Sixth Grade Mathematics

**Unit 3: Fractions, Decimals, & Percents II**

Resource: *Bits and Pieces II, Connected
Mathematics*

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.

* *

__Investigation 2: More About Percents__

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.

* Example
– Adding and subtracting fractions (the traditional method) *

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
6^{th} 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 2^{nd} 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.

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