Unit Title: Probability Models and Geometric Reasoning
Grade Level: 7
Timeframe: Marking Period 4
Unit Focus & Essential Questions
Unit Focus:
(1) Understanding that the entire 7th grade mathematics course revolves around the study of relationships between things we count and measure quantities.
(2) Understand the quantities measured in the study of chance in the experiments we find in probability models.
(3) Develop a probability model for an experiment and use it to determine the probabilities of events.
(4) Represent probability models.
(5) Understand the types of questions we ask and answer about probability model.
(6) Continue to build fluency with the values that appear in probability models.
Rubric for Learning (Probability Models)
- Experiment
- Possible Outcome, Sample space
- Successes
- Probability of Success
- Lines withPoints of Intersection and the 360º they form
- Lines and the 180º they form
- Relationships between angles formed by intersecting lines
- Polygons formed by intersecting lines
Essential Questions:
(1)Can we become confident in our knowledge of quantities, units, and values used in experiments we find in probability models?
(2)Can we become effective and efficient at creating probability models?
(3)Can we become effective and efficient at representing probability models?
(4)Can we become effective and efficient at asking and answering questions about probability models?
(5)Can we become effective and efficient at talking about probability models and questions asked about probability models?
(6)Can we become effective and efficient at reading and understanding others’ probability models and questions asked about probability their models?
New Jersey Student Learning Standards
7. SP.5Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
7. SP.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7. SP.7Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
7. SP. 8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
7.G.2Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
Instructional Plan / Reflection
Standards & Objectives SWBAT
7. SP.5Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
7. SP.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7. SP.7Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
7. SP. 8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
SWBAT…
CREATE & REPRESENT
- Choose an Experiment
- List the possible outcomes
- Define successes
- Perform the experiment
- Record the outcomes
- Discuss all aspects of the Learning Focus
- Compare all aspects of the Learning Focus
- Interpretprobability models provided in textbook problems
- Communicate those models through other representations
FLUENCY
Choose numbers that students struggle with when they are working with the probability models.
CREATE & REPRESENT
Allow students to spend most of their time working in groups as they create the relationships in this fourth marking period in order to observe them working and learn their strengths.
Have the group work on the experiments together, each one choosing a different success to record and talk about.
Make sure to cover coin flipping, dice rolling, spinning spinners, ball picking from a hat, surveys, and other experiments that show up in the theoretical problems as well as combinationsof them.
Each time a group talks about their experiment, make sure to let them know that since they CONDUCTED the experiment, this is an Experimental probability.
After a few experiments, teach the vocabulary of Sample Space for the Possible Outcomes, Event for the Successes, …
FLUENTLY TALK ABOUT
Make sure students can differentiate between the different bullets of the learning rubric. Don’t just allow them to talk through the whole process without denoting which part of each experiment is described by each bullet.
FLUENTLY READ
When given a probability scenario, students will create a model, and ask and answer questions about the model /
CREATE & REPRESENT
Use Rubric for Learning to check for each of the bulleted items as students work through their probability experiment
Move towards students using the rubric on each other’s and their own experiments
FLUENTLY TALK ABOUT
Use Rubric for Learning to check for each of the bulleted items as students talk about their probability experiments.
Move towards students using the rubric on their own and each other’s talking.
FLUENTLY READ
Use Rubric for Learning to check for each of the bulleted items as student’s read/work the problems from typical textbooks and assessments.
Move towards students using the rubric on their own and each other’s reading.
DIFFERENTIATION
Providing feedback, according to a rubric of the bullets listed to the left, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.
As you see each student is able to do the work, celebrate the learning of that individual student, eye-to-eye establishing that they can learn, in this class, from you.
Grouping students who struggle talking about each of the mathematical foci listed or fluency above as well as those not struggling with anything.
Comparing models using the same and different representations of each. / FLUENCY
Count around the room and make combinations with numbers that appear in probability problems.
CREATE & REPRESENT an Experimental probability
- Choose an Experiment
- List the possible outcomes
- Define a success for each student in the group
- Conduct the experiment 100 times
- Record the outcomes
- Each student count their successes from the outcomes
- Each student record the probability of their success
Students practice talking about the math of their own and their group members’ successes.
Students talk about the math in other group’s experiments.
Have students record their quantities publicly on the Quantities Word Wall.
FLUENTLY READ
Have students read scenarios from sample problems and create representations of the probability experiments.
SUMMARY
Sum up each relationship discussing how the focus of the unit showed up in that particular problem/activity. Reworking the problem usually includes particular outcomes and successes, whereas, summing up usually includes the words like “possible outcomes” and “successes” comparing and contrasting it to other models that are already recorded.
