Learning Math: Patterns, Functions, and Algebra
Functions and Algorithms Homework
Session 3, Homework
Invent a number game like the one from Problem D1. Draw a network that lets you figure out the output for any number. Draw another network that lets you find someone’s original number if you know his or her final result. Try out your game on someone else.
Here’s a number game:
- Pick a number
- Add 3
- Double your answer
- Subtract 4
- Finally, multiply by 3
Tell whether each example below is a function, and explain how you decided.
- Input: a circle. Output: the ratio of the circumference to the diameter.
- Input: a soccer team. Output: a member of the team.
- Input: a CD. Output: a song on the CD
Tip: Remember, a function is any relationship in which each input leads to exactly one output, and the same output may be repeated more than once for different inputs. A rule is not a function when the same input can lead to multiple outputs.
Gabriela and Ben are trying to decide whether the rule y = x4 is a function. Represented as an algorithm, the equation y = x4 is equivalent to starting with a number, then multiplying that number by itself four times. For example:
Input: x = 2
Output: y = 2 x 2 x 2 x 2 = 16.
Who is correct, Ben or Gabriela? Is y = x4 a function? Explain how you know.
Tip: Pick a number. Add one. Add one again. Again. Again. By doing this, you’re iterating the “add one” function. The “his father before him” function is frequently iterated in conversation.
Problems H4-H7 involve iteration, a process that can be done to any function where the output is the same type as the input. When you iterate a function, you apply it again and again, each time using the previous output as the new input. Iteration is a very important technique for solving equations approximately when typical algebraic methods can’t be used, and it is also used to model many real-world problems like population growth and the change of weather.
Here’s an example:
Input: a real number
Output: half that number
Start with an input of 20. The first output is 10. Now, use that output as the new input. The second output is 5. Use 5 as the next input, and you get an output of 2.5. And so on.
Use the function described above, starting with an input of -16. What are the first five outputs as you iterate the function? What will happen to the value if the iteration continues forever?
Tip: The first output is -8, then use -8 as the next input.
Build a network of function machines that adds 1 to an input and then divides by 3. Now iterate the function. Try three different original inputs, and iterate the function at least 10 times for each input. What is going on?
Tip: What happens to the values “in the end,” after many iterations? For what numbers does this happen? Can you explain why?
A “fixed point” of an iteration is a value of the input that produces itself as an output. For example, if your algorithm subtracts 3 from an input, then multiplies by 2, the input value 6 has output 6, so it is a fixed point.
Take It Further
Find any fixed points for these algorithms. There may not be any, or there may be more than one.
- Add 1, and divide by 3
- Subtract 6, and multiply by 3
- Add 6
- Square the input.
- Add 12, and divide by 4
- Square the input, then subtract 6
Tip: If a fixed point’s input is n, what would its output have to be?
You can also iterate functions that act on geometric shapes. Here’s a new function:
Start with a right-side-up triangle; that is, a triangle with one base horizontal. Find the midpoints of each side of the triangle, and connect them to each other. Repeat this process on each right-side-up triangle in the output.
Here’s the start of the iteration for this function:
Draw your own triangle. Iterate the function above at least three times. Describe how the outputs relate to each other.
- Yes, the output is always Π, but there is exactly one output for any input.
- No, the same team can have lots of different members.
- No, the same CD can have many different songs.
Gabriela is correct. The number of matching outputs is not important (witness the “3” function!), only that there is exactly one output each time.
The outputs are -8, -4, -2, -1, and -1/2. The iterates get closer and closer to zero.
No matter what you start with, the inputs will get closer and closer to 1/2.
In each case, the output value must equal the input, so writing the rule and making it equal n is one way to solve the problem.
Each output is three half-sized copies of the input, with a hole in the middle. The pattern can repeat itself indefinitely.
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.