Learning Math: Patterns, Functions, and Algebra
Linear Functions and Slope Homework
Session 5, Homework
In the Achilles and the tortoise problems in Part C, Achilles runs at a constant rate of 9 miles per hour, and the tortoise moves at 1 mile per hour. Suppose that the speeds of Achilles and the tortoise are unchanged but Achilles catches up to the tortoise in 1 1/2 hours. How much of a head start did the tortoise get?
Tip: Using a spreadsheet can help solve this problem.
The tortoise has taken some “turtle speedup potion” and can now walk at 2 miles per hour. If Achilles still runs at 9 miles per hour and catches up to the tortoise in 3 hours, how much of a head start did the tortoise get?
Here’s a trick that master carpenter Norm Abram might use when building supports for roofs. He knows he’ll need evenly spaced supports along the roof. He carefully measures what length he needs for the 1st one, and finds that it’s 12 feet. Then he measures what he’ll need for the 2nd, and finds it is 9 feet. He calls to his assistant: “Don’t measure the others, just make them 6 and 3 feet long!” Why does Norm’s trick work?
You’ve worked with undoing functions. Take a moment to think about undoing a linear function. If given the formula d = 3t + 2 for distance traveled in terms of time, what would you do to express time in terms of distance? When undoing a linear function, will the result always be a new function? If so, will the new function always be a linear function?
Earlier we found that Achilles ran 13 1/2 miles in 1 1/2 hours. Therefore, we have to find a way to get the tortoise to the 13 1/2-mile mark after 1 1/2 hours. The tortoise walks at 1 mile per hour, so it can walk 1 1/2 miles in that time. The remaining distance must be its head start: 13 1/2 – 1 1/2 = 12 miles.
Use the same logic as in Problem H1. Achilles runs 27 miles in the 3 hours, therefore the tortoise needs a head start that will get it to the 27-mile mark after 3 hours. Because it walks at 2 miles per hour, it can walk 6 miles in 3 hours, so the head start is 27 – 6 = 21 miles. An algebraic equation could also be used for this problem.
It works because the roof is a straight line, and therefore it has a constant rate of change. By carefully measuring the 1st and 2nd support, the carpenter has calculated a rate of change: (change in height of support) / (distance between supports). Since this rate is constant, and the distance between supports stays the same, the change in the support’s height must also be constant. This is identical to predicting the next number in the output of a linear function; in this case, the output drops by 3 for every new support.
It can be done by solving the algebraic equation for the other variable, using the technique of “undoing” that was first used in Session 3. For the equation d = 3t + 2, start by subtracting 2 from each side to produce d – 2 = 3t. Then divide both sides by 3, so that the equation is (d – 2) / 3 = t. If a linear function can be undone, the result will always be a new, linear function. The only linear functions which cannot be undone are constant functions like y = -7. See Session 5, Problem D7 and Session 3, Problems E9-E12.
Session 1 Algebraic Thinking
In this initial session, we will explore algebraic thinking first by developing a definition of what it means to think algebraically, then by using algebraic thinking skills to make sense of different situations.
Session 2 Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of what algebra is.
Session 3 Functions and Algorithms
In Session 1, we looked at patterns in pictures, charts, and graphs to determine how different quantities are related. In Session 2, we used patterns and variables to describe relationships in tables and in situations like toothpicks and triangles. This session extends the exploration of relationships to include the concepts of algorithm and function. Note1
Session 4 Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation.
Session 5 Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations.
Session 6 Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.
Session 7 Nonlinear Functions
Continue exploring functions and relationships with two types of non-linear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions and explore exponential and quadratic functions in real-life situations.
Session 8 More Nonlinear Functions
Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.
Session 9 Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the K-2 grade band.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 3-5 grade band.
Session 12 Classroom Case Studies, Grades 6-8
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 6-8 grade band.