One of the central goals of algebra is describing and reasoning about relationships between quantitative variables. Beginning in grade six, the students compare sides and angles in regular polygons and similar figures. They explore many relationships in grade 7 and 8 covering linear patterns and relationships.
This algebra unit brings to the students the concept of useful nonlinear relationships, ones that model exponential growth and exponential decay. Studies of various types of population growth model the exponential growth patterns; while examples of radioactive substances and medication effectiveness provide examples of exponential decay where the students can see the pattern of decline over time.
The unit starts out with situations that involve repeated doubling, tripling, and quadrupling of data. The students make tables and graphs while describing the patterns that they are observing. Students make equations for this rapid growth while conceptualizing their understanding of the general form of an exponential equation. Continuing with the rapid growth patterns, the unit takes the students to the exploration of the y-intercepts that are less than and greater than 1. Throughout the unit, exponential relationships are also being compared to linear relationships as students compare and contrast these representations. Patterns of decay are subsequently introduced where the students will discover strategies for finding the decay factor, which is between 0 and 1, and for describing this non-constant rate of change.
The conclusion of the unit focuses on the patterns with exponents. Extended lessons were added that concentrate on operations with exponents, scientific notation, operations with scientific notation, and cube roots.
PA.CCSS.Math.Content.CC.2.2.HS.D.7. Create and graph equations or inequalities to describe numbers or relationships. (HSA-CED.A.1)
PA.CCSS.Math.Content.CC.2.2.8.B.1. Apply concepts to radical and integer exponents to generate equivalent expressions. (8.EE.1, 8.EE.2, 8.EE.3, 8.EE.4)
PA.CCSS.Math.Content.CC.2.1.8.C.1. Define, evaluate, and compare functions. (8.F.1, 8.F.2, 8.F.A.3,)
PA.CCSS.Math.Content.CC.2.1.8.C.2. Use concepts of functions to model relationships between quantities. (8.F.B.5)
PA.CCSS.Math.Content.CC.2.2.HS.C.2. Graph and analyze functions and use their properties to make connections between the different representations. (HSF-F.IF.C.7)
PA.CCSS.Math.Content.CC.2.2.HS.C.3. Write a function or sequence that model relationships between two quantities. (HSF-F.BF.A.1)
PA.CCSS.Math.Content.CC.2.2.HS.C.5. Construct and compare linear, quadratic, and exponential models to solve problems. (HSF-F.LE.A.1, HSF.F.LE.A.2, HSF.F.LE.A.3)
#1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.
#2 Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.
#3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
#4 Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
#5 Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, or a calculator. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
#6 Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.
#7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Mid-level students will see 7 × 15 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.
#8 Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
As mentioned previously, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
Representing and reasoning about patterns of change has been the main focus up to this point in the development of algebra. Students have used tables, graphs, and symbols to represent relationships and to solve equations to find information or make predictions. The properties of real numbers such as the Commutative and Distributive properties were first introduced in Accentuate the Negative and then used again in Moving Straight Ahead and Growing, Growing, Growing. The Distributive Property was expanded in Frogs, Fleas, and Painted Cubes to include multiplication of two binomials.
Major misconceptions by and struggles for students in this unit include:
- Adding rate of change to y-intercept in the equation
- Trying to solve exponential equation by adding or subtracting, or by using square root
- Solving for the base of exponents by dividing
- Aligning the y-scale on a graph to the numbers on the table; therefore creating a straight line
- Finding the y-intercept in a table