Williamstown Elementary School - WatsonMath.com€¦ · Web viewThe word decimal is derived from...

Williamstown Elementary SchoolGrade 5 Unit # 3

Base Ten Fractions, Decimals, and PercentsVermont Mathematics GEs covered: M5:1, M5:2, M5:4

Base-Ten Fraction Models...................................................................................................3Lesson # 1 – Fractions with powers of 10 in the denominator............................................3

Activity # 1a:...................................................................................................................3Lesson # 2 – Place Value: An Infinite Progression of Squares and Strips..........................8

Activity # 2a: Physical models of whole numbers.........................................................9Lesson # 3 – Place Value and Deimals..............................................................................11Lesson # 4– The Decimal Point Indicates the Unit of Measure........................................13

Activity # 4a: Naming by moving decimal point to change the unit.............................14Lesson # 5 – Making the Fraction-Decimal Connection...................................................16

Activity # 5a: Base-ten fractions to Decimals and Vice Versa.....................................17Activity # 5b: Use of the Calculator to Teach the Concept of Decimals......................17

Lesson # 6 – Developing Decimal Number Sense: Familiar Fractions Connected to Decimals............................................................................................................................18

Activity # 6a with 10 x 10 Grid.....................................................................................18Activity # 6b: Hundredths Disk – Estimate, then Verify...............................................19

Lesson # 7 - Repeating Decimals: Finding Decimal Equivalent for 1/3 and 2/3 as a Class...........................................................................................................................................21Lesson # 8: Decimals and Fractions on a Number Line....................................................23

Activity # 8a: Decimals on a Friendly Fraction Line...................................................23Lesson # 9 - Approximation with a Nice Fraction............................................................24

Activity # 9a: Close to a Friendly Fraction..................................................................24Activity # 9b: Best Match..............................................................................................24

Lesson # 10 – Ordering Decimal Numbers.......................................................................25Activity # 10a: Line ‘Em Up........................................................................................25Activity # 10b: Close “Nice” Numbers.........................................................................25

Lesson # 11 – Introducing Percents...................................................................................26Set Model...................................................................................................................26Area Model................................................................................................................26Linear Model.............................................................................................................26Example using set model...........................................................................................27

Lesson # 12 – Realistic Percent Problems and Nice Numbers..........................................28Activity # 12a................................................................................................................28

Grade 5 Unit # 3 – Fractions, Decimals, PercentsWilliamstown Elementary School – developed by Elaine Watson, Ed.D.Primary Source: John A. VandeWalle (2004), Elementary and Middle School Mathematics: Teaching Developmentally of 65

Williamstown Elementary SchoolGrade 5 Unit # 3

Fractions, Decimals, and Percents

Resource: John A. VandeWalle (2004), Elementary and Middle School Mathematics: Teaching Developmentally (Chapters 15 and 17)

Big Ideas: Decimal numbers are simply another way of writing fractions.

o Decimal numbers in the form of 0.234, for example, can be expressed as proper fractions (fractions between 0 and 1)

o Decimal numbers in the form of 3.157, for example, can be expressed as improper fractions or mixed numbers.

The base-ten place value system extends infinitely in two directions. There is a ten-to one ratio between the values of any two adjacent places.

o Moving from left to right the place value on the right is one-tenth smaller than the place value on the left. (example: tenths to hundredths)

o Moving from right to left, the place value on the left is ten times larger than the place value on the right. (example: hundreds to thousands)

There are two major underlying concepts for students to understand when using a number line:

o A number line shows an infinite number of integers (positive and negative whole numbers). When looking at a number line, to the left the integers approach negative infinity and to the right the integers approach positive infinity.

o Between any two integers on a number line, we can zoom in an infinite number of times. When using the base ten system, if we zoom in to the space between two integers once, we get tenths. If we zoom into the space between two tenths, we get hundredths, etc.

The decimal point is a convention that has been developed to indicate the units position. The position to the left of the decimal point is the unit that is being counted as singles or ones. (If no unit is indicated, the unit is understood to be ones.)

Percents are simply hundredths and are a third way to write fractions and decimals.

o Percents between 0% and 100% can be expressed as proper fractionso Percents greater than 100% can be expressed as mixed numbers.

Addition and subtraction of decimals are based on the fundamental concept of adding and subtracting the numbers in like position values – a simple extension from whole numbers.

