After students had many chances to roll two number cubes
and record the sums, the teacher gave the following instructions
for this multipart 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.
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 standardsetting
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. 

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
studentcreated 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.


