Algebra 1 Workbook — Chapter 19: The Data Cycle: Scatterplots and Curve of Best Fit

SOL A.ST.1 (a–i) · Companion to Textbook Chapter 19

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 124.

PAGE 1 — Chapter opener

Chapter 19 · The Data Cycle: Scatterplots and Curve of Best Fit

Standard A.ST.1 (a–i)

In this chapter you will:

Apply the data cycle to bivariate questions
Formulate investigative questions that need two paired measurements
Choose variables and an appropriate sampling method (including SRS)
Analyze associations in a scatterplot
Use technology to choose a linear or quadratic curve of best fit (≤30 points)
Predict with a linear model and interpret slope and $y$ -intercept in context
Communicate conclusions without claiming causation from association alone

Words to know: data cycle · bivariate · investigative question · independent variable · dependent variable · simple random sample · scatterplot · association · curve of best fit · linear regression · quadratic regression · interpolation · extrapolation · lurking variable

Technology is required for regression. Coefficients are rounded to the nearest hundredth. Association is not causation.

PAGE 2 — The data cycle

19.1 The Data Cycle and Bivariate Questions

FIGURE: fig1-data-cycle-bivariate-questions.png (full width)

Fill in the blanks.

The four stages are: ____________ questions · ____________ or acquire data · organize and ____________ data · ____________ data and communicate results.

A data set is bivariate when ______ quantitative measurements are recorded for each item and kept ____________.

Name the four stages in order.
1. _______________ 2) _______________ 3) _______________ 4) _______________
What makes a data set bivariate? _______________________________________________
Which stage is writing a question that requires two paired measurements?

PAGE 3 — Bivariate questions

Writing Investigative Questions

Is "How tall are the players on the basketball team?" bivariate? ______

Explain: _______________________________________________
Rewrite "Does advertising work?" as a bivariate investigative question:
Correct the statement. "The data cycle ends once technology produces a regression equation."
Name the stage for each action.

Action	Stage
a) Measuring length and mass of 16 rods
b) Writing a seedlings days-vs-height question
c) Displaying a scatterplot on a calculator
d) Reporting a linear model from technology

Put in order: organize and represent; analyze and communicate; formulate questions; collect or acquire.

PAGE 4 — Practice · questions

Practice · Univariate or Bivariate

Univariate or bivariate? If univariate, rewrite as bivariate.

Question	Type	Rewrite if needed
a) Minutes students exercise each day
b) Resting heart rate vs age (clinic)
c) Mass of each backpack
d) Mass vs number of books in backpacks

Apply it. Write one bivariate question your class could answer in a week.
Reasoning. Why does "happiness and success" fail as an A.ST.1a question?
Error analysis. Heights in one column, arm spans shuffled in another, then linear regression. Error:

PAGE 5 — Exit ticket 19.1

Exit Ticket 19.1

Four stages: _______________ · _______________ · _______________ · _______________
Bivariate question about practice time and free throws:
Choosing an SRS of 20 students and recording two measurements — which stage?
Why must the two measurements stay paired?

PAGE 6 — Variables and sampling

19.2 Choosing Variables and Sampling

FIGURE: fig2-simple-random-sample-diagram.png (full width)

Fill in the blanks.

The ____________ variable ( $x$ ) goes on the horizontal axis. The ____________ variable ( $y$ ) is the response.

In a simple random sample, every member of the population has the ____________ chance of being chosen.

Car age and resale value: horizontal axis variable _______________ Name: _______________
Two variable pairs for "more sleep helps school performance":

a) _______________ b) _______________
What is a simple random sample?
Teacher picks first 20 students who walk in. SRS? ______ Why?

PAGE 7 — Sampling practice

Sampling Methods

Club of 60 wants SRS of 12. How, using technology?
Why does A.ST.1 care whether a sample is representative?
Name $x$ and $y$ for each.

