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Digging Deeper Into Mathematics

Clayton County Summer Math AcademyAlgebra 1 – Curves of Best Fit

Sarah LedfordJune 15, 2015

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Falling DominoesWhat mathematical questions can you ask about falling dominoes?

How could you design experiments to answer your questions?

Do you think your experiment would answer your question?

Can you use your experiment to make predictions & test them?

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Falling Dominoes

Our experiment will investigate the relationship between the number of dominoes spaced one inch apart in a straight line and the time it takes for all the dominoes to fall down.

We will need timers!!

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Falling DominoesWe’ll begin by sticking a piece of masking tape (about 3 feet) to the floor or the table, and marking it off every inch.

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Falling DominoesNext, line up dominoes at every mark to match each row of the data collection table.

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Falling DominoesComplete the table in groups.

Why is there a column for “mean time?”

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Falling DominoesGraph the data

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Falling DominoesShould your graph go through the origin? Why or why not?

Use a piece of raw spaghetti to place on your graph. Try to determine the line of best fit. This line should go through as many of your data points as possible, and should have roughly the same number of points below the line as above the line. Use a ruler to draw your line of best fit.

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Spaghetti Regression

Why do we say that the line in the top graph fits the points better than the line in the bottom graph?

Can we say that some other line might fit them better still?

Usually we think of a close fit as a good fit. But, what do we mean by close?

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Falling DominoesDetermine the equation of your line of best fit.

Use the equation of your line of best fit to predict how long it would take to knock down a line of 60 dominoes.

Test your prediction by building a line of 60 dominoes and timing how long it takes for all of them to knock down. How good was your prediction?

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Calculator

Let’s find the line of best fit using our calculators.

We are going to enter the data on our scatterplot so we need the ordered pairs.

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Calculator

To enter data in lists:• “stat”• “edit”• enter data in L1 (inputs) & L2 (outputs)

• To show data on scatterplot:• “y=“ highlight “Plot1”

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Calculator

To do regression on data:• “stat” “calc” Choose the kind of

regression you want to do.• We want linear so “Lin Reg(ax + b)” is a good

option for us.Let’s talk about what we see on our screens.

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Calculator

In order to be able to graph equation on the scatterplot as an example, we want to type “LinReg L1, L2, Y1”• L1 can be typed with “shift” “1”; L2 is “shift”

“2”; etc.• Comma is beside x2 button• Y1 is found in “vars” ”y-vars” “function”

”Y1”

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Calculator

To get the correlation coefficient:• go to “catalog” • scroll down to “diagnostic on,” • hit enter • make sure your calculator says “done”

The closer to 1, the better the fit!

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Hmmm…

I surveyed my students by asking them how long they studied for an exam. Then I made a scatter plot showing the relationship between minutes

spent studying for an exam and the exam results.

What do you think this scatter plot looked like?

Good questions for math teaching: Why ask them and what to ask grades 5–8. Math Solutions Publications: Sausalita, CA. Schuster, L. and Andeson, N.C. (2005). Pg. 127.

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Texting and Grades IIhttps://www.illustrativemathematics.org/illustrations/1028

Medhavi suspects that there is a relationship between the number of text messages high school students send and their academic achievement. To explore this, she asks a

random sample of 52 students at her school how many text messages they sent yesterday and what their grade point

average (GPA) was during the most recent marking period. Her data are summarized in the scatter plot on the next

slide. The least squares regression line is also shown.

17REFERENCE THE source (GaDOE

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Texting and Grades II

18REFERENCE THE source (GaDOE

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Texting and Grades II

The equation of the least squares regression is GPA = 3.8 – 0.005(Texts sent).

Interpret the quantities -0.005 and 3.8 in the context of these data.

19REFERENCE THE source (GaDOE

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Task Review

• What GSE (content) was addressed?– Domain, cluster, and standards

• What SMPs were addressed?

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Algebra I GSE

S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S.ID.8 Compute (using technology) and interpret the correlation coefficient “r” of a linear fit.

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6-8 GSE6.SP.4 Display numerical data in plots on a number line, including dot plots (line plots), histograms, and box plots.

8.SP.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.

ADD THESE!!

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6-8 GSE8.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 a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

ADD THESE!!23

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Price vs. Quantity

A manufacturer wants to examine the relationship between the price of one of its products (x) and the quantity demanded (y).

The scatterplot here shows the results and the line of best fit y = -5.8x + 105.7.

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Price vs. Quantity

Using the equation for this line of best fit, what is the number of units demanded at a price of $15?

y = -5.8x + 105.7

A. 25,000 B. 20,000

C. 15,000 D. 30,000

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Price vs. Quantity

Which equation best describes this data?

A. y = 7.2x – 14.4

B. y = 7.2x + 1.6

C. y = 0.12x – 8.16

D. y = 0.12x + 0.28

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EOC like question

A positive correlation always exists on a scatter plot when A. y remains unchanged as x increases.B. y changes randomly as x increases.C. y decreases as x increases.D. y increases as x increases.

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Non-linear data?

What if our data isn’t linear?

We can use regression to find several different kinds of equations & then determine which we think is best based on the correlation coefficient, r.

Using our calculators we can do exponential, quadratic, cubic, trigonometric, logarithmic, etc., regression.

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Water Fountain

We can find an equation that couldtell us the trajectoryof the water.

What does the shapeof the water remind you of?

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Water Fountain

We need to find a few ordered pairs that lie on the stream of water, put them in a table, & do regression…

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Model with Mathematics…

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Model with Mathematics…

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Model with Mathematics…

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Model with Mathematics…

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Model with Mathematics…

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Model with Mathematics…

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Model with Mathematics…

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Olympic Pole Vault

Let’s look at the data.

