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# 5. BINARY CHOICE

“I believe that we do not know anything for certain, but everything probably.”

― Christiaan Huygens (mathematician & physicist)

BINARY CHOICE MENU | MATH + CODING MENU |

5. Binary choice — In the classroom 5A. In grades 1-3 classrooms 5B. Tossing a coin 5 times 5C. Puzzle 1 – tossing 1 coin with Python 5D. Puzzle 2 – tossing 2 coins with Python 5E. Binary choice 5F. Mathematical surprise 5G. Sums of paths 5H. Calculating theoretical probability 5I. Statistician Interview 5J. Puzzle 3 – modelling outcomes with Scratch | Home page 1. Squares and spirals — introducing variables [gr. 3-9] 2. Making 10 — random coordinates line up [gr. 1-10] 3. Inequalities — in 1D, 2D & 3D [gr. 1-10] 4. Infinity — it’s big but it fits in your hand [gr. 3-12]5. Binary choice — probability, Pascal’s triangle & algebra [gr. 1-12] |

#### IN THE CLASSROOM

###### Ontario Curriculum

**Gr. 1-8**- D2 – describe the likelihood that events will happen, and use that information to make predictions
- D2.1 – describe the likelihood of events happening and use that likelihood to make predictions and informed decisions
- C1.3 – determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in patterns
- C3.1

**Gr.5-8**- D2.2 – determine and compare theoretical and experimental probability

**Gr. 9**- B1.1 – research an algebraic concept to tell a story
- B2.2
- C2 – apply coding skills to represent mathematical concepts and relationships dynamically, and to solve problems, in algebra and across the other strands

**Gr. 11-12**- the Binomial Theorem

###### Implementation

**Gr. 1-3**- Students conduct experiments to determine the likelihood of outcomes when tossing a coin once and twice.
- Using a tree diagram they explain that getting a H and T combination is twice as likely (there are 2 paths to destination B).

**Gr. 4-8**- Students also investigate connections to Yang Hui’s (aka Pascal’s) triangle.
- They also use code to model coin flips.

**Gr. 9-1**2- Students also investigate connections to expansions of binomial powers, & the Binomial Theorem.

#### 5a. IN GRADES 1-3 CLASSROOMS

###### Tossing 1 coin

**Q1. **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?”

###### Tossing 1 coin twice

**Q2.** 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?”

#### 5B. TOSSING A COIN 5 times

**Q5. **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?”

#### 5C. PUZZLE 1 – Tossing 1 coin with PYTHON

**Q3. **The code below models tossing a coin 100 times and counting the number of heads and tails. Enter and run the code at https://cscircles.cemc.uwaterloo.ca/console/

- How does the code work?

Q4. Alter the code to model:

- 20 trials and 10 trials
- calculate the percents of H and T

- 1000 trials and 10,000 trials
- calculate the percents of H and T

- what is the effect of doing more or fewer trials?

#### 5D. PUZZLE 2 – Tossing 2 coinS with PYTHON

**Q4. **The code below models tossing 2 coins 100 times and counting the number of times we get HH, HT and TT.

- Notice that the code is missing some components.
- How is the code different from the code for tossing 1 coin?
- How is it similar?
- What are the missing components?
- Enter and complete the code at https://cscircles.cemc.uwaterloo.ca/console/

- Run and test your code.

#### 5E. 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.

**Q6. **Do you want to know more? Yes or No?

#### 5F. MATHEMATICAL SURPRISE

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 mathematical **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 x
^{2}+ 2xy + y^{2} - 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

#### 5G. **Sums of paths**

**Q8.** The diagram on the right shows the number of paths leading to each 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 = ?

#### 5H. **Calculating theoretical probability**

**Q9. **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?

#### 5I. Statistician Interview

Interview with Dr. Bethany White, University of Toronto.

###### Tossing a coin twice

Bethany White discusses coin tossing (and computational thinking) activities explored by Grade 1 students.

**Q10.** 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?

###### Tossing a coin 5 times

**Q11. **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?

###### All possible outcomes

**Q12.** Let’s look at all the possible outcomes.

- What pattern do we notice?

###### Probability & algebra

**Q13.** 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.

###### 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.

#### 5J. PUZZLE 3 -Modelling OUTCOMES with SCRATCH

**Q14. **The Scratch code below simulates tossing of 2 coins. You may access, run and edit this code at https://scratch.mit.edu/projects/402742403/editor

- What does the code do?
- How does it do it?
- Alter the code to also show the percents of HH, HT and TT.

BINARY CHOICE MENU | MATH + CODING MENU |

5. Binary choice — In the classroom 5A. In grades 1-3 classrooms 5B. Tossing a coin 5 times 5C. Puzzle 1 – tossing 1 coin with Python 5D. Puzzle 2 – tossing 2 coins with Python 5E. Binary choice 5F. Mathematical surprise 5G. Sums of paths 5H. Calculating theoretical probability 5I. Statistician Interview 5J. Puzzle 3 – modelling outcomes with Scratch | Home page 1. Squares and spirals — introducing variables [gr. 3-9] 2. Making 10 — random coordinates line up [gr. 1-10] 3. Inequalities — in 1D, 2D & 3D [gr. 1-10] 4. Infinity — it’s big but it fits in your hand [gr. 3-12]5. Binary choice — probability, Pascal’s triangle & algebra [gr. 1-12] |

by George Gadanidis, PhD