A1 Pre-Race Warm Up

Introduction

A1 Pre-Race Warm Up is the first investigation in a unit on acceleration. In my 9th grade math class, this unit is preceded by an inaugural unit on lines and systems, but as I am introducing this material to the blog in November I'll skip lines for now. In any case, I think the acceleration unit is rather more interesting and actually stands well on its own no matter where you are in your year of work with your students. Indeed, you might use it in any high school math class.

As you may have guessed, this unit is followed by a quadratics unit. You might think that students would need to understand quadratics before modeling accelerating objects, but actually the converse is true. The acceleration unit, driven by tables and graphs, eventually draws students to develop a quadratic function. As students approximate solutions to problems, they invariably ask me what algebraic tools they can use to get more precise results. Algebraic tools are introduced when a student has developed a need for the tool. 

Paintbrushes and wrenches, words and hammers, these tools of the trade are powered by the simple desire to order the world. For the toddler, before the hammer had any known application, hammering for the sake of hammering sufficed. But even the toddler's first word, "water," was arguably uttered for want of a solution to a problem, thirst. The question, "Why do we have to learn this," is evidence of a tool lacking a purpose. This acceleration unit flips the question to "Do you have any tools I can use to solve this more precisely?"

General Teaching Notes

My classes are 40 minutes long and we meet 4 or 5 times a week. There is a group test, an individual test and several group and individual quizzes in the unit. The unit takes about four weeks, given our time constraints. My students' only homework is completed online on the weekends--they complete and correct the week's investigations and watch optional videos and do any needed extra practice posted on the school's moodle website. In this way I am giving them considerable personal responsibility as well as giving them more time to read and write for other classes for which independent, solitary work is arguably more essential. In 9th grade we explore lines, acceleration, quadratics, polynomials, rational functions, exponential functions and logarithms. In 10th grade we explore trigonometry, trigonometric models and transformations, sequences and series, and probability and statistics. 

Specific Teaching Notes for This Investigation

This probably takes about 80 minutes for everyone to complete. Students who finish early move on to the next investigation, which I will post soon. I always introduce an investigation that's early in the unit with open ended questions to generate interest. How fast does a car on the highway go? How fast can a person run in kilometers per hour? How fast is that in meters per second? If your students are not familiar with kilometers and meters you can ask these questions in miles per hour and feet per second to start with, but the acceleration unit uses the metric system. In general, the first investigations of a unit require a bit more intervention from me, and sometimes mini-lectures, which you will see below. Later investigations are more student directed.

1) The first question on the investigation is equally open ended. Students will want to know the correct answer to the question about negative velocity or a velocity of zero--I encourage you to let them struggle and resolve and write their ideas as the question will be implicitly answered later in the unit.

2) Allow students to struggle with the conversion. Most of them can figure this out. I discourage using the factor label method as I feel that for students it is mechanical and relatively mindless. I prefer them to continually re-think whether to multiply or divide when they are doing a simple conversion until it becomes second nature. Some students may use proportions, which I consider acceptable.

3) This question presumes an understanding of how to create a function machine for a line. It also assumes the use of the language of a function machine. (This is well covered in the lines unit, which I've not yet posted.) If your students can't do this, I would pause the class after 5 to 10 minutes of work on #1 and #2, and show them how to make table and distance graph for a car going at 80 km/hr. I would then ask them to tell me how far the car goes in 10 hours, and their resulting multiplication justifies the machine y = 80x. This may seem trivial and obvious, but if students mechanically put the velocity in front of x without understanding the logic of multiplication, they've missed the point. You should also ask them to imagine a graph for a car with a speed of 100 km/hr and how this would be different. They should understand how to draw slope triangles and how this represents the speed of each car. This should only take 10 minutes. Leave the work on the board for students who want to take notes before returning to the investigation.

4) Please note that they are graphing speed as a function of time. This graph will be a horizontal line. You may need to pause the class to explain, but this should be not more than 3 minutes. Usually I just explain that this graph is a speedometer, and I show how to make it for a car that goes at variable speeds in the first hour, second hour, etc. 

6) You may want to pause after some students complete #5 to play a 5 minute game with the class. I call it "going on a journey." Students give you a velocity, and a time you are driving. You graph distance on the board, roughly. Then something happens, students tell you you are suddenly going slow in traffic, so the line flattens out a bit for an hour. Then you stop for gas, so your graph goes horizontal. Then you realize you forgot something, so your distance graph drops down. Then you get on the highway and complete your journey, whatever they want you to do. This is the game they are playing in number 6, using the story in #5. There are a lot of unknown times, speeds and distances which they have to invent. The graph is meant to be a rough sketch, discourage students from trying to be overly precise.

7) This is a more precise table version of what they did in #6. There are still some things they have to invent.

9) Speed here is like a bar graph. Some students may ask if they need to show speed increasing or decreasing, or if it just leaps from one speed to another in an instant. This is a great question and one that you could pause the class to answer, or you could just let that group resolve the question on their own, as it will be revisited in later investigations.