(c) 2021 George Gadanidis
“The ideas in this workshop were used many years ago in my secondary school math classroom, when teaching about binomial expansions, the Binomial theorem, and the connections among probability, algebra and binary choice.
More recently, I have used some of the same ideas, in a low floor, high ceiling manner, to engage grades 1-2 students with concepts of probability, as well as computational modelling.”George Gadanidis
Which result is more likely? A, B, or C?
- What does HH + 2HT + TT represent?
- What does this have to do with x2 + 2xy + y2?
ABOUT THE BINARY CHOICE WORKSHOP
- For students (enrichment; remediation), teachers (professional learning; online teaching resource) & parents (math therapy)
- Self-serve workshops: pick and choose activities to complete
|A. In grades 1-2 classrooms|
B. Solving an artistic puzzle
C. Counting outcomes
D. Statistician interview
E. Modelling with code
|— probability with coins|
— binary choice
— Yang Hui’s (aka Pascal’s) triangle
— Binomial theorem
— history of mathematics
— coding extensions
A.1. TOSSING A COIN
Working in pairs, students place a red/yellow (heads/tails) plastic counter (or a coin) in a cup.
- they cover the cup with one hand, and shake the cup
- they look at the counter (or coin)
- they use this “algorithm” below to decide if the result on the path is A or B
- using a show of hands, each pair indicates whether their result is A or B
- the teacher records a tally of the class results
- the teacher comments: “B won! It has the most tallies.”
- then asks: “Which letter would you predict will win next time? A or B? Why?”
A.2. TOSSING A COIN TWICE
The teacher presents this new algorithm and the new diagram.
- Using pair/share, students discuss in pairs, and then share with the whole class, how this algorithm is different from the previous one, and how the new diagram would be used.
- The teacher models the algorithm by tossing a coin twice and following the path to one of the letters A, B or C.
- The teacher asks: “If we did this as a whole class, and tallied the results, which letter would win? A, B or C? Why?”
What do you think?
- Which letter is most likely to win? A, B or C? Why?
A.3. TOSSING A COIN 5 times
The teacher presents the following paths diagram.
The teacher asks: “How do we alter the algorithm (on the right), so we can get to the letters at the bottom of the path?”
The teacher then asks: “If we did this as a whole class, which letters would be more likely to win? Which would be more likely to lose? And, can you explain why?”
What do you think?
- How would you alter the algorithm?
- Which letters are more likely to win? Why?”
- Which letters are more likely to lose? Why?”
A.4. BINARY CHOICE
When you toss a coin, you are making a binary choice. A choice between 2 options: heads or tails.
A light switch also offers a binary choice: on or off.
Digital devices, like smartphones, make decisions through complex combinations of ON and OFF, or 1 and 0. This is why the binary number system is especially well suited to how they function.
The idea of binary choice is powerful, and can be seen in probability (tossing coins) and in algebra (x + y), as well in the binomial theorem, which is a really powerful connection between probability and algebra.
Do you want to know more?
Yes or No?
[PS – Aunt Athanasia used to say: In life you’ll have to make many many choices between paths. The right path will always be the most difficult one.]
B.1. STORIES OF DOING MATHEMATICS
In 2004, we invited Apostolos Doxiadis, a math prodigy who eventually became a filmmaker and author, to a Symposium hosted by Western University and funded by the Fields Institute for Research in Mathematical Sciences.
Doxiadis said to us that mathematics education will not change unless what counts as mathematics first changes—what counts as mathematics needs to include the stories of doing mathematics.
How do we solve the artistic puzzle of designing math experiences as good math stories; so students can and want to share stories of doing mathematics?
B.2. GOOD STORIES
Good math stories incorporate mathematics that:
- is complex enough to capture their attention
- has a low floor (so everyone can engage) and a high ceiling (offers opportunities to engage with varied representations)
- incorporates mathematical surprise & conceptual insight
These are the elements that are in Sections A.1- A.3 above.
The introductory activities in A.1 – A3 above, engage grades 1-2 students with the probabilities of tossing a coin once, twice and 5 times, and consider what are the possible outcomes.
Low floor, high ceiling
The use of a paths diagrams provides a low floor by giving students a visual anchor for considering possible outcomes.
