standard_description: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
problem_activity: Optional activity
The mathematical purpose of this activity is to compare probabilities calculated through experimentation to probabilities calculated by using the sample space. The activity is optional to offer students additional practice with creating sample spaces, calculating probability, and comparing hands-on experiments to theoretical probabilities.
Digital
Launch
Arrange students in groups of 2. Consider demonstrating the random number generator applet for students. Each box randomly generates a number 1 through 6, simulating rolling two standard number cubes. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with whole-class discussion.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support small group discussion. After students have completed a large portion of the task, provide students with sentence frames to support their conversation comparing their answers and determining if they are both correct. For example, "I agree because . . .”, "I disagree because . . .”, “Why do you think . . . ?” This will help students understand that probabilities calculated using sample space and probabilities calculated by experimentation are related, but not necessarily equivalent.
Design Principle(s): Support sense-making
Student Facing
Roll the double number cube simulator twenty times. Record your results in the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "first cube"}, {"_attributes": {"scope": "col"}, "_value": "second cube"}]}]}], "tbody": [{"tr": [{"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "first cube"}, {"_attributes": {"scope": "col"}, "_value": "second cube"}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}]}]}]}
List all the possible outcomes in each event. The first one is done for you.
The first cube is a 6.  6,1;  6,2;  6,3;  6,4;  6,5;  6,6
The cubes have a 4 and a 6.
The cubes are doubles. (Doubles means that the number on the two number cubes is the same.)
The cubes are doubles and the first cube is a 6.
The cubes are doubles or the first cube is a 6.
The first cube is not a 6.
The cubes are doubles and the first cube is not a 6.
The cubes are not doubles.
Use the information in the table to answer the questions.
What percentage of the rolls have a 6 on the first cube?
What percentage of the rolls have a 4 and a 6?
What percentage of the rolls are doubles?
What percentage of the rolls are doubles and have a 6 on the first cube?
What percentage of the rolls are doubles or have a 6 on the first cube?
What percentage of the rolls do not have a 6 on the first cube?
What percentage of the rolls are doubles and do not have a 6 on the first cube?
What percentage of the rolls are not doubles?
The sample space has 36 outcomes. Use this and the number of outcomes in each event to find the actual probability for each event in the previous problem. Compare your answers.
Why is the actual probability different from the percentage of rolls you made for each event?
Student Response
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Student Facing
Are you ready for more?
Roll the triple number cube simulator forty times. Record your results in the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"td": [{"p": [{"strong": [{"_value": "first cube"}]}]}, {"p": [{"strong": [{"_value": "second cube"}]}]}, {"p": [{"strong": [{"_value": "third cube"}]}]}]}, {"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "first cube"}, {"_attributes": {"scope": "col"}, "_value": "second cube"}, {"_attributes": {"scope": "col"}, "_value": "third cube"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "first cube"}, {"_attributes": {"scope": "col"}, "_value": "second cube"}, {"_attributes": {"scope": "col"}, "_value": "third cube"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "first cube"}, {"_attributes": {"scope": "col"}, "_value": "second cube"}, {"_attributes": {"scope": "col"}, "_value": "third cube"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}]}]}]}
List all the possible outcomes in each event. The first one is done for you.
The first cube is a 6.  6,6,1; 6,6,2; 6,6,3; 6,6,4;  6,6,5; 6,6,6; 6,5,1; 6,5,2; 6,5,3;  6,5,4; 6,5,5; 6,4,6; 6,6,1; 6,4,2; 6,4,3;  6,4,4; 6,4,5; 6,4,6; 6,3,1; 6,3,2; 6,3,3; 6,3,4;  6,3,5; 6,3,6; 6,2,1; 6,2,2; 6,2,3; 6,2,4; 6,2,5; 6,2,6; 6,1,1;  6,1,2; 6,1,3; 6,1,4; 6,1,5; 6,1,6.
The cubes have a 4, a 5, and a 6.
The cubes are triples. (Triples means that the number on the three number cubes is the same.)
Use the information in the table to answer the questions.
What percentage of the rolls have a 6 on the first cube?
What percentage of the rolls have a 4, a 5, and a 6?
What percentage of the rolls are triples?
The sample space has 216 outcomes. Use this and the number of outcomes in each event to find the actual probability for each event in the previous problem. Compare your answers.
If you rolled 500 times, do you think that the difference between the actual probability and the percentage of rolls you made for each event would increase or decrease? Explain your reasoning.
Describe a method for recording the 216 outcomes in the sample space. Do not actually record all 216 outcomes.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Print
Launch
Arrange students in groups of 2. Distribute 2 standard number cubes to each pair of students. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support small group discussion. After students have completed a large portion of the task, provide students with sentence frames to support their conversation comparing their answers and determining if they are both correct. For example, "I agree because . . .”, "I disagree because . . .”, “Why do you think . . . ?” This will help students understand that probabilities calculated using sample space and probabilities calculated by experimentation are related, but not necessarily equivalent.
