One of Carlo’s favorite things to do is to make different Piles of his Blocks based on color: one pile of green ones, one for blues, one for reds. Carlo is experimenting with ordering objects based on attributes like color, shape and size. Children like to organize items, even at a very young age. It is also fun for them to compare different groups, or compare what they have to what others have. Carlo might play his block game with a friend, noticing that he has more green Blocks than Jimmy, but Jimmy has more blue ones. Without even knowing it, and in a very basic sense, Carlo is making deductions based on a set of data. Sorting things like toys and food is a great way for Carlo to organize data. Asking him questions like, “which color do you have more of?” or “do you have the same amount of each object?” are great ways to help Carlo connect number sense to data gathering.
As Carlo’s thinking and processing develops, he should be able to sort and categorize with more complexity. He can sort by more than one attribute and discuss what the Groups have in common. He can also express not only which one is larger, but how much larger it is. This is a good time to introduce simple bar and line graphs to represent the quantities and comparative attributes of various objects. It is important that Carlo understands this grouping context from various angles. So he not only can chose an attribute and group objects by it, but he can also look at Groups of objects and determine what characteristics they have in common and what characteristics are dissimilar. He will start to be able to organize things and discuss attributes with more and more detail. He may enjoy running experiments and recording results.
At the end of 2nd grade, Carlo should be able to solve this odds and order problem:
Ask classmates which is their favorite ice cream flavor: chocolate, vanilla, strawberry, or mint. Collect the results, then make a bar or picture graph to represent the totals. Discuss the results: which flavor was the most popular? What about the least popular? How much more popular was one than another?
Carlo should be practicing how to read and interpret data, expressing conclusions based on observations. Line graphs are a great tool to understand measurements, and line plots to understand fractions. It is important that Carlo can look at a graph and talk about what it means, as well as taking a set of data and creating a line or bar graph from it. He should be collecting more and more complex sets of data and processing them for meaning based on his knowledge of their attributes. This is also when Carlo will start to understand probability. A simple dice game involving rolling two dice and noting the total will illustrate probability. Let’s say Carlo is looking for 10’s, and Jamaal is looking for 7’s. After many rolls, Carlo might observe that the game is not “fair”, and that Jamaal is winning. This is showing that there is a greater probability that the sum of two dice rolled is 7 than that it is 10. Carlo loves looking through the Guinness Book of World Records, which is a great opportunity to talk about the probability of world record events happening.
At the end of 5th grade, Carlo should be able to solve this odds and order problem:
1.Watch your favorite television show from beginning to end. Time each of the commercials and determine the percentage of commercial time for the entire show. Now determine the percentage of time the actual show is.
2. List the final score of your favorite basketball team for the last 5 games. What was their mean score? What was the median? What was the mode?
Odds and Order and Probability are very important foundational concepts at this age. Carlo should be able to create and interpret more complicated graphs, including graphs based on statistics and probability. With his firm understanding of how to read data, there will be a natural progression to using and manipulating the Cartesian coordinate system (graphs with an x and y axis) which involves plotting data points to create number lines. Carlo should be able to both plot and interpret each data point, and use basic graphing formulas to calculate attributes like the slope of a number line. The more concrete these skills can be, the better Carlo’s understanding of more complex graphs and 3D models. It will help Carlo stay focused to relate these graphs to real world scenarios, using examples that involve politics, business advertising, or sales to reinforce concrete skills and deductive reasoning. Having him create the data through experiments will also help him apply and retain the information.
Playing probability-based games like Yahtzee is a great way for Carlo to apply the likelihood of a specific roll. For example, the likelihood of a player rolling a Yahtzee on the first roll is 1/1296. Discussing this with other game play like carnival games or the lottery will increase Carlo’s interest. Having students create their own fair and unfair games is an interesting addition to the practice.
At the end of 8th grade, Carlo should be able to solve these odds and order problem:
1. 11 people are racing. What are the odds of being 1st, 2nd or 3rd?
2. A bag contains 3 red, 4 green and 3 yellow marbles. Three marbles are randomly drawn from the bag. What is the probability that they are all
(i) the same color?
(ii) different colors (one of each color)?