HSS.ID.B.6Common CoreMathStatistics and ProbabilityGrades 9-12
HSS.ID.B.6: Scatter Plots, Fitted Functions and Residuals
In plain English: HSS.ID.B.6 is the Common Core statistics standard that asks students to show two quantitative variables on a scatter plot and describe how they are related. Students fit linear, quadratic or exponential functions to the data, use them to solve problems in context, and judge the fit by plotting residuals. It is usually taught in Algebra I, and quadratic and exponential fits often return in Algebra II.
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.
Common Core State Standards for Mathematics · Domain: Interpreting Categorical and Quantitative Data (ID) · Cluster: Summarize, represent, and interpret data on two categorical and quantitative variables Also written as HSS-ID.B.6 or S-ID.6 · Official standard
Students move from a table of paired data to a model they can use. They draw scatter plots, describe the direction, form and strength of the relationship, and fit functions to the data: lines by eye and with technology, and quadratic and exponential functions when the context or the plot calls for them. Each model is then used to answer a question about the situation, such as a prediction or the time something lands.
Residuals give students a way to judge a fit. By computing observed minus predicted values and plotting them, students see whether a model captures the pattern in the data or misses a curve. The lesson emphasizes the three model families the standard names, linear, quadratic and exponential, and the reasons from context for choosing each one.
Learning Objectives
By the end of this lesson, students will be able to:
Make a scatter plot of two quantitative variables and describe the direction, form, strength and unusual points of the relationship
Fit a linear function to data that suggest a linear association, by eye and with technology
Choose a linear, quadratic or exponential function suggested by the context or the data, or use a given function, and use it to solve problems in context
Compute residuals, make a residual plot and use it to assess informally whether a model fits
Prior Knowledge Required
Students should already be comfortable with:
Constructing and interpreting scatter plots for bivariate measurement data 8.SP.A.1
Fitting a straight line informally to a scatter plot that suggests a linear association 8.SP.A.2
Distinguishing linear and exponential growth by equal differences and equal factors HSF.LE.A.1
Evaluating linear, quadratic and exponential functions and solving simple quadratic equations
Show three small invented tables without graphs and ask students to predict the shape of each scatter plot before they draw anything.
Warm-Up Prompt
"Table A: a phone battery at 100%, 92%, 84%, 76%, 68% after 0, 1, 2, 3, 4 hours of video. Table B: a population of fruit flies of 40, 60, 90, 135, 203 on days 0, 1, 2, 3, 4. Table C: a paper airplane at heights 1.5, 2.3, 2.5, 2.1, 1.1 meters at 0, 0.5, 1, 1.5, 2 seconds. Which one grows or shrinks by the same amount each step, which by the same factor, and which goes up and then down?"
Answers: Table A drops by 8 percentage points each hour (constant differences, linear). Table B is multiplied by 1.5 each day (constant ratios, exponential). Table C rises and then falls, and its second differences are constant (-0.6 each step), which is the quadratic pattern. Record the three words linear, exponential and quadratic on the board: the lesson is about choosing among these three shapes and fitting them to data that are not perfect.
Direct Instruction25 minutes
From scatter plot to model. Put the explanatory variable on the x-axis and the response on the y-axis. Describe the relationship by its direction (positive or negative), form (linear or curved), strength (how tightly the points follow the form) and any unusual points. Then fit a function and check it with residuals:
Choose a family. Use the context and the plot: constant change per unit suggests a linear function, a constant percent change suggests an exponential function, and a rise-then-fall (or a physical law such as gravity) suggests a quadratic function. Sometimes the model is given.
Fit it. For a line, draw a line through the middle of the points and write its equation from two points on it, or use technology (LinReg). For curves, use QuadReg or ExpReg, or build the function from the pattern.
Compute residuals. Residual = observed y - predicted y. Plot the residuals against x.
Assess the fit. Residuals scattered randomly around 0 with no pattern mean the model captures the form. A curved pattern (for example a U shape) means the model has the wrong form.
Use the model. Answer the question in context, with units, and be careful when predicting far outside the data.
