In this chapter, students are introduced to the square root function and learn how shifts, flips, and stretches change the shape of the graph. Students use the properties of radicals to solve radical equations and problems involving the Pythagorean theorem, distance and midpoint formulas. The chapter ends with an introduction to descriptive statistics and graphical displays.
Graphs of Square Root Functions -
Radical Expressions -
Radical Equations -
The Pythagorean Theorem and Its Converse -
Distance and Midpoint Formulas -
Measures of Central Tendency and Dispersion -
Stem-and-Leaf Plots and Histograms -
Box-and-Whisker Plots -
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Problem-Solving Strand for Mathematics
Along with the four-step problem-solving plan presented in the text, students are prompted to reflect back on strategies, evaluate their effectiveness and analyze their usefulness for future problems. This chapter, Algebra and Geometry Connections; Working with Data, continues to promote such an analysis.
Recognizing patterns, drawing connections, and considering extraneous solutions are all featured. When dealing with extraneous solutions, point out to students where in the procedures the potential for a false solution has been introduced. This awareness can prepare them for more sophisticated mathematics ahead.
Encourage students to think about everyday applications of the Pythagorean Theorem (students might interview carpenters, plumbers, architects, and/or structural engineers, for example) and emphasize that the distance and midpoint formulas are extensions of the basic Pythagorean theorem and “taking an average” respectively. Similarly, with Stem-and-Leaf Plots, Histograms, and Box-and-Whisker Plots, be sure students understand the focus and strengths of each style of graph. The point with each of these tools is to represent and/or be able to interpret data in ways that make sense and communicate clearly.
Alignment with the NCTM Process Standards
This chapter, focused on algebra and geometry connections and working with data, aligns with many of the NCTM process standards. Throughout students are asked to recognize and use connections among mathematical ideas (CON.1), to appreciate how mathematical ideas interconnect and build on one another to produce a coherent whole (CON.2) and to recognize and apply mathematics in contexts outside of mathematics (CON.3). When dealing with radical equations and extraneous solutions, students apply and adapt a variety of appropriate strategies to solve problems (PS.3) and monitor and reflect on the process of mathematical problem solving (PS.4). In working with the Pythagorean theorem and its converse, students use representations to model and interpret physical and mathematical phenomena (R.3). In applying the distance and midpoint formulas students select, apply, and translate among mathematical representations to solve problems (R.2).
In the lessons which produce and analyze data communication tools, students create and use representation to organize, record, and communicate mathematical ideas (R.1); communicate their mathematical thinking coherently and clearly to peers, teachers, and others (COM.2); analyze and evaluate the mathematical thinking and strategies of others (COM. 3); and use the language of mathematics to express mathematical ideas precisely (COM.4).
- COM.2 - Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
- COM.3 - Analyze and evaluate the mathematical thinking and strategies of others.
- COM.4 - Use the language of mathematics to express mathematical ideas precisely.
- CON.1 - Recognize and use connections among mathematical ideas.
- CON.2 - Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
- CON.3 - Recognize and apply mathematics in contexts outside of mathematics.
- PS.3 - Apply and adapt a variety of appropriate strategies to solve problems.
- PS.4 - Monitor and reflect on the process of mathematical problem solving.
- R.1 - Create and use representations to organize, record and communicate mathematical ideas.
- R.2 - Select, apply, and translate among mathematical representations to solve problems.
- R.3 - Use representations to model and interpret physical, social, and mathematical phenomena.