Download SFUSD Unit S.1 Categorical and Quantitative Data

1
SFUSD Mathematics Core Curriculum Development Project
2014–2015
Creating meaningful transformation in mathematics education
Developing learners who are independent, assertive constructors of their own understanding
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Algebra 1
S.1 Categorical and Quantitative Data
Number
of Days
1
8
Lesson
Reproducibles
Entry Task
Lesson Series 1
1
6
Apprentice Task
Lesson Series 2
1
Expert Task
Scatter it! Predict Billy’s Height (2 pages)
CPM CCC1 Lesson 8.1.1 (3 pages)
Lesson 4: Creating a Histogram
Creating a Histogram Exercises 1-9 (2 pages)
CPM CCC1 Lesson 8.1.2 (6 pages)
CPM CCC1 Lesson 8.1.3 (3 pages)
Introduction to Standard Deviation (2 pages)
Making a Boxplot (2 pages)
CPM CCC1 Lesson 8.1.4 (3 pages)
Lesson 8.1.4 Resource Page: High-Temperature Data
CPM CCC1 Lesson 8.1.5 (2 pages)
Counters Game (4 pages)
Heights of Basketball Players (2 pages)
Z-Score Table (2 pages)
Matching Histograms and Boxplots (2 pages)
A Sweet Task (3 pages)
A Sweet Task Exit Ticket
Two-Way Table Practice (2 pages)
CPM CCA Lesson 10.1.1 (2 pages)
How High Can You Jump (4 pages)
Pizza Party
CPM CCA Lesson 6.1.1
CPM CCA Lesson 6.1.2 (2 pages)
CPM CCA Lesson 6.1.2 Homework (2 pages)
How Strong is the Association between Income and Race?
Number of
Copies
1 per student
1 per pair
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1 per student
1 per pair
1 per student
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Materials
Graph paper, spaghetti
Dice
M&Ms and Skittles
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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8
Lesson Series 3
1-2
Milestone Task
CPM CCA Lesson 6.1.4 (6 pages)
Lesson 6.1.4 Resource Page: El Toro Basketball
CPM CCA Lesson 6.2.1 (6 pages)
CPM CCA Lesson 6.2.2 (6 pages)
CPM CCA Lesson 6.2.3 Methods and Meanings (2 pages)
CPM CCA Lesson 6.2.3 (4 pages)
CPM CCA Lesson 6.2.5 (3 pages)
CPM CCA Chapter 6 Review (2 pages)
Who Stole the iPhone?
1 per pair
1 per student
1 per pair
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1 per student
1 per student
Graphing calculators or software
Graphing calculators or software
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Unit Overview
Big Idea
Representing and analyzing categorical and quantitative data in univariate and bivariate data and using linear models to establish
and explain correlation and/or causation in a set of data.
Unit Objectives
● Students will be able to represent univariate data with dot plots, histograms and boxplots and bivariate data with scatterplots
and two-way frequency tables; Students will be able recognize associations and trends in the two-way tables
● Students will be able to perform numerical analyses of the data using shape, center and spread and interpret the significance
of those elements within context; students know that the median is a more appropriate measure of center for skewed data
● Students will be able to estimate population percentages using the area under the normal curve and z-scores
● Students will be able to choose appropriate linear, quadratic and exponential functions to model a set of data and solve
problems in context
● For a linear function, Students will be able to interpret slope and y-intercept in context
● For a linear function, Students will be able to compute (with technology) and interpret the correlation coefficient, r, using
strength and direction in context
● Students will be able to compare two distributions of data and determine which distribution is most likely to contain a
particular data point
● Students will be able to distinguish between correlation and causation (correlation does not always imply causation)
Unit Description
Students will begin by displaying data in one variable using dot plots, box plots and histograms. Then, they will analyze shape and
spread by calculating measures of center (mean, median), measures of spread (IQR and standard deviation) and discussing shape
(skewed right/left, symmetric, bimodal) and the effect of outliers. Then, students will represent data on 2 quantitative variables using
a scatterplot and describe the relationship in context. Given different scatterplots, student will decide if a linear, quadratic or
exponential model would be the best fit and use that model to solve for the valuable of a variable. Then, students will focus on linear
models and use graphing calculators to find the line of best fit and the correlation coefficient (r). Students will interpret the slope and
y-intercept in context and discuss the strength of the model given the “r” value and by calculating and plotting the residuals. Lastly,
students will determine if correlation implies causation or if lurking variables may be a factor in the relationship.
In the final section of the unit, students will display categorical data using a two-way table and describe possible trends. Students will
also be able to pull relative frequencies from a two-way table.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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CCSS-M Content Standards
S- ID Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on a single count or measurement variable.
1. Represent data with plots on the real number line (dot plots, histograms, and box plots).
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.
3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data
points (outliers).
4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.
Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to
estimate areas under the normal curve.
Summarize, represent, and interpret data on two categorical and quantitative variables.