Multiplication of two numbers will produce the same digits, regardless of the positions of the decimal point. As a result, for most practical purposes, there is no reason to develop new rules for decimal multiplication and division. Rather, the computations can be performed as whole numbers with the decimal placed by way of estimation.


Connecting two different representational systems (fractions and decimals)The symbols 3.75 and 3 ¾ represent the same quantity, yet on the surface they appear very different to most students. Many people think of fractions as sets or regions, and think of decimals as more like numbers. A major goal of this unit is to teach students that fractions and decimals (and later percents) are two systems of representation that represent the same concept.

To help students see the connection between fractions and decimals, we will: Use familiar fraction concepts and models to explore rational numbers that are

easily represented by decimals: tenths, hundredths, and thousandths Help them see how the base-ten system can be extended to include small positive

numbers less than one (between 0 and 1 on the number line) as well as large numbers.

Help students use models to make meaningful translations between fractions and decimals.

Base-Ten Fraction Models This unit will use the following models to teach the concept of decimals:

Circular disks to model tenths and hundredths Meter stick to model tenths, hundredths, and thousandths Number line between 0 and 1 (Note: We will discuss the whole number line as a

model and then focus on the number line between 0 and 1 for this unit.) Base-ten blocks where the large square is 1 unit, making the strip 1/10 or .1 and

the small square is 1/100 or .01Note: While money is a good application of decimals, it only illustrates the hundredths place and therefore is not a good model to teach the concept of decimals.

Lesson # 1 – Fractions with powers of 10 in the denominator Before starting the unit, give students the pre-assessment. Ask them to do their best, but if they don’t know how to answer something, it is okay. Once they finish the pre-assessment, give the students an overview of the unit by introducing the big ideas stated above. They are not expected to fully understand the big ideas. The unit will lead them through them.

In the first lesson, they will be using the following models: Circular disks to model tenths and hundredths Meter stick to model tenths, hundredths, and thousandths Number line between 0 and 1

Before they use the models, explain to them that each model represents one whole.


Circular DiskUsing the circular disk, students need to understand that one complete circle represents one whole or “1”. If you ask them to show 7/10 with the blue, ask them what the orange represents. (3/10). Do the same with hundreds. Ask them to show 34/100 with the blue. They should also see that the orange is 66/100.

Meter StickWhen using the meter stick, students need to understand that the whole stick represents one whole and the whole is one meter.

Two Fundamental Concepts that can be Modeled on the Number Line Integers (positive and negative whole numbers) go to negative infinity to the

left and positive infinity to the right. Between any two integers, you can zoom in an infinite number of times to

represent a very small number.


-2 -1 0 1 2 3

-10 - 5 0 5 10

To negative infinity

To positive infinity

Between any two integers, you can zoom in an infinite number of times to represent a very small number.

On this number line, the space between each integer has been zoomed in to make ten divisions, making tenths. If you zoom in between any two of the tenths marks, making ten divisions, you will have hundredths. These small numbers are hard to draw, but can be visualized in your mind.

If this spot represents 0 or 1 (or 10/10 or 100/100)…

Then this spot represents 5/10 or 50/100

Proper Fractions and Mixed Numbers

Proper Fractions are fractions that lie between 0 and 1 on the number line. All proper fractions have a numerator that is smaller than the denominator. (Negative proper fractions lie between 0 and –1 on the number line.)

Improper Fractions are fractions that are to the right of one on the number line. All improper fractions have a numerator that is larger than the denominator. (Negative improper fractions lie to the left of –1 on the number line.)

Improper Fractions can be expressed as Mixed Numbers. A mixed number consists of a whole number and a proper fraction.

It is important to spend some time giving students a general overview of the number line and what it models. Once they see the overview, explain to them that for this unit, we will be looking at positive proper fractions, which means that we will be looking at the number line from 0 to 1.

Introducing the DecibunnyIn Activities 1A, 1B, and 1C, students are first introduced to the idea of naming base ten fractions as a sum of other base ten fractions. This can be confusing to some students. One way to introduce this is to introduce the “decibunny” that hops from zero to the identified number using different lengths of hops. He is called the decibunny because he hops in base ten lengths, such as 1, 1/10, 1/100, 10, 100, etc.


