Comparing Bits & Pieces
Standards & Objectives | |
File Size: | 218 kb |
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Goals of Comparing Bits & Pieces
Students should...
- Understand fractions and decimals as numbers that can be located on the number line, compared, counted, partitioned, and decomposed
- Understand ratios as comparisons of two numbers
- Understand equivalence of fractions and ratios, and use equivalence to solve problems
Investigation 1
Investigation 1 Vocabulary | |
File Size: | 222 kb |
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Students explore fractions and ratios in this Investigation, in the mathematical context of understanding and making comparison statements. Understanding equivalence of fractions, and equivalence of ratios, is a major emphasis. This Investigation treats fractions as numbers, as locations and distances on a number line, and as part-to-whole relationships. It treats ratios as comparisons between numbers. Students use manipulatives (fraction strips), visual models (thermometers and other diagrams), word names, and symbols for fractions. They use visual models, symbols, and language to express ratio comparisons. The measuring of progress in a school fundraiser focuses students' attention on the part-to-whole nature of fractions, while comparing the goals and progress of different grades focuses their attention on ratios.This Problem introduces comparisons through the context of a school fundraiser. Students examine each comparison statement to see if the statement is valid. From the discussion, students explore both multiplicative and difference comparisons.
Investigation 2
Investigation 2 Vocabulary | |
File Size: | 158 kb |
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Using what they know about comparison statements and equivalent fractions, students will develop strategies for finding equivalent ratios. The context of sharing chewy fruit worms allows students to partition while keeping track of a relationship between shares of segments of a worm. Students write equivalent ratios with an associated unit rate. Rate tables provide an additional strategy for writing equivalent ratios. Students use ratio and rate reasoning to solve problems.
Investigation 3
Investigation 3 Vocabulary | |
File Size: | 229 kb |
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Students extend their work with the number line to the right of 1 and to the left of 0, so that the number line forms the foundation for understanding improper fractions, and negative rational numbers. Students use partitioning strategies, and the idea that the numerator counts same-sized pieces, to locate numbers to the right of 1 and to the left of 0 on the number line. Students take advantage of two new ideas, opposites and absolute value, to help locate numbers on the number line. Students partition the units on a number line and think about fractions as both locations on a number line and distances between locations on a number line.In addition to developing decimal and fraction benchmarks, the last three Problems in Investigation 3 use estimation and linear models (fraction strips and number lines) to introduce students to the decimal place value system. Students also learn that the decimal place value system is a way to interpret and compare decimal numbers.
In the second Problem, students investigate subdividing a 100-square grid to show 1,000 parts or 10,000 parts. This process of subdividing and naming the new parts is important mathematically, as is developing strategies to find a decimal that falls between two given decimals.
In the second Problem, students investigate subdividing a 100-square grid to show 1,000 parts or 10,000 parts. This process of subdividing and naming the new parts is important mathematically, as is developing strategies to find a decimal that falls between two given decimals.
Problem 3.1:
In this Problem, students learn about improper fractions and negative rational numbers as they relate to a location on the number line. Students use partitioning strategies and the idea that the numerator counts same sized pieces to move past 1 and past 0 on the number line. The intention is to get at the idea of "total number of same-sized pieces" and "number of whole units plus leftover number of same-sized pieces". Students also learn about opposites and absolute value as they relate to locations on the number line. Although most of the Problem focuses on locating rational numbers on a number line, part E moves to recognizing rational numbers as distance. |
Problem 3.2:
In this Problem, students compare and order fractions using benchmarks, estimation, equivalence, and distances. Students use the benchmark values of −112, –1, −12, 0, 12, 1 and 112 to estimate the value of rational numbers. After estimating the size of the rational numbers, they locate them on a number line. The number line can help students understand fractions as both location and distance. This Problem offers ample opportunity for students to practice finding equivalent fractions using strategies developed in preceding problems. |
Problem 3.4:
For decimals to be useful to us, we need to be able to estimate how large a number is and to make comparisons between numbers. In this Problem, we build on what we know about fractions, fraction benchmarks, tenths and hundredths strips and grids, and number lines as a way to make sense of decimals. Students begin by thinking about what happens when a strip or grid is partitioned into increasingly smaller subdivisions, such as hundredths and thousandths. This promotes sense of patterns as students think about what would be the next decimal place, what decimals might fall between two decimals, and then how decimals can be represented. |
Investigation 4
Investigation 4 Vocabulary | |
File Size: | 216 kb |
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The percent bar model is used as a strategy for relating ratio, fraction, and percent. Students consider percents as a way of making comparisons among basketball players' free-throw data and among the frequencies of various personal characteristics in a class of sixth-grade students.
Students learn how to write percents for situations that are not based on 100. As strategies for finding a percent of a given quantity, they use
These strategies are also used to find the percent that one given quantity is of another. From knowledge about the percentage of a given total, students also make predictions using the strategies listed above.
Students learn how to write percents for situations that are not based on 100. As strategies for finding a percent of a given quantity, they use
- percent bars
- rate tables
- rates
- equivalent fractions
- ratios.
These strategies are also used to find the percent that one given quantity is of another. From knowledge about the percentage of a given total, students also make predictions using the strategies listed above.
Problem 4.1: In this problem, percent is defined as meaning “out of 100.” However, some ratios are not easily scaled to be “out of 100” because not all ratios are easily scaled to percents. Students use percent bar models and benchmark percents to reason proportionally. They translate percents, such as 25%, into a ratio of successful shots out of a known total. Then they compare free throws made to respective total attempts by two NCAA teams using a bar model. Students can use the virtual number line to aid in their exploration.
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