I'm brainstorming / laying out my end-of-year plans for my 8th-graders. After their end-of-year exam in late May, we will close grades, but we will still have about 3 or so weeks of instruction, which is enough time to do something very rich and not have to coordinate with other classes (since we use the May test to do placement for Grade 9). Last year, I used this extra time to let the 8th-graders define their own math projects, which were plenty of fun, but I wasn't entirely happy with the rigor of their mathematical results. This year, I'm toying with the idea of doing an exploratory quadratic functions unit. (Technically, quadratic FUNCTIONS are a Grade 9 topic for us, but previewing it in Grade 8 is always beneficial.)
I'm thinking of making it largely exploratory, since by then pacing won't be much of an issue and I can let them really take the time to develop their conceptual understanding of quadratic functions, which is the essential access point to a lot of higher-level algebra analysis down the road.... The timing is tight (as it was last year with my other end-of-year projects), but I think it's still doable and has a lot of potential!!!
Let me know what you think. Is it an OK approach for intro to quadratic functions / basic function transformations?? This is based on my rumination about a different way to think about flexible factorization of quadratic functions.
Day 1: Developing the understanding of how to graph y = x2 + bx.
Plan - In pairs, kids will be given y = x2 + 2x, y = x2 + 5x, y = x2- 3x, y = x2 - 7x. to graph on the calculator. They will sketch results in their notes, recording all intercepts in the form (x, y), and writing a one-sentence hypothesis about what the graph of y = x2 + bx will look like.
Then, they will be reminded that in a function, we can solve for the x-intercept(s) by setting the height of the point, y, equal to zero. They will algebraically show that their hypothesis works for all b values.
Day 2: Developing the understanding of how to graph y = ax2 + bx, which is a more general version of the quadratic function.
Plan - In pairs, kids will be given y = x2 + 6x, y = 2x2 + 6x, y = 3x2 + 6x, y = 12x2 + 6x. They will again sketch graphs, noting x-intercepts, and then make a hypothesis about the effect of the leading coefficient on the graph. They will show how to solve for the x-intercepts using algebra only.
Day 3: Developing the understanding of how the graph is affected by the sign of its leading coefficient.
Plan - In pairs, kids will be given y = -x2 + 6x, y = -2x2 + 6x, y = -3x2 - 6x, y = -12x2 - 6x. They will again sketch graphs, noting x-intercepts, and then make
a hypothesis about the effect of the leading coefficient on the shape of the graph.
They will show how to solve for the x-intercepts using algebra.
As part of Day 3, they will do some matching between equations and pictures of graphs and to justify their choices orally.
By the end of Day 3, they should also be able to explain in writing how to graph y = ax2 + bx.
Day 4: Developing the understanding of the effect of the constant term c.
Plan - In pairs, the kids will put in a function like y = x2, and look at its table in the calculator. They will be asked to generate a second function that would increase all y-values by 1. They will prove their new equation works, by showing the values of both functions side by side in the calculator (y1 and y2), and copying down the table values. Then, they will write down the formula for the new function.
They will then look at the graphs of the two functions to determine "what happened" visually to the original graph when the equation got changed that way.
They will keep playing around with this idea, translating upwards and downwards and checking both the table and the graph to observe/verify the effect of c.
By the end of Day 4, they will be given a graph of two functions. One of the functions will have an accompanying formula, and they will be able to see visually what happened to the points on the graph. They will then need to "guess" at the equation of the other, vertically shifted function, and verify it in the calculator.
Day 5: Putting the algebra pieces altogether
Plan - One partner of the pair will have a set of sequenced instructions, to be given to their partner one step at a time. The first step will sound like, "Sketch a graph of y = x2 - 9x, labeling x-intercepts with values." Then, after that has been successfully completed, the next instruction will be given: "Now sketch the result of shifting that graph vertically up 4 units, labeling the resulting images of those original points you knew." After that has been done, the partner gives the third instruction: "Now, write the formula for this new graph." Once the partner is finished, they verify their results using the graphing calculator's graphing and table features and write a brief explanation of how they checked their results. Then, they switch, and the new partner has instructions that has to do with a downwards facing function like y = -x2 - 6x, and repeat a similar sequence of instructions to generate a new graph, a new / related equation, and to verify all results against the calculator.
Both partners will then work together to complete problems starting with functions of the form
y = ax2 + bx and translating those graphs vertically to get new graphs.
By the end of Day 5, they should be able to explain the connection between y = ax2 + bx and
y = ax2 + bx + c, and explain how to use this connection to graph any standard-form quadratic function quickly in under 1 minute.
Day 6: Practicing/drilling the connection between quadratic function equation and graphs
Plan - In pairs, they will start with a function y = x2 - 9x + 1, highlight the first two terms, sketch that function using dashed lines, and then sketch in the "real" final function using solid line. They will repeat this a few times with different functions, until they can fluidly graph any y = ax2 + bx + c function. On this day, they'll also learn to visualize the axis of symmetry and to write its equation by inspection of graph.
Day 7: Going backwards from a graph to an equation
Plan - In pairs, they will be given one quadratic graph with two "nice", symmetric integer points being emphasized on the graph, one of the points being on the y-axis. They will be asked to sketch using dashed lines what this function would look like if you shifted those two points down to the x-axis, and be asked to write the function equation of both graphs. They will practice this a few times.
At the end of Day 7, they will be given a quadratic graph whose two "nice", symmetric integer points are both not on the y-axis. This tests them to see if they can figure out that the translated graph would have an equation that looks like y = (x - m)(x - n) + p instead of y = x(x - n) + p
Day 8: Playing around with the idea of adjusting "a".
Plan - In pairs, they will import Dan Meyer's basketball photo into GeoGebra. We will discuss as a class the need to find a modeling equation in order to fully predict whether the ball will make it into the hoop. From there, they will choose two nice integer points, write the equation, and graph. If they notice that the curve goes through those two points but doesn't have the correct steepness desired in order to fit the photo, then they will create a slider value in GeoGebra and toggle the value of "a" until they get a good "fit" around the graph, and record their results.
As a class, we will then go over the idea of solving for "a" using an unused point (x, y) and link it to solving for the y-intercept in linear functions. They will solve for "a" this way to compare analytical results against the technology results.
Day 9: Modeling Individually
Plan - Following a discussion of examples of parabolic applications, each pair will find and import their own photos of "real-life" parabolic shapes from the web. They will then model the function in Geogebra both using technology and using algebraic analysis.
Day 10: Creating posters
Plan - Each pair will create two posters, one with the modeled functions overlaying the photos, and one poster explaining the general process of graphing y = ax2 + bx + c and the general process of fitting an equation to a parabolic graph.
Day 11: Practice presentations
Day 12: Math fair for other classes / parents?!