / percentages (CONNECTION)
Tuning In
Does Luck Exist? - There are many times in our life where we talk about being lucky, or unlucky. There are many numbers, symbols and items, across all different cultures that represent luck, or are said to give us luck - just as there are many things that are said to be unlucky. But what is luck? Does luck even exist?
Think about where you stand on the question. Do you think it does exist? If so, why? Do you think there is no such thing as luck? What makes you think that? What are things that you know that are associated with luckiness, or unluckiness? Do you think they work? If so, how do you think they work? Can luck be measured?
Look at some of the items below. What do they have in common?
Does Luck Exist? - There are many times in our life where we talk about being lucky, or unlucky. There are many numbers, symbols and items, across all different cultures that represent luck, or are said to give us luck - just as there are many things that are said to be unlucky. But what is luck? Does luck even exist?
Think about where you stand on the question. Do you think it does exist? If so, why? Do you think there is no such thing as luck? What makes you think that? What are things that you know that are associated with luckiness, or unluckiness? Do you think they work? If so, how do you think they work? Can luck be measured?
Look at some of the items below. What do they have in common?
Thoughts on the Line - As a class, create a line with a YES at one end and a NO on the other. Using a Post-it Note, write down your thoughts about Luck and if it exists. Once you've written down your thoughts, post your Note onto the line at a place where you feel your thoughts are related to luck. If you are resolutely positive there is no such thing as luck, place your note by the NO. If you are certain there is luck, then put your note at YES. If you are somewhere in between yes and no, then place your note where you think you feel your opinions lay.
Lucky vs. Unlucky - What do you think is lucky? What do you think is unlucky? Create something that shows what you know about the two words. Think about the things you associate with luckiness and unluckiness. What other words do you know that you would associate with the chance of something happening? Draw pictures, write words, use numbers and show what you know.
Impossible to Certain - On a line, with Impossible on one end and Certain on the other, write down the words that you know that have to do with the chance of something happening. Place those words on the line to show their meaning. For example, if I say something is a 50-50 chance, where do you think it might go on the line?
Finding Out
Lucky vs. Unlucky - What do you think is lucky? What do you think is unlucky? Create something that shows what you know about the two words. Think about the things you associate with luckiness and unluckiness. What other words do you know that you would associate with the chance of something happening? Draw pictures, write words, use numbers and show what you know.
Impossible to Certain - On a line, with Impossible on one end and Certain on the other, write down the words that you know that have to do with the chance of something happening. Place those words on the line to show their meaning. For example, if I say something is a 50-50 chance, where do you think it might go on the line?
Finding Out
"Instead of luck, mathematicians think of probability.
BrainPop Jr. - There are many different places where we can find out about probability. One of the places we go to often is BrainPop Jr. When it comes to probability, there are three great videos which can help us to understand the basics. Click on the pictures below to watch the videos. Do not forget to use your research sheet and your best visual note-taking to record your thoughts.
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Math is Fun - Another great website that we can use for many different things. Math is Fun has some great information on probability. Click on the link here to find out more about chance and probability.
Concrete
All of the concrete materials below can be used to help solve problems, test ideas and play games with chance. Some are more complicated than others. Listed below are some example games and lessons where you might use these tools.
Concrete
All of the concrete materials below can be used to help solve problems, test ideas and play games with chance. Some are more complicated than others. Listed below are some example games and lessons where you might use these tools.
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Coins
- can be used for very basic probability - good for games |
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Spinner
- all kinds of different spinners can be made or used - useful when discussing probability - good for number games |
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Dice
- wide range of games can be played using dice - useful when discussing probability |
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Cards
- wide range of games can be played using cards - useful when discussing probability |
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Counters
- useful in probability in the same way that bears can be used - can be used for keeping track of different probability games |
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Bears
- anything that can be counted, really (counters, money, shells, buttons, etc.) - can be used for probability - chance can change depending on how many and what colour |
Pictorial
Tally Marks - There are many different different ways to keep track of results from experiments and games involving chance. Some are more effective than others, and some are only effective in certain situations. When might you use tally marks? How would you use tally marks?
Tree Chart - A Tree Chart, or Diagram, is a way of showing probability over multiple events. Like the branches on a tree though, things can can pretty big, pretty quickly, even if you're drawing something simple, like a coin flip. Check out the tree chart below, which shows the possible outcomes of a coin being flipped just three times. How might you draw a tree chart if there were more flips or if you were rolling a 6 sided dice?