Scatter plot and description
A cocoa stand records the afternoon temperature x (°F) and cups sold y on 8 days (invented, Diagram 1): (40, 63), (45, 58), (50, 51), (55, 47), (60, 40), (65, 35), (70, 31), (75, 25). Describe the relationship.
Equation: Strong, negative, linear association with no outliers: warmer days go with fewer cups sold
Fitting a linear function
Fit a line to the cocoa data and use it to predict sales on a 58°F afternoon.
Equation: LinReg: ŷ ≈ -1.09x + 106.2; at x = 58, ŷ ≈ 43 cups (about 1.09 fewer cups for each extra degree)
Residuals
Find the residual for the 60°F day and describe the residual plot for the cocoa line (Diagram 2, left).
Equation: Predicted ≈ 41.0, observed 40, residual ≈ -1.0 cup; the residuals scatter between about -1 and 1 with no pattern, so a line fits well
Using a given quadratic function
A ball is tossed upward from a 5 m balcony. Invented heights at t = 0, 0.5, 1, 1.5, 2 and 2.5 s are 5.1, 10.6, 14.3, 14.8, 13.5 and 9.2 m. A physics model gives h(t) = -4.9t² + 14t + 5. When does the model say the ball lands?
Equation: -4.9t² + 14t + 5 = 0 gives t ≈ 3.18 s (the negative root does not fit the context); the model peaks at t ≈ 1.43 s, h = 15 m
Choosing an exponential function
A used car is worth $24,000 new and, in invented data, $20,500, $17,200, $14,800, $12,400 and $10,700 at ages 1-5 years. Choose and fit a model, then estimate the value at age 8.
Equation: Year-to-year ratios are all near 0.85, so V(t) = 24,000(0.85)ᵗ; V(8) ≈ $6,540
Use Diagram 1 with Examples 1-3 and point to the short vertical gaps between each point and the line: points above the line have positive residuals and points below have negative ones. In Example 4, stress that the model was given by the context (gravity) and that only the positive solution makes sense. In Example 5, have students compute the ratios 20,500 / 24,000 ≈ 0.854, 17,200 / 20,500 ≈ 0.839 and so on: the car loses about the same percent of its value each year, which is why an exponential model fits better than a line, and why the model never predicts a negative value.
Guided Practice15 minutes
Pairs use this invented table of car stopping distances on dry pavement.
Invented stopping distances
Speed x (mph)
20
30
40
50
60
70
Distance y (ft)
42
73
122
172
243
313
Steps and answers:
Scatter plot: strong, positive and slightly curved upward.
LinReg gives ŷ ≈ 5.47x - 85.4. Its residuals are about 18, -6, -11, -16, 0 and 15 feet: positive at both ends and negative in the middle. That U shape (Diagram 2, right) says a line is the wrong form, even though the points look close to a line.
Braking distance grows with the square of the speed, so the context suggests a quadratic. A given model is d(x) = 0.05x² + x. Its residuals are 2, -2, 2, -3, 3 and -2 feet, small and with no curved trend, so it fits much better.
Use the quadratic to estimate the stopping distance at 55 mph: 0.05(55)² + 55 ≈ 206 feet.
Ask pairs why the line predicts a negative stopping distance at 10 mph (5.47 × 10 - 85.4 ≈ -31 ft), and what that says about extrapolating with the wrong model.
Independent Practice10-15 minutes
A school video gets these invented total views on days 1-6: 200, 318, 515, 822, 1,305 and 2,100. Students (1) make a scatter plot, (2) show that the ratios of consecutive values are close to 1.6, (3) write an exponential model and (4) estimate total views on day 8. Answers: the ratios are about 1.59, 1.62, 1.60, 1.59 and 1.61; a model through the day-1 value is V(d) = 125(1.6)d (ExpReg gives a similar model); V(8) ≈ 5,369 views. Ask students whether the model can keep working for day 30 (it would predict about 166 million views, far more than any school audience), a reminder that fitted models describe the data range, not the distant future.