5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the
data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a
function suggested by the context. Emphasize linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Interpret linear models
7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
8. Compute (using technology) and interpret the correlation coefficient of a linear fit.
9. Distinguish between correlation and causation.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Progression of Mathematical Ideas
Prior Supporting Mathematics
Current Essential Mathematics
Students should have developed an
understanding of statistical variability of
data and be able to analyze the information
by finding the center (mean and/or median),
variability (interquartile range and/or mean
absolute deviation), spread, and overall
shape. They are able to graphically display
such data using box plots, histograms, and
scatterplots. While investigating patterns in
bivariate data, students are able to use
linear equations to model relationships
between two quantitative variables and to
interpret the slope and intercept in context
of the problem. Students will be able to
construct and interpreting two-way tables
by summarizing data on two categorical
variables.
Students revisit displaying and analyzing
data in one and two variables. In 9th
grade, students will investigate the
standard deviation of a set of data for the
first time and use this to calculate z-scores
and the area under the normal curve.
Students also revisit fitting a linear model
to a set of data, but use the correlation
coefficient, r, for the first time to discuss
the legitimacy of the model. The word,
“residual”, is formalized for the first time
and used to defend the fit of the linear
model.
Future Mathematics
Students use z-scores and the normal
curve to report the probability of getting a
certain value given a set of data.
Use the mean and standard deviation to fit
it to a normal distribution and estimate
populations percentages.
Students are making inferences and
justifying conclusions. They will understand
and evaluate random processes underlying
statistical experiments and observational
studies.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Unit Design
All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both
formative assessments of student learning. The tasks are designed to address four central questions:
Entry Task:
Apprentice Task:
Expert Task:
Milestone Task:
1 Day
What do you already know?
What sense are you making of what you are learning?
How can you apply what you have learned so far to a new situation?
Did you learn what was expected of you from this unit?
8 Days
1 Day
6 Days
1 Day
8 Days
1 Day
Total Days: 26
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Entry Task
Apprentice Task
Expert Task
Milestone Task
Predict Billy’s Height
Matching Histograms and Boxplots
How Strong is the Association Between
Race and Income?
Who Stole the iPhone?
CCSS-M
Standards
8. SP. 1, 8. SP. 2, 8. SP. 3
S-ID 1,S-ID 2, S-ID 3
S-ID 7, S-ID 8, S-ID 9
S-I D 2,S-I D 3, S-ID 5, S-ID 8
Brief
Description
of Task
Assesses students knowledge of
independent variables, graphing
univariate data, numerical
analysis, and justification of
conclusions.
Assesses student understanding
of shape, center and spread and
differing methods of displaying
and comparing data sets.
Assesses student understanding of
how to represent and analyze
univariate vs. bivariate data and
determine the validity of a linear
model based on the residuals. Also
assesses interpretation of slope
and y-intercept in context and
analysis of association in a two-way
table.
Assessing student understanding of
the appropriateness of the least
square regression line through the
use of the correlation
coefficient,residual plots and the
equation of the least square
regression line. Also student
understanding of the process of
comparing two data sets and
selecting who belongs in one of the
two given data sets.
Source
adapted from
http://www.amstat.org/education/stew/
pdfs/ScatterIt!PredictBillysHeight.pdf.
adapted from:
https://math.la.asu.edu/~saldanha
/STP420Webpage/attachments/G
raphMatchingTasks.pdf
SFUSD Teacher Created
data from:
http://www.sfplanning.org/Modules/ShowDocum
ent.aspx?documentid=8501
adapted from “If the Shoe Fits”
http://www.amstat.org/education/st
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Lesson Series 1
Lesson Series 2
Lesson Series 3
CCSS-M
Standards
S-ID 1, S-ID 2, S-ID3, S-ID4
S-ID 5, S-ID 6 a, b ,c
S-ID 7, S-ID 8, S-ID 9
Brief
Description
of Lessons
Students will review how to generate
histograms and the mean and median of a
given set of data. Students will be able to find
the center, spread, range, and IQR of a set of
data. Students will be able to graph a boxplot,
graph a normal curve, find the median
absolute deviation, standard deviation, and zscore of a set of data.
Students will be able to construct two way
tables, find the equation of a linear Line of
Best-Fit, and calculate residuals. There will be
continued spiraling of content from Lesson
Series 1 such as boxplots and numerical
analysis.
Students will learn how to use graphing
calculators to graph the Least Squares
Regression Line, graph Residual Plots, find r
(Correlation Coefficient), determine the
association between two variables, and find a
non-linear line of best-fit.
Sources
Heights of Basketball Players - DOE of
Virginia
http://www.doe.virginia.gov/testing/solsearch/s
ol/math/A/m_ess_a-9_4.pdf
Histogram Mini Lesson - from
http://www.engageny.org/sites/default/files/res
ource/attachments/math-g6-m6-teachermaterials.pdf
Making a Boxplot - SFUSD Grade 9
Mathematics Core Curriculum Development
Team, June 2014
A Sweet Task (adapted from STEW,
http://www.amstat.org/education/stew/)
A Sweet Task Exit Ticket (from STEW,
http://www.amstat.org/education/stew/).
