This unit contains two main ideas: interpreting data using measures of center and spread, and modeling data
with an emphasis on linear models. Students make comparisons between graphs, lists, and tables of multiple
data sets by describing the shape, center, spread, and extreme values. Linear models are emphasized, but
quadratic and exponential models are mentioned. Students should develop a conceptual understanding of
correlation and causation and recognize that correlation does not imply causation.
How do we use evidence to support arguments?
How do we interpret evidence in order to support arguments?
Goal:​ Support students with strategies for finding the mean and
median from a set of data.
Standards​: 6.SP.A.3
Materials Needed:
●
●
●
Copies of ​Student Handout
Document Camera (optional)
Contemplate then Calculate
○
○
​Warm-Up: Find the mean/median
Posters (​Task 1​, T
​ ask 2​)
Slides (​Task 1​, ​Task 2​)
5 – 7 min
Students find the mean and median of a small set of numbers. The goal of this
activity is to activate students’ prior knowledge of mean and median and to see
how many students know how to do this already.
●
●
●
Independent work: ​Students write responses.
Pair-share:​ Have students share their justification with a partner.
Teacher ​circulates​, read what students write, and if necessary selects a couple
students to share their processes with the class.
Look fors in student responses:
● Students use a procedure to find the mean/median.
● Students have a shortcut already to find the mean/median.
Exploration & Whole Class Discussion: Contemplate then Calculate
30 – 40 min
​Click here for more information about how to use Contemplate then Calculate.
EXPLORATION:
● Use the ​Contemplate then Calculate instructional routine​ with the following two
tasks, one after another.
○ Find the Mean
○ Find the Median
DISCUSSION:
● Display student work that surfaces content or strategies you want the class to
discuss. ​Annotate student work​ as students describe their thinking, and push
for clarity and consistent use of mathematical and academic language.
Tip: ​Use a document camera or create posters so that all students will be
able to see.
​Application & Reflection: Reflect on Learning
Included in Routine
Students reflect on what is important to pay attention to either when calculating the
mean from a bar graph or calculating the median from a set of data.
●
●
●
Individual Work: ​Students write a response to the reflection question (1-2
minutes)
Partner Work:​ Students share what they wrote with a partner.
Teacher ​circulates​ to observe how students apply what they learned and selects
reflection responses to share with the class.
Algebra I, Unit 7 Big Idea 1 - Measures of Center - Lesson 0 Teacher Notes
Measures of Center Part 1: Solutions
Note that every one of these problems has multiple solutions. These first six examples are provided
as samples only!
Puzzle 1:
Puzzle 2:
Create a ​symmetrical​ data set with a ​mean​ of
5.
Create a ​non-symmetrical​ data set with a
mean​ of 5.
Puzzle 3
Puzzle 4
Create a ​non-symmetrical​ data set where the
mean​ and ​median​ are the same.
Create a data set where the ​mean​ and ​median
are different.
Algebra I, Unit 7 Big Idea 1 - Measures of Center - Lesson 1 Sample Solutions
Puzzle 5
Puzzle 6
Create a data set where the ​mean​ is 5 and the
median​ is 8.
Create a data set where the ​median​ is larger
than the ​mean​.
Observations students may have​ (likely using different words):
● For some distributions, the median can be quite unstable. The mean tends not to move as much as the
median when numbers are changed in some ways; the median tends not to move as much as the
mean in other ways.
○ What types of moves will keep the median stable while the mean changes?
○ What types of moves will keep the mean stable while the median changes?
● If a distribution is symmetric, the mean and median are always the same.
● The mode doesn’t give you very much information about what the distribution of data looks like, except
where some common values are. Sometimes, there can be two numbers that are the mode.
● The mean is less useful as a measure of the center of the data when the data has two or more clumps.
● Changing values of points below or above the median almost never changes the value of the median.
● The more you pull the points out to one side or the other (away from the center), the more the mean
and median separate.