Closure10 minutes
Exit ticket: a line of fit for invented data on study time x (hours) and quiz score y is ŷ = 6x + 58. Two students studied 3 hours and 5 hours and scored 79 and 85. (1) Find both residuals. (2) If a residual plot for the whole class showed an upside-down U, what would you conclude? Answers: (1) 79 - 76 = 3 and 85 - 88 = -3. (2) The line has the wrong form; the scores may level off, so a curved model would fit better.
Differentiation Strategies
For Struggling Students
Provide pre-scaled axes for every scatter plot so students spend their time on the pattern, not the scale
Give a residual table with columns x, observed y, predicted y and residual, and have students fill it one column at a time
Use a single "clue card" for choosing a model: same difference means linear, same ratio means exponential, up-then-down or a square in the context means quadratic
For Advanced Students
Ask students to fit both a quadratic and an exponential model to the stopping-distance data with technology, compare the residual plots, and argue from the context which model is better
Have students find the least-squares line for the cocoa data by hand from the formulas for slope and intercept and compare it with the calculator result
Ask students to invent a data set whose linear residual plot shows a clear pattern even though the correlation is above 0.95
Assessment Guidance
What to Look For
Check that scatter plots have labeled axes with units and the explanatory variable on the x-axis. Descriptions should mention direction, form and strength, not only "it goes down". When students fit a model, ask them to justify the family from the context or from differences and ratios, not only from how the graph looks. For residuals, look for the correct order (observed minus predicted) and a conclusion based on the pattern of the residual plot, not on the size of a single residual. Every prediction should be stated in context with units.
02
Classroom Activities
3 Activities
1
Spaghetti Lines and Residuals
20 minPairs
Pairs fit a line by eye with a strand of uncooked spaghetti, write its equation, and test it against the calculator line with residuals.
Procedure
Each pair gets one data card (invented): arm span and height for 10 students, or shoe length and height, or practice minutes and free throws made out of 20
Pairs make a scatter plot on graph paper, describe it, and lay a spaghetti strand where they think the line fits best
Pairs pick two points on the strand (not necessarily data points), write the equation of their line and interpret its slope in context
Pairs compute the residual for every data point, make a residual plot and add up the squared residuals
Pairs then use LinReg on a calculator, repeat the residual computations, and compare the two sums of squared residuals
Discussion Questions
Why is the calculator line called the least-squares line? Did any pair beat it?
Do the residuals of a good line add up to about 0? Why?
Which point had the largest residual, and what might explain it?
Modification for Distance Learning
Pairs use an online graphing tool with a movable line (slider for slope and intercept), then compare with the regression line the tool computes.
2
Bouncing Ball Model
20 minGroups of 3-4
Groups collect data from a real bouncing tennis ball and decide which model family fits the heights of successive bounces.
Procedure
One student drops a tennis ball from 2 meters next to a wall marked in 10 cm steps; another records the peak height of each of the first 5 bounces, repeating each drop twice and averaging
Groups plot bounce number against peak height and compute the differences and the ratios of consecutive heights
Groups choose linear or exponential, justify the choice from the context (each bounce keeps about the same fraction of the height), and fit a model, for example h(n) = 2(0.6)n if the ratio is near 0.6
Groups compute residuals and use the model to predict the bounce on which the peak first falls below 10 cm
Discussion Questions
Why would a linear model eventually predict a negative bounce height?
How much did your ratio change from bounce to bounce? What could cause the variation?
Would a different ball give the same model? What would change?
Challenge Variation
Film one bounce with a phone at a high frame rate, read the height every 0.1 s, and fit a quadratic function to the height of the ball over time within a single bounce.
3
Model Match-Up
15 minGroups of 3-4
Groups match 6 invented data cards to a model family and a residual plot, then use each model to answer one question.