Pizza Party (adapted from Statistics Through
Applications 2nd Edition, W.H. Freeman and
Company, 1996, Chapter 4, page 173.)
Methods and Meanings Resource Pages
(adapted from Statistics Supplement: CPM
Common Core Algebra , 2013, Chapter 6,
pages 1-92)
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Entry Task
Predict Billy’s Height
What will students do?
Mathematics Objectives and Standards
Framing Student Experience
Math Objectives:
● Students will identify the independent variable.
● Students will construct a graph to model the relationship between two
variables and describe the relationship between them.
● Students will find the mean and median of the set of data.
● Students will be to informally identify a line of best-fit and estimate the
slope.
Scatter It! Predict Billy’s Height.
Launch:
● Formulate a hypothetical situation.
● Billy is an eighth-grade student who loves to play basketball. Because
of this, he wants to know how tall he will be when he is in tenth-grade –
the first year he will be eligible to play on his high school basketball
team. Billy was looking through his school pictures and noticed that he
seemed to grow in height about the same amount each year and wants
to know if he can use his height from each year to predict how tall he
will be in two years.
●
Ask students to write some questions that they would be interested in
investigating about students’ age and height. Some possible questions
might include:
● What are some typical heights of students?
● Is there a relationship between age and height? If so, what kind of
relationship?
● Can we use the relationship between age and height to predict the
height of a student in the future? If so, for how many years is this
relationship valid?
During:
● Students are given data.
● Students will work in group.
● Students are asked to graph the points.
● Students will find the mean and median of data.
● Students will find possible equation of line of best-fit.
● Estimate the slope the line of best-fit.
● Groups will predict height at age 15.
CCSS-M Standards Addressed:
8. SP. 1. Construct and interpret scatterplots 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. 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. 3. Use the equation of the linear model to solve problems in the context
of bivariate measurement data, interpreting the slope and intercept.
Notes (* questions should be asked 2015-2016 and beyond).
● Students will identify the slope and intercept of the line of best-fit.
● Students will be able to predict a value.
Closure/Extension:
● Groups will come up to present estimates of line of best-fit and
predictions.
● Groups will explain how they found their line of best-fit.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Predict Billy’s Height
How will students do this?
Focus Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them.
4. Model with mathematics.
Structures for Student Learning:
Academic Language Support:
Vocabulary: Independent variable, Mean, Median, Line of Best Fit, Scale
Sentence frames:
The _______ is the independent variable because __________________.
We believe the line of best fit is because __________________________.
We think that Billy’s height is _________ at age 15 because ___________________.
Participation Structures (group, partners, individual, other):
● This activity can be done in pairs or groups of four. The whole class discussion at the end pulls out the mathematics.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Lesson Series #1
Lesson Series Overview: Students review how to construct histograms, dot plots, and calculate the mean and median from a set of data. Students find the
mean absolute deviation, shape, spread, range, construct a boxplot, and find the IQR of a set of data. Students estimate population percentages under a normal
curve and find the z-score. Students compare two distributions of data and determine which is likely to contain a certain point.
CCSS-M Standards Addressed: S-ID 1, S-ID 2, S-ID 3, S-ID 4
Time: 8 days
Lesson Overview – Day 1
Description of Lesson:
Take a census of the class and have them record with dots on a large graph in front of the
class. Once all dots are placed, have a class discussion about the questions in 8-1.
Students should record ideas from the class as the teacher records them on the board.
Resources
Core Connections, Course 1, CPM 8.1.1,
Problems 8-1 to 8-4 and 8-6.
Students should work in pairs on 8-2 to 8-4. If they finish early, the challenge is 8-6.
When all groups are done with 8-4, bring class back together to link the conceptual idea of
mean (from 8-3) to the algorithm most of them should remember from middle school (add
all values and divide by the number of value). Put this in a toolkit for “mean”. Also, put
“median” and “range” in the toolkit. Add “outlier” to the toolkit if any of the data points are
far award from the other point.
Exit Ticket: Give student a set of data and have them calculate the mean, median and
range of the data.
Lesson Overview – Day 2
Resources
Description of Lesson:
Reteach students how to make a histogram using the assignment in the resources folder
(students should have learned this in 8th grade).
Lesson 4: Creating a Histogram – from EngageNY:
http://www.engageny.org/sites/default/files/resource/attachments/
math-g6-m6-teacher-materials.pdf
Afterwards, students should work in groups on 8-12 to 8-15. For 8-12, students should
Core Connections, Course 1, CPM 8.1.2
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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draw the histogram by hand. If histograms seem new for most students, then you can give
them the width of the bins for each set of data. Also, put the definition of “outlier” on the
board (if not already in the toolkit) so students can remember what it means for problems 813 and 8-15.
Problems 8-12 to 8-15 and 8-21 to 8-26
After most groups are done with 8-15, put the idea that outliers affect the mean much more
than the median in the toolkit.
