Decimal Ops
Standars & Objectives | |
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Goals for Decimal Ops
Students should...
- Understand that estimation can be used as a tool in a variety of situations, including as a way to check answers and make decisions
- Revisit and continue to develop meanings for the four arithmetic operations on rational numbers, and practice using algorithms to operate on decimals
- Use variables to represent unknown values and number sentences to represent relationships between values
- Develop understanding of percents through various contexts, such as sales tax, tips, discounts, and percent increases
Investigation 1
Vocabulary | |
File Size: | 215 kb |
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This Investigation has three Problems. The Problems help students recognize problem structures that lead to certain operations in given situations. Students also develop estimation strategies and work to understand unit rates as expressions of ratios. While the remainder of the Unit requires that students use algorithms to compute with decimals, this Investigation focuses on helping students understand decimal numbers and how these numbers might be applied in real-world situations. Because of this, students should have access to scientific calculators to perform the arithmetic. They should estimate beforehand to predict the solutions and check their work.
In Problem 1.1, students enhance and test their abilities to recognize when each basic arithmetic operation is appropriate for solving problems. Students identify when addition, subtraction, multiplication, or division of decimals will answer the question(s) posed. In some cases, a combination of two or more operations is needed. In Problem 1.2, students work to develop common-sense strategies for estimating results of decimal operations. They also use their estimates to determine whether or not an answer is reasonable. Problem 1.3 addresses unit rates and their connection to ratios. Students work with decimal numbers to find and scale unit rates. They continue to build their understanding of operations with decimals as they work through real-world problems. |
Investigation 2
Vocabulary | |
File Size: | 155 kb |
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This Investigation has three Problems. Problem 2.1 reviews ideas of place value for decimals (from Comparing Bits and Pieces ) using a card-sorting activity. Next, students analyze student work and develop a standard algorithm for adding decimals. Problem 2.1 also introduces the expanded form of decimals. Expanded form serves as an explicit representation of place value, which will be revisited later in the Unit for the division algorithm. Problem 2.2 uses a similar structure to develop a standard algorithm for subtraction of decimals, using paper and pencil. Problem 2.3 brings back the idea of addition and subtraction fact families, in which students examine fact-family relationships. These fact-family relationships are designed to be an early introduction to algebra.
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Investigation 3
Vocabulary | |
File Size: | 158 kb |
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This Investigation has five Problems. Problem 3.1 leads students to develop an algorithm for multiplying decimals with aid of, at most, paper and pencil. Problem 3.2 reviews and extends student understanding and skill in multiplication of multi-digit whole numbers and connects this with multi-digit decimal multiplication. Problem 3.3 leads students to develop an algorithm for dividing decimals with aid of, at most, paper and pencil. Problem 3.4 and Problem 3.5 review and extend student understanding and skill in division of multi-digit whole numbers and connect this to multi-digit division of decimals. As in Investigation 2, students use fact families to check the results of division calculations. The goal of this Investigation is to provide guidance for students to make sense of these standard algorithms.
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Investigation 4
Vocabulary | |
File Size: | 215 kb |
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This Investigation has four Problems. Problems 4.1, 4.2, and 4.3 deal with applications of percents as taxes, gratuities, and discounts—one application in each Problem. Problem 4.4 acts as a reprise and variation on the theme of Problem 1.1. It asks students to choose appropriate operations, estimate and find exact results, and report answers in complete sentences.Percent bars, also called tape diagrams, are useful in picturing relationships and in writing and solving mathematical sentences. Fact families again come into play when rewriting sentences to find an unknown quantity.
Throughout this Investigation, we assume that students have access to at least a scientific calculator to perform the arithmetic. The focus is on the recognition of problem structures that lead to various operations on percents and decimals. In all four Problems of this Investigation, students are asked three fundamental questions about percents: (1) What is a given percent of some base quantity? (2) What is the base quantity if a percent increase or decrease (as rate and amount) is known? and (3) How can a given increase or decrease in a quantity be expressed as a percent? |