The 6 Data Cards
Card 1: a gym membership costs $40 to join plus $25 a month, total cost for months 1-6
Card 2: a savings account balance of $1,000 growing 4% a year, years 0-5
Card 3: the height of a model rocket every second for 6 seconds, rising and then falling
Card 4: the area of square tiles with side lengths 2-7 inches
Card 5: the number of infected plants in a greenhouse, roughly doubling each week for 5 weeks
Card 6: the depth of water in a pool being drained at a steady 3 inches per hour
Procedure
Groups classify each card as linear (Cards 1 and 6), exponential (Cards 2 and 5) or quadratic (Cards 3 and 4), and explain the clue in the context
For each card, groups sketch what the residual plot would look like if someone fit a line to it: random for linear cards, curved for the others
Groups write a model for Cards 1, 2 and 6 and use each to answer: the gym cost for 12 months ($340), the balance after 10 years (about $1,480), and when a 60-inch-deep pool is empty (20 hours)
Discussion Questions
Which clue words in a context point to each family?
Card 4 is quadratic but always increasing. How can you tell it from an exponential card?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Scatter Plot with a Fitted Line and Its Residuals
The 8 invented cocoa-stand days from Worked Examples 1-3, drawn to scale, with the least-squares line ŷ ≈ -1.09x + 106.2. The association is strong, negative and linear. Each residual is the vertical distance from a point to the line, positive when the point is above the line; here every residual is about 1 cup or less, which Diagram 2 shows on a larger scale.
Diagram 2: Reading Residual Plots
Left: the residuals of the cocoa line scatter above and below 0 with no pattern, so the linear model captures the form. Right: the residuals of a line fit to the invented stopping-distance data from Guided Practice are positive at both ends and negative in the middle. That U shape shows the data curve and a quadratic model is a better choice.
04
Homework Assignment
~30 min
HSS.ID.B.6 Homework: Fitting Functions to Data
Directions: All data are invented. Draw every scatter plot on graph paper with labeled axes and units. Show the model you use, the residuals you compute (observed minus predicted), and answer each question in a sentence with units.
Part 1: Scatter Plots and Linear Fits (Problems 1-2)
Ten students report hours of sleep x and their reaction time y in milliseconds on a computer test: (5, 330), (5.5, 318), (6, 305), (6.5, 300), (7, 284), (7, 290), (7.5, 276), (8, 270), (8.5, 262), (9, 255). (a) Make a scatter plot. (b) Describe the direction, form and strength of the relationship. (c) Can you conclude from these data that more sleep causes faster reactions? Explain.
A candle is measured every 10 minutes while it burns: at 0, 10, 20, 30, 40 and 50 minutes its length is 20.0, 18.6, 17.3, 15.8, 14.6 and 13.1 cm. (a) Fit a linear function, either through two points on a line you draw or with LinReg. (b) Interpret the slope in context. (c) Use your model to estimate when the candle will burn out.
Part 2: Residuals (Problems 3-4)
A trainer uses ŷ = 9.5x + 12 to predict calories burned y from minutes on a treadmill x. Invented data for one client: (10, 104), (15, 158), (20, 205), (25, 254), (30, 300), (35, 348). (a) Compute the six residuals. (b) Make a residual plot. (c) What does the plot say about the model? Suggest a small change to the equation that would fit better.
A drama club sets different ticket prices p (dollars) for five performances and records the revenue R: $4 gives $950, $6 gives $1,215, $8 gives $1,270, $10 gives $1,190 and $12 gives $975. (a) Explain why a linear model cannot fit these data. (b) A consultant gives the model R(p) = -20p² + 320p. Compute its residuals. (c) Use the model to find the price with the greatest revenue and the revenue at a $9 price.
Part 3: Choosing a Model (Problems 5-6)
A student measures the caffeine in her body after a drink with 200 mg of caffeine (invented data): 200, 152, 115, 88 and 66 mg at 0, 2, 4, 6 and 8 hours. (a) Compute the ratios of consecutive values. Which family of functions do they suggest? (b) Explain why the model C(t) = 200(0.87)ᵗ, with t in hours, fits the data. (c) Use it to estimate the caffeine after 5 hours. (d) Estimate when the amount first falls below 50 mg.