Exit Ticket or individual work: 8-21 and 8-26
Lesson Overview – Day 3
Description of Lesson:
Do CPM 8.1.3 problems 8-30 to 8-36. Tell students that they are learning about 2 new
ways to represent data, shape and spread.
Resources
Core Connections, Course 1, CPM 8.1.3
Problems 8-30 to 8-36
Introduction to Standard Deviation worksheet
For the section about shape, add “skewed right and skewed left” to the descriptions.
Provide the CPM glossary as a resource for students who are not familiar with certain
words. Also, add this to the toolkit and remind students that skewed data should use the
median as a measure of center since the extreme large and small values will impact the
mean more than the median.
Tell student to get a checkpoint after 8-35 so they have the appropriate understanding to
do 8-36.
After students are done, review meaning of absolute mean deviation. Then, pass out the
standard deviation worksheet to teach students how to find the standard deviation.
You can take the students through this as a whole class or have students work in pairs to
decipher how to find the standard deviation. After this, Students will be able to calculate the
standard deviation from the calculator, but this teaches them where it comes from.
At the end of this worksheet, student should realize that balanced and symmetric
distributions have a mean and median that are pretty close in value while skewed data will
have different values (connect back to the effect of an outlier).
At the end of the lesson, put “standard deviation” in the toolkit.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Notes:
Teaching the standard deviation may require 2 days or a block day.
Lesson Overview – Days 4-5
Description of Lesson:
For day 4, Start with mini-lesson “How to Make a Boxplot” to introduce students to the
notion of constructing a boxplot and how to find quartiles and the IQR.
Do CPM 8.1.4 problems 8.44 to 8.46. Students should work in groups on these problems.
The objective is to represent data graphically in a different way utilizing the boxplot that
they’ve just learned about.
Resources
Making a Boxplot - SFUSD Grade 9 Mathematics Core Curriculum
Development Team, June 2014
Core Connections, Course 1, CPM 8.1.4
Problems 8-44 to 8-49
Lesson 8.1.4 Resource Page: High-Temperature Data
For day 5, do CPM 8.1.4 problems 8-47 to 8-49. These set of problems allows more
practice on how to construct a boxplot and how to interpret data from a box plot.
These two lesson days focus mainly on the construction of boxplots and the interpretation
of data. During the conclusion of the lessons, put boxplots in their toolkits or learning logs.
Notes:
The mini-lesson may require some direct-instruction.
Lesson Overview – Day 6
Description of Lesson:
Do CPM 8.1.5 problems 8-61 to 8-64. Focus on Histograms and boxplots. It is up to the
teacher’s discretion on how much focus is emphasized on stem-leaf plots.
Resources
Core Connections, Course 1, CPM 8.1.5
Problems 8-61 to 8-64
The objective of this lesson is to help students develop a sense to use the appropriate tool
to best represent the data. Students will either use a boxplot or histogram to solve the
problem and interpret the data.
Notes: This website can be used to give students more practice matching histograms with
boxplots
http://higheredbcs.wiley.com/legacy/college/mann/0470444665/applets/applet_01_v4.html
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Lesson Overview – Day 7
Description of Lesson:
Introduce students to the idea of a normal curve with the following video:
https://www.youtube.com/watch?v=Sqq4k50dxbI
Resources
Counters Game Worksheet
Dice
Time: 6:56-9:03
Have students write individually on what they think a “normal curve” is and predict what
things in the real world might create a normal distribution. Have a few students share out.
Pass out Counters Game and have students bet on numbers 2-12 in their group when
adding the sum of 2 dice. Once groups have chosen a strategy, play a game as an entire
class and roll dice as a teacher and record outcomes on the board (students should also
record on their paper and remove pennies). Stop recording when there is a winner.
Students should make a histogram from the data and then find the theoretical probability of
rolling sums of 2-12. This is the first opportunity to introduce probability very briefly, but
students should have background from 7th grade. The definition is explained in the task,
but you may need to spiral this earlier or present it in the do now.
For the debrief, teach students how to find the standard deviation and the mean from the
calculator. Then, take notes on how to find a z-score given a value, the standard deviation
and mean. Use this formula to find the probability (using the table of values) of getting a
sum of 3 or less and 11 or more. Students should discuss why these are identical.
Notes: You do not have to introduce probability here, but it will come up again in lesson
series 2 and again in Advanced Algebra. The standards do not explicitly state that we need
to discuss probability in the context of relative frequencies, but we decided to use this
language in order to prepare students for future statistics content.
Lesson Overview – Day 8
Resources
Description of Lesson:
The objective of this lesson is to have students be able to find the z-score of a value in a
population. Have students work in groups to work through the activity “Height of Basketball
Height of Basketball Players (adapted from
http://www.doe.virginia.gov/testing/solsearch/sol/math/A/m_ess_a9_.pdf
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Players.” Once most teams have reached the conclusion, this would be an ideal time to
lead a class discussion about what it means for a value to be certain number of standard
deviations away from the mean. From there, direct students to a z-score table to lead a
discussion on to read a z-score table and what do the numbers mean (percentage of the
area under the normal curve).
