welcome to common core high school mathematics leadership

5.1

WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIPSUMMER INSTITUTE 2014

SESSION 5 • 20 JUNE 2014SEEING PATTERNS AND TRENDS IN BIVARIATE DATA

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TODAY’S AGENDA Homework review and discussion Grade 8, Lesson 7: Patterns in Scatter Plots Grade 8, Lesson 8: Informally Fitting a Line

Reflecting on CCSSM standards aligned to lessons 7 & 8 Break Grade 8, Lesson 10: Linear Models Grade 8, Lesson 11: Using Linear Models in a Data Context

Reflecting on CCSSM standards aligned to lessons 10 & 11 Group presentation planning time Homework and closing remarks

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ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

Table discussion

Discuss your write ups for the Day 4 homework tasks: Compare your strategies with others at your table Reflect on how you might revise your own solution and/or

presentation

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Day 4 homework: Complete the problem set problems in grade 6, Lesson 20. Extending the mathematics:

Write a 3 to 5 sentence summary of the perch data that was presented in Lesson 20. Pretend you are a report for the Milwaukee Journal. Develop a headline that would go along with your short article.

Reflecting on teaching:Conducting a genuine statistical study with students in the middle to high school ages poses some challenges. Explain how you would develop a statistical study with your students that would involve collecting numerical data from other students. Be attentive in your plan to how you would supervise students’ process in collecting the data.

ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

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LEARNING INTENTIONS AND SUCCESS CRITERIA

We are learning to… Recognize and describe patterns and trends in dataUse those patterns and trends to make predictionsDistinguish between linear and nonlinear trends in data

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LEARNING INTENTIONS AND SUCCESS CRITERIAWe will be successful when we can:Recognize a pattern or trend in a scatter plot of a data set,

and describe that pattern or trend using correct statistical and/or mathematical language

Informally fit a line to data that shows a linear trendUse a line we have fit to data to make predictions about

the context from which the data was derived

5.7

ACTIVITY 2 LESSON 7: PATTERNS IN SCATTER PLOTSLESSON 8: INFORMALLY FITTING A LINESEEING PATTERNS IN BIVARIATE NUMERICAL DATA

ENGAGENY/COMMON CORE GRADE 8, LESSONS 7 & 8

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ACTIVITY 2 LESSON 7: PATTERNS IN SCATTER PLOTS

Prior knowledge

In earlier lessons of this module, students will have encountered Linear relationships (Lesson 1) Interpreting rate of change and initial value (Lesson 2) Representations of a line (Lesson 3) Increasing and decreasing functions (Lessons 4 & 5) Scatter plots (Lesson 6)

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What does the word “relationship” mean to you? What does it mean to say there is a relationship between two

numerical variables?

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Three questions to ask when you look at a scatter plot: Does it look like there is a relationship between the two variables

used to make the scatter plot? If there is a relationship, does it appear to be linear? If the relationship appears to be linear, is the relationship a positive

linear relationship or a negative linear relationship?

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In pairs, discuss Exercises 1, 4, 5, 6, 8, 10.

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ACTIVITY 2 LESSON 8: INFORMALLY FITTING A LINE

Size (square feet)

Pric

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rs)

6000500040003000200010000

1,200,000

1,000,000

800,000

600,000

400,000

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Example 1: Housing CostsThis is a scatter plot of data from one midwestern city that indicates the sizes and sale prices of various houses sold in this city

Data Source: http://www.trulia.com/for_sale/Milwaukee,WI/5_p accessed 7/13/2013

http://www.trulia.com/for_sale/Milwaukee,WI/5_p

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ACTIVITY 2 LESSON 8: INFORMALLY FITTING A LINE

In pairs, work through Example 2

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Review the following CCSSM Grade 8 content standards:8.SP.A.18.SP.A.2

Where did you see these standards in the lesson you have just completed? What would you look for in students’ work to suggest that they have made

progress towards these standards?

ACTIVITY 2LESSONS 7 & 8: SEEING PATTERNS IN DATA

Reflecting on CCSSM standards aligned to lessons 7 & 8

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8.SP.A.1: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

8.SP.A.2: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

ACTIVITY 2 LESSONS 7 & 8: SEEING PATTERNS IN DATA

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Read MP7, the seventh CCSSM standard for mathematical practice. Recalling that the standards for mathematical practice describe student

behaviors, how did you engage in this practice as you completed the lesson? What instructional moves or decisions did you see occurring during the

lesson that encouraged greater engagement in MP7? Are there other standards for mathematical practice that were prominent as

you and your groups worked on the tasks?