Algebra I, Unit 7 Big Idea 1 - Measures of Center - Lesson 1 Sample Solutions
Goal:​ Describe how changing the distribution of a set impacts
the mean and median of that set of data.
Materials Needed:
Standards​: S.ID.A.1, S.ID.A.2, S.ID.A.3
​Warm-Up: Find the mean and the median
●
●
●
Copies of ​Student Handout
Laptops or tablets
Document Camera (optional)
5 – 7 min
Students decribe connections between a frequency dot plot and a given set of
data, then find the mean and median of the set of data.
●
●
●
Independent work: ​Students write responses.
Pair-share:​ Have students share their work with a partner.
Teacher ​circulates​, read what students write, and selects a couple students to
share their justifications with the class.
Look-fors in student responses:
● Students accurately describe how a frequency distribution
works.
● Students apply shortcuts for finding the mean and median from
the previous day.
Exploration & Whole Class Discussion: Using an Applet
20 – 25 min
​Here is an alternate no-tech activity​ if you do not have access to laptops or tablets.
EXPLORATION:
● Independent Think Time:​ Provide all students with 3-4 minutes to try the first
puzzle before they work with a partner.
●
Partner/Small Group Work​: Students use an applet to solve 12 puzzles
related to different distributions that correspond to values of the mean and
median. ​Sample solutions here​ ​and some ​observations to look for are here​.
●
Teacher ​circulates​ ​to observe student approaches, and select student work to
display during full group discussion. Remind students that each should be
ready to explain/share their reasoning for classification​.
DISCUSSION:
● Display student work that surfaces content or strategies you want the class to
discuss. ​Annotate student work​ as students describe their thinking, and push
for clarity and consistent use of mathematical and academic language.
Tip: ​Tell students to record responses to the prompts in the application
portion of the student materials as they work through the puzzles.
​Application & Reflection: Writing Statements about Mean and Median
5 min
Students apply what they learned to describe features of the mean and median.
●
●
Individual Work: ​Students complete the application and then write a response
to the reflection question (5-7 min)
Teacher ​circulates​ to observe how students apply what they learned and selects
reflection responses to share with the class.
Reflection:​ Describe similarities and differences between the mean and the median.
Algebra I, Unit 7 Big Idea 1 - Measures of Center - Lesson 1 Teacher Notes
Connections/relationships students may see:
● The stem and leaf diagrams, frequency tables, and histograms are really just three very similar displays
of the same basic distribution of data.
● Stem and leaf diagrams show the whole number part and the decimal parts of numbers separately.
● The class intervals for the frequency table correspond to the left and right parts of the histogram.
● Box and whisker plots are a completely different display than any of the other four representations.
They rely on the calculations we did. We can estimate/read the minimum value, first quartile, median,
third quartile, and maximum from the box and whisker plots. The interquartile range doesn’t show up
directly, but must be the length of the box.
● Stem and leaf diagrams, frequency tables, histograms all had a part in the middle with more data,
which must mean that the box for the box and whisker plots shows where there is more data.
Algebra I, Unit 7 Big Idea 1 - Measures of Center - Lesson 2 Connections Students May See
Goal:​ Use calculators to calculate statistics related to univariate
data, and determine connections between different
representations of the same data.
Standards​: S.ID.A.1, S.ID.A.3
Materials Needed:
●
●
●
●
Copies of ​Student Handout
Copies of the Calculator
Guide
Calculators
Document Camera (optional)
​Warm-Up: Finding Mean and Median
5 min
Students find the mean and median of a frequency distribution bar graph.
●
●
●
Independent work: ​Students write responses.
Pair-share:​ Have students share their responses with a partner.
Teacher ​circulates​, read what students write, and selects a couple students to
share their justifications with the class.
Look fors in student responses:
● Students use shortcuts based on understanding mean and
median to find the values of the mean and median.
● Students who use procedures that can be connected to the
shortcuts other students used.
Exploration & Whole Class Discussion: Calculating and Using Univariate Statistics
20 – 25 min
EXPLORATION:
● Independent Think Time:​ Provide all students with 4-5 minutes to work
through the calculator guide before starting.