Z-score table (from
http://www.utdallas.edu/dept/abp/zscoretable.pdf
Have students calculate the z-scores for the elements 69 and 78, using the formula. Ask
why one z-score is positive and one is negative. If students used a positive z-score for the
last question on the activity sheet, go back and address this now
Have students look at the data and determine how many z-scores would be positive and
how many would be negative. Have them explain why this makes sense.
Ask students whether they could figure out an element of a data set if they knew only its zscore.
Ask what additional information they would need. Help them see how to use the formula
and solve for x.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Apprentice Task
Matching Histograms and Boxplots
What will students do?
Mathematics Objectives and Standards
Math Objectives:
● Students
●
●
●
will be able represent univariate data with dot
plots, histograms and boxplots and bivariable data with
scatterplots and two-way frequency tables
Students will be able perform numerical analyses of the
data using shape, center and spread and interpret the
significance of those elements within context; students
know that the median is a more appropriate measure of
center for skewed data
Students will be able to estimate population percentages
using the area under the normal curve and z-scores
Students will be able to compare two distributions of data
and determine which distribution is most likely to contain a
particular data point
Framing Student Experience
Launch: Today you will use your knowledge of distributions to match different
graphical representations of data and to discuss shape, center and spread. You
may use your notes, toolkits and any previous work to help you.
We will be using a participation quiz to make sure that you are staying together
and giving justifications for your answers.
During:
Record justifications that you hear in each group. Give points for use of
vocabulary and sticking together before moving to a different match.
At the 1st checkpoint, tell students if there is an error, but walk away so they
can correct it. If correct, do a shuffle quiz to ask one person from the group to
defend a match.
S-ID 1, S-ID 2, S-ID 3, S-ID 4
For the 2nd checkpoint, do another shuffle quiz to have one person defend
which measure of center and spread they used depending on the shape. Make
sure students are creating parallel boxplots and labeling their scale as they go.
Again, write down when groups use vocabulary.
Potential Misconceptions
Closure/Extension:
CCSS-M Standards Addressed:
●
●
●
Students may combine the data and create 1 boxplot for both classes.
Instead of subtracting the percent from 100% to find the area to the
right, students may just use the percent reported in the table, which is
the percent of student who would be shorter.
Students may forget that skewed data required the use of IQR and
median since extreme values will affect the mean more significantly
As an extension, ask students to find the height that would be taller than 89.8%
of the population.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Matching Histograms and Boxplots
How will students do this?
Focus Standards for Mathematical Practice: 2) Reason abstractly and quantitatively; 7) Look for and make use of structure
Structures for Student Learning: Toolkits, roles, previous work
Academic Language Support:
Vocabulary: outliers, shape, center, spread, quartile, boxplot, IQR, standard deviation, mean, median
Sentence frames: The shape of the distribution is...because…; I will use the median and IQR because…; I will use the mean and the standard deviation
because…; The value...is an outlier because….
Differentiation Strategies:
Participation Structures (group, partners, individual, other):
Group of 4 with roles and checkpoints
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015
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Lesson Series #2
Lesson Series Overview: Students find joint, marginal and conditional relative frequencies from a 2-way table, fit a function to linear and quadratic data, and
find and analyze the residuals in order to determine the fit of a function
CCSS-M Standards Addressed: S-ID 5, S- ID 6a, S- ID 6b, S-ID 6b
Time: 7 days
Lesson Overview – Day 1-2
Description of Lesson:
Resources
A Sweet Task
(adapted from STEW, http://www.amstat.org/education/stew/)
Do Now: Create a one-way table for the neighborhoods students live in (or you can choose
another identifying characteristic for which each student only belongs to one category) for your
classroom. Include the total, but leave one of the neighborhoods blank. Ask students to find the
ratio of students who live in certain neighborhoods.
A Sweet Task Exit Ticket
(from STEW, http://www.amstat.org/education/stew/).
In the debrief of the Do Now, talk about how the ratio also represents the probability if a student
from the class is chosen at random. (Probability is not mentioned in this unit’s standards, but it
can be introduced here so students are more familiar with it in Advanced Algebra).
Show reported statistics for the distribution of colors for M&Ms and Skittles using the website:
http://www.exeter.edu/documents/mandm.pdf
We are going to use statistics to see if this claim is true.
Pass out Sweet Task, 1 per student. Groups show work together to fill out the table and sort
candy. Teacher needs to collect data from all group and report class data for the 2-way table.
Put joint, marginal and conditional probability in toolkit towards the end of class. Ask groups to
determine if they think the M&M company’s claim is true based on their individual vs. whole
class data.
Pass out exit ticket and give 10 minutes for students to work individually.
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Notes:
This lesson may take 2 days or one block period. A good place to stop would be before the
conditional probabilities.
Lesson Overview – Day 3
Description of Lesson:
Students will need more practice analyzing two-way tables and joint, marginal and conditional
frequencies. Tell them that this is a new skill, so mistake and confusion are ok and help with the
learning process.