For example, in A.2, students notice that B is more likely than A or than C. Grades 1-2 students are able to explain this phenomenon by noticing that there are 2 paths to B and only 1 path to A and only 1 path to C.
Similarly, when considering tossing a coin 5 times, they predict that A and F are very unlikely, as they is only 1 path that leads to each of them. They also predict that the letters the centre, C and D, will be more likely as they look like they have many more paths leading to them.
Also offering a low floor is the algorithm used, as it provides a consistent, repeatable and extendable way of representing the experiments with tossing a coin once, twice and 5 times.
The paths diagram naturally offers a high ceiling, as it leads to more complex mathematics (as we will see below), such as:
- Yang Hui’s (aka, Pascal’s) triangle
- counting possible outcomes for different numbers of tosses of a coin
- coefficients of binomial expansions
- and the Binomial theorem
Surprise & insight
The first activity, where students toss a coin once, and where the outcomes of A and B are equally likely, sets the stage for young students to expect that outcomes A, B and C will also be equally likely when tossing a coing twice.
This leads to a mathematica surprise, as it turns out in their experiments that bB is more likely than A or than B. As well, the paths diagram allows students to make sense of the situation. leading to the conceptual insight that there are more paths that lead to B.
For higher grades that are also these surprises and insights:
- the patterns in Yang Hui’s triangle represent the sums of the paths leading to each intersection, and also represent the possible outcomes when tossing a coin repeatedly
- the possible outcomes represent the numeric coefficients of binomial expansions: HH + 2HT + TT is equivalent to x2 + 2xy + y2
- two seemingly disparate topics, probability of tossing a coin and the algebra of binomial expansions, can overlap due to the common underlying idea of binary choice: H or T and x or y
B.1. POSSIBLE OUTCOMES WHEN TOSSING A COIN
If you flip a coin 5 times, one possible outcome is HHHHH (or, 5 Heads). How likely is this outcome?
What other outcomes are possible? How likely are they?
B.2. Sums of paths
The diagram on the right shows the number of paths leading to each green node.
Notice a pattern? What are the missing numbers below?
- 1 + 1 = 2
- 1 + 2 + 1 = 4
- 1 + 3 + 3 + 1 = 8
- 1 + 4 + 6 + 4 + 1 = ?
- 1 + 5 + 10 + 10 + 5 + 1 = ?
B.3. Calculating theoretical probability
The theoretical probability of getting 2 heads and 1 tail (HHT) tossing 3 coins is 3/8. Can you see how 3/8 can be derived from the information shown above?
What is the theoretical probability of each of the following?
- HHHT when tossing 4 coins
- HHHTT when tossing 5 coins
- HHTTTT when tossing 6 coins?
B.4. POSSIBLE OUTCOMES
If you flip a coin, which is more likely? Heads (H) or tails (T)?
Each additional coin flip doubles the possible outcomes?
Is this true? Can you prove it?
Interview with Dr. Bethany White, University of Toronto.
D1. tossing a coin twice
Bethany White discusses coin tossing (and computational thinking) activities explored by Grade 1 students.
Let’s toss a coin to decide whether we walk left or right.
- If we toss the coin twice, where will we end up?
- Which path is more likely?
D.2. tossing a coin 5 times
Let’s toss a coin 5 times.
- Which path are we more likely to follow?
- And what does this have to do with Pascal’s Triangle?
D.3. all possible outcomes
Let’s look at all the possible outcomes.
- What pattern do we notice?
D.4. probability & algebra
There is a link between probability and algebra!
If you imagine H and T as X and Y, you may discover a powerful link between probability, algebra and Pascal’s triangle.
D.5 weighted outcomes
If you plot the frequency of one of two equally likely events as a graph, its plot looks like a “bell”.
If the events are not equally likely, the graph skews to one side of the other.
The Scratch code shown on the right simulates tossing of 2 coins.
You may access, run and edit this code at https://scratch.mit.edu/projects/402742403/editor
- H = 0 and T = 1
- SUM = H + T
- if SUM = 0, we must have HH, and a green cat is stamped
- if SUM = 2, we must have TT, and an orange cat is stamped
- if SUM = 1, we must have HT, and a blue cat is stamped
- sample output is shown below