Reflection:
FLUENCY
Choose numbers that students struggle with when they are working with the probability models.
CREATE & REPRESENT
Allow students to spend most of their time working in groups as they create the relationships in this fourth marking period in order to observe them working and learn their strengths.
Have the group work on the experiments together, each one choosing a different success to record and talk about.
Each time a group talks about their experiment, make sure to let them know that since they DID NOT CONDUCT the experiment, this is a theoretical probability.
FLUENTLY TALK ABOUT
Make sure students can differentiate between the different bullets of the learning rubric. Don’t just allow them to talk through the whole process without denoting which part of each experiment is described by each bullet.
FLUENTLY READ
When given a probability scenario, students will create a model, and ask and answer questions about the model / .
CREATE & REPRESENT
Use Rubric for Learning to check for each of the bulleted items as students work through their probability experiment
Move towards students using the rubric on each other’s and their own experiments
FLUENTLY TALK ABOUT
Use Rubric for Learning to check for each of the bulleted items as students talk about their probability experiments.
Move towards students using the rubric on their own and each other’s talking.
FLUENTLY READ
Use Rubric for Learning to check for each of the bulleted items as student’s read/work the problems from typical textbooks and assessments.
Move towards students using the rubric on their own and each other’s reading.
DIFFERENTIATION
Providing feedback, according to a rubric of the bullets listed to the left, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.
As you see each student is able to do the work, celebrate the learning of that individual student, eye-to-eye establishing that they can learn, in this class, from you.
Grouping students who struggle talking about each of the mathematical foci listed or fluency above as well as those not struggling with anything.
Comparing models using the same and different representations of each. / FLUENCY
Count around the room and make combinations with numbers that appear in probability problems.
CREATE & REPRESENT a Theoretical probability
- Choose an Experiment
- List the possible outcomes
- Define a success for each student in the group
- Each student count their successes from the outcomes
- Each student record the probability of their success
- Compare the theoretical and experimental probabilities of any experiments the students already conducted themselves.
Students practice talking about the math of their own and their group members’ successes.
Students talk about the math in other group’s experiments.
Have students record their quantities publicly on the Quantities Word Wall.
FLUENTLY READ
Have students read scenarios from sample problems and create representations of the probability experiments.
SUMMARY
Sum up each relationship discussing how the focus of the unit showed up in that particular problem/activity. Reworking the problem usually includes particular outcomes and successes, whereas, summing up usually includes the words like “possible outcomes” and “successes” comparing and contrasting it to other models that are already recorded.
Reflection:
Standards & Objectives SWBAT
7.G.2Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
SWBAT…
CREATE & REPRESENT
- Draw lines, parallel lines and intersecting lines at various angles.
- Measure all angles that are formed by intersecting lines.
- Discuss combinations that make up the 360º of angles formed at every intersection.
- Discuss combinations that make up the 180º of angles formed by every line.
- Discuss polygons that are created by lines.
- Discuss all aspects of the Learning Focus
- Compare all aspects of the Learning Focus
- Interpret geometric ideas provided in textbook problems
- Communicate those ideas through other representations
FLUENCY
Choose geometric shapes that students struggle with when they are working with the geometry models.
CREATE & REPRESENT
Allow students to spend most of their time working in groups as they create the relationships in this fourth marking period in order to observe them working and learn their strengths.
Have the group work on the geometry models together, each one choosing a different geometric shapes to record and talk about.
Make sure to coversupplementary, complementary, vertical, and adjacent angles. Construct triangles and other geometric shapes that show up in the geometry problems.
After a few construction of geometric shapes, teach the vocabulary of supplementary, complementary, vertical, and adjacent angles.
FLUENTLY TALK ABOUT
Make sure students can differentiate between the different bullets of the learning rubric. Don’t just allow them to talk through the whole process without denoting which part of each experiment is described by each bullet.
FLUENTLY READ
When given a geometric model scenario, students will create a model, and will ask and answer questions about the model /
CREATE & REPRESENT
Use Rubric for Learning to check for each of the bulleted items as students work through their probability experiment
Move towards students using the rubric on each other’s and their own experiments
FLUENTLY TALK ABOUT
Use Rubric for Learning to check for each of the bulleted items as students talk about their probability experiments.
Move towards students using the rubric on their own and each other’s talking.
FLUENTLY READ
Use Rubric for Learning to check for each of the bulleted items as student’s read/work the problems from typical textbooks and assessments.
Move towards students using the rubric on their own and each other’s reading.
DIFFERENTIATION