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CCSSM MP.7MP.7 Look for and make use of structure

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

engageny MP.7MP.7 Look for and make use of structure

Students identify pattern or structure in scatter plots. They fit lines to data displayed in a scatter plot and determine the equations of lines based on points or the slope and initial value.

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Why do you think it is a good idea to look at a scatter plot when you have data on two numerical values?

What should you look for when you are looking at a scatter plot?

What is the difference between predicting an outcome by looking at a scatter plot and predicting the outcome using a line that models the trend?

In a scatter plot, which variable goes on the horizontal axis and which variable goes on the vertical axis?


Closing questions for lessons 7 & 8

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ACTIVITY 3 LESSON 10: LINEAR MODELSLESSON 11: USING LINEAR MODELS IN A DATA CONTEXTMAKING USE OF PATTERNS IN BIVARIATE NUMERICAL DATA

ENGAGENY/COMMON CORE GRADE 8, LESSONS 10 & 11

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Vocabulary

What do we mean when we speak of a dependent variable?

An independent variable?

In statistics, a dependent variable is often called a response variable, or a predicted variable.An independent variable is often called an explanatory variable, or a predictor variable.

ACTIVITY 3 LESSON 10: LINEAR MODELS

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At your tables, discuss Exercises 1 & 2.


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At your tables, discuss Exercises 3-9, 10 and 11.


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What questions come to mind as you watch this video?

http://www.youtube.com/watch?v=LWrklFuYnb0

ACTIVITY 3 LESSON 11: USING LINEAR MODELS IN A DATA CONTEXT




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Work through Exercise 1 in pairs.


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At your tables, discuss Exercise 2.


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Read the following CCSSM Grade 8 content standards:

8.SP.A.1, 8.SP.A.2, 8.SP.A3

Where did you see these standards in the lesson you have just completed? What would you look for in students’ work to suggest that they have made

progress towards these standards?

ACTIVITY 3 LESSONS 10 & 11: MAKING USE OF PATTERNS IN DATA


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8.SP.A.1: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

8.SP.A.2: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

8.SP.A.3: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.


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Read MP4, the fourth CCSSM standard for mathematical practice. Recalling that the standards for mathematical practice describe student

behaviors, how did you engage in this practice as you completed the lesson? What instructional moves or decisions did you see occurring during the

lesson that encouraged greater engagement in MP4? Are there other standards for mathematical practice that were prominent as

you and your groups worked on the tasks?



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CCSSM MP.4MP.4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

engageny MP.4MP.4 Model with

mathematics.Students model relationships between variables using linear and nonlinear functions. They interpret models in the context of the data and reflect on whether or not the models make sense based on slopes, initial values, or the fit to the data.

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Why would we want to make a mathematical model of a real-world situation?

When is it appropriate to use a linear model?

Closing questions for lessons 10 & 11


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LEARNING INTENTIONS AND SUCCESS CRITERIA

We are learning to… Recognize and describe patterns and trends in dataUse those patterns and trends to make predictionsDistinguish between linear and nonlinear trends in data

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LEARNING INTENTIONS AND SUCCESS CRITERIAWe will be successful when we can:Recognize a pattern or trend in a scatter plot of a data set,

and describe that pattern or trend using correct statistical and/or mathematical language

Informally fit a line to data that shows a linear trendUse a line we have fit to data to make predictions about

the context from which the data was derived

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During Week 2 of the institute, you will present (in groups of no more than three) one of the following Engage NY lessons: Grade 6, Lessons 2, 3, 4, 5, 16 Grade 8, Lesson 6 Grade 9 (Algebra 1), Lessons, 3, 17

For the rest of our time today, you should continue to study and plan your chosen lesson with your group.

ACTIVITY 4 GROUP PRESENTATION PLANNING TIME

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Complete Lesson 7, Problem 4 and Lesson 11, Problem 1. Extending the mathematics:

Describe one set of bivariate data from your own experience, or of your own devising, in which you would expect a linear association between the dependent and independent variables. Explain why you would expect the association to be linear. Repeat, with a set of bivariate data in which you would expect an association, but one which is nonlinear.

Reflecting on teaching:How might working with linear models in the context of data help students to build “basic skills” in other areas of mathematics?

ACTIVITY 5 HOMEWORK AND CLOSING REMARKS

welcome to common core high school mathematics leadership

Documents

line lesson

scatter plots lesson

initial value lesson

lineseeing patterns

scatter plots grade

data set

scatter plotswhat

scatter plotslesson