Beginning a lesson on probability after teaching students about fractions and proportions is a great way to promote students continual understanding of these concepts because probabilities are written as fractions and odds are written as ratios.
In order for students to undestand the concept of probability and how to account for the probability an event will occur, a student must have prior knowledge on the concept of a numerator and denominator and what each represents in a fraction. When solving for probabilities, it is crucial for students to understand the break down of a fraction and where each number within the fraction comes from. When it comes to probability, students must be able to decide what their numerator will be and what their denominator will be. If students do not understand what numbers to use for their numerator and what numbers to use for their denominator, students will not be able to solve for probability problems. Therfore, it is essential that students have a good grasp on the concept of fractions before beginning a segment on probability.
When we talk about the odds of an event, we represent this as a ratio, which was highlighted in the unit on proportional reasoning. Ratios can be found within fractions, which is why it makes sense to follow the fraction unit with the proportional reasoning unit and then the probability unit. All three of these units/concepts build off of one another, making it easier for students to understand concepts and how they connect. A ratio can be found within a fraction and then broken down into ratio form- (ex: 1/2 is a 1:2 ratio, meaning that one of the two things we are comparing is half as large as the other. This comes in handy when we begin to deal with odds because we write odds as ratios.
For example, if the probability of pulling a blue marble is 1/10, we would say the odds of pulling a blue marble are 1:9, because, for every ten pulls, we would expect to draw one blue, so for every one blue, we would expect 9 pulls where we did not get blue.
Ideas and concepts of probability build off of concepts taught in the fraction and proportional reasoning unit, making it a smart move to introduce and tie probability in at the end of teaching these units.
--> A strength of tying in these topics is that each topic builds off of the other. In order to understand probability, one must understand how to write the probability out as a fraction and understand what numbers they use for their numerator and what numbers they use for their denominator, if a student is confused on where numbers within a fraction come from and what they represent, the student will not be able to succeed in identifying probability. Therefore, it extrememly benefits students to learn about probability after learning about fractions, because the concept of fractions is still fresh in their minds and they are able to deepen their understanding on what fractions are representing and how probability can be represented through a fraction.
When discussing probability we also talk about the odds of an event happening. Odds are represented as ratios, therefore talking about proportional reasoning prior to probability and then tying the two concepts together is extremely beneficial to students.
--> Another strength of tying in the sequence of these topics is that students are able to see how concepts connect to one another. When students are able to see that a concept is used to solve for multiple things, it makes students feel that what they are learning is actually beneficial and something that will be used in many different areas of life/mathematical situations.
--> A major weakness that could be found in tying together these sequences is students struggling with the concepts and feeling lost throughout the entire unit. When one understands a concept and the concept continues to build, it is great! When students are confused from the basics of the unit however, it can be a VERY long unit that students feel will never end. If students struggle with fractions, the concept of proportional reasoning and probability may also present them with struggles. If the child's confusions are not adressed, the student may end up in a downward spiral they feel they can not get out of. As topics continue to build off of one another, a student may begin to overcomplicate and overthink concepts, becoming discouraged and feeling there is no hope for them and giving up on the unit as a whole.
"Can you Beat the Odds?"
The Can you Beat the Odds video showed a very energetic and interactive way to introduce kids to the idea and concepts of probability, equally likely probabilities, independent events, and odds.
The host of the video uses donuts, coins, soccer balls, and dice to interactively engage and visually show students how probability problems work.
To demonstrate the concept of an influencing factor, the host had a boy eat a donut without licking his lips and then had another girl come in and eat a donut, licking her lips. The host explained that when the girl came into the demonstration, she acted as an influence and the chances of him licking his lips were no longer independent.
The host then elaborated on the idea of independent events by talking about the flipping of a coin and asking probability problems relating to the flipping of a coin. When an individual flips a coin, the outcome of the flip before the next flip in no way affects the outcome of the next flip, making it an independent event. A coin flip has two equally likely outcomes- heads or tails. The probability an individual has of getting heads or tails on a single flip will always stay the same and be independent of other events due to a coin flip having equally likely outcomes.
The host also sets up an interactive exercise with soccer balls to resemble the probability one has of winning the lottery. After working through the simulation, the children solve for the probability of winning the lottery. The host then asks what the odds of winning the lottery were and explains how the fractional probability can be represented as a ratio when speaking in terms of odds.
The host of this video did an excellent job at finding different ways to interactively involve students in probability questions through the use of many manipulatives. The host was able to explain and demonstrate general probability problems, equally likely probability problems, independent events, and odds.
"Probability of Dependent & Independent Events"
This video gives us a great glimpse at an effective lesson plan for independent and dependent events. The teacher of this video helps students identify key words like replace and no replacement to determine whether or not an event is independent or dependent.
Once students have grasped an understanding for independent and dependent events, students are given manipulatives to use to create a word problem that resembles either an independent or dependent event.
Through this, students are able to interactively engage in the creation of an independent or dependent event and share with the class their word problems and how one would be able to know off of key words that the event was independent or dependent.
--> Both of these videos show great ways that one could utilize manipulatives to have students interactively engage with probability.