ET and Haleemah explore a new situation of three friends purchasing game apps for a phone or iPad. They come up with a method for finding how much the friends spend altogether, then apply their method for apps of different prices. They generalize their first method, then create a different, shorter method. Finally, they compare their methods and explain why the equations are equal.
ET and Haleemah make sense of the game app context and think of a way to find the total amount of money spent by a group of friends on several game apps.
The students generalize their method and write an algebraic equation that shows the relationship between the cost per game and the total amount spent by the friends.
Haleemah and ET generalize new methods for finding the total amount spent by the friends on game apps. They write an equation that shows the relationship between the cost per app and the total amount spent by the friends.
ET and Haleemah resolve a struggle from the previous episode and The students work to explain the relationship between the two algebraic expressions they have written.
Haleemah and ET work to explain why the two methods are equal by explaining what each part of their equations mean in the gaming app context.
Mathematics in this Lesson
Targeted Understandings
Understand the transitivity of equality. For example, if 3c + 4c + 2c = T and 9c = T, then 3c + 4c + 2c = 9c.
View situations in which one re-expresses an algebraic expression using a “shorter” or “more compact” expression (e.g., expressing 3x + 5x as 8x) as involving variability in the x-values.
Understand algebraic expressions from both process and product perspectives and in terms of the quantities in the context. Specifically, in the Game App Task, 3c can mean the 3 game apps purchased by Danyal times the cost per app (c), or the amount of money spent by Danyal on game apps.
Common Core Math Standards
CCSS.M.HSA.SSE.B.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
Throughout this lesson, ET and Haleemah are exploring a context in which three friends each purchase a different number of apps, all of which cost the same amount. Haleemah and ET create different equations to find the total amount of money spent by the three friends, when the price per app is given [e.g., Episode 2, 1:13 and Episode 5, 5:40] and when it is unknown [e.g., Episode 3, 1:20; Episode 4; 1:33 and 3:40]. Notably, they produce the following three equations for the general case:
3c + 2c + 4c = T
9c = T
(3 + 2 + 4)c = T
In each equation, c represents the cost of one app and T represents the total amount of money spent by the three friends. Haleemah and ET use the first two equations to create a fourth equation, 3c + 2c + 4c = 9c [Episode 5, 3:50], which illustrates the transitive property of equality and provides an equation for future exploration.
CCSS.M.HSA.SSE.A.1. Interpret expressions that represent a quantity in terms of its context.
In Episode 3, ET and Haleemah generalize a context in which three friends are purchasing apps. They initially create the equation 3c + 2c + 4c = T, where c is the cost of an app and T is the total amount spent by the three friends [1:20]. In Episode 6, the students are asked to explain the meaning of parts of their new equation, 3c + 2c + 4c = 9c. They provide two different ways of contextualizing. On the one hand, the students are able to interpret the equation from a process perspective: 3 apps bought by Danyal times the cost per app plus the 2 apps bought by Suado times the cost per app, plus the 4 apps bought by Kiaan times the cost per app is the same as the 9 apps the friends bought altogether times the cost per app [see Episode 6, 1:34 – 3:46]. On the other hand, the students also interpret the equation from a product perspective: The sum of the amounts of money each friend spent is the same as the amount of money spent by the three friends altogether [see Episode 6, 6:18 – 8:25].
According to the CCSSM, “Mathematically proficient students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.” After writing equations for three specific contexts [Episode 1, 5:30; Episode 2, 1:13; and Episode 2, 4:40], ET and Haleemah write an equation to represent the total cost spent by three friends on apps that cost c per app [Episode 3, 1:20; Episode 4; 1:33 and 3:40]. In creating these equations, Haleemah and ET were able to decontextualize the problem scenario, using abstract mathematical symbols to model the real-world situation. Later, they contextualized these abstract symbols by reinterpreting what the expressions mean in the game app context. For example, the pair explain that 3c represents the total money spent by Danyal and 9c is the total money spent by the three friends [Episode 6, 6:30].
According to the CCSSM, “Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately.” In the previous lesson, ET and Haleemah used a variable without defining it. They had to be prompted to do so by the teacher [Lesson 2, Episode 3, 1:52]. In this lesson, notice how Haleemah and ET use more precision when defining their variables [e.g., Episode 3, 0:44 and Episode 4, 5:16]. They even push each other to be more precise, as is seen in Episode 6 at 1:35, when Haleemah says the 15 is the total amount spent and ET says he agrees, but that they “should specify it to Danyal.”
