After students had many chances to roll two number cubes and record the sums, the teacher gave the following instructions for this multi-part probability activity:

1. Think of a way to figure out all the combinations (sums) for two number cubes rolled together. Write a title and explain what you did. We will discuss the various strategies you came up with, and look at what works well.

2. Use a 6 x 6 grid to figure out the combinations for two number cube sums to check that you found all the ways. (The teacher provided an example of a 6 x 6 grid, without supplying the numbers in the inner cells.)
When you are finished, show how each sum can be shown as a fraction of the total number of combinations that are possible.

3. Play this game with a partner several times:

 • Draw a chart with spaces under the sums for two number cubes. • Draw 11 circles in any of the 11 spaces in any combination you want. • Write the reason why you put the circles where you did. Try to use fractions to explain your choices. • With a partner, take turns rolling the number cubes. Each time you roll a sum, put X’s in all the circles you have for that sum. You win when all your circles are crossed off.

This game is based on an activity from A Collection of Math Lessons, 3 - 6, Marilyn Burns, Math Solution Publications, 1987.

Circumstances of performance
This sample of student work was produced under the following conditions:

- alone - in a group
- in class as homework
- with teacher feedback - with peer feedback
timed opportunity for revision

Students had prior experience making their own charts and using conventional fractional notation, and had explored probability using number cubes and other materials. The activities were spread out over two days.

 This work sample illustrates a standard-setting performance for the following parts of the standards: d Arithmetic and Number Concepts: Describe and compare quantities by using simple fractions. e Statistics and Probability Concepts: Predict results and find out why some results are more likely, less likely, or equally likely. f Statistics and Probability Concepts: Find all possible combinations. g Mathematical Skills and Tools: Read, create, and represent data. b Mathematical Communication: Show mathematical ideas in a variety of ways. c Mathematical Communication: Explain solutions to problems clearly and logically.

What the work shows
d Arithmetic and Number Concepts: The student describes and compares quantities by using concrete and real world models of simple fractions; that is, finds simple parts of wholes….
All possible sums for paired number cubes are represented and compared as simple fractions.

e Statistics and Probability Concepts: The student predicts results, analyzes data, and finds out why some results are more likely, less likely, or equally likely.
The student organized the possible outcomes in order to make predictions about the likelihood of rolling particular two number cube sums.

 f Statistics and Probability Concepts: The student finds all possible combinations and arrangements within certain constraints involving a limited number of variables. The student created a way to find all possible combinations for the sums of two number cubes. g Mathematical Skills and Tools: The student reads, creates, and represents data on…charts, tables…. The student-created charts, especially the horizontal tables under “Combinations” and the fractions chart, are appropriate, clear, and complete. b Mathematical Communication: The student shows mathematical ideas in a variety of ways, including words, numbers, symbols,…charts,… tables…. The student used words to explain how the chart works. Numbers, symbols (“A” and “B”), and a table were used to communicate the combinations. c Mathematical Communication: The student explains solutions to problems clearly and logically, and supports solutions with evidence, in… written work. The written explanations are sufficiently clear and logical for the elementary level. The student provided evidence in support of the game strategy by referring back to the data contained in the tables. The work includes several errors in usage and grammar, e.g., “two dices,” “loosed” for “lost,” and “come” for “came.” This was a class assignment and was not intended to be further edited.