Key visualizations: Algebra I

Technology is a powerful tool for students as they explore and create models for data. However, it is important that students have some understanding of the underlying mathematics behind the technology so that they thoughtfully evaluate the results. As students move from creating linear models by hand to creating linear models using regression techniques with technology, it can be helpful to “demystify” the result so that students understand what is unique about the linear regression equation the technology produces.

 Connections to the TEKS [−]
This animation addresses the following Algebra I TEKS:

(1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
(C)
select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;
(D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

(4) Linear functions, equations, and inequalities. The student applies the mathematical process standards to formulate statistical relationships and evaluate their reasonableness based on real-world data. The student is expected to:
(C) write, with and without technology, linear functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems.

In this animation, students connect what they know about how changes in slope and y-intercept affect the graph of a line in order to fit a linear model to a set of data. By manipulating the sliders to set different values for these parameters, they can visualize how well different lines fit the data, investigating the error squares formed by lines that appear to be a “good” fit for a set of data to see how these are minimized with the line of best fit.

The equation predicts that a boy 67 inches tall would wear a size $9\frac{1}{2}$ shoe. But the only boy in the class who is 67 inches tall actually wears size 11 shoes! In this case, did the line underestimate or overestimate the shoe size?
Here is one trend line modeling height and shoe size that you found earlier. Using this equation, what is the predicted shoe size for a boy who is 67 inches tall?
In a line of good fit, the sum of the overestimates and underestimates should be close to 0. Squaring the amount of error is one way to ensure that the "overs" and "unders" will no longer cancel each other out, so you can get an idea of the total amount of error.
The blue lines indicate the difference between the actual data values and the values that are predicted by the equation. Some indicate overestimates while others indicate underestimates.
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Each blue line illustrating the amount of error is now the side of a square. The total area of the squares gives you a visual image of the amount of error.
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This time, set the sliders to match the slope and y-intercept from the regression line you found with your calculator. Does the sum of the areas of the error squares look larger or smaller than the other lines you tried?
Manipulate the slope slider to get a different line that you think might better model the data points.
Did the total blue area increase or decrease? Either try another slope, or this time, try the y-intercept slider.
Did the total blue area increase or decrease? Try another y-intercept.
Do you think you found the best fit line? You can try another slope or y-intercept. Click the Show error squares button to see the total error.
Keep adjusting the sliders until you find a line with the smallest amount of blue area.