● Partner/Small Group Work​: Students use their calculators to calculate
univariate statistics, then look for relationships between different
representations of univariate data.
● Teacher ​circulates​ ​to observe student approaches, and select student work to
display during full group discussion.
DISCUSSION:
● Display student work that surfaces content or strategies you want the class to
discuss. ​Annotate student work​ as students describe their thinking, and push
for clarity and consistent use of mathematical and academic language.
● Here are some connections​ students may make.
Tip: ​Use a document camera or an image on an interactive whiteboard so
that all students will be able to see each others’ reasoning.
​Application & Reflection: Fluency Practice
10 min
Students apply what they learned to calculate univariate statistics and represent the
same data using a variety of representations.
●
●
Individual Work: ​Students complete the application question (5-7 min)
Teacher ​circulates​ to observe how students apply what they learned and selects
responses to share with the class.
Reflection:​ N
​ one provided.
Algebra I, Unit 7 Big Idea 1 - Measures of Center - Lesson 2 Teacher Notes
Materials Needed:
Goal:​ Students make connections between different frequency
distributions and how these distributions are represented with a
box and whisker plot.
Standards​: S.ID.A.1, S.ID.A.2, S.ID.A.3
●
●
●
●
●
Copies of ​Student Handout
Laptops or tablets
Document Camera (optional)
Calculators
Optional: Posters for the
Connecting Representations
routine
​Warm-Up: Write down what you remember
5 min
Students write down what they know already about box and whisker plots.
● Independent work: ​Students write responses.
● Pair-share:​ Have students share their justification with a partner.
● Teacher ​circulates​, read what students write, and selects a couple students to
share their justifications with the class.
Look fors in student responses:
● Students are able to connect the parts of a box and whisker plot
with the median, Q1, Q3, minimum, and maximum of a data
set.
Exploration & Whole Class Discussion:
30 – 35 min
​Here is an alternate no-tech activity​ if you do not have access to laptops or tablets.
EXPLORATION:
● Independent Think Time:​ Provide all students with 3-4 minutes to try one of
the puzzles independently.
● Partner/Small Group Work​: Students work together to solve a variety of
puzzles related to box and whisker plots.
● Teacher ​circulates​ ​to observe student approaches, and select student work to
display during full group discussion. Remind students that each should be
ready to explain/share their reasoning for classification​.
DISCUSSION:
● Display student work that surfaces content or strategies you want the class to discuss. ​Annotate student work​ as
students describe their thinking, and push for clarity and consistent use of mathematical and academic language.
Tip: ​ ​Encourage students to record observations or generalizations in the application portion of their student
materials as they work through the puzzles.
​Application & Reflection:
2 min
Students record observations they have while they work through the puzzles.
●
●
Individual Work: ​Students complete the application (5-10 min)
Teacher ​circulates​ to observe how students apply what they learned and selects
reflection responses to share with the class.
Reflection:​ N
​ one provided.
Algebra I, Unit 7 Big Idea 1 - Measures of Center - Lesson 3 Teacher Notes
Materials Needed:
Goal:​ Make connections between the mean, standard
deviation, and the distribution of a data set.
●
●
●
●
●
Standards​: S.ID.A.1, S.ID.A.2
​Warm-Up:
Copies of ​Student Handout
Copies of the ​Calculator
Guide
Calculators
Laptops or tablets
Document Camera (optional)
20 – 25 min
Students use their calculators and what they have learned about the one variable
statistics to solve two Regents problems.
● Independent work: ​Students write responses.
● Pair-share:​ Have students share their justification with a partner.
● Teacher ​circulates​, read what students write, and selects a couple students to
share their justifications with the class.
Look fors in student responses:
● For question 1, which of the different terms do students
remember?
● For question 2, look for students who are able to solve this
question without their calculators and students who can solve it
using their calculator.
