Learning Math: Patterns, Functions, and Algebra
Proportional Reasoning Part A: Two Different Meanings of “More” (15 minutes)
Session 4, Part A
Suppose there are two classes in a school, one with 20 students and one with 25. If the first class has 10 girls and the second class has 12, which class has more girls?
In this video segment, participants share their answers to Problem A1 with Professor Cossey. Watch the segment after completing Problem A1, and compare your answer with those of the onscreen participants. If you get stuck on the problem, you can watch the video segment to help you.
What different definitions of “more” are used here?
You can find this segment on the session video, approximately 3 minutes and 8 seconds after the Annenberg Media logo.
If you said that the second class has more girls, you’re making an absolute comparison. You probably thought that 12 is 2 more than 10, so there are more girls in the class with 12.
If you said that the first class has more girls, you’re making a relative comparison. You probably thought 10 is half of 20, and 12 is less than half of 25, so there are more girls in the class with 10.
Clearly these are two different interpretations of “more.” Although both interpretations are correct, in some cases it is more appropriate to look at relative rather than absolute comparisons. For example, compare an all-girl class of 20 students with a class of 25 students, 22 of whom are girls. In a sense, there are “more” girls in the class with 20.
Describe the numerical calculations you would do with two numbers to make an absolute comparison.
Describe the numerical calculations you would do with two numbers to make a relative comparison.
Give some examples of situations where it is more useful to make an absolute comparison. Give some examples of situations where it is more useful to make a relative comparison.
Suppose the average height of eighth-grade students is greater than the average height of seventh-grade students. Is this an absolute or relative comparison? Why?
Groups: Begin Part A by discussing the meaning of “more.” Read through the exercise and share ideas about which class has “more” girls. Consider the difference between absolute and relative comparisons. Answer Problems A1-A5 in pairs or in small groups.
Either answer is defendable. The first class has a higher percentage of girls — half its students are girls — while girls make up less than half of the second class. The second class, however, has a larger number of girls. Essentially, the meaning of “more” is the real issue in this problem.
An absolute comparison is done by counting: 22 is more than 20, because you count to 20 before counting to 22. Or it can be done by subtraction: 22 is more than 20 because 22 – 20 = 2, a positive number.
A relative comparison is done by finding percentages or fractions or by finding a rate. On a quiz, 22 out of 25 is worse than 18 out of 20, even though 22 is larger in absolute terms.
You might use absolute comparisons when looking at annual salaries, but a relative comparison when looking at per-hour wages. An absolute comparison might tell you that extended cable TV is more expensive than basic cable, but a relative comparison might tell you that you get more channels per dollar on extended cable. A truck may be able to travel further than a compact car before needing to be refueled (an absolute comparison), but the compact car may travel more miles per gallon (a relative comparison).
This is a relative comparison, because it compares heights “per student” by using the average. An absolute comparison might compare the total height of all eighth graders to the total height of all seventh graders.
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.