Unit 3: Fractions, Decimals, & Percents II

Resource:  Bits and Pieces II, Connected Mathematics

# I.   Unit Overview

In Unit 3, your child will continue to explore fractions, decimals, and percents by acquiring the following skills:

§       Describe, create, solve, and justify real-life problems that involve money, unit pricing, percents, discounts, sales price, and sales tax with one or more operations.

§       Estimate and verify the percent of a number by using a benchmark.

§       Express ratios as fractions.

§       Calculate a missing term in a proportion and explain the process.

§       Develop, create, solve, and justify real-life problems that involve ratios and proportions.

§       Construct and interpret frequency tables, circle graphs, and scatterplots that involve whole numbers, decimals, and fractions, using technology when appropriate.

§       Estimate sums, differences, products, and quotients of whole numbers and decimals using estimation techniques such as round and compatible numbers.

§       Describe and justify situations when an estimate answer is sufficient and when an exact answer is required.

§       Estimate and calculate sums, differences, products, and quotients of fractions and mixed numbers.

§       Calculate and describe how to calculate division involving decimals.

II.   Vocabulary

In Unit 3, the following vocabulary words are essential to understanding the concept of rational numbers:

algorithm                                         fraction

base ten number system                    numerator

benchmark                                       percent

decimal                                             reciprocal

denominator                                     unit fraction

equivalent fractions

These terms can be found in the “Descriptive Glossary” located in front of the index in the back of your child’s “Bits and Pieces II” booklet.  These descriptions are not formal definitions, but describe the concepts in a friendly manner and provide examples that should help students make sense of these important terms.

III.    Investigations

In the Bits and Pieces II unit, your child will be completing seven investigations.  Each investigation consists of daily problems designed for students to explore the key concepts of fractions, decimals, and percents.  Students should be completing homework assignments on a daily basis.  Homework practice is an essential part of learning mathematics and is crucial for reinforcing the skills and concepts taught in class.  Students are usually assigned homework problems from the ACE (Applications, Connections, and Extensions) section at the end of each investigation in their Bits and Pieces II booklet.  These problems are based on work completed in class.  Your child should have notes and practice problems from their mathematics class that will help them complete the assigned homework problems.  In addition, the following information can serve as a guide for completing homework assignments:

Investigation 1: Using Percents

In Investigation 1, students build on the concepts taught in Bits and Pieces I (Unit 1).  Students will use what they learn about percents to calculate discounts and tax on various items.

Examples – changing decimals to percents

We read 0.35 as “thirty-five hundredths”.  Write this as a fraction, .  Since percent means out of 100, the answer is 35%.

We read 0.9 as “nine tenths”.  Write this as a fraction, .  Since percent means out of 100, we rewrite this fraction as an equivalent fraction out of 100:  = .  0.9 = 90%.

Students learn the “short-cut” for changing decimals to percents: move the decimal point 2 places to the right.

Examples – changing percents to decimals

7% means 7 out of 100 or .  As a decimal this is 0.07 – “seven hundredths”

16% means 16 out of 100 or .  As a decimal this is 0.16 – “sixteen hundredths”

The “short-cut” for changing a percent to a decimal is move the decimal point two places to the left.

Example - changing fractions to percent:

means 3 ÷ 8 which can be typed in a calculator or calculated by hand.  If typing it in a calculator, type “3 ÷ 8 =”.  The calculator should display 0.375.  If calculating this by hand do:.  Keep adding zeros to 3.0 using the division process until you get a remainder of 0.  This will give you 0.375.  Now move the decimal point 2 places to the right to get the percent.  = 37.5%.

Examples – changing percents to fractions

8% means 8 out of 100 or .  in lowest terms is.

18% means 18 out of 100 or .    in lowest terms is .

Examples – creating equivalent fractions

To create a fraction equivalent to a given fraction you multiply or divide both the numerator (top number) and denominator (bottom number) of the given fraction by the same number.

OR

( is only one fraction equivalent to .  There are infinitely many more)

Example – calculating the percent of a number      What is 5% of \$24.75?

Convert 5% to a decimal see page 2.  5% = 0.05.

Multiply 0.05 x 24.75 (At this point, students use a calculator to do the multiplication).

0.05 x 24.75 = 1.2375

Depending on the context of the problem this would either be \$1.23 or \$1.24.  If you are calculating a discount, stores will not round up.  The discount would be \$1.23.  However if they are calculating tax, they would round up.  The tax would be \$1.24.