Exploration & Whole Class Discussion:
20 – 25 min
​Here is an alternate no-tech activity​ if you do not have access to laptops or tablets.
EXPLORATION:
● Independent Think Time:​ Provide all students with 1–2 minutes to read
through the different puzzles.
● Partner/Small Group Work​: Students solve the puzzles given.
● Teacher ​circulates​ ​to observe student approaches, and select student work to
display during full group discussion. Remind students that each should be
ready to explain/share their reasoning for classification​.
DISCUSSION:
● Display student work that surfaces content or strategies you want the class to
discuss. ​Annotate student work​ as students describe their thinking, and push
for clarity and consistent use of mathematical and academic language.
Tip: ​ ​Encourage students to record observations or generalizations in the
application portion of their student materials as they work through the
puzzles.
​Application & Reflection:
5 - 10 min
Students solve a Regents problem related to mean and standard deviation.
●
●
Individual Work: ​Students complete the application and then write a response
to the reflection question (5-7 min)
Teacher ​circulates​ to observe how students apply what they learned and selects
reflection responses to share with the class.
Reflection:​ Describe what the mean and standard deviation tell you about the
distribution of a set of data.
Algebra I, Unit 7 Big Idea 1 - Measures of Center - Lesson 4 Teacher Notes
Lesson 1: Analyze Contextual Data
Goal:
Interpret and explain trends in contextual statistical bivariate
statistical data.
○ construct two way frequency tables and summarize data for each
category
○ interpret frequencies in context and describe possible trends
Materials Needed:
● Student handout
● Document camera to display
student work (optional)
● Calculators for students
A primary goal of this lesson is that students see that organizing data can
make questions much easier to do, and that students understand how a
two-way frequency chart is constructed.
Warm - up
5 min
1. Individual Work: ​Students answer questions based on a context
2. Whole Class Discussion:​ Share some responses by students. Use
annotation​ on the table to highlight how students quickly found students
that drink more than the recommended amount.
Exploration & Whole Class Discussion
Questions based on context
30–35 min
EXPLORE:
1. Individual Work Time:​ Students answer the questions about the context
independently.
2. Partner Work: ​Students share and compare their work with a partner.
Teacher ​circulates to observe student approaches​, selects student work to display.
DISCUSSION:
Select and display student matches for discussion. ​Annotate the examples​ to highlight how students have made use of
measures of center and/or spread to make sense of the data.
Possible/suggested prompts for discussion:
● How did [this person] organize their data to make answering the question easier?
● What evidence supports [this person]’s conclusion?
Tip: ​Focus on sharing student strategies that emphasize the goal of the lesson rather than sharing a
variety of different approaches.
​Application & Reflection:
5–10 min
1. Individual Work: ​Students apply what they learned to solve an example
Regents Problem (3 - 4 min) then write a response to the reflection
prompt (2 - 3 min).
2. Reflection:
a. Students share reflections with a partner.
b. Select 2-3 reflections to be read to the class. Highlight
connections between reflections and the goal for today’s lesson.
Algebra I, Unit 7 Big Idea 2 - Interpreting Data - Lesson 1 Teacher Notes
Lesson 2: Reading Two-Way Frequency Charts
Goal:
Construct two way frequency tables and summarize data for each
category.
○ interpret relative frequencies in context and describe possible trends
○ identify joint, marginal and conditional relative frequencies
Materials Needed:
● Student handout
● Document camera to display
student work (optional)
● Connecting Representations
○
○
○
Task
Slides
Posters​ (either print as
large as possible or copy
these to chart paper)
Warm - up
5 min
1. Individual Work: ​Students answer contextual questions based on a table. This is
in part to give you as the teacher information about what students understand
already about probability.
2. Pair-share:​ Students share their responses to the contextual problems with a
partner while the teacher circulates to gather responses to share with the room.
EXPLORATION & WHOLE CLASS DISCUSSION: Connecting Representations
25 min
This task should help students both interpret two-way frequency charts and recognize that
these charts can help us answer questions that begin with “how likely…”.
