I begin this lesson by making an analogy between leftovers from a meal and the remainder you get when dividing some numbers. I say that a remainder tells us whether or not a number "m" is divisible by "n" (is n a divisor of m?) and leftovers tell us whether or not meal was tasty (no leftovers!) I start with integer examples like 24/3 and 25/3 to review how to write a remainder, then post a polynomial division problem on the board and ask my students to divide using any algorithm they choose. (MP1) I expect that some students will try to factor while others will use long division to determine if the given polynomials divide without remainder. I ask for volunteers to demonstrate how they got their answera and review long division, synthetic division and factoring. I close this section by telling my students that today's activity will let them quickly check several polynomial expressions for "leftovers"!
For the first part of this section I tell my students that they will be working with their left-shoulder partner but that each student should complete the handout so they have a copy for reference. I distribute the Leftover Division handout and check for understanding using fist-to-five. If most of the students are at 4 or 5 I let them all begin working and help those who need additional support. If the majority are still indicating confusion (fist, 1, or 2) I review the directions with the class and work another example if needed. While my students are working, I walk around offering encouragement and redirection as needed. (MP1) When all the teams have completed the problems, I randomly select students to share the results for one problem (selected by the roll of a ten-sided die) with the class and invite the class to critique the answers (appropriately). (MP3) As we work through the problems I ask my students to self-check their papers and make corrections as needed so they have an accurate reference. I continue to select students to post problems until we've checked the entire handout.
Now that my students are comfortable finding remainders, I tell them that they get to combine two techniques to find a specific value for a function. I walk them through two examples, one that would be easy to work out by substituting the value directly into the dividend and the second that has simpler numbers using synthetic division. For practice I challenge my students to divide into three or four teams of approximately equal members and send one member to the board. While they're figuring out their teams, I divide my front board into four equal sections. When my students are ready I tell them that they get to use synthetic division and the remainder theorem or direct substitution to find given values for the problems they've just solved. I roll a die to generate values and allow teammates to read the original dividends for each problem, rotating through players until everyone has had an opportunity to test their skill. (MP2) This activity helps students see a value in using synthetic division and also builds a better understanding of the Remainder Theorem and its possible applications.
To wrap up this lesson I use a whole-class discussion of what remainder means when we're talking about polynomials. I am particularly interested in having my students recognize that a remainder is not a factor! I explain this further in my Leftover video. I also hope that my students recognize that the Remainder Theorem combined with synthetic division give us a quick way to find a specific value of the function: f(a).