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Appendix A: Completed Classification of Triangles Chart

Juli K. Dixon Solution Tree Press ePub
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Chapter 3: Geometry

Nolan, Edward C.; Dixon, Juli K.; Safi, Farshid; Haciomeroglu, Erhan Selcuk Solution Tree Press ePub

CHAPTER 3

Geometry

Geometry often provides the first opportunity for a formal exploration of proof. Students make sense of and apply both inductive and deductive reasoning in this exploration. Inductive proof could be thought of as connecting to recursive thinking in the study of arithmetic sequences in algebra, while deductive thinking links more closely to determining explicit rules for those sequences. Thinking back to the bridges problem from chapter 2 (see figure 2.5, page 42), the recursive rule would be that the number of beams in the next bridge is equal to the number of beams in the current bridge with four additional beams added. The number of beams in each subsequent bridge is based on the previous example. An explicit rule for the number of beams in any bridge is not based on the number of beams in a prior bridge but is rather determined using the length of the bridge.

Specifically, inductive reasoning is using observations and examples to reach a conclusion. For example, when you recognize a pattern in a number of different isosceles triangles that the base angles are congruent, you may inductively conclude that base angles of isosceles triangles are congruent. However, this type of reasoning is always subject to finding an example that does not follow the pattern. For example, in the bridges problem if you only examine a bridge of length 4 (see figure 2.6, page 43), you might conclude that a rule for the number of beams in a bridge of any length is 3 + n(n − 1). However, upon further examination, you would see that this rule cannot be generalized to bridges of other lengths.

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Chapter 5 Geometry

Juli K. Dixon Solution Tree Press ePub

The focus of this chapter is the mathematics for teaching plane and solid geometry with depth so that you and your students develop a strong foundation for the study of geometry. What you need to know about the study of geometry is sometimes beyond what you will address with students in grades 3–5. This is in large part to be sure that you do not teach rules that will expire as students learn geometry in later grades.

Geometry is the study of space, objects in space, and the movement of objects in space. School geometry includes a focus on objects with zero, one, two, and three dimensions. Consider the images in figure 5.1.

Figure 5.1: Dimensions of plane and solid geometry.

An example of an object with zero dimensions is a point. A point does not have dimensions such as length, width, and height. Although the geometric object of a point seems very simple, the point is quite vital to the subsequent dimensions in geometry. For instance, it takes two distinct points to create a line segment. A line segment is an example of an object with one dimension; it has length. By connecting line segments that do not exist on the same line, you can create objects with two dimensions, such as a rectangle. A rectangle has the dimensions of length and width. Zero-, one-, and two-dimensional geometry (commonly described as plane geometry) includes abstract representations of the real world. When you hold up an attribute block that is the shape of a rectangle and say, “This two-dimensional shape is a rectangle,” you are not actually correct. The shape you are holding is actually three-dimensional because it has length, width, and height to it. Even when you draw a representation of a rectangle on a sheet of paper, the drawn lines have a thickness, even though it is quite small. It is understood, by most adults, that you are ignoring the third dimension, the height, in order to represent the rectangle in a way that makes sense in the real world.

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Appendix: Hypothetical Weight Loss Study Data

Nolan, Edward C.; Dixon, Juli K.; Safi, Farshid; Haciomeroglu, Erhan Selcuk Solution Tree Press ePub

Appendix: Hypothetical Weight Loss Study Data

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Chapter 4 Fraction Operations

Juli K. Dixon Solution Tree Press ePub

This chapter focuses on mathematics for teaching addition, subtraction, multiplication, and division of fractions with depth. You can develop students’ fraction operation sense by embedding operations in context through word problems. To do so, first use visual models to solve the problems, then represent the contexts and solutions with equations, and finally, make sense of procedures for solving the equations more efficiently. Check the results through estimation to be sure solutions are reasonable.

The initial task in this chapter (see figure 4.1) begins this process by providing word problems to be solved with visual models. These three problems may be challenging if you have not previously explored representing fraction operations with drawings. The key is to act out the context of each problem with pictures. The discussion that follows will be much more meaningful if you make an attempt to solve each problem using a picture and then write the situation equation before proceeding.

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