Let’s Be Rational

The goal of Let’s Be Rational is to develop meaning for and skill with computations involving fractions. Your child will

have had some experiences during upper elementary grades with developing fraction operations. Typically, however,

they will not have developed a sense of which operation to use in a particular problem situation. This unit begins by

extending your child’s knowledge of the material learned in the previous unit, Comparing Bits and Pieces.

Students can learn to carry out an algorithm (steps to solve) as a procedure, but the real power of this Connected

Mathematics Unit lies in the experiences that foster your child’s abilities to examine a problem situation and determine

which operation or operations are needed to solve that problem. In middle school, our goal is to help students develop

the deep understanding of rational numbers that comes from experiencing operations when solving a variety of

problems, including those that require more than one operation.

This Unit does not explicitly teach a specific or preferred algorithm for working with rational numbers. Instead, it helps

create a classroom environment in which students work on problems and generate strategies that make sense to them.

At a point in the development of each operation, students are asked to pull together their strategies into an algorithm

that works for all situations involving that operation on fractions. As they work individually, in groups, and as a whole

class on the problems, students practice the algorithms to develop skill and fluency in carrying them out. This

development process allows students to recognize special cases that can be easily handled and also provides students

with an efficient general algorithm that works for all cases within an operation.

You child is coming into this unit with a strong foundation in adding and subtracting fractions from elementary school.

As a result, our main focus will be on multiplication and division with fractions. Students will have a review of addition

and subtraction of fractions, followed by more in depth learning of multiplication and division of fractions.

Let’s Be Rational Investigation 1.4-Adding and Subtracting Fractions and/or Mixed Numbers-REVIEW

Your child has a lot of experience from fifth grade in adding and subtracting fractions. In this investigation, we will review their learning from elementary school. We will be focusing on using the methods learned in elementary school.

There is one aspect of adding and subtracting fractions and mixed numbers that MUST occur. This is that the denominators (the bottom number of the fractions) have to be the same for adding and subtracting. Once a problem has common denominators, there are many methods for adding or subtracting. A couple of options are illustrated below; these by no means cover every possibility but are to be used as a guide:

Adding Two Mixed Numbers Strategy 1

If you have quantities with different denominators, you can rename the fractions as equivalent fractions with a common denominator. Then add the parts, and then add the whole numbers.

1 56+2 14=1 10

12+2 312

=3 1312

=4 112

Adding Mixed Numbers Strategy 2

You can change each mixed number to an improper fraction, then find a common denominator and add.

1 56+1 14=116

+ 54=2212

+ 1512

=3712

=1 112

Subtracting Mixed Numbers Strategy 1

You can regroup one whole from the first mixed number in the subtraction sentence. That way, you will be able to subtract easily by finding a common denominator, subtracting the fractions and subtracting the whole numbers.

5 13−2 56=5 2

6−2 56=4 8

6−2 5

6=2 36=2 1

2

Subtracting Mixed Number Strategy 2

You can change each mixed number to an improper fraction, then find a common denominator and subtract.

5 13−2 56=163

−176

=326

−176

=156

=2 36=2 12

Let’s Be Rational Investigation 2 (Multiplying Fractions) Overview

Investigation 2 focuses on developing computational skill with and understanding of fraction multiplication. Various

contexts and models are introduced to help students make sense of when multiplication is appropriate.

In Problem 2.1, students develop an understanding of multiplication with simple fractions. Problems 2.2 and 2.3 focus

on multiplication with fractions, mixed numbers, and whole number combinations.

Estimation is used across the Problems so that students can determine the reasonableness of their answers. Also,

students develop the idea that multiplication does not always lead to a larger product. Within these Problems, students

form a general algorithm for fraction multiplication.

Investigation 2.1-Finding Parts of Parts

This investigation introduces students to finding the part of a part (multiplying a fraction by a fraction). We do not go

straight to the multiplication algorithm; instead we focus on using an area model to demonstrate how much is left after

taking part of a part. Your child will not be seeing the multiplication symbol until the next investigation; instead we will

use the word “of” to imply multiplication. The problem is presented with the context of serving brownies from a pan

that starts out partially full. An example is below, including how to illustrate it with an area model:

Example: Mr. Williams asks to buy 12 of a pan of brownies that is

23 full. What fraction of the whole pan has Mr.

Williams bought?

The models below show the thinking behind solving this problem. Part I shows the initial pan before Mr.

Williams makes his purchase, you can see that the entire pan is split into three equal parts, two of which are

shaded to represent the 23 . Since Mr. Williams is buying

12 of the pan, which is illustrated in part 2. When you

combine them, the gold color represents the overlap of the grey and blue, you can see that Mr. Williams is

buying 26 or

13 of the entire pan.

Investigation 2.2-Modeling Multiplication Situations

In this investigation, students extend their understanding of multiplication by modeling situations that involve fractions, mixed numbers, and whole numbers. A model is simply a drawing or diagram to illustrate the problem. This is where students will transition from using the word “of” to the traditional multiplication symbols and begin to develop the algorithm for multiplying fractions, mixed numbers and whole numbers. A series of examples are below.

Example 1: Multiplying a part by a whole

A recipe calls for 23 of a 16-ounce bag of chocolate chips. How many ounces are needed?

Below is a possible diagram for solving and the answer. You can see there are 16 squares that are then split into

three equal amounts. The shaded part represents the 23 of 16.

Example 2: Multiplying a part by a mixed number

Mr. Flansburgh buys a 2 12 pound block of cheese. His family eats 13 of the block. How much cheese has Mr.

Flansburgh’s family eaten?

Below is a possible diagram for solving and the answer.

Example 3: Multiplying a Mixed Number by a Mixed Number

2 13×1 12

Below is a possible diagram for solving and the answer

Investigation 2.3-Multiplication of Fractions, Mixed Numbers and Whole Numbers

In this investigation, we bring together all of the ideas from the first two to develop an algorithm for multiplying fractions, mixed numbers and whole numbers. The most common strategy that most students will use will be to change all mixed numbers to improper fractions and then multiply the numerator by the numerator, then the denominator by the denominator and finish by simplifying. An example is below.