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Making Sense of Mathematics for Teaching Grades 6-8: (Unifying Topics for an Understanding of Functions, Statistics, and Probability)

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Develop a deep understanding of mathematics. This user-friendly resource presents grades 6–8 teachers with a logical progression of pedagogical actions, classroom norms, and collaborative teacher team efforts to increase their knowledge and improve mathematics instruction. Make connections between elementary fraction-based content to fraction operations taught in the middle grades. Explore strategies and techniques to effectively learn and teach significant mathematics concepts and provide all students with the precise, accurate information they need to achieve academic success.

Benefits

  • Dig deep into mathematical modeling and reasoning to improve as both a learner and teacher of mathematics.
  • Explore how to develop, select, and modify mathematics tasks in order to balance cognitive demand and engage students.
  • Discover the three important norms to uphold in all mathematics classrooms.
  • Learn to apply the tasks, questioning, and evidence (TQE) process to grow as both learners and teachers of mathematics.
  • Gain clarity about the most productive progression of mathematical teaching and learning for grades 6–8.
  • Access short videos that show what classrooms that are developing mathematical understanding should look like.

 

Contents

Introduction

1    Fraction Operations and Integer Concepts and Operations

2    Ratios and Proportional Relationships

3    Equations, Expressions, and Inequalities

4    Functions

5    Measurement and Geometry

6    Statistics and Probability

Epilogue:   Next Steps

References and Resources

Index

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Introduction

ePub

The only way to learn mathematics is to do mathematics.

—Paul Halmos

When teaching, much of the day is spent supporting students to engage in learning new content. In mathematics, that often means planning for instruction, delivering the planned lessons, and engaging in the formative assessment process. There are opportunities to attend conferences and other professional development events, but those are typically focused on teaching strategies or on administrative tasks like learning the new gradebook program. Opportunities to take on the role of learner of the subject you teach are often neglected. As you read Making Sense of Mathematics for Teaching Grades 6–8, you will have the chance to become the learner once again. You will learn about the mathematics you teach by doing the mathematics you teach.

There is a strong call to build teachers’ content knowledge for teaching mathematics. A lack of a “deep understanding of the content that [teachers] are expected to teach may inhibit their ability to teach meaningful, effective, and connected lesson sequences, regardless of the materials that they have available” (National Council of Teachers of Mathematics [NCTM], 2014, p. 71). This lack of deep understanding may have more to do with lack of exposure than anything else.

 

Chapter 1 Fraction Operations and Integer Concepts and Operations

ePub

This chapter connects fraction-based content from the elementary grades to fraction operations in the middle grades. We begin with explorations that are grounded in elementary-level topics to help you support learners who have likely learned fractions in ways unlike how you may have developed your own understanding. These experiences are connected to topics in the middle grades to help you link these explorations to middle-grades content. This chapter also links early work with whole numbers to solving problems involving integers.

Fraction and integer operation sense is developed by embedding these operations in context through word problems. The word problems become valuable sense-making tools when visual models are used to solve them. Those visual solutions are then connected to equations, and finally, this process is connected to procedures by making sense of steps for solving the equations more efficiently. Results of following the procedures are checked through estimation to be sure solutions are reasonable.

 

Chapter 2 Ratios and Proportional Relationships

ePub

This chapter transitions the focus on rational numbers in terms of fractions to rational numbers as ratios and proportions. A fraction is a rational number because it describes a ratio of two numbers. In a fraction , the numerator, a, describes the number of equal-size pieces of the whole, and the denominator, b (where b is equal to 0), indicates the number of those pieces needed to make the whole. A rational number that is not a fraction can also be described as ; however, a and b (b ≠ 0) do not describe a part-to-whole relationship. While the focus of rational numbers was on fractions in the previous chapter, ratios are explored in this chapter.

Comprehending ratio and proportionality concepts empowers students to solve problems that include a number of different real-world applications. In providing problems to solve involving ratios and proportions, you support students to develop proportional reasoning (Kilpatrick et al., 2001). Proportional reasoning includes the understanding of the interrelationship of two quantities and how a change in one connects to a change in the other.