Tally Marks - There are many different different ways to keep track of results from experiments and games involving chance. Some are more effective than others, and some are only effective in certain situations. When might you use tally marks? How would you use tally marks?
Tree Chart - A Tree Chart, or Diagram, is a way of showing probability over multiple events. Like the branches on a tree though, things can can pretty big, pretty quickly, even if you're drawing something simple, like a coin flip. Check out the tree chart below, which shows the possible outcomes of a coin being flipped just three times. How might you draw a tree chart if there were more flips or if you were rolling a 6 sided dice?
Table - In the image above, you can see how someone has shown all the possible combinations of throwing two six sided dice using a table to keep track of each possibility. This is a great way to see just how many different options there are. Does it remind you of something? How might you use a table to keep track of other possible outcomes for different games of chance.
Abstract
Probability in Words - If you were to draw a number line from 0 to 1, where ZERO meant that the chance of something happening was impossible and ONE meant that the chance of something happening was certain, then were would you put these other expressions and words along that line? Is "a small chance," less likely than "probably?" Are there any words that mean the same thing?
Abstract
Probability in Words - If you were to draw a number line from 0 to 1, where ZERO meant that the chance of something happening was impossible and ONE meant that the chance of something happening was certain, then were would you put these other expressions and words along that line? Is "a small chance," less likely than "probably?" Are there any words that mean the same thing?
Probability as a Fraction - When it comes to games of chance, we can write down the chance of something happening using fractions. Just like when we write fractions we will need a numerator and a denominator. In the spinning wheels below, what is the chance of landing on a ONE for each of the different wheels? Is it the same for each? How would you write the number as a fraction? Can you apply this knowledge to other games of chance?
Probability as a Percentage - Games of chance can also be described a percentage. We know the term 50-50 chance means it's an even chance of one thing or another, but why do we use the number 50? Where does it come from? If you think about the spinners below, is there one that has a 50% chance? What are the chances for the other spinners?
Probability as a Percentage - Games of chance can also be described a percentage. We know the term 50-50 chance means it's an even chance of one thing or another, but why do we use the number 50? Where does it come from? If you think about the spinners below, is there one that has a 50% chance? What are the chances for the other spinners?
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Applying
Hangman - Everyone knows the game Hangman, but did you know there was a strategy behind it? If I guess the letter "m" to start, or if I guess the letter "e," which letter do you think is more likely to come up? Are there certain words that are harder to guess than others? Is it better to have a long word or a short word? |
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Guess Who - A classic guessing game, like Hangman. Like all games of chance, let's try to think if there is a better way to play the game so that we increase our odds of winning. Click on the icon to the right to go to the online version of the game, or try your hand at the original. |
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Lucky Numbers - What are the lucky and unlucky numbers? Why do you think they are considered luck or unlucky? If we know that mathematicians talk about probability instead of luck, what numbers do you think they might think are "lucky?"
First Out Wins - When we roll two dice, we can get any result from 2 - 12. Is there are number that is more or less likely to come up than any other? In this game, take 11 counters and place them on any number you'd like from 2-12. You can put as many or as few counters on each number as you want. Roll the dice. If a number comes up where you have a counter, you can remove the counter. Play this game with a friend. Whoever removes all their counters first is the winner. What is your strategy? Are there better numbers to choose?
First Out Wins - When we roll two dice, we can get any result from 2 - 12. Is there are number that is more or less likely to come up than any other? In this game, take 11 counters and place them on any number you'd like from 2-12. You can put as many or as few counters on each number as you want. Roll the dice. If a number comes up where you have a counter, you can remove the counter. Play this game with a friend. Whoever removes all their counters first is the winner. What is your strategy? Are there better numbers to choose?
Who Wants to Be a Millionaire? - Let's imagine that you are on a TV show. You are given a choice between playing 4 different games. If you win the game you select, you will receive $1,000,000! Which game would you choose and why?
Game A
Flip 2 coins and if they both land on heads, you win. |
Game B
Pull a royal card out of a pack of cards, you win (52 card deck, shuffled). |
Game C
Choose 1 cup, if the ball is beneath it, you win (2 balls, hidden under 6 cups). |
Game D
Roll two dice, if you roll a double, you win. |
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Data Handling
An inquiry into...