For each invented table, x = 0, 1, 2, 3, 4. Table A: y = 5, 8, 11, 14, 17. Table B: y = 2, 6, 18, 54, 162. Table C: y = 0, 7, 12, 15, 16. (a) Decide whether each is best modeled by a linear, quadratic or exponential function, using differences or ratios. (b) Write a function for each table. (c) Describe a real context that could produce Table C.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Scatter Plots and Descriptions
Labeled, scaled plots; direction, form and strength described
Plot correct but description incomplete
Plot missing or axes wrong
Fitting and Choosing Models
Correct family justified by context or by differences and ratios; equation correct
Correct family with an error in the equation
Family missing or unjustified
Residuals
Residuals computed as observed minus predicted, plotted and interpreted
Computed correctly but not interpreted, or sign reversed
Missing
Solving in Context
Predictions correct, with units and a comment on reasonableness
Correct value without units or context
Missing or unreasonable
05
Quiz: 20 Questions
Interactive, with answers
Instructions
All data in this quiz are invented. Answer each question before you open its explanation. The score at the top counts your multiple-choice answers, and Reset quiz starts the whole set again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
On a scatter plot of invented data, the points fall close to a straight line that slopes down from left to right. Which description fits best?
Answer: C
Points close to a line mean a strong association, a straight-line pattern means the form is linear, and "down from left to right" means negative. Choice A gets both the strength and the direction wrong. Choice B would need points that rise along a curve.
Question 2 of 20 · Multiple Choice
A coach wants to know whether minutes of practice predict the number of free throws made. On a scatter plot, which variable belongs on the x-axis?
Answer: B
The explanatory variable, the one used to predict, goes on the x-axis, and the response goes on the y-axis. Choice C is wrong because swapping axes changes the fitted line and its meaning. Choice D confuses quantitative data with categorical data, which is when a two-way table is used.
Question 3 of 20 · Multiple Choice
A line of fit for invented data on weeks of training x and a runner's weekly miles y is ŷ = 2.4x + 31. What does the model predict for week 15?
Answer: D
Substitute x = 15: 2.4(15) + 31 = 36 + 31 = 67 miles. Choice A leaves out the intercept 31. Choice B multiplies 2.4 by the sum 15 + 31. Choice C adds 2.4, 15 and 31 instead of multiplying 2.4 by 15.
Question 4 of 20 · Multiple Choice
A model for invented data is ŷ = 1.8x + 12. One data point is (10, 27). What is its residual?
Answer: A
The predicted value is 1.8(10) + 12 = 30, and residual = observed - predicted = 27 - 30 = -3: the point is 3 units below the line. Choice B reverses the subtraction. Choice D is the predicted value, not the residual.
Question 5 of 20 · Multiple Choice
A residual plot for a linear model shows residuals that are positive for small x, negative in the middle, and positive again for large x. What should you conclude?
Answer: B
A U-shaped residual plot shows that the line is too high in the middle and too low at the ends: the data curve and the line misses that form. Choice A is a common trap, because residuals of a least-squares line always average about 0, even when the form is wrong. Choice D would flip the U upside down, but the conclusion would be the same.
Question 6 of 20 · Multiple Choice
Which residual plot pattern best supports using a linear model?
Answer: C
Random scatter around 0 means the line captures the form of the data and what is left over is noise. An upside-down U (choice A) or a steady trend (choice B) shows a pattern the model missed. All positive residuals (choice D) mean the line sits below the data and should be moved up.
Question 7 of 20 · Multiple Choice
A lab culture of bacteria grows by 20% every hour (invented). Which type of function does the context suggest?
Answer: D
Growing by 20% means multiplying by the same factor, 1.2, in each equal time step, which is exponential growth: N(t) = N₀ · 1.2ᵗ. Choice A would mean adding the same number of bacteria each hour. Choice B fits situations such as projectile heights or areas.
Question 8 of 20 · Multiple Choice
Which invented situation is best modeled by a quadratic function?
Answer: A
The height of a kicked ball rises and then falls because of gravity, which gives a quadratic function of time. Choice B adds the same $2 per mile, which is linear. Choice C multiplies by 1.03 each year, which is exponential.
Question 9 of 20 · Multiple Choice
A hiker walks down a mountain trail, losing elevation at a steady 400 feet per hour (invented). If you fit a function to data on her elevation over time, which family should you choose, and why?