Once students are describing their strategies, you can introduce vocabulary like joint
relative frequency, marginal relative frequency, and conditional relative frequency as
students describe the parts of the two-way frequency chart to which these terms refer.
1. Launch the routine: ​Tell students what, why, and how they will work today.
2. Make Connections:​ Students share connections with a partner while the teacher
circulates to select which pairs will share (2-4 min)
3. Share and study connections:​ Students discuss connections as a group (15 min)
DISCUSSION: ​Annotate​ representations to highlight connections as students describe their
thinking; push for clarity and for consistent use of mathematical and academic language.
Prompts for discussion:
Select from the following based on the needs of your students and the time available.
● How did you connect the two representations?
● How did you interpret this part of the representation?
● How are the solutions (or lack of a solution) to these equations represented in the graphs?
Tip: ​For this task, display each set of representation types separately for the ask yourself questions since there
is otherwise a lot of information which may be overwhelming for students.
Click here for more information about Connecting Representations.
​Application & Reflection:
15 min
4. Create a representation: ​Students create a representation with a partner and few
partner pairs will share their work with the whole group (6-8 min)
5. Reflect: ​Students reflect on learning and the teacher selects a few students to read
their reflections with the whole group​ ​(3-5 min)
Algebra I, Unit 7 Big Idea 2 - Interpreting Data - Lesson 2 Teacher Notes
Goal:
Explain what the line of best fit represents for a scatter plot.
● describe the relationship between variables for the graph of a scatter
plot
● describe how several points in a data set are related
● interpret a single data point and explain its relationship to the data set
Materials Needed:
●
●
Copies of ​Student Handout
Document Camera (optional)
​Warm-Up: Interpret a scatter plot
5 – 7 min
Students interpret a scatter plot and answer contextual questions about the
scatter plot.
●
●
●
Independent work: ​Students write responses.
Pair-share:​ Have students share their work with a partner.
Teacher ​circulates​, read what students write, and selects a couple students to
share their justifications with the class.
Look-fors in student responses:
● Are students using the legend when answering the questions?
● Do students see a general trend represented in the scatter
plot?
Exploration & Whole Class Discussion: Interpret lines of best fit
30 – 35 min
EXPLORATION:
● Independent Work:​ Students complete the contextual questions on their own.
●
Partner/Small Group Work​: Students work together to ​compare and improve
their responses to the contextual questions and then work together to interpret
the scatter plots and the lines of best fit.
●
Teacher ​circulates​ ​to observe student approaches, and select student work to
display during full group discussion.
DISCUSSION:
● Display student work that surfaces content or strategies you want the class to
discuss. ​Annotate student work​ as students describe their thinking, and push
for clarity and consistent use of mathematical and academic language.
Tip: ​During the whole group discussion, we suggest helping students draw
connections between the mean of a data set, how lines of best fit are
constructed, and the vertical/horizontal lines given the last activity.
​Application & Reflection: Draw lines of best fit
5 – 7 min
Students apply what they learned to create lines of bet fit.
●
●
Individual Work: ​Students complete the application and then write a response
to the reflection question (5-7 min)
Teacher ​circulates​ to observe how students apply what they learned and selects
reflection responses to share with the class.
Reflection:​ Describe a strategy for drawing a line of best fit on a
scatter plot.
Algebra I, Unit 7 Big Idea 2 - Interpreting Data - Lesson 3 Teacher Notes
Goal:
Compute and interpret the correlation coefficient.
● create a scatter plot and describe the correlation between data points in a
set
● explain what the line of best fit represents for a scatter plot.
● use the line of best fit to predict and justify other possible data points or
solve situations
​Warm-Up: Match Correlation Coefficients to Scatter Plots
Materials Needed:
●
●
●
●
●
Copies of ​Student Handout
Calculators for students
Calculator guides
Document Camera (optional)
Calculator Emulator (optional)
5 – 7 min
Students decribe how they can match correlation coefficients with scatter plots
given.