Tuning In Survey Says (Prior Knowledge) - One way to find out about our friends is to survey them about things they like and dislike. Using your knowledge of surveys and graphs, ask some of your friends about their favourite things. Decide how you want to record their answers and how you would display what you find out. |
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Pictograph - What does the graph to the right tells us? Is it a Good Graph? Why or Why not? When might someone use a graph like this one? Can this type of graph be used any time we need to make a graph, or does it have limitations?
Form and Perspective (Prior Knowledge) - What makes a good graph? How can we find information that is important from the different types of graphs? Why do we have different types of graphs to begin with. Look at the graphs below and fill in the information showing our two concepts - FORM (What Data does this Graph Provide?) and PERSPECTIVE (Your Thoughts on This Type of Graph)
Form and Perspective (Prior Knowledge) - What makes a good graph? How can we find information that is important from the different types of graphs? Why do we have different types of graphs to begin with. Look at the graphs below and fill in the information showing our two concepts - FORM (What Data does this Graph Provide?) and PERSPECTIVE (Your Thoughts on This Type of Graph)
Finding Out
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Bad Graph - To the right are a series of graphs, but nothing about them is perfect. What is wrong with them though? Based on what we see, can we establish some rules about what makes a graph a Good Graph and what makes a graph a Bad Graph.
BrainPop Jr. and Math is Fun - Just like when it comes to probability there are many different places where we can find out about Surveys and Graphs. BrainPop Jr. is a great place to find out about the different types of graphs. Click on the pictures below to watch the videos. As always, do not forget to use your research sheet and your best visual note-taking to record your thoughts. Additionally, Math is Fun has great info on Data. Click on the link to find out more. |
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Concrete
Many concrete tools can be used when helping to collect Data. With a little creativity, some of the tools we use in math every day can also be used to help create different graphs. Below are just two of the possible ways that graphs can be made using tools we have in the classroom.
Many concrete tools can be used when helping to collect Data. With a little creativity, some of the tools we use in math every day can also be used to help create different graphs. Below are just two of the possible ways that graphs can be made using tools we have in the classroom.
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Unifix Cubes (or Blocks)
- can be used to create bar graphs - can also be used to count, or keep track of results in a survey |
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Bears (or Gummy Bears)
- like counters, they can be used to keep track of survey results - helpful for showing how pictographs can be created |
Pictorial
Choose the Right Tools - We have learned a little bit about how to make a good graph. We have created success criteria. Now it's time to actually start making some graphs to show the information we have found. Now, what if you were given just a blank piece of paper and a pencil? How difficult might your job be. To make a graph, we might need to first get certain tools. What tools will you need? How will you use them?
Try making two graphs, one with just a paper and pencil and one with the correct tools. See how they turn out. What made things more easy? What was challenging?
Abstract
Khan Academy - To practice what you have learned about Data, a great place to do practice questions is Khan Academy. There are many different practice lessons as well as videos to help you find out more. Practice solving problems with Bar Graphs, or Picture Graphs, or Line Plots, by clicking on the each of the links. You can also practice making Picture Graphs, Bar Graphs, or Line Plots, again, by clicking on the links. Each one of the linked words is a separate practice session.
Applying
Units of Inquiry - The great part about finding data and representing it in graphs, is that we can do so with so many different things. Our units of inquiry will be full of information that we can put into graphs. Think about how the learning we have done may be graphed. Think about how things you have learned in your daily life can be represented. A good mathematician will be able to choose the best method to show what they have found.
Choose the Right Tools - We have learned a little bit about how to make a good graph. We have created success criteria. Now it's time to actually start making some graphs to show the information we have found. Now, what if you were given just a blank piece of paper and a pencil? How difficult might your job be. To make a graph, we might need to first get certain tools. What tools will you need? How will you use them?
Try making two graphs, one with just a paper and pencil and one with the correct tools. See how they turn out. What made things more easy? What was challenging?
Abstract
Khan Academy - To practice what you have learned about Data, a great place to do practice questions is Khan Academy. There are many different practice lessons as well as videos to help you find out more. Practice solving problems with Bar Graphs, or Picture Graphs, or Line Plots, by clicking on the each of the links. You can also practice making Picture Graphs, Bar Graphs, or Line Plots, again, by clicking on the links. Each one of the linked words is a separate practice session.
Applying
Units of Inquiry - The great part about finding data and representing it in graphs, is that we can do so with so many different things. Our units of inquiry will be full of information that we can put into graphs. Think about how the learning we have done may be graphed. Think about how things you have learned in your daily life can be represented. A good mathematician will be able to choose the best method to show what they have found.