Answer: C
A constant rate of change, the same 400 feet every hour, is the defining feature of a linear function. Choice A is the error of thinking that every decrease is exponential: exponential decay removes the same percent, not the same amount, each step. Choice B confuses the shape of the trail with the shape of the graph of elevation against time.
Question 10 of 20 · Multiple Choice
Invented data on the value of a collectible card fit an exponential model through (0, 500) and (1, 400). What value does the model predict at x = 3?
Answer: B
The model is y = 500(0.8)ˣ because 400 / 500 = 0.8. Then y(3) = 500(0.8)³ = 500(0.512) = $256. Choice A subtracts $100 each step, which is a linear model. Choice C is y(2), one step too few.
Question 11 of 20 · Multiple Choice
A given model for the height of a ball tossed from a platform is h(t) = -16t² + 48t + 4, with h in feet and t in seconds. What maximum height does the model predict?
Answer: D
The vertex is at t = -48 / (2(-16)) = 1.5 s, and h(1.5) = -16(2.25) + 48(1.5) + 4 = -36 + 72 + 4 = 40 ft. Choice C is the time of the maximum, not the height. Choice A adds 48 and 4 and ignores the -16t² term.
Question 12 of 20 · Multiple Choice
A student draws a line of fit through the points (2, 15) and (10, 55) on a scatter plot. What is the equation of the line?
Answer: A
The slope is (55 - 15) / (10 - 2) = 40 / 8 = 5, and 15 = 5(2) + b gives b = 5. Choice B uses the y-value 15 as the intercept. Choice D inverts the slope, dividing the change in x by the change in y.
Question 13 of 20 · Multiple Choice
A model fitted to invented data for ages 8-16 predicts height from age. Why is it risky to use the model to predict height at age 40?
Answer: B
Predicting far outside the range of the data (extrapolation) assumes the pattern continues, but people stop growing, so a line would predict unrealistic adult heights. Choice C goes too far: models are meant for predictions between data values, just not far beyond them.
Question 14 of 20 · Multiple Choice
Invented data: x = 0, 1, 2, 3 and y = 6, 9, 13.5, 20.25. Which function fits the data exactly?
Answer: B
Each y-value is 1.5 times the one before (9 / 6 = 13.5 / 9 = 20.25 / 13.5 = 1.5), and the value at x = 0 is 6, so y = 6(1.5)ˣ. Choice A fits only the first two points: its differences stay at 3. Choice D swaps the starting value and the factor.
Question 15 of 20 · Short Answer
A line of fit for invented data is ŷ = 3x + 10. The data points are (1, 12), (3, 20), (5, 26), (7, 30) and (9, 36). (a) Compute the five residuals. (b) Does the line fit well? Explain.
(a) Predicted values 13, 19, 25, 31, 37, so the residuals are -1, 1, 1, -1, -1. (b) Yes, reasonably. The residuals are small compared with the y-values and show no curved pattern.
Question 16 of 20 · Short Answer
Invented data on a puppy's weight: a student draws a line of fit through (2, 3.2) and (8, 6.8), where x is the age in weeks and y is the weight in kilograms. (a) Write the equation of the line. (b) Interpret the slope. (c) Predict the weight at 10 weeks.
(a) Slope = (6.8 - 3.2) / (8 - 2) = 0.6, so ŷ = 0.6x + 2. (b) The puppy gains about 0.6 kg per week. (c) 0.6(10) + 2 = 8 kg.
Question 17 of 20 · Short Answer
A laptop bought for $1,200 loses 25% of its value each year (invented). (a) Write a function for its value after t years. (b) Find the value after 3 years. (c) After how many whole years is the value first below $300?
(a) V(t) = 1,200(0.75)ᵗ. (b) 1,200(0.75)³ = $506.25. (c) 0.75ᵗ < 0.25 first holds at t = 5 years: V(4) ≈ $379.69 and V(5) ≈ $284.77.
