Comparing bits and pieces parent letter

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

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

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.

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.