SYSTEMS, STATS AND ASSESSMENTS At Home in College

COMPASS TOPICS Pre-AlgebraAlgebra Basic operations with integersSubstituting values into algebraic expressions Basic operations with fractionsSetting up equations for given situations Basic operations with decimalsBasic operations with polynomials Exponents, Square Roots, and Scientific NotationFactoring of polynomials Ratios and Proportions Linear equations in one variable(using integers, fractions, and decimals as coefficients) PercentagesRational Expressions Averages (means, medians, and modes)Exponents and Radicals Linear Equations in Two Variables

COMPASS QUESTION LEVELS BASIC  Solved by performing a sequence of basic operations. APPLICATION  Applying sequences of basic operations to novel settings or in complex ways. ANALYSIS  Demonstrate conceptual understanding of the principles and relationships relevant to particular mathematical operations

Two Pointers and Three Pointers

We can complete a table to make sure that our graph is correct:

Pia’s Graph

What is the equation that represents the graph of the translated line?

Sandy and Charlie

Going Through Systems Solve the tasks in as many different ways possible. Anticipate student responses. Anticipate questions that you would ask students in response to certain difficulties or understandings.

Card Set A 1. Share the cards between you and spend a few minutes, individually, completing the cards so that each has an equation, a completed table of values and a graph. 2. Record on paper any calculations you do when completing the cards. Remember that you will need to explain your method to your partner. 3. Once you have had a go at filling in the cards on your own: Explain your work to your partner. Ask your partner to check each card. Make sure you both understand and agree on the answers. 1. When completing the graphs: Take care to plot points carefully. Make sure that the graph fills the grid in the same way as it does on Cards C1 and C3. Make sure you both understand and agree on the answers for every card.

Card Set B 1. You are going to link your completed cards from Card Set A with an arrow card. 2. Choose two of your completed cards and decide whether they have no common solutions, one common solution or infinitely many common solutions. Select the appropriate arrow and stick it on your poster between the two cards. 3. If the cards have one common solution, complete the arrow with the values of x and y where this solution occurs. 4. Now compare a third card and choose arrows that link it to the first two. Continue to add more cards in this way, making as many links between the cards as possible.

Evaluating Sample Responses to Discuss What do you like about the work? How has each student organized the work? What mistakes have been made? What isn’t clear? What questions do you want to ask this student? In what ways might the work be improved? P-21

Alex’s solution P-22

Danny’s solution P-23

Jeremiah’s solution P-24

Tanya's solution P-25

Planning a Lesson As a table, please discuss the following reflecting on the prior lessons use in your classroom:  How will you organize the classroom and the resources?  How will you introduce the questioning session?  Which ground rules will you establish?  What will be your first question?  How will you give time for students to think before responding?  Will you need to intervene at some point to refocus or discuss different strategies they are using?  What questions will you use in plenary discussions during or towards the end of the lesson?

Descriptive Statistics-the standards S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points(outliers).

Concepts and Skills Concepts  Dot Plot  Histogram  Box Plot  Median  Mean  Interquartile Range  Standard Deviation  Outlier Skills  Represent Data  Approximate Center  Approximate Spread  Interpret differences in shape, center, and spread  Account for outliers

GAISE Statements Steps in the statistical problem-solving process Formulate questions Collect data Analyze data Interpret results Guiding principles for teaching statistics Conceptual understanding takes precedence over procedural skill. Active learning is key to the development of conceptual understanding. Real-world data must be used wherever possible in statistics education. Appropriate technology is essential in order to emphasize concepts over calculations. All four steps of the investigative problem-solving process should be encountered at each grade level. The illustrative investigations should show situations in which the statistics is essential to answering a question, not just an add-on. Such investigations should be tied to the mathematics they illustrate, motivate, and emphasize. Source: GAISE Report

Getting Started Assessment to gather prior knowledge of mean. Activity to engage and interest students in ideas around measures of center and error.

Earlier Assessment Conversations… Make the objective of the lesson explicit Assess groups as well as individual students Watch and listen before intervening Use divergent assessment methods (“Show me what you know about…”) Give constructive, useful feedback

Guessing Ages PhotoGuessActualPercent Error

Collecting Data Do girls text more than boys?  Think pair share on how to figure this out?  Collect data from class.  Organize.  Analyze relying on measures of center.  Conclusions.

Open Questions The mean of a set of numbers is 8. What might the numbers be? Create a set of data in which the mean is greater than the median.

Lesson One Students use informal language to describe the shape, center, and variability of a distribution based on a dot plot, histogram, or box plot. Students recognize that a first step in interpreting data is making sense of the context. Students make meaningful conjectures to connect data distributions to their contexts and the questions that could be answered by studying the distributions.

What’s the Story?

Exit Ticket Sam said that a typical flight delay for the sixty BigAir flights was approximately one hour. Do you agree? Why or why not?

Sam said that 50% of the twenty-two juniors at River City High School who participated in the walkathon walked at least ten miles. Do you agree? Why or why not?

Lesson Two Students construct a dot plot from a data set. Students calculate the mean of a data set and the median of a data set. Students observe and describe that measures of center (mean and median) are nearly the same for distributions that are nearly symmetrical. Students observe and explain why the mean and median are different for distributions that are skewed. Students select the mean as an appropriate description of center for a symmetrical distribution and the median as a better description of center for a distribution that is skewed.

Data Set 1: Pet owners Students from River City High School were randomly selected and asked, “How many pets do you currently own?” The results are recorded below: Data Set 2: Length of the east hallway at River City High School Twenty students were selected to measure the length of the east hallway. Two marks were made on the hallway’s floor, one at the front of the hallway and one at the end of the hallway. Each student was given a meter stick. Students were asked to use their meter sticks to determine the length between the marks to the nearest tenth of a meter. The results are recorded below: Data Set 3: Age of cars Twenty-five car owners were asked the age of their cars in years. The results are recorded below:

How is the choice made between mean and median to describe the typical value related to the shape of the data distribution? Sketch a dot plot in which the median is greater than the mean. Could you think of a context that might result in data where you think that would happen?

Lesson 3 Students estimate the mean and median of a distribution represented by a dot plot or a histogram. Students indicate that the mean is a reasonable description of a typical value for a distribution that is symmetrical but that the median is a better description of a typical value for a distribution that is skewed. Students interpret the mean as a balance point of a distribution. Students indicate that for a distribution in which neither the mean nor the median is a good description of a typical value, the mean still provides a description of the center of a distribution in terms of the balance point.

Computer Games: Ratings P-45 Imagine rating a popular computer game. You can give the game a score of between 1 and 6.

Bar Chart from a Frequency Table P-46 Mean score Median score Mode score Range of scores

Matching Cards 1. Each time you match a pair of cards, explain your thinking clearly and carefully. 2. Partners should either agree with the explanation or challenge it if it is unclear or incomplete. 3. Once agreed stick the cards onto the poster and write a justification next to the cards. 4. Some of the statistics tables have gaps in them and one of the bar charts is blank. You will need to complete these cards. P-47

Parallel Tasks A set of nine pieces of data, all of which are different, has a mean of 30 and a median of 10. What could the data values be? A set of nine pieces of data, all of which are different, has a median of 10. What could the data values be? P- 48

Thought Experiment What do you think the mean height of all 15 year old boys in the United State is? Explain. What do you think the standard deviation of a set of data representing income in the United States? Explain.

Standard Deviation Find the mean, median, and mode of the three sets of data below:  5, 5, 5, 5, 5  3, 4, 5, 6, 7  1, 3, 5, 7, 9