Using Negative and Decimals as Values for Algebraic Symbols
Haleemah and ET work to model a scooter trip by relating the scooter’s start location, trip time, velocity, and end location. Using this context, they begin to think about the different kinds of values that an algebraic symbol can take on.
ET and Haleemah make sense of the scooter applet and create a conjecture for the relationship between velocity, trip time, starting location, and ending location.
The students apply their method for finding the ending location for a scooter trip that starts at 4 meters and lasts for 3 seconds to two more trips with different velocities.
ET and Haleemah reflect on their generalized algebraic expression and think about the values their variable could represent.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Write a function to represent a situation where the independent and dependent variables can both take on negative values. For example, in the scooter context, students create a function that relates velocity and a starting location of the scooter.
Realize that an algebraic letter, such as x, can take on values other than natural numbers, For example, x could equal −9 or 0.26.
Understand that putting a negative sign in front of x does not mean that the term necessarily represents a negative number. For example, if x = −5, then -x is positive.
Common Core Math Standards
CCSS.M.HSN.QA.2. Define appropriate quantities for the purpose of descriptive modeling.
In Lesson 4, Haleemah and ET investigate a context in which someone named Hector rides a scooter. Using an applet, they manipulate three different quantities—trip time, starting location, and velocity—that have an impact on Hector’s end location, which is a fourth quantity. These quantities can take on different values. For example, ET and Haleemah investigate what happens if:
the starting location is a negative value (meaning a location that is west of Hector’s home, Episode 2, 1:05),
the velocity is negative [Episode 4, 3:28], and
different fractions and decimals are used (e.g., 6.5 seconds for the trip time in Episode 2, 1:26 and –4/3 for velocity in Episode 6, 6:20).
CCSS.M.HSF.BF.A.1.Write a function that describes a relationship between two quantities.
Throughout this lesson, ET and Haleemah work to generalize a context in which the end location of a scooter ride is determined by three quantities: the starting location of the ride, the velocity of the ride, and the length of time (in seconds) of the ride. As described above, Haleemah and ET begin this process by exploring the quantities involved in this motion context. They also work to make sense of relationships between these quantities, for example noting that the distance traveled by the scooter is equal to the product of the velocity and time [Episode 1, 4:23]. They make several predictions, and sometimes revise their thinking after checking those predictions against the data they collected [e.g., Episode 4, 5:25]. Finally, they generalize the context for when the starting location is 4 and the trip time is 3 seconds, which results in the function 3v + 4 = L, which relates the two quantities velocity and end location[Episode 5, 1:18]. They also compare this to a similar function, 3(–v) + 4 = L, which helps them better understand how a variable such as v can stand in for both positive and negative values [Episode 6, 3:19].
According to the CCSSM, mathematically proficient students “are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions.” This lesson features ET and Haleemah making sense of a motion context by using a dynamic applet to help them identify quantities and relationships between [e.g., Episode 1, 1:56; Episode 2]. Based on their observations and the data they collect, they formulate an initial model for the situation: the end location can be found by adding the starting location to the product of the velocity and the trip time [Episode 2, 1:57; Episode 3, 1:31]. When the context changes slightly so that the start location is always 4 and the trip time is always 3 seconds, they create a function to model this new specific context: 3v + 4 = L, where v is the velocity and L is the end location [Episode 5, 1:18].
