5.1
WELCOME TO COMMON CORE HIGH SCHOOL
MATHEMATICS LEADERSHIP
SUMMER INSTITUTE 2014
SESSION 5 • 20 JUNE 2014
SEEING PATTERNS AND TRENDS IN BIVARIATE DATA
5.2
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
5.3
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
5.4
ACTIVITY 1
HOMEWORK REVIEW AND DISCUSSION
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.
5.5
LEARNING INTENTIONS AND SUCCESS CRITERIA
We are learning to…
 Recognize and describe patterns and trends in data
 Use those patterns and trends to make predictions
 Distinguish between linear and nonlinear trends in data
5.6
LEARNING INTENTIONS AND SUCCESS CRITERIA
We 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 trend
 Use 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 PLOTS
LESSON 8: INFORMALLY FITTING A LINE
SEEING PATTERNS IN BIVARIATE NUMERICAL DATA
ENGAGENY/COMMON CORE GRADE 8, LESSONS 7 & 8
5.8
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)
5.9
ACTIVITY 2
LESSON 7: PATTERNS IN SCATTER PLOTS
 What does the word “relationship” mean to you?
 What does it mean to say there is a relationship between two numerical
variables?
5.10
ACTIVITY 2
LESSON 7: PATTERNS IN SCATTER PLOTS
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?
5.11
ACTIVITY 2
LESSON 7: PATTERNS IN SCATTER PLOTS
 In pairs, discuss Exercises 1, 4, 5, 6, 8, 10.
5.12
ACTIVITY 2
LESSON 8: INFORMALLY FITTING A LINE
Example 1: Housing Costs
This is a scatter plot of data from one midwestern city that indicates the sizes and sale prices of various
houses sold in this city
1,200,000
Price (dollars)
1,000,000
800,000
600,000
400,000
200,000
0
0
1000
2000
3000
4000
Size (square feet)
5000
6000
Data Source: http://www.trulia.com/for_sale/Milwaukee,WI/5_p accessed 7/13/2013
5.13
ACTIVITY 2
LESSON 8: INFORMALLY FITTING A LINE
 In pairs, work through Example 2
5.14
ACTIVITY 2
LESSONS 7 & 8: SEEING PATTERNS IN DATA
Reflecting on CCSSM standards aligned to lessons 7 & 8
Review the following CCSSM Grade 8 content standards:
8.SP.A.1
8.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?
5.15
ACTIVITY 2
LESSONS 7 & 8: SEEING PATTERNS IN DATA
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.
5.16
ACTIVITY 2
LESSONS 7 & 8: SEEING PATTERNS IN DATA
Reflecting on CCSSM standards aligned to lessons 7 & 8
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?
5.17
ACTIVITY 2
LESSONS 7 & 8: SEEING PATTERNS IN DATA
CCSSM MP.7
engageny MP.7
MP.7 Look for and make use of structure
MP.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.
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.
5.18
ACTIVITY 2
LESSONS 7 & 8: SEEING PATTERNS IN DATA
Closing questions for lessons 7 & 8
 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?
Break
5.20
ACTIVITY 3
LESSON 10: LINEAR MODELS
LESSON 11: USING LINEAR MODELS IN A DATA CONTEXT
MAKING USE OF PATTERNS IN BIVARIATE NUMERICAL DATA
ENGAGENY/COMMON CORE GRADE 8, LESSONS 10 & 11
5.21
ACTIVITY 3
LESSON 10: LINEAR MODELS
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.
5.22
ACTIVITY 3
LESSON 10: LINEAR MODELS
 At your tables, discuss Exercises 1 & 2.
5.23
ACTIVITY 3
LESSON 10: LINEAR MODELS
 At your tables, discuss Exercises 3-9, 10 and 11.
5.24
ACTIVITY 3
LESSON 11: USING LINEAR MODELS IN A DATA CONTEXT
What questions come to mind as you watch this video?
http://www.youtube.com/watch?v=LWrklFuYnb0
5.25
ACTIVITY 3
LESSON 11: USING LINEAR MODELS IN A DATA CONTEXT
Work through Exercise 1 in pairs.
5.26
ACTIVITY 3
LESSON 11: USING LINEAR MODELS IN A DATA CONTEXT
At your tables, discuss Exercise 2.
5.27
ACTIVITY 3
LESSONS 10 & 11: MAKING USE OF PATTERNS IN DATA
Reflecting on CCSSM standards aligned to lessons 10 & 11
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?
5.28
ACTIVITY 3
LESSONS 10 & 11: MAKING USE OF PATTERNS IN DATA
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.
5.29
ACTIVITY 3
LESSONS 10 & 11: MAKING USE OF PATTERNS IN DATA
Reflecting on CCSSM standards aligned to lessons 10 & 11
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?
5.30
ACTIVITY 3
LESSONS 10 & 11: MAKING USE OF PATTERNS IN DATA
CCSSM MP.4
MP.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.4
MP.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.
5.31
ACTIVITY 3
LESSONS 10 & 11: MAKING USE OF PATTERNS IN DATA
Closing questions for lessons 10 & 11
 Why would we want to make a mathematical model of a real-world situation?
 When is it appropriate to use a linear model?
5.32
LEARNING INTENTIONS AND SUCCESS CRITERIA
We are learning to…
 Recognize and describe patterns and trends in data
 Use those patterns and trends to make predictions
 Distinguish between linear and nonlinear trends in data
5.33
LEARNING INTENTIONS AND SUCCESS CRITERIA
We 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 trend
 Use a line we have fit to data to make predictions about the
context from which the data was derived
5.34
ACTIVITY 4
GROUP PRESENTATION PLANNING TIME
 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.
5.35
ACTIVITY 5
HOMEWORK AND CLOSING REMARKS
 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?