In this Investigation students are asked to think about percents in situations in which the number of objects is greater than or fewer than 100.  Students use percents to help construct and make sense of circle graphs.

Example –determining what percent one number is of another.

5 is what percent of 25?

Create a fraction from the two numbers:.  The denominator of the fraction is always the “of” number.  Convert this fraction to a decimal (see page 2).  = 0.2.  Covert this decimal to a percent: 20%.   = 20%.

Investigation 3: Estimating Fractions and Decimals

In Investigation 3, students use what they learned in Bits and Pieces I as well as skills learned earlier in this unit to estimate sums of fractions and decimals.

Investigation 4: Adding and Subtracting Fractions

In this investigation, students use equivalent fractions and conversions between fractions and decimals to develop their own algorithm for adding and subtracting fractions.  Students are not given the traditional algorithm (shown below) for adding and subtracting fractions, however by the end of this investigation they should have an efficient way to add and subtract fractions.  The traditional algorithm (using equivalent fractions) could be the method they choose.

Step 1 – rewrite the fractions as equivalent fractions with the same denominator.

=

- =

Step 2 – add or subtract the fractions.

Step 3 – “reduce” or rewrite the fraction in lowest terms (if necessary)

Investigation 5: Finding Areas and Other Products

In Investigation 5, students use the area model and two other methods to help make sense of multiplication of fractions.  In the last problem, students use their experience with these models to develop their own algorithm (possibly the traditional one shown below) for multiplying fractions.

Example – multiplying fractions (the traditional method)

To multiply fractions, multiply the numerators and multiply the denominators.  Rewrite the fractions in lowest terms (if necessary).

Note: whole numbers can be written as fractions by writing them with a denominator of 1.  Ex: 16 = .

Mixed numbers can be written as fractions as well:.

Investigation 6: Computing with Decimals

In this investigation, students further develop their computation skills with decimals.  Students play a game and apply decimal addition, subtraction, multiplication, and some division to real-world situations.  While division of decimals is not a major focus of this Investigation, it is a skill 6th grade students are expected to master.  Your child’s teacher may provide extra lessons and practice in addition to what is provided in the Bits and Pieces II booklet to assist students with dividing decimals..

Example – addition & subtraction of decimals.

To add or subtract decimals, line up the decimal points, fill in any missing digits with zeros (0), add or subtract the numbers as if they were whole numbers, then bring the decimal point straight down in the answer.

3.5 – 1.76

3.50

- 1.76

1.74

Example – multiplication of decimals.

To multiply decimals, multiply the numbers without the decimal points as if they were whole numbers then count the total number of decimal places to the right of the decimal point in the problem and place the decimal point in the answer so there are that many numbers to the right of the decimal point.

4.76 x 7.8

476                    4.76 x 7.8 = 37.128

x  78

3808                    4.76 has 2 decimal places to the right of the decimal

33320                    7.8 has 1 decimal place to the right of the decimal

37128                    2 + 1 = 3.

The answer needs to have 3 decimal places to the right of the decimal point.

Example – division of decimals.

To divide decimals, move the decimal point on the outside number (the divisor) to the right until the number is a whole number.  Move the decimal point on the inside number (the dividend) the same number of places.  Bring the decimal point straight up in the answer (the quotient).  Divide the numbers as if they were whole numbers.  (Add zeros to the dividend until you get a repeating pattern or you get a 0 remainder).

5.05 ÷ .025                                              5.05 ÷ .025 = 202

Investigation 7: Dividing Fractions

In Investigation 7, students develop the meaning of division with fractions and the strategies and algorithms for dividing them.  Other operations with fractions are also reviewed.

Example – division of fractions.

To divide fractions, take the reciprocal of the 2nd fraction (“flip it”) and then multiply the two fractions.

=  = = .  (Note:  means 100 ÷ 4 which is 25).

IV.    How Can You Help?

1.       When shopping let your child help in determining discount amounts and sale prices of items.

2.       When shopping for items that have sales tax, ask your child to estimate or calculate the amount of tax on an item.

3.       If you have the opportunity to dine out, share with your child how you determine the amount of the tip (usually about 15%) for the waiter or waitress.

4.       Look for newspaper or magazine articles that contain fractions, decimals, and percents.  Discuss the article with your child and what the numbers in the article mean.  The financial section is often a good section to find these numbers.

5.       Check your child’s assignment notebook for homework each night.

6.       Assist your child with their homework when needed.