Design Principle(s): Support sense-making
Student Facing
Roll two standard number cubes twenty times. Record your results in the table.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "first cube"}]}, {"strong": [{"_value": "second cube"}]}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "first cube"}]}, {"strong": [{"_value": "second cube"}]}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}, {"td": [{}, {}]}]}]}]}
List all the possible outcomes in each event. The first one is done for you.
The first cube is a 6.  6,1;  6,2;  6,3;  6,4;  6,5;  6,6
The cubes have a 4 and a 6.
The cubes are doubles. (Doubles means that the number on the two number cubes is the same.)
The cubes are doubles and the first cube is a 6.
The cubes are doubles or the first cube is a 6.
The first cube is not a 6.
The cubes are doubles and the first cube is not a 6.
The cubes are not doubles.
Use the information in the table to answer the questions.
What percentage of the rolls have a 6 on the first cube?
What percentage of the rolls have a 4 and a 6?
What percentage of the rolls are doubles?
What percentage of the rolls are doubles and have a 6 on the first cube?
What percentage of the rolls are doubles or have a 6 on the first cube?
What percentage of the rolls do not have a 6 on the first cube?
What percentage of the rolls are doubles and do not have a 6 on the first cube?
What percentage of the rolls are not doubles?
The sample space has 36 outcomes. Use this and the number of outcomes in each event to find the actual probability for each event in the previous problem. Compare your answers.
Why is the actual probability different from the percentage of rolls you made for each event?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Roll three standard number cubes forty times. Record your results in the table.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "first cube"}]}, {"strong": [{"_value": "second cube"}]}, {"strong": [{"_value": "third cube"}]}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "first cube"}]}, {"strong": [{"_value": "second cube"}]}, {"strong": [{"_value": "third cube"}]}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "first cube"}]}, {"strong": [{"_value": "second cube"}]}, {"strong": [{"_value": "third cube"}]}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "first cube"}]}, {"strong": [{"_value": "second cube"}]}, {"strong": [{"_value": "third cube"}]}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}]}]}]}
List all the possible outcomes in each event. The first one is done for you.
The first cube is a 6.  661, 662, 663, 664, 665, 666, 651, 652, 653, 654, 655, 656, 641, 642, 643, 644, 645, 646, 631, 632, 633, 634, 635, 636, 621, 622, 623, 624, 625, 626, 611, 612, 613, 614, 615, 616
The cubes have a 4, a 5, and a 6.
The cubes are triples. (Triples means that the number on the three number cubes is the same.)
Use the information in the table to answer the questions.
What percentage of the rolls have a 6 on the first cube?
What percentage of the rolls have a 4, a 5, and a 6?
What percentage of the rolls are triples?
The sample space has 216 outcomes. Use this and the number of outcomes in each event to find the actual probability for each event in the previous problem. Compare your answers.
If you rolled 500 times, do you think that the difference between the actual probability and the percentage of rolls you made for each event would increase or decrease? Explain your reasoning.
Describe a method for recording the 216 outcomes in the sample space. Do not actually record all 216 outcomes.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may not know how to represent probability as a percentage. Prompt students to multiply the value they obtained to represent probability (between 0 and 1 inclusive) by 100 to get the percentage. Emphasize that a percentage is a quantity described by a rate per 100.
Activity Synthesis
The goal of this discussion is to make sure students understand that probabilities calculated using sample space and probabilities calculated by experimentation are related, but not necessarily equivalent. A secondary goal of the discussion is to informally assess student understanding of how “and,” “or,” and “not” are used when finding probabilities.
Here are some questions for discussion.
“Why is the sample space 36 outcomes for rolling two number cubes?” (It is 36 outcomes because there are 6 choices for the first cube and 6 choice for the second cube, 6 times 6 is 36. You can see this in a tree diagram from a previous lesson.)
“According to the sample space there are 36 different outcomes for rolling two number cubes. You only rolled the cubes 20 times. Raise your hand if you rolled the same outcome more than once when you rolled 20 times. Why is this possible?” (It is possible to repeat the same roll more than once because rolling the dice is random. Just because you rolled double sixes on one turn does not reduce your chance of rolling double sixes on the next turn.)
“How are the events “rolling doubles” and “rolling doubles and rolling a 6 on the first cube” different? Explain your reasoning.“ (The difference between the two questions is that the first question included all of the double rolls, but the second question only counts the rolls when both cubes are six.)
“What is the difference between the probability of “rolling doubles” and the probability of “rolling doubles and rolling a 6 on the first cube”? Explain your reason” (It is 0.15. The probability of “rolling doubles” was 0.25 and the probability of “rolling doubles and rolling a 6 on the first cube is 0.10, 0.25 minus 0.10 equals 0.15.)
“For your 20 rolls, what percentage of them have the first die equal to 1 or 2?” (Sample response: 0.4 or 40%)
“Using the sample space, what is the probability of rolling a 1 or a 2 on the first die?” (
\(\frac{12}{36}\)
or 33.3%)


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