●
●
●
Independent work: ​Students write responses.
Pair-share:​ Have students share their work with a partner.
Teacher ​circulates​, read what students write, and selects a couple students to
share their justifications with the class.
Tip:​ We recommend using student explanations of the matches to
define the correlation coefficient of the graph and to watch for as
precise language as possible decribing the match. A good follow-up
question to ask is, “What do you think a negative correlation would look
like when graphed?”
Exploration & Whole Class Discussion: Calculating Lines of Best Fit
25 – 30 min
​ ​EXPLORATION:
● Independent Reading Time:​ Provide all students with 2-3 minutes to read
through the calculator guide before working with a partner.
● Partner/Small Group Work​: Students work with a partner to follow the steps
given in a calculator guide for finding the line of best fit and the correlation
coefficient with a partner, and then try to repeat that process without using the
calculator guide.
● Teacher ​circulates​ ​to observe student approaches, and select student work to
display during full group discussion.
DISCUSSION:
● Display student work that surfaces content or strategies you want the class to
discuss. ​Annotate student work​ as students describe their thinking, and push
for clarity and consistent use of mathematical and academic language.
Tip: ​You can either use a document camera to display a calculator when
students are describing steps or purchase/download a calculator emulator.
​Application & Reflection: Solving Regents Problems
5 – 10 min
Students apply what they learned to solve related Regents exam problems.
Individual Work: ​Students complete the application and then write a response
to the reflection question (5-7 min)
● Teacher ​circulates​ to observe how students apply what they learned and selects
reflection responses to share with the class.
Reflection:​ Explain how you can estimate the value of the correlation coefficient for a
scatter plot.
●
Algebra I, June 2014, Q11
Algebra I, August 2016, Q6
Algebra I, Unit 7 Big Idea 2 - Interpreting Data - Lesson 4 Teacher Notes
Goal:
Describe a strong or weak correlation, positive or negative
correlation, for a given data set.
Materials Needed:
● compute and interpret the correlation coefficient
● distinguish correlation and causation
​Warm-Up: Find the mean and the median
●
●
●
●
Copies of ​Student Handout
Calculators
Document Camera (optional)
Matching activity for each pair
of students​.
5 – 7 min
Students decribe how two data sets can be correlated but that this does not
mean that there is an actual relationship (causation) between the two data sets.
●
●
●
Independent work: ​Students write responses.
Pair-share:​ Have students share their work with a partner.
Teacher ​circulates​, read what students write, and selects a couple students to
share their justifications with the class.
Tip: ​The objective of this portion of the lesson is for students to distinguish
correlation from causation. More examples of spurious relationships are
available here: ​http://www.tylervigen.com/spurious-correlations
Exploration & Whole Class Discussion: Using an Applet
20 – 25 min
​Click here for more information on using matching activities/card sorts.
EXPLORATION:
● Independent Think Time:​ Provide all students with 2 minutes to make a
match on their own before working with their partner.
●
Partner/Small Group Work​: Students work with a partner or in a small group
to make matches between the graphs and the situations with data.
●
Teacher ​circulates​ ​to observe student approaches, and select student work to
display during full group discussion.
DISCUSSION:
● Display student work that surfaces content or strategies you want the class to
discuss. ​Annotate student work​ as students describe their thinking, and push
for clarity and consistent use of mathematical and academic language.
Tip: ​Students should feel free to use their calculators to assist them when
making matches. Also, remind students that the graphs are not exact
matches for the data.
​Application & Reflection: Writing Statements about Mean and Median
5 – 10 min
Students apply what they learned to solve a Regents problem related to describing the
correlation of a set of data.
●
●
Individual Work: ​Students complete the application and then write a response
to the reflection question (5-7 min)
Teacher ​circulates​ to observe how students apply what they learned and selects
reflection responses to share with the class.
Reflection:​ Describe similarities and differences between the mean and the median.
Algebra I, Unit 7 Big Idea 2 - Interpreting Data - Lesson 5 Teacher Notes