What kind of model do you think will best fit the data?

This is REAL LIFE MODELING!!

https://www.nsa.gov/academia/_files/collected_learning/high_school/algebra/beginning_modeling.pdf

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Olympic Pole Vault

Let’s look at the data.

What kind of model do you think will best fit the data?

This is REAL LIFE MODELING!!

https://www.nsa.gov/academia/_files/collected_learning/high_school/

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Spaghetti Regression

From GaDOE CA Unit 4 Framework

We need to discuss “good fit.”

Handout & 3-5 pieces of spaghetti.

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Spaghetti Regression

On your own:On the scatter plot, tape a piece of spaghetti to represent the line of best fit.

In your group: Decide who in your group has the best line of best fit.

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Spaghetti Regression

On your own:Using another piece of spaghetti, measure the distances between each data point and your line. Break off the spaghetti to represent each distance.

In your group: Decide how your group will measure this distance.

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Spaghetti Regression

There are at least three ways to measure the space between a point and the line:

vertically in the y direction,

horizontally in the x direction, and

the shortest distance from a point to the line (on a perpendicular to the line.)

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Spaghetti Regression

Jfd

In regression, we usually choose to measure the space vertically. These distances are known as

residuals.

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Why would you want to measure this way? What do the residuals represent in relation to our function?

Consider the purpose of the line and the following diagram.

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The vertical distance represents how far off the prediction is from the actual data poin.t Residual = actual – predicted = yi −f(xi)

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Spaghetti Regression

On your own: Measure the total length of your spaghetti pieces that represent the residuals.

In your group:Determine who actually has the closest fit.

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Spaghetti Regression

Formative Assessment Questions:

• How did you decide where to put your line of best fit on the scatter plot?

• How did you choose measure the distance from each point to the line of best fit? Why?

• What do the residuals tell you about your line of best fit?

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GA DOE Implementation Resourceshttps://www.georgiastandards.org/Common-Core/Pages/Math.aspx

– Common Core Tools – Schedule of Unit webinars– Recordings:• Standards for Mathematical Practices• Grade level overviews• Unit webinars

– List Serve (subscribe to receive emails from DOE)

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GA DOE • GaDOE Wiki – Grades 6-8• Please visit http://ccgpsmathematics6-8.wikispaces.com/ to

provide feedback, ask questions, and share ideas and resources.

• Multi-Grade Resources: http://ccgpsmathematics6-8.wikispaces.com/Multi+Grade+Resources

• Secondary Mathematics Specialist• Brooke Kline – bkline@doe.k12.ga.us

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GA DOE • GaDOE Wiki – Coordinate Algebra & Analytic Geometry• Please visit http://ccgpsmathematics9-10.wikispaces.com/ to

provide feedback, ask questions, and share ideas and resources.

• Multi-Grade Resources: http://ccgpsmathematics9-10.wikispaces.com/Multi+Grade+Resources

• Secondary Mathematics Specialists• Brooke Kline – bkline@doe.k12.ga.us

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Implementation Resources

• Progressions – “offer explanations for the sequence of the standards, potential cognitive difficulties, and pedagogical solutions which may be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks.”http://ime.math.arizona.edu/progressions/

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Implementation Resources

• Illustrative Mathematics Project – “illustrates the range and types of mathematical work” http://illustrativemathematics.org/

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Implementation Resources

• The Mathematics Common Core Toolbox – “This site is a resource designed to support districts working to meet the challenge and the opportunity of the new standards. Here you will find tools and instructional materials that help you to better understand and to implement the CCSSM.

• …created through a collaboration of the Charles A. Dana Center at the University of Texas at Austin and Agile Mind with partial funding from the Bill & Melinda Gates Foundation.”

http://www.ccsstoolbox.org/

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Implementation Resources

• Inside Mathematics – “a professional resource for educators passionate about improving students' mathematics learning and performance. This site features classroom examples of innovative teaching methods and insights into student learning, tools for mathematics instruction that teachers can use immediately, and video tours of the ideas and materials on the site.”

http://insidemathematics.org/index.php/home

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Implementation Resources

• Engage NY – A comprehensive collection of units of instruction and assessments to support the implementation of the Common Core State Standards in New York State.

• https://www.engageny.org/

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Implementation Resources

• Mathematics Assessment Project – “The project is working to design and develop well-engineered assessment tools to support US schools in implementing the Common Core State Standards for Mathematics.”

• http://map.mathshell.org/

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Implementation Resources

• Math Common Core Coalition – “The Mathematics Common Core Coalition works to provide expertise and advice on issues related to the effective implementation of the Common Core State Standards for School Mathematics (CCSSM).”www.mathccc.org

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Implementation Resources

Achieve the Core• Free resources • Founded by one of the contributing authors of

the CCSS (Student Achievement Partners), now devoted to successful implementation

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Assessment Resources

• Initial Set of Test Items and Task Prototypes Released

• http://www.parcconline.org/initial-set-test-items-and-task-prototypes-released

• http://www.smarterbalanced.org/sample-items-and-performance-tasks/

• Georgia Online Formative Assessment Resource (GOFAR)

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Outcomes:• Dig into the domains to determine the

development of the content across the grade levels

• Work tasks that help develop conceptual understanding

• Discuss the integration of the Standards for Mathematical Practice (SMPs)

• Identify teacher actions to develop content and mathematical practices while addressing TKES

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Now What?• What does this mean for me and my students?

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