20 Chapters

Chapter 6 Measurement

Juli K. Dixon Solution Tree Press ePub

In grades 3–5, topics central to measurement include concepts related to perimeter, area, volume, elapsed time, angles, and conversion within systems. Contexts related to measurement offer opportunities for students to engage in problem solving.

The initial task in this chapter (figure 6.1) provides an opportunity for you to engage in problem solving related to measurement. This task involves the use of the geoboard and geoboard dot paper. If you do not have access to a geoboard, the activity can be completed using virtual geoboard manipulatives or the geoboard dot paper provided.

Make nine different (simple) polygons, each with an area of four square units. Record them on geoboard dot paper so that each polygon is on its own board.

Visit go.solution-tree.com/mathematics for a free reproducible version of this figure.

In order to solve this task, you also need to define different in terms of the problem. In this problem, different means not congruent. Therefore, the polygons in figure 6.2 would not qualify as different because they are transformations of one another. In figure 6.2, you can see that the second figure was just a translation of the first, and the third figure was a rotation of the first figure. Polygons are the same if they are congruent, regardless of their location or orientation on the geoboard (see chapter 5 for more on polygons).

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Chapter 6 Measurement

Juli K. Dixon Solution Tree Press ePub

In kindergarten through grade 2, topics in measurement are primarily related to linear measurement, time, and money. Contexts related to measurement offer opportunities for students to engage in problem solving in real-world contexts.

The initial task in this chapter (see figure 6.1) provides an opportunity for you to engage in problem solving related to linear measurement, time, and money, respectively—each of the primary measurement topics in K–2. Be sure to solve each problem before proceeding. While solving the task, think about how you might use the problems, or modify the problems, with your students.

1. Sergei and Lois each measure the length of the same object. Sergei says the length is 5 units. Lois says the length is 15 units. How can they both be correct?

2. What time is it when the hour hand is on the 24th minute mark?

3. Jackie had 45 cents in her purse, but she has no dimes. What coins could she have in her purse? What process did you use to determine the coins?

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Appendix B: Completed Diagram for Classifying Quadrilaterals

Juli K. Dixon Solution Tree Press ePub

Chapter 3 Addition and Subtraction Using Counting Strategies

Juli K. Dixon Solution Tree Press ePub

This chapter connects your understanding of number concepts and word problem structures with the operations of addition and subtraction using counting strategies. In the following pages, you will consider how to develop understanding of the concepts of addition and subtraction. The addition and subtraction learning progression begins with learners using strategies to solve real-world problems. We then blend the use of these strategies with the learning of facts to develop a deep meaning for fluency.

Our initial task for you provides a context for the addition of two two-digit addends (see figure 3.1).

Jamila has a total of 28 playing cards in her set. Jocelyn has 37 playing cards in her set. If the girls combine their two sets, how many playing cards would be in the combined set?

What strategies can be used to solve the problem in figure 3.1? Make a list of different ways to solve the problem, and provide a brief description for each strategy. What was the first strategy you considered? Did you begin by using the standard algorithm? Other strategies demonstrate how to make sense of the task in meaningful ways, including the modeling of place value. The use of concrete materials such as base ten blocks can aid in representing the place value connection (see figure 3.2).

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Chapter 1 Place Value, Addition, and Subtraction

Juli K. Dixon Solution Tree Press ePub

Place value and the operations of addition and subtraction on whole numbers, with a connection to decimals, are important topics in grades 3–5 mathematics instruction. In this chapter, we focus on creating a deep understanding of the base ten number system and place value to assist in developing an understanding of invented and standard algorithms. You will explore how to use manipulatives such as base ten blocks to build students’ understanding of how to compose and decompose numbers in order to help with the process of regrouping. We also include multiple strategies for developing fluency to add and subtract within 1,000, which will lead to success with the standard algorithm.

In the first task (see figure 1.1), consider how to add two three-digit numbers using an invented algorithm. Imagine that you don’t yet know the standard algorithm of lining up the addends vertically, adding the ones and regrouping if necessary, then adding the tens, and so on. As with all tasks in this book, take the time to work the task out before reading on. The time you take with the task makes the text that follows more meaningful.

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