Dice: Probability, Problem Solving, and Critical Thinking

Dice: Probability, Problem Solving, and Critical Thinking
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PDF

(4 MB|42 pages)
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Standards
  • Product Description
  • StandardsNEW

This problem-set is all about experimenting with probability through both standard and non-standard dice sets. What happens to the probability of each sum, 2- 12, when we change the numbers on the dice? What happens to the probabilities when we use dice that have 4 sides, rather than 6? How can we make our own dice with our own numbers, but keep the probabilities the same?

To complete these problems, your kids will do hands-on experiments with each set of dice, think carefully about the resulting probabilities, and make comparisons across the sets. They’ll also be challenged to create dice sets of their own, where the numbers are different, but the probabilities of each sum are the same – and then explain not just why their set works, but why it’s the only way to do it!

Challenge your students to use probability, experimentation, logical thinking, and problem-solving skills to not just understand how the probabilities work, but how they connect across sets of dice!

This download includes a complete problem-solving journey to get your students thinking, including 3 unique, engaging, problem-solving tasks; the Warm-Up, the Problem, and its Exstemsion.

Created by teachers, with teachers and parents in mind, each task is built to challenge students to use their prior knowledge, and think creatively as they strive to solve the context-driven problems.

This download includes the following additional teaching tools:

•Thinking skills (what kind of thinking are kids building here?)

•Problem/Solution

•Exstemsion/Solution

•Supporting Questions (ideas for the questions you might as a student when they are stuck, place where they are most likely to get stuck!)

•Big Ideas (what are the math ideas built through this problem?)


Log in to see state-specific standards (only available in the US).
Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (𝑦 – 2)/(𝑥 – 1) = 3. Noticing the regularity in the way terms cancel when expanding (𝑥 – 1)(𝑥 + 1), (𝑥 – 1)(𝑥² + 𝑥 + 1), and (𝑥 – 1)(𝑥³ + 𝑥² + 𝑥 + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression 𝑥² + 9𝑥 + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(𝑥 – 𝑦)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers 𝑥 and 𝑦.
Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents-and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
Total Pages
42 pages
Answer Key
Included
Teaching Duration
N/A
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