Situation	$x$	$y$
a) Fertilizer and tomato yield
b) Temperature and hot chocolate cups
c) Weeks of advertising and weekly sales

Survey only soccer teammates about phone use and sleep, generalize to all teens in the state. Critique:

PAGE 8 — Sampling application and exit

Apply · Sample Design · Exit 19.2

Apply it. 50 seedlings; measure 20 for days vs height. Sampling method:
Apply it. One strength and one weakness of using a public data set:

Strength: _______________ Weakness: _______________
Reasoning. Why is "any sample of size 30 is automatically representative" false?
Error analysis. "Independent variable is whichever column the calculator plots first." Error:
Hours charged and battery percent: $x$ = _______________ $y$ = _______________
Define SRS in one sentence: _______________________________________________
One reason a cafeteria convenience sample may fail: _______________________________________________
One appropriate method other than SRS: _______________________________________________

PAGE 9 — Positive and negative association

19.3 Analyzing Relationships in a Scatterplot

FIGURE: fig3-scatterplot-linear-association.png (half width)
FIGURE: fig4-scatterplot-negative-association.png (half width)

Study-time association: _______________________________________________
Car-age association: _______________________________________________
Two points that support the study-time positive trend: _______________ and _______________
Car-age result without claiming causation:

PAGE 10 — Curved and none

Curved Patterns and No Association

FIGURE: fig5-scatterplot-curved-association.png (half width)
FIGURE: fig6-scatterplot-no-association.png (half width)

HVAC pattern: _______________ Why is "positive linear" incomplete?
Last-name association: _______________________________________________
Four pattern descriptions: _______________ · _______________ · _______________ · _______________
Car-age: association _______________ Context sentence:
Why might a curved pattern suggest a quadratic model?
"Association is not causation" in your own words + example:

PAGE 11 — Plant data plot

Practice · Plant Heights

FIGURE: fig12-blank-scatterplot-practice-grids.png (full width)

Days ( $x$ )	2	4	6	8	10	12	14	16	18	20
Height ( $y$ )	5	8	12	15	19	22	26	29	33	36

Number of pairs: ______ Within 30-point limit? ______
Plot on Grid A. Label both axes with quantities and units.
Association: _______________________________________________
Justification (evidence + context):
Apply it. Correct: "Any U-shaped cloud is no association."
Reasoning. Why can two students with $x = 3$ appear as two stacked points?
Error analysis. "Scatterplots must pass the vertical-line test." Confusion:

PAGE 12 — Technology chooses the fit

19.4 Technology: Linear or Quadratic Fit

FIGURE: fig7-linear-fit-on-curved-data.png (half width)
FIGURE: fig8-quadratic-curve-of-best-fit.png (half width)

Remember: Enter the pairs · display the scatterplot · compare fits with technology · record the equation · round to the hundredth.

What must you use to obtain a curve of best fit? _______________________________________________
Maximum number of points: ______
HVAC linear (technology): $y =$ _______________ Why weak?
HVAC quadratic (technology): $y =$ _______________
If the scatterplot shows no association, conclude:
One acceptable technology tool: _______________________________________________

PAGE 13 — Practice · choosing fits

Practice · Study and Car Fits

Study data — technology linear: $y = 6.99x + 55.27$

Number of points: ______ Within limit? ______
Is linear appropriate? ______ Explain:

Car data — technology: $y = -12.74x + 196.87$

Is linear appropriate? ______ Explain:
Apply it. Two sentences: state HVAC quadratic; reject HVAC linear.
Reasoning. Why "use available technology" instead of hand computation?
Error analysis. Eye-sketch $y = 7x + 55$ labeled "curve of best fit." Skipped:

PAGE 14 — Exit ticket 19.4

Exit Ticket 19.4

30-point limit in your own words:
HVAC quadratic (to the hundredth): $y =$ _______________
When do you report neither linear nor quadratic?
Why might two calculators differ in the hundredths place?

PAGE 15 — Regression models

19.5 Regression Models and Predictions

FIGURE: fig9-linear-regression-equation.png (half width)
FIGURE: fig10-quadratic-regression-equation.png (half width)

Study linear model: $y =$ _______________
Strength: _______________ Weakness: _______________
Advertising quadratic: $y =$ _______________

FIGURE: fig11-prediction-from-linear-model.png (full width)

Phone model $y = 21.88x + 8.67$ . $\hat{y}$ at $x = 3$ : _______________
Interpolation or extrapolation? _______________ Why?
Why is $\hat{y} = 118\%$ at $x = 5$ not valid?

