See my Do Now in my Strategy folder that explains my beginning of class routines.
Often, I create do nows that have problems that connect to the task that students will be working on that day. Today I want students to compare data in a table and a graph. Some students may have difficulty reading the graph, and that is okay.
I ask students to share their thinking. Some students may share that each line on the y-axis is counting by $200 and therefore Adrian’s Bikes charges $900 for 30 bike rentals. Other students may use another data point, like 20 bikes for $600 to figure out the cost of 30 bike rentals. Compared to Rocky’s cost of $975 for 30 rentals, Adrian’s is the better deal. Students are engaging in MP2: Reason abstractly and quantitatively and MP3: Construct viable arguments and critique the reasoning of others.
I want students to begin to work with input-output tables. For today I will stick to one-step equations. I present these input-output tables to students as a puzzle. I have a rule for how x and y are related in my head and their job is to use the clues to determine my rule. I begin by giving students a series of three x and y-values that follow my rule. I make sure that my x-values are not consecutive so students must focus on the relationship between the x and y value. Then I call out a fourth x value and ask for volunteers for guesses for the corresponding y-value. I emphasize to students that it is okay if they haven’t figured out the rule yet, they need to keep trying. I call out a fifth y-value and ask for volunteers for guesses of the corresponding x-value. To finish the table I call on volunteers to share corresponding x and y-values that they think match my rule. If they work, we add them to our table. Then students participate in a Think Pair Share to brainstorm how to find y if they are given the x-value. Students share out their thinking and then we talk about how to represent that rule with an equation. Students are engaging in MP2: Reason abstractly and quantitatively and MP8: Look for and express regularity in repeated reasoning.
We watch the Learn Zillion video on independent and dependent variables and fill in the notes. I want students to start using the sentence frame “the _______________ depends on the _____________” to determine which variable is which.
Students work independently on the practice problems. I call on students to share and explain their thinking.
We work on the example about Carlos saving money together. I mention that our calculations do not take into account interest that Carlos will earn. The amount of money Carlos has saved depends on the number of months he saves. Students participate in a Think Pair Share to brain storm an equation that fits Carlos’ situation. Some students may propose y = 100x . If students don’t mention this, I present it as a possibility. I want students to realize that although Carlos is saving 100 dollars each month he starts off with $50, so that equation doesn’t work. If you substitute in zero for x, y = 100(0) = 0. We must add in what he starts with at month zero. Another common mistake would be the equation y = 150x. This equation only works for the first month. It is important for students to see that the equation must work for all of the values in the table. We use substitution to check that they equation y = 100x + 50 works for eac of our values. Students are engaging in in MP2: Reason abstractly and quantitatively, MP4: Model with mathematics, and MP8: Look for and express regularity in repeated reasoning.
Students then work on the practice problem independently. I walk around and monitor student progress and behavior.
If students are stuck, I may ask one of the following questions:
I Post A Key so that students can check their work once they are finished. If students successfully complete the practice problem, they can work on the challenge problems. These problems are more difficult because the equation involves two steps.
Some of my students were struggling to generate an equation to match Carlos’ situation. I chose to offer students a variety of equations. On the board I wrote:
a) y = 100 + x
b) y = 50x + 100
c) y = 100x
d) y = 100x + 50
e) y = 150x
Students worked in partners to determine which of these equations matched Carlos’ situation. When we came back together, I asked students to identify one equation that did not match and explain why. Students used the data in the table and graph to prove their ideas. When a student mentioned (b), I declared that(b) did work since for the first month 50 x 1 + 100 = $150. I wanted students to tell me that even thought it worked for the first month, the equation did not work for the other months. I also did this for equation (e). Then we talked about why (d) correctly modeled Carlos’ situation. Students were engaging MP3: Construct viable arguments and critique the reasoning of others.
I ask students to return to the two bike shops in the do now. I ask students to write an equation that represents the cost from renting bikes from each shop. I want students to realize that they cannot create an equation for Rocky’s Cycle Center because it is not linear. I ask students to share how they created an equation for Adrian’s Bike Shop. Students are engaging in MP2: Reason abstractly and quantitatively and MP3: Construct viable arguments and critique the reasoning of others.