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Integrated Algebra I Function Families 1 & 2 Teacher Edition Notes on Exploring Functions with Fiona INTRODUCTION: The Georgia Performance Standards for Grade 8 Mathematics include: recognize a relation as a correspondence between varying quantities, recognize a function as a correspondence between inputs and outputs where the output for each input must be unique, distinguish between relations that are functions and those that are not functions, recognize functions in a variety of representations and contexts, identify relations and functions as linear or non-linear, and translate among verbal, tabular, graphic, and algebraic representations of functions. (Standard M8A3, parts a, b, c, d, h, i) This learning task launches the unit by giving students an opportunity to work with a variety of relations, most of them functions. Each of these relations is presented in a real-world context since most of the students’ prior work with relations and functions has been in such contexts. Function notation is introduced as a new topic. Through work with the variety of examples given, students should develop understanding and skill in using standard algebraic function notation while revisiting the concepts and skills inherent in the Grade 8 standards listed above. Items 1 and 2 below explore some relations involving heights of various fictional human beings. The two different contexts allow review of the important distinction between continuous and discrete input values, and exploration of possible alternative perspectives in item 2 lead to review of the distinction between relations that are and are not functions. Function notation is introduced at the end of Item 2; here common English usage for describing the context of the function closely parallels standard terminology for the notation. The remaining items allow students extensive practice with function notation in meaningful contexts. Throughout the task, there is a deliberate emphasis on correct terminology in service of conceptual understanding. Supplies Needed: graph paper Exploring Functions with Fiona 1. One vacation when visiting his grandmother, Todd found markings on the inside of a closet door showing the heights of his mother, Julia, and her brothers and sisters on their birthdays growing up. From the markings in the closet, Todd wrote down his mother’s height form ages 2 to 16. His grandmother found the measurements at birth and the one year by looking in his mother’s baby book. The data is provided in the table below, with heights rounded to inches. Integrated Algebra I Age (yrs.) Height (in.) x y 0 21 1 30 2 35 Function Families 1 & 2 3 39 4 43 5 46 6 48 7 51 8 53 9 55 10 59 11 62 12 64 13 65 14 65 a. Which variable is the independent variable, and which is the dependent variable? Explain your choice. Comments: It is important to note that the information given states that the heights are measured at the same point each year, on Julia’s birthday. This property is important for this question in clarifying that each x-value value corresponds to an exact age in years; that the input values are unambiguous. In question c below, it allows us to conclude that connecting the dots corresponds to representing points in time between Julia’s birthdays. Solution: The age in years is the independent variable, and the height in inches is the dependent variable. Since height was measured once each year on Julia’s birthday, there is a unique height as output for each age in years used as input. The height cannot be used as the independent variable, because, for some heights, there is more than one associated age. b. Make a graph of the data. Julia's height Height in inches 70 60 50 40 30 20 10 0 0 2 4 6 8 10 Age in years 12 14 16 c. Should you connect the dots on your graph? Explain. Comments: This is the first of many questions in this unit asking students whether the data points graphed should be connected. These questions are designed to focus 15 66 16 66 Integrated Algebra I Function Families 1 & 2 students on the differences between situations where the domain is continuous and where the domain consists of discrete numbers. The important issues are (i) whether the points created by connecting the dots are both meaningful and (ii) whether they provide reasonably accurate representations of the functional relationship. Note that, given the nature of this data, allowing students to use technology to create the graph would be quite appropriate. Excel scatter plot was used to create the graph; however, any similar statistical graphing tool would also be a good choice. Solution: Yes, the dots for the (age, height) pairs should be connected. Julia had a height at every moment from the time she was 0 years old until 16 years old, and her height at any time between her birthdays would be approximated by the values shown on the line joining the measured heights. d. Describe how Julia’s height changed as she grew up. Comments: Students will likely apply varying amounts of prior knowledge about growth to answering this question. Those students whose families have carefully tracked the students’ growth patterns will be aware that periods of no growth are often followed by growth spurts and may give a more detailed description of the changes in growth rate. Thus, teachers should anticipate and accept varying levels of specificity for this answer. The correct term to describe Julia’s height function is that it is a nondecreasing function since, on any subinterval in the interval [0, 16], the function is either increasing or constant. For most groups of students, this context allows an informal introduction to this important term used in describing characteristics of functions. While student answers may not include all the details given above, students should (i) describe the nondecreasing nature of the growth (in their own words, use of the term “nondecreasing” is not expected) and (ii) note that the rate of growth in height is not constant, with the most rapid growth rate during the first year. Solution: Julia’s height increased continuously as she grew up until age 13, then did not change by age 16 except for an increase of one inch during her 15th year. Her height increased most rapidly during the first year. From age 1 to 6, the number of inches she grew each year decreased, going from an increase of 5 inches in her second year to an increase of only 2 inches in her sixth year. From ages 6 to 13, she grew 2, 3, or 4 inches per year with the numbers going up and down again twice. She did not increase in height during her 14th year, grew an inch in her 15th year, and did not increase in height during her 16th year. e. How tall was Julia on her 11th birthday? Explain how you can see this in both the graph and the table. Solution: Integrated Algebra I Function Families 1 & 2 Julia was 62 inches tall at age 11, as shown by the y-value corresponding to x = 11 in the table and by the point (11, 62) of the graph. f. What do you think happened to Julia’s height after age 16? Explain. How could you show this on your graph? Comments: The graph shown includes values for Julia’s age through 35 years. It is reasonable to assume that the mother of a high school student is about this age or older. Some students may extend the graph further and predict a slight decrease in height in Julia’s senior years since many women lose some height in their later years due to osteoporosis. The important point is understanding how to represent on the graph that the height values remain constant after age 16, but other discussions can be beneficial in providing additional depth of understanding for interpreting graphs. Solution: Most females reach their adult heights by age 16, so it is likely that Julia’s height remained the same after age 16. The graph could be extended as a horizontal line at y = 66, as shown in the graph below. Julia's height Height in inches 70 60 50 40 30 20 10 0 0 5 10 15 20 Age in years 25 30 35 Integrated Algebra I Function Families 1 & 2 Integrated Algebra I Function Families 1 & 2