For Warm Up problems today, I have included questions from the previous unit on scientific notation. Some of my students continue to struggle with solving problems in this notation, so I will continue to provide Spiraled Review during Warm Up problems for additional practice.
For my students, the biggest stumbling block is with expressions that involve the addition or subtraction of terms with unlike exponents. When I encounter students who are making incorrect calculations, I ask them if they have another strategy to use when solving these problems. My goal is to surface their current conceptions, and, to identify any misconceptions. For these problems, the prompt usually results in performing operations by expanding the numbers first, then adding them, and then converting the result into scientific notation. While this is not the most efficient method, it is one that makes sense to the students, and, it gives them the opportunity to see the structure of the number in two different forms. My goal is for all students to be comfortable with at least one go-to strategy. If they persevere, my students generally make progress learning more efficient algorithms.
Before diving into this unit on functions, I want to provide my students a strong anchor for understanding. Today, I make a first step in this direction by introducing the Rule of 5 Star. I display the graphic and then I distribute a copy for each student to glue in their math journals.
The Rule of 5 Star represents my thinking about the essential elements of understanding a function completely:
In coming up with this model, I added picture to a 4-step model that I had previously used. I find it helps my students to visualize the situation, such as by sketching a picture of what is going on in the story. We will refer to this graphic organizer over and over as we work with functions and multi-step problems set in real world (or imaginary) contexts.
Once students have glued the Rule of 5 Stars in their journals, I reference our day's learning objective (written on the board) and ask for four volunteers to help act out a story. I display the story of the race of Turtle & Snail and read through it with the class. We then act out the story using students as the trees and main characters.
Teacher's Note: I have marked off 12 one-foot increments on the floor with masking tape to prepare the race course.
I place place my 'trees' on either end of the course. I then put my 'turtle' and 'snail' at the first tree and say, "Raise your hand if you know who is going to win this race before we even begin." The majority of students raise their hands, so I select one to tell me. When the student responds "Turtle," I ask how she knows. The student invariably answers, "Because he walks faster." I ask the class to give a thumbs up signal if they agree.
I insist that we act out the story to be sure. I say, "So the race has begun. One minute has passed, so how far will Turtle have traveled?" The students respond, "Three feet", so I encourage my actor to move to the three feet mark. I then ask, "And where is Snail after 1 minute?" The students respond, "Two feet", so Snail moves two feet. As we continue, I call attention to what is happening to the distance between the two animals as the race continues. Typically, students notice that the distance between the two is growing over time. I ask why this is happening and call on a student to explain: "Snail is traveling slower, so after each minute, she falls further behind turtle."
Once we have acted out the entire scenario and confirm that Turtle did, indeed, win the race, I explain that we have just experienced the story from our Rule of 5 Star. I then direct students to the model and ask, what comes next? I display a picture that represents the story and ask my students to create one in their journal. I remind them to include the important information, especially the length of the race.
After 2 minutes, I ask what comes next? I then display two tables with time and distance as the independent and dependent variables. I explain that these tables will display the data that was represented in the story. We then fill in the missing data. I ask, "At zero minutes, how far had Turtle traveled?" Typically, I hear both 'zero' and 'three', so I ask a student to clarify. This usually helps anyone who initially said, 'three' correct their thinking. We continue filling in the table for Turtle and then do the same for Snail.
I recap, "So we've told the story, drawn the picture, created tables...What comes next?" Students respond, so I move the next slide for the rule/equation. I explain that we must find a rule that would describe what is happening in the table, much like we did when we played "What's My Rule" function game. A student immediately called out, "Add three!". I'm always grateful when this misconception appears because it allows me the opportunity to address it early in the lesson sequence. I write down d = t + 3 as the rule and ask for a thumbs up from students who agree. Several always do, so I continue by explaining we must check our rule to make sure it fits. I begin with t = 0. I ask, "Does 0 = 0 + 3?" Several students answer 'no', so I continue, knowing I have created some cognitive dissonance for my students. "Who has another idea for a rule?" Another student suggests 'multiply by 3', so I write the rule d = 3t, and ask, "Does this rule fit? Let's see!" I then begin substituting the values from the table into the equation. Students nod in agreement that this rule fits. We then move on to the rule for Snail, which comes easily now that we have done Turtle's rule.
To reinforce the Rule of 5, I recap again, "We told the story, drew a picture, created tables, determined the rules for those tables...so what's left?" We must create a graph that represents the story. I explain that we can do this in one of two ways. We can use the data from the tables, or we could use the rules we created. I remind the students that the tables represent independent (x) and dependent (y) values, which we can plot as coordinate pairs on a graph. I move to the graph slide and quickly rewrite the two tables of values. I ask for a volunteer to graph Turtle's values in green. As the student plots points, I recap with statements like, "So you start at (0,0), the origin, which makes sense since the animals had not gone anywhere yet." and "At one minute, you went to three on the graph, since that is how far Turtle traveled in one minute." This commentary helps students who may have forgotten how to plot points while also reinforcing the relationship between the table and the graph.
Once the student has finished graphing the Turtle's data, I ask, "What do you notice about the Turtle's graph?" The student notes, "It's a straight line.", so I add, "So the data is linear. Does that fit with what we already know about graphing? Did the Turtle travel at a constant rate? Now let's go to our rule and see if it fits. If I change the d= 3t to y = 3x, does the rule fit this line? Is the equation of this line y = 3x? Some students nod in agreement, but I can see that many are unsure. Most students have not seen slope-intercept form since the year before, so I write y = mx + b to remind them. At this point, I am not concerned if they do not have full understanding as we will be revisiting this concept many more times in the coming days.
I ask another student volunteer to graph Snail's data on the same graph as Turtle's but using a different color. When she finishes, I ask, "What do you notice about Snail's graph?" She replies, "It's linear." I ask, "What is different about Snail's line than Turtle's?" She shrugs in response, so I ask the class. A student responds, "It's lower than Turtle's." I ask the student to describe the line in terms of steepness, so he says, "It's less steep." I want to make connections to the story so I say, "So Turtle's line is steeper. How does that connect to the story?" Another student responds, "He's going faster, so his line is steeper."
In a typical sequence 30 minutes has now passed, so I take time to congratulate the students on completing their first Rule of 5 activity. I explain that we will continue to use the Rule of 5 with other stories in the coming days.
To give me feedback on student's depth of understanding of today's lesson, I ask for a Ticket Out the Door (see Turtle & Snail Part 1 TOTD). I give each student an index card on which to respond to the prompt, "Explain how the numbers from the table relate to the graph." I collect these as students exit class.