8 Chapters
Medium 9781942496458

Chapter 1 Fraction Operations and Integer Concepts and Operations

Edward C. Nolan Solution Tree Press 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.

See All Chapters
Medium 9781942496458

Introduction

Edward C. Nolan Solution Tree Press 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.

See All Chapters
Medium 9781942496458

Chapter 2 Ratios and Proportional Relationships

Edward C. Nolan Solution Tree Press 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.

See All Chapters
Medium 9781942496458

Epilogue Next Steps

Edward C. Nolan Solution Tree Press 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.

See All Chapters
Medium 9781942496458

Chapter 3 Equations, Expressions, and Inequalities

Edward C. Nolan Solution Tree Press 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.

See All Chapters

See All Chapters