Resources
CPM CCA 10.1.1
Problems 10-8 to 10-9
Do a Do Now that review changing fractions to percents since students will need this for the
task.
Pass out Two-way table practice. Students should work in pairs. Remind students of the given
notation, where the given comes after the line. Students should use toolkits and notes from the
previous day to help.
The second part of the task ask students to calculate percents and determine if there is an
association between the variables. Popular has no association since the percents remain pretty
much the same. Students may think that grades and athletic have an association, even though
there is no clear pattern. Let me share their ideas and compare to the high school table where
the association is much more obvious.
When student are finished, they should work on problems 10-8 and 10-9 from CPM Core
Connections Algebra Statistic Supplement.
Lesson Overview – Day 4
Resources
Description of Lesson:
The objective of this lesson is an introduction to determine association between two variables
of data and informally find the line of best-fit.
How High Can You Jump Worksheet from STEW http://www.amstat.org/education/stew/pdfs/HowHighCanYouJ
ump.pdf
Prior to this lesson, designate an area big and tall enough for students to jump. Teacher may
want to place markers on the wall before this activity so that students can get a more accurate
CPM CCA Chapter 6 Methods and Meanings Resource
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reading for their data. Students will work in groups on this worksheet. Groups will gather data
about their jump heights. After each group has completed gathering data, have the class come
together, and write down the totals for every as a class total. Have each student/group
construct boxplots, compare the data, and have them draw conclusions.
Have each student/group draw a scatterplot and complete the worksheet. Draw the class back
together and discuss what kind of relationship exists between the student heights and jump
heights. Is that relationship positive? Negative? Neutral? Can they fit a line through the data
points? How can they find that line? How many different lines can there be?
Teacher may want to model this.
Lesson Overview – Day 5
Resources
Description of Lesson:
This lesson begins with groups working on the the Pizza Party problem. This problem
introduces the concept of a negative association between two variables. Students should try to
find the equation of the line of best-fit. After groups have completed the worksheet, lead a
discussion about the slope and y-intercept of the line of best-fit. Why are they important? What
do groups think the slope and intercept mean? Can they use this information to predict a value?
Pizza Party (adapted from Statistics Through Applications 2nd
Edition, W.H. Freeman and Company, 1996, Chapter 4, p.
173)
CPM CCA 6.1.1
Problem 6-4
The second half of the day is from CPM problem 6-4. This problem gives more practice on
finding the line of best fit, interpreting the data, and predicting a value.
Lesson Overview – Day 6
Description of Lesson:
The objective of this lesson is to study residuals. Have groups work on CPM problems 6-10
through 6-14. Residuals are defined from the beginning and students will have the opportunity
to see the physical representation of residuals as the distance of a point from the line of best fit.
Resources
CPM CCA 6.1.2
Problem 6-10 through 6-14
Recommended HW: CCA 6-16 and 6-24
6-13 gives the students practice with residuals and continues to spiral in the interpretations of
slope and y-intercepts.
6-14 brings in the notion of negative residuals and what do they mean.
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Lead a class discussion at the end to summarize the definition of residuals and to ensure that
all students know how to calculate them.
Notes:
A 6-page Methods and Meanings handout that includes all the notes from the CPM Statistics
Supplement for Core Connections Course 1 is included in the Resources folder. This might be
a resource you provide to students.
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Expert Task
How Strong is the Association Between Income and Race?
What will students do?
Mathematics Objectives and Standards
Framing Student Experience
Math Objectives:
● Students will be able to represent univariate data with dot plots,
histograms and boxplots and bivariate data with scatterplots and twoway frequency tables; Students will be able to recognize associations
and trends in the two-way tables
● Students will be able to perform numerical analyses of the data using
shape, center and spread and interpret the significance of those
elements within context; students know that the median is a more
appropriate measure of center for skewed data
● For a linear function, Students will be able to interpret slope and yintercept in context.
Launch: Today we will investigate our own neighborhoods in San Francisco in
order to determine if there is an association between median household income
and race.
CCSS-M Standards Addressed:
S-ID 1, 2, 3, 5, 6b, 7
*Since this is the end of the year, hopefully there will be norms and ground
rules that allow students to share their ideas safely, without judgement. If not, it
may be helpful to set explicit ground rules before this conversation:
confidentiality, respect the experience of others, one mic, etc.
Remind students about the different graphical representations--boxplot,
scatterplot, histogram--and the different parts of the equation of the line of best
fit.
Potential Misconceptions
●
Do Now: Have students do a quick write about their impression of the
relationship between race and income. They should pair share with a partner
and then ask for volunteers to share with the whole class.
Students may switch the values of x and y for the line of best fit.
Students may use the wrong representation for univariate vs bivariate
data. Students may read the given incorrectly when finding conditional
probabilities.
Students should be able to use their toolkits and any previous assignments that
will help them with the two-way tables or residual plotting.
During:
Similar to the apprentice task, the teacher can conduct this as a participation
quiz where groups are given points for using their role and staying together.
Pay particular attention to vocabulary usage and justifications.
