Algebraic ideas and techniques are powerful tools for reasoning about geometric shapes on a coordinate grid. Conversely, geometric images are useful aids to algebraic reasoning about linear equations and inequalities. This unit is designed to capitalize on the strong connections between algebra and geometry in order to extend students’ understanding and skill in several significant aspects of those two key strands in the middle grades curriculum. These are the key ideas developed by problems in The Shapes of Algebra unit. They extend earlier work with the Pythagorean Theorem by connecting it to the standard equation for circles; with properties of polygons by connecting parallel and perpendicular lines to slopes of lines and linear functions; and with solutions of linear equations by considering solutions of linear systems and equations in standard ax + by = c form, and solutions of linear inequalities. These topics are standard parts of traditional Algebra I syllabi and they are included in many standard algebra examinations. However, the Connected Mathematics approach to the topics exploits the rich connections between algebra and geometry to strengthen student understanding of problems and solution methods that are often taught and learned in quite formal and rote ways. Since students who proceed to mathematically oriented academic specialties will undoubtedly study these topics in greater detail in high school mathematics, it is important to develop the sort of conceptual understanding that will provide a solid base for future work, not simply to settle for short-term rote learning of procedures that will be quickly forgotten.
#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 × 8 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.
This unit helps students learn important questions to ask themselves about any situation that involves data analysis, such as What is the population? What is the sample? What kinds of comparisons or relationships can I explore using data from the sample? Can I use my results to make predictions or generalizations about the populations? Students have seen many of these data gathering measures before in earlier grades, and they will continue to use them in future endeavors. This unit will help students continue to practice their statistical investigation process. The process involves four parts: pose a question, collect the data, analyze the distribution, and interpret the analysis in light of the question. When completed, students need to communicate the results. Students will encounter situations in which different samples can produce different data and different characteristics of the data. Natural variability is inherent both within and between different samples taken from the same population.
Teachers need to be ready to facilitate understanding of plotting data on various forms such as scatter plots, box plots, and histograms. Teachers will need to be familiar with all of these concepts before they can delve into the instruction part. Questioning techniques are important in this unit. One question may be, “What does it mean to say one variable is related to another variable?”
Major misconceptions by and struggles for students in this unit include:
- Drawing the line of best fit and realizing that some lines may vary.
- Realizing the difference between a simulation and a real-life experiment.
- Distinguishing how two variables are related to one another.
- Writing an equation for a line through two pairs of given points (using already learned concepts)
- Predicting other data points on a scatter plot based on information given.