Experience First

In this lesson students look at various Panera orders to determine the price of a tub of cream cheese and a bagel. Students realize in question 1 that having one order is insufficient to determine the cost of each order. Choosing any price of bagel would allow students to solve for the necessary price of a tub of cream cheese, or vice versa.

In questions 2 and 3 students get a second order (Kelly’s), which is a scaled version of Peyton’s order. Students reason that fair pricing means charging consistently for each good for every customer, which is the exact definition of a consistent system--the idea that there exist values for the variables that satisfy both equations (prices that work for both orders). Students walk away with a much firmer grasp of dependent systems, because they see Kelly’s order as equivalent to Peyton’s order and thus the cost of her order would be exactly 1.5 times the cost of Peyton’s order. Nevertheless, there is still not enough information to determine the cost of a bagel or tub of cream cheese.    

Finally, in question 4, students receive Carter’s order which is an independent equation. The question is worded intentionally so they will compare Carter’s order to twice Peyton’s order. The difference in price between twice Peyton’s order and Carter’s order must be the price of 3 bagels, since otherwise the orders are the same! This is the idea of elimination--scaling the equations so that the only difference in price can be attributed to one variable. It’s important that students understand this conceptually instead of just going through the rote procedure of multiplying equations by a scalar and then adding or subtracting equations. While students leave Algebra 2 feeling pretty confident using elimination as a strategy, we want students to be able to connect this method with important ideas about equivalence.

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Formalize Later

The Important Ideas section ties together graphical and analytical representations of dependent, independent, and inconsistent systems. Students should be able to reason about systems of linear equations from the perspective of slopes and y-intercepts, as well as equivalent equations and scalar multiples. Questions like 3 and 5 on the Check Your Understanding encourage students to strategically assess what conditions are needed to classify a system as independent, dependent, or inconsistent. This understanding is a critical piece of the checkpoint open middle task on day 5.