Lesson 5 of 15
Objective: SWBAT develop equations using calendars, graph these, then reason with the results.
Accessing Prior Knowledge
I begin class by pairing up students randomly and handing out a monthly calendar. I tell the students to look at their calendars and choose any four days that lie in a square, such as:
I then ask a student volunteer from any group to add his/her four dates together and tell me the sum of the dates (only the sum, not the actual dates). Then, I tell them that I will guess what dates they chose.
Teacher's Note: I calculate the dates by dividing the sum by 4 and then subtracting 4. Using the example above, the sum is 52. Dividing 52 by 4, and then subtracting 4 gives me 9, which is the first date. The rest are easily calculated.
After "concentrating" for a minute, I tell the class the four days the student chose and confirm with the student volunteer. I repeat the game with another student.
After giving the class a little time to be amazed, I let students know how I did it. I tell students that the answer can be calculated easily using algebra.
In this section I encourage the students to find the formula for finding four dates from a monthly calendar. Although students work in pairs, for this activity I will encourage sharing between pairs. I write on the board:
Let n represent the first date chosen
I intend to allow students to struggle with this task for a few minutes. I will offer the following hint, if necessary: "if the first date is n, the rest must be written using an expression that includes n."
After a few minutes, I expect students to recognize that:
n = first date
n + 1 = second date
n + 7 = third date
n + 8 = fourth date
From this point, my students have usually worked quickly to realize that
n + n + 1 + n + 7 + n + 8 = sum of dates
At this point, I will demonstrate how algebra offers them power in a situation like this by helping them to simplify the expressions to 4n + 16 = Sum of dates, and, to see that 4n + 16 equals 4(n + 4) when factored. Then, I will be able to employ my number trick more easily using mental math..
When students begin to show understanding, I plant to ask the question: "Could this same formula be used to find the dates that were given by student volunteers at the beginning of the activity or any four dates grouped in the same way, for that matter?"
Each group of students will now devise their own Calendar Clever formula. I encourage them t be creative in how they select their calendar arrangement. I let them vary the number of dates in their arrangement as well as the position of the dates. As they are working, I walk around assist students if they need it.
Each group should show:
- The arrangement of dates they’ve created on their corresponding calendar
- The formula associated with it
- The graph that represents the equation
Instructional Note: If I find students who are having difficulty with this task, I scaffold their work by asking them to begin by using only two dates on the calendar. I also spend some time teaching them how to develop the formula for two dates.
As we prepare for the end of class, I call on one of the pairs to come to the board and show the class their calendar pattern and formula. I ask the pair to simplify the formula to make it easier to use mentally. I then have this pair of students try their formula with any other pair in the class. When the chosen pair calls out their sum I ask everyone else in the class to use the simplified version to mentally find the dates. Allow calculators, pencil, and paper. They should all get the right dates and feel “calendar clever”.