Quadratic patterns

For this lesson I hand students a slip of paper as they enter in the door, one that has a quadratic pattern printed on it.  Their 1st task is to make a guess, no calculations, how many squares/circles/shaded squares with the 50th term have?

All the patterns used for this lesson were taken from http://www.visualpatterns.org/

Then I will take a few answers and put them up on the board to incite some competition and buy-in from the students.

At this point I may show them how to do it, or remind them if they already know.

They have some time to work, most of the students can accurately find the equation using the table and the process I trained them with to find the equation from the table, but fewer of them can look at the pictures and write an equation relating x to side lengths and so forth.

Here are the patterns that we have been using:

Here is some student work that shows the process of finding the equations:

some students can look as the figures and relate the parts to x, but it is more common that they use the table approach as it is more formulaic.

Drawing the next picture is crucial to getting enough data for using a table.  It allows students to get the 2nd differences and work backwards to find the y-intercept.

Most of the questions that I received, or the students that I helped that were stuck had miscounted the squares, thus finding no pattern.

After they have done the work to generate the equation they use it to find the value of the 5th term, and at the end of the hour we look to see whose guess was the closest.

Students then go home with this practice Hw.  The first pattern is actually linear, and although it confuses students because that's not the unit we're in it is a good connection to make.