PAGE 16 — Predictions practice

Practice · Predict and Critique

Plant model (technology): $y = 1.74x + 1.33$

Height at $x = 10$ : _______________ Work:
At $x = 30$ : _______________ Interp / extrap? _______________ Validity:
Strength: _______________ Weakness: _______________
Car model $y = -12.74x + 196.87$ . Predict at age $4$ : _______________
Apply it. Manager claims huge sales at advertising week $40$ using the quadratic. Critique:
Reasoning. Good sample fit, poor new prediction — why?
Error analysis. Student gets $6.99 + 55.27 = 62.26$ at $x = 3$ . Correct $\hat{y}$ : _______________ Mistake:

PAGE 17 — Exit ticket 19.5

Exit Ticket 19.5

Phone linear model: $y =$ _______________
$\hat{y}$ at $x = 2$ : _______________ Interpolation? ______
One weakness of any sample regression model:
Why does A.ST.1f focus on linear predictions?

PAGE 18 — Slope and intercept in context

19.6 Interpreting Parameters and Communicating Conclusions

Slope of $y = 6.99x + 55.27$ in context (with units):
$y$ -intercept of that model in context:
Slope of $y = -12.74x + 196.87$ in context:
Caution about the car model's $y$ -intercept:
Three ingredients of a complete conclusion:
1. _______________ 2) _______________ 3) _______________
Rewrite honestly: "Our scatterplot proves that studying causes better grades."

PAGE 19 — Interpret and conclude

Practice · Interpret · Conclude

$y = 1.74x + 1.33$ : slope _______________ intercept _______________
$y = 21.88x + 8.67$ : slope _______________ intercept _______________
Apply it. Full conclusion for the car-age study:
Apply it. Full conclusion for HVAC (include why quadratic over linear):
Reasoning. Why is "the slope is $6.99$ " incomplete for A.ST.1g?
Error analysis. "Car loses $\$12.74$ per year." Correct units:

PAGE 20 — Exit ticket 19.6

Exit Ticket 19.6

Study-model slope in one sentence: _______________________________________________
Phone-model $y$ -intercept in one sentence: _______________________________________________
Sentence to include with an observational association:
Standard letter for communicating conclusions: ______

PAGE 21 — Review · design and scatterplots

Chapter Review · A.ST.1a–c and h

Bivariate question (temperature, lemonade sales):
For that question: $x$ = _______________ $y$ = _______________
SRS of 15 days from a 40-day season:

_______________________________________________

Critique surveying only rainy days:

_______________________________________________

Study-time association + evidence:

_______________________________________________

HVAC pattern + suggested model type:

_______________________________________________

Last-name association: _______________________________________________

PAGE 22 — Review · technology and predictions

Chapter Review · A.ST.1d–g

Study linear (technology): $y =$ _______________
HVAC quadratic: $y =$ _______________ Beats linear because:

_______________________________________________

Advertising quadratic: $y =$ _______________
Study model — strength: _______________ weakness: _______________
Phone model: $\hat{y}$ at $x = 1$ : _______________ Interpolation? ______
Study model: $\hat{y}$ at $x = 4$ : _______________
Validity of study prediction at $x = 15$ :

_______________________________________________

Car model — slope: _______________ intercept: _______________
Plant model — slope: _______________ intercept: _______________

PAGE 23 — Review · communicate and mixed

Chapter Review · Communicate · Mixed

Four-sentence conclusion · phone charging:

_______________________________________________

Four-sentence conclusion · last-name study:

_______________________________________________

Apply it. Outline technology steps for a 30-pair positive linear set:

_______________________________________________

Apply it. 18-pair U-shape — how to choose linear vs quadratic with technology:

_______________________________________________

Reasoning. Why refuse calculator-free regression?

_______________________________________________

Error analysis. Preferring $y = 7x + 55$ over technology's $6.99x + 55.27$ . Violation:

_______________________________________________

PAGE 24 — Review · cumulative

Chapter Review · Cumulative Performance

Plant: model $y = 1.74x + 1.33$ . $\hat{y}$ at day 12: _______________

Slope meaning: _______________  Intercept meaning: _______________

Car: $y = -12.74x + 196.87$ . $\hat{y}$ at age 6: _______________

No-causation caution: _______________________________________________

HVAC linear: $y =$ _______________ HVAC quadratic: $y =$ _______________

Report as curve of best fit: _______________

New bivariate follow-up question from the advertising result:

_______________________________________________

Canva production notes

Page size: Letter (8.5 × 11 in), portrait; margins ≥ 0.5 in.
Figures: Full-width unless marked half width (side-by-side on one page).
Answer lines: Keep underline length generous for context sentences and conclusions.
Do not flatten technology equations into "guess the slope" tasks — blanks that ask for regression equations expect the chapter's stated technology results.
Item parity: Workbook item numbers match the textbook exactly (1–124).