 

Chapter 3 Equations, Expressions, and Inequalities

ePub

In the elementary grades, students explore numbers and unknowns to make sense of operations; in the middle grades, students extend that work to include variables, where unknown values can be replaced by numbers to make equations true. Focus is placed on making sense of independent and dependent variables, equivalent expressions, and order of operations. Students find ways to describe sequences by using patterns. For example, when determining the nth term in a sequence, students begin by describing the first few terms in the sequence with the ultimate goal of finding a generalization to describe any term.

The initial task in this chapter (see figure 3.1) examines a series of staircases to generalize how many 1 × 1 squares are in any staircase. Before reading on, determine the number of 1 × 1 squares in a 5-step staircase and a 10-step staircase, and then determine a generalized expression for the number of 1 × 1 squares in an n-step staircase. Pay particular attention to how you developed your expression. Challenge yourself to either justify your expression in another way or to create a new expression that represents an n-step staircase.

 

Chapter 4 Functions

ePub

This chapter emphasizes understanding functions by unifying topics such as ratio, rate, proportionality, expressions, and equations explored in previous chapters. A fundamental understanding of function is that it is a quantitative relationship in which each input, x, results in a unique output, f(x). In the middle grades, the function notation, f(x), may not be required; a function can be notated using y or any other variable. For the purpose of this text and to connect to mathematics content explored in high school, function notation will be used in this chapter.

The initial task in this chapter (see figure 4.1) challenges you to explore your understanding of functions by asking you to create a word problem that could be modeled by the given function.

Describe a quantitative relationship that could be modeled with the function f(x) = 2x + 20, describe the relationship as a story, then draw a graph to represent the context you describe.

 

Chapter 5 Measurement and Geometry

ePub

In this chapter we unpack big ideas in measurement and geometry in order to make sense of concepts as well as formulas for area, surface area, and volume of various simple and complex shapes. Teaching geometric measurement with depth involves making sense of why these formulas work, connecting them to algebraic reasoning within the coordinate plane, and applying geometric measurement to real-world scenarios. Geometry concepts also include examining angles, transformations, congruency and similarity, and the Pythagorean theorem. As learning progresses through grades 6–8, the contexts and applications of measurement and geometry become more complex.

Take a few minutes to make sense of a formula for the area of a trapezoid (see figure 5.1). Pay particular attention to how you develop your formula.

A trapezoid is shown below. Using any combination of rectangles, parallelograms, and triangles, determine a formula for the area of this trapezoid. Justify why your formula works.

 

Chapter 6 Statistics and Probability

ePub

How do you define statistics and probability? Statistics is the collection, analysis, and interpretation of data, often in numeric form. Probability involves examining the fraction of successful outcomes out of the total number of possible outcomes and is an important factor in understanding random processes. This chapter explores how to build conceptual understanding of statistics and probability in the middle grades. You will reason with problem-solving situations that involve statistical investigations, data analysis, and probability.

One important aspect of statistical reasoning in the middle grades is comparing data sets. What are the different ways that you can compare data sets? How do you know which way to use for particular data sets? Think about your answers to these questions as you complete the task shown in figure 6.1.

Mr. Richard and Ms. Chutto decide to compare the grades in their two science classes on the last quiz.

The grades in Mr. Richard’s class on the twenty-point quiz were:

 

Epilogue Next Steps

ePub

An important role of mathematics teachers is to help students understand mathematics as a focused, coherent, and rigorous area of study, regardless of the specific content standards used. To teach mathematics with such depth, you must have a strong understanding of the mathematics yourself as well as a myriad of teaching strategies and tools with which to engage students. Hopefully, by providing the necessary knowledge, tools, and opportunities for you to become a learner of mathematics once more, this book has empowered you to fill this role.

Now what? How do you take what you learned from doing mathematics and make good use of it as the teacher of mathematics?

Our position is that you first need to apply what you learned to your lesson planning. Are you planning for instruction that focuses on teaching concepts before procedures? How is your planning aligned to developing learning progressions? How will you ensure that your lessons do not end up as a collection of activities? What follows are strategies that will help you use what you experienced as learners and apply it to what you do as teachers.

 

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