Before the first checkpoint, make sure the student choose an appropriate
graphical display. At the first checkpoint, do a shuffle quiz to ask students to
defend one of the following: shape center, spread. Before the second
checkpoint, make sure the scale of the scatterplot will allow students to see the
association. Since the income value may be larger than typical value, you can
also give them a hint or provide the scale for them. At the second checkpoint,
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do another shuffle quiz to have students defend the strength and direction of
the association using the scatterplot. Before the third checkpoint, make sure
students are plugging in percent (1 instead of 0.01) for x and calculating the y,
median household income. You can also provide the formula for residual
(actual - predicted) or have them reference their notes if they forget. At the third
checkpoint, do a final shuffle quiz for the interpretation of the residual plot or the
outliers--whichever you think needs the most emphasis for your particular class.
The last part goes back to two-way tables in context and is not the most
important part of this task since student knowledge was assessed at the
beginning of lesson series 2. Think of this as more of an extension and review
of previous material.
Closure/Extension:
Compare the statistical analysis of the data with the pre-write at the beginning
of class and students experience with different neighborhoods of San
Francisco. You can also have students hypothesize about the association
between median household income and other races in San Francisco--Black,
Asian, Latino.
You can also show this video to highlight the dire state of wealth inequality in
the USA: https://www.youtube.com/watch?v=LlYojsi3Zqw
It uses good vocabulary like “quintile” (connect to quartile) and can be related
back to the state of wealth inequality in San Francisco.
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How Strong is the Association Between Income and Race?
How will students do this?
Focus Standards for Mathematical Practice: 3) Construct viable arguments and critique the reasoning of others (Mayor Lee); 6) Attend to precision
Structures for Student Learning: Think-pair-share before task; participation quiz during assessment; shuffle quiz at checkpoints
Academic Language Support:
Vocabulary: residual, line of best fit, slope, y-intercept, association, shape, center, spread, quartile, conditional probability, validity,
Sentence frames: We chose to use a ... (graphical representation) because…; % White and median household income have a ….association
because…; The slope means…; The y-intercept means…; The residuals do/do not provide convincing evidence for the linear model because…; The
outliers are...because…; The real world supports this conclusion because…; The table is/is not correct because…; Mayor Lee’s conclusion is/is not
correct because...
Differentiation Strategies: The last part of the task can be used as an extension for the students who finish early; students can also be asked to plot the nonwhite percent vs. median household income
Participation Structures (group, partners, individual, other): individual and pair for pre-write; group for task
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Lesson Series #3
Lesson Series Overview: The objective of this lesson series is to go more in-depth with the statistical analysis of a set of data. Students find the Least Squares
Regression Line, do Residual Plots, find the Correlation Coefficient, differentiate between causation and correlation, and fit a parabolic line to a scatterplot.
CCSS-M Standards Addressed: S-ID 7, S-ID 8, S-ID 9
Time: 8 days
Lesson Overview – Day 1-2
Description of Lesson:
Day 1 - Students will use graphing calculators to find the equation of a Least Squares
Regression Line. Begin with problems CPM 6-30 to 6-31. These problems should be done in
groups. After giving them time to work on the problems, lead a class discussion on residuals
and the notion about how residuals is the distance from the predictions are to the actual
observed value. Keep in mind about the sum of the residuals, positive residuals, and
negative residuals. Ensure that every group finds the sum of the residuals for this part of the
discussion to happen. If possible, continue on to 6.32.
Resources
Core Connections, Algebra/Integrated 1, CCA CPM 6.1.4
Problems 6-30 through 6-33.
Lesson 6.1.4 Resource Page
Day 2 - Have groups work on CPM problems 6-32 to 6-33. The objective of this lesson is to
show students how to use graphing calculators to find the LSRL and residuals.
Notes:
It is recommended that on Day 2, to spend a few minutes to review the commands and
functions of the graphing calculator.
Lesson Overview – Day 3-4
Description of Lesson:
Day 3 - Have groups work on CPM CCA 6-47 to 6-50. The objective of this lesson is to
practice generating scatterplots and LSRL’s with a graphing calculator. Students will be able
to interpret these graphs and be able to connect that with a separate graph that shows the
distances of the residuals from the LSRL. If you can not fit 50 into Day 1, do it on Day 2.
Resources
Core Connections, Algebra/Integrated 1, CCA CPM 6.2.1
Problems 6-47 through 52, 6-55 a, d, e, f.
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Day 4 - Have groups work on CPM CCA 6-51 to 6-52 and 6-55 a, d, e, f. Students will
calculate residuals by hand and sum them up, which should equal zero. This will solidify the
concept that was eluded to on Day 1. Model how to generate a residual plot for students. The
last few problems are practice.
Notes:
Ensure that students read or summarize methods and meanings on LSRL
Lesson Overview – Day 5-6
Description of Lesson:
Resources
Core Connections, Algebra/Integrated 1, CCA CPM 6.2.2
Problems 6-67 to 6-69, 6-71, 6-73, 6-85, 6-86.
