Samples and ProbabilityType of Unit: ConceptualPrior KnowledgeStudents should be able to:Understand the concept of a ratio.Write ratios as percents.Describe data using measures of center.Display and interpret data in dot plots, histograms, and box plots.Lesson FlowStudents begin to think about probability by considering the relative likelihood of familiar events on the continuum between impossible and certain. Students begin to formalize this understanding of probability. They are introduced to the concept of probability as a measure of likelihood, and how to calculate probability of equally likely events using a ratio. The terms (impossible, certain, etc.) are given numerical values. Next, students compare expected results to actual results by calculating the probability of an event and conducting an experiment. Students explore the probability of outcomes that are not equally likely. They collect data to estimate the experimental probabilities. They use ratio and proportion to predict results for a large number of trials. Students learn about compound events. They use tree diagrams, tables, and systematic lists as tools to find the sample space. They determine the theoretical probability of first independent, and then dependent events. In Lesson 10 students identify a question to investigate for a unit project and submit a proposal. They then complete a Self Check. In Lesson 11, students review the results of the Self Check, solve a related problem, and take a Quiz.Students are introduced to the concept of sampling as a method of determining characteristics of a population. They consider how a sample can be random or biased, and think about methods for randomly sampling a population to ensure that it is representative. In Lesson 13, students collect and analyze data for their unit project. Students begin to apply their knowledge of statistics learned in sixth grade. They determine the typical class score from a sample of the population, and reason about the representativeness of the sample. Then, students begin to develop intuition about appropriate sample size by conducting an experiment. They compare different sample sizes, and decide whether increasing the sample size improves the results. In Lesson 16 and Lesson 17, students compare two data sets using any tools they wish. Students will be reminded of Mean Average Deviation (MAD), which will be a useful tool in this situation. Students complete another Self Check, review the results of their Self Check, and solve additional problems. The unit ends with three days for students to work on Gallery problems, possibly using one of the days to complete their project or get help on their project if needed, two days for students to present their unit projects to the class, and one day for the End of Unit Assessment.

Students begin to formalize their understanding of probability. They are introduced to the concept of probability as a measure of likelihood and how to calculate probability as a ratio. The terms discussed (impossible, certain, etc.) in Lesson 1 are given numerical values.Key ConceptsStudents will think of probability as a ratio; it can be written as a fraction, decimal, or a percent ranging from 0 to 1.Students will think about ratio and proportion to predict results.Goals and Learning ObjectivesDefine probability as a measure of likelihood and the ratio of favorable outcomes to the total number of outcomes for an event.Predict results based on theoretical probability using ratio and proportion.

Learn Alberta offers a real-world scenario of skateboarding to introduce the concept of square roots. This multimedia site offers two videos, printable exercises and interactive exercises of finding the square root through visual prompts of squares.

Learn about ratios with this handy reference site. Detailed explanations and examples are used to explain ratios. Also highlights common mistakes that can be made when learning about ratios.

The Math Page tutorial explains and defines ratio and proportion and includes this practice section at the end of the lesson.

Learn about ratios by watching an Atlantean Dodgeball game! Ratio errors confuse one of the coaches as two teams face off in an epic dodgeball tournament. See how mathematical techniques such as tables, graphs, measurements, and equations help to find the missing part of a proportion. A teaching guide with printable resources is included. [5:27]

Learn how understanding distance--both lunge distance and the distance between you and your opponent--is an important strategy in the art of fencing. This clip is from the Center for Asian American Media. In the accompanying classroom activity, students measure their own lunge distance, write and solve equations, and use ratio language to compare their numbers to that of their classmates. This resource is part of the Math at the Core: Middle School Collection. [2:21]

Learn about the mathematics of fencing, including the ratio of touches to time remaining that an athlete needs to mentally calculate in order to stay in the match, in this video from Center for Asian American Media. In the accompanying classroom activity, students reflect on the mathematics involved and analyze the ratios and calculations involved in this numerical relationship. This resource is part of the Math at the Core: Middle School Collection. [3:26]

The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories? This activity provides a real-life context for working with proportionality, speed, rates, and units of measurement and is aimed at Key Stage 3 students (age 11-14).

In this video, musician Jake Shimabukuro uses the pitch of musical notes and vibrations in hertz to illustrate the concept of ratios. This clip is from Center for Asian American Media. In the accompanying classroom activity, students watch a portion of the video in which Shimabukuro explains the 3/4 time signature. After a brief introduction to musical pitch, students watch the rest of the video. They then use a chart listing the frequency of musical notes to identify ratios among notes that are one or more octaves apart. To get the most from the activity, students should have some familiarity with physics of sound.

Musician and composer Jake Shimabukuro explains how ratios of notes have an inverse relationship to the string length of a ukulele, allowing musicians to produce various notes and pitches, in this video from the Center for Asian American Media. In the accompanying classroom activity, students engage in discussions before and after the video in order to better understand the mathematics involved in describing this inverse relationship.

Composer and ukulele artist Jake Shimabukuro describes the concept of transposing notes in music and how ratios are applied in this video from Center for Asian American Media. Shimabukuro explains why it is necessary to transpose music written for a piano, or other instruments with a large range, to the corresponding notes on an instrument with a small range, such as a ukulele. In the accompanying classroom activity, students use ratios to transpose a short piece written for the ukulele back to a lower key for the piano.

Learn how composer and ukulele artist Jake Shimabukuro uses ratios and fractions as he creates and plays music in this video from Center for Asian American Media.

Composer and ukulele artist Jake Shimabukuro describes the concept of transposing notes in music and how ratios are applied in this video from Center for Asian American Media. Shimabukuro explains how to transpose music written in the key of C to the key of F using ratios. The accompanying classroom activity helps students transpose a popular children's song from the key of C to the key of F.

In this lesson on ratios, students explore how ratios can be used to demonstrate rates by exploring proportional relationships.

In this lesson, students will use ratios, rates, and unit rates to solve problems.

See if the King, Queen, or one of their loyal subjects can figure out how to solve the problem of their crummy bridge. This animated video from Math Snacks focuses on using what you know about converting measurements and reading a line graph to help the engineers solve their construction problems, taking math out of the classroom and into the kingdom.

An interactive simulation that teaches about ratios, proportional reasoning, and unit rate by making predictions and testing proportions while designing a necklace, throwing balloons filled with paint, playing billiards, or shopping for apples. This simulation can either be downloaded or played online and includes handouts, lesson plans, and additional materials.

An interactive simulation that teaches about ratios, proportional reasoning, and using double number lines to look for patterns through interactions while shopping for fruits, vegetables, and candy and by comparing cars with different rates on a race track. This simulation can either be downloaded or played online and includes handouts, lesson plans, and additional materials.

At the end of this lesson about ratio/proportion, students will be able to recognize and apply ratios and proportions to solve real-life problems and solve for any unknown component.