Question 18 of 20 · Short Answer
A bakery fits the model P(x) = -2x² + 60x - 250 to invented data on its daily profit P (dollars) when a dozen cookies costs x dollars. What price gives the greatest profit, and what is that profit?
The vertex is at x = -60 / (2(-2)) = $15, and P(15) = -450 + 900 - 250 = $200.
Question 19 of 20 · Short Answer
Invented data on the age (years) and trunk diameter (cm) of 7 trees: (5, 12), (10, 21), (15, 32), (20, 41), (25, 48), (30, 22), (35, 71). Describe the relationship shown by a scatter plot, including any unusual point.
A strong, positive, linear association: diameter increases by about 2 cm per year. The point (30, 22) is an outlier, far below the pattern; it could be a measurement error or a tree that grew in poor conditions, and it should be checked before fitting a model.
Question 20 of 20 · Short Answer
Invented data: x = 0, 1, 2, 3, 4 and y = 3, 6, 11, 18, 27. (a) Which family of functions fits best? Justify with differences. (b) Find a function that fits the data exactly.
(a) The first differences are 3, 5, 7, 9 and the second differences are all 2, so a quadratic fits. (b) y = x² + 2x + 3; for example, x = 4 gives 16 + 8 + 3 = 27.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSS.ID.B.6 mean?
HSS.ID.B.6 means students can plot two quantitative variables on a scatter plot, describe how they are related, and fit a function to the data. They use the function to solve problems in context, choose among linear, quadratic and exponential models, and check the fit with residuals.
Is HSS.ID.B.6 Algebra 1 or Algebra 2?
It is usually taught in Algebra I. The Common Core appendix on course design places it in Algebra I, where linear fits and residuals get the most attention, and many courses return to quadratic and exponential models in Algebra II as students study those functions in more depth.
What is a residual?
A residual is the observed y-value minus the y-value the model predicts. A positive residual means the point is above the model and a negative residual means it is below. Residuals measure how far off the model is for each data point.
How do you use a residual plot to judge a fit?
Plot each residual against its x-value. If the residuals scatter randomly above and below 0 with no pattern, the model has the right form. A curved pattern, such as a U shape, means the model misses a curve in the data, and a different family of function should be tried.
How do I know whether to use a linear, quadratic or exponential model?
Look at the context and the data. A constant amount of change per step suggests linear, a constant percent change or a constant ratio suggests exponential, and a rise-then-fall or constant second differences suggests quadratic. The residual plot then confirms or rejects the choice.
What is the difference between a line of fit and the least-squares regression line?
A line of fit can be drawn by eye through the middle of the points. The least-squares regression line is the specific line that makes the sum of the squared residuals as small as possible, and calculators and spreadsheets compute it with LinReg or a trendline. Both are valid ways to fit a linear function under this standard.
What are common mistakes with scatter plots and lines of fit?
Common mistakes include putting the variables on the wrong axes, computing residuals as predicted minus observed, judging the fit by one large residual instead of the pattern, and using a model far outside the range of the data. Another is claiming that one variable causes the other because the association is strong.
How do you find a line of best fit on a TI-84?
Enter x-values in L1 and y-values in L2, then choose STAT, CALC, LinReg(ax+b). The calculator gives the slope a and intercept b. QuadReg and ExpReg in the same menu fit quadratic and exponential models, and the RESID list can be graphed to make a residual plot.
Does a strong correlation mean a linear model is right?
No. Data that curve gently can still have a correlation close to 1 or -1, and a residual plot is what reveals the curve. Correlation is studied in HSS.ID.C.8; for this standard, the residual plot is the main tool for judging whether a linear form fits.
How does HSS.ID.B.6 connect to later math?
It leads directly to interpreting the slope and intercept of a linear model (HSS.ID.C.7) and to computing and interpreting the correlation coefficient (HSS.ID.C.8). The work with exponential models connects to HSF.LE, and on the digital SAT, scatter plots and lines of best fit appear in the Problem-Solving and Data Analysis domain.
07
Related Standards
5 standards
These standards connect to HSS.ID.B.6: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.SP.A.1Prerequisite
Construct and interpret scatter plots to investigate patterns of association