Day 5 - Have groups work on CPM prbol6-67 to 6-69, 6-71, 6-73.
Students will be able to calculate the correlation coefficient and observe the scatter for
various extremes of r. Students will describe an association between two variables in more
mathematical terms.
The correlation coefficient, r, is an “arbitrary” computation of the strength of linear
dependence between the variables—the strength of the linear association. Unlike slope,
which has a real-world meaning as the rate of growth, the correlation coefficient does not
have an obvious interpretation in an everyday context.
As a wrap up of ideas or when circulating, check in with students to see if they noticed that as
the data points get closer to the line of best-fit, the value of r gets closer to 1 or -1.
Day 6 - Have groups work on CPM problems 6-85 and 6-86. These problems give students
more practice with LSRL, Residual plots, and finding r.
Lesson Overview – Day 7
Description of Lesson:
Have groups work together on CPM 6-79 through 6-83. The objective of this lesson is to have
students better understand that cause and effect can’t be determined from a study that
reports an association. In other words, association is not causation.
Resources
Core Connections, Algebra/Integrated 1, CCA CPM 6.2.3
Problems 6-79 through 6-83
Correlation is a computation of the mathematical coefficient. The word “association” is akin to
relation. For example: “There is a negative association between the number of cars on the
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road and the quality of the air. The correlation coefficient is -0.4” A non-example would be
“There is a negative correlation between the number of cars on the road and the quality of
the air.”
Notes:
EL students may struggle with the vocabulary and discussion starters for these activities. It is
highly suggested that sentence frames/starters be made available or discussed at the
beginning of this lesson.
Recommended HW 6-99 and 6-100.
Lesson Overview – Day 8
Description of Lesson:
Have groups work together on CPM 6-105 a, b, c to 6-106 and 6-111. Students will explore
an example of a non-linear line of best-fit. You may want to have students try to fit a line of
best-fit by hand first before going to the technology. They will quickly realize that a linear line
will not always be the line of best fit. The teacher may pose the question, “If a linear line of
best-fit does not work, what shape should the line of best-fit be?” The checkpoint is for
students to see that a quadratic function is ideal for that particular situation.
Resources
Core Connections, Algebra/Integrated 1, CCA CPM 6.2.5
Problems 6-105 a, b, c to 6-106
Review Problems
6-111, 6-122, 6-123 a, b, 6-124, 6-127
Problem 6-111 is a review on causation.
If you have time suggested review problems are 6-122, 123 a,b, 6-124, and 6-127.
Notes:
Ensure that all TI graphing calculators are set to FLOAT to prevent excessive rounding of
numbers.
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Milestone Task
Who Stole the iPhone?
What will students do?
Mathematics Objectives and Standards
Framing Student Experience
Math Objectives:
● Students will be able to perform numerical analyses of the data using
shape, center and spread and interpret the significance of those
elements within context; students know that the median is a more
appropriate measure of center for skewed data
● For a linear function, Students will be able to interpret slope and yintercept in context
● For a linear function, Students will be able to compute (with
technology) and interpret the correlation coefficient, r, using strength
and direction in context
● Students will be able to compare two distributions of data and
determine which distribution is most likely to contain a particular data
point
Launch: Today, you will use your knowledge of linear models and shape, center
and spread to figure out who stole an iPhone!
CCSS-M Standards Addressed:
For part 2, the teacher should remind students to construct parallel boxplots so
they can compare the distributions.
S-I 2, 3, 5, 8.
Potential Misconceptions
●
●
As part of a do now, review different types of association--strong, weak, positive
and negative. It may also be a good idea to review how to input data in the
graphing calculator (or other device) and calculate the r value and line of best
fit.
During:
Since this is an individual task, students should be working quietly. The teacher
should make sure students are providing at least 3 justifications for part 1 and
pushing them to explain their reasons in a clear, thorough way.
Closure/Extension:
Students may think that all students could be the thief, but there are
only 4 possibilities--Pat, Pablo, Quan and Maurice--since these
backpack weights are unknown.
Students may think it is sufficient to state the strength and direction of
the association, but must use the actual line or the equation to
determine which students owns the backpack.
After collecting the task, take a poll of the class to see who chose which thief.
Or, share methods that you saw students using during class to expose all
students to different approaches.
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Who Stole the iPhone?
How will students do this?
Focus Standards for Mathematical Practice: 1) Make sense of problems and persevere in solving them; 2) Reason abstractly and quantitatively
Structures for Student Learning: toolkits, word wall, visual aids (posters)
Academic Language Support:
Vocabulary: positive/negative, strong/weak association, scatterplot, boxplot, correlation, residual, slope, y-intercept, distribution
Sentence frames:
I think _________ stole the iPhone because _______________.
My conclusion makes sense because ___________________.
Differentiation Strategies:
The number of justifications can vary per student.
Participation Structures (group, partners, individual, other): individual
SFUSD Mathematics Core Curriculum, Algebra 1, Unit S.1: Categorical and Quantitative Data, 2014–2015