-2 -1 0 1 2 3

6/10 is a proper fraction

23/10 is an improper fraction that can be expressed as a mixed number:2 and 3/10 2 + 3/10or 2 3/10

Proper fractions

Improper fractions, mixed numbers, or whole numbers

Negative Proper fractions

NegativeImproper fractions, mixed numbers, or

whole numbers

-1 is a negative whole

number

Decibunny on the Circular Disk

Using the circular disk, imagine that the decibunny is hopping different ways to show on the orange a fraction of 7/10.

Starting at the border between the orange and the blue and hopping into the orange region along the edge of the circle, the bunny could take two lengths of hops. The longer hop would cover 1/10 of the circle and the smaller hop would cover 1/100 of the circle. To get to 7/10, the decibunny could hop seven 1/10ths OR could hop seventy 1/100ths. (7/10 or 70/100)

To get to 62/100, the decibunny could first take large hops of 6/10 and then take smaller steps of 2/100, so 62/100 = 6/10 + 2/100. Another way to get there would be to take 62 smaller steps, making 62/100.

Decibunny on the Meter Stick

On a meter stick, the decibunny has four lengths of hops. The largest hop would cover the entire meter. A medium hop would cover one decimeter (1/10 of a meter). The small hop would cover one centimeter (1/100 of a meter) and the teeny hop would cover one millimeter (1/1000 of a meter). It helps if you use your voice to describe the hops…raising your voice as the hops get smaller…to the teeny-weeny millimeter steps.

To show 8/10, the decibunny could take the following set of hops: 8 medium hops of 1/10 each (8/10) 80 small hops of 1/100 (80/100) 800 teeny hops of 1/1000 (800/1000)

To show 54/100, the decibunny could take the following set of hops: 5 medium hops of 1/10 each (5/10) and 4 small hops of 1/100 each (4/100),

therefore 54/100 = 5/10 + 4/100 54 small hops of 1/100 each (54/100) 540 teeny hops of 1/1000 each (540/1000)

To show 972/1000, the decibunny could take the following set of hops: 9 medium hops of 1/10 each (9/10) and 7 small hops of 1/100 each (7/100) and 2

teeny hops of 1/1000 each (2/1000). Therefore, 972/100 = 9/10 + 7/100 + 2/1000 97 small hops of 1/100 each (97/100) and 2 teeny hops of 1/1000 each (2/1000).

Therefore 972/1000 = 97/100 + 2/1000. 972 teeny hops of 1/1000 each (972/1000)


Decibunny on the Number LineUse the model for a number line between 0 and 1. When the hundredths and thousandths are not visible, the students should imagine the hops. Use the same hops as on the meter stick: From 0 to 1 is a large hop. A medium hop covers 1/10. A small hop covers 1/100. A teeny hop covers 1/1000. That is far enough for them to get the idea. You might want to go one more to a teeny-tiny hop that covers 1/10000.

The idea is similar as above, but you will have to have the students imagine the thousandths and ten-thousandths (if you go there). They can imagine them by referring back to the meter stick and millimeters.

Activity # 1A:See Activity # 1A Worksheet at end of unit. Practice some examples as a whole class, then have them work independently.

Activity # 1B:See Activity #1B Worksheet at end of unit. Practice some examples as a whole class then have them work independently.

Activity # 1C:When the students do the activity, they will be looking at the number line between 0 and 1. However, we will first introduce the number line as a whole and then make it clear that proper fractions and decimals less than one are represented on the number line between 0 and 1.See Activity # 1C Worksheet at end of unit.


Lesson # 2 – Place Value: An Infinite Progression of Squares and StripsReview whole number place value. One of the most basic ideas is the 10-to-1 relationship between the value of any two adjacent positions. In terms of the base-ten blocks, 10 of any one piece will make 1 of the next larger and vice-versa. Also, note that this can be modeled as a series of alternating square and strips, with each getting larger as one increases from one (small square) to ten (small strip) to hundred (large square) to thousand (large strip). (Note: although the base ten blocks model 1000 to be a cube of 10x10x10 units (or 10 large squares placed in a stack) for ease, it could also be represented as 10 large square placed end to end to form a long strip.)

Have the students imagine the squares and strips alternating as the numbers get larger to the right. Will there ever be a largest piece?


This continues down to form a long strip of 100 units

Onesquare

Ten strip

Hundred square

Thousandstrip

Imagine 10,000 to be a square with 10 columns of 1000 strips

What is the numerical representation of the following model? Is there more than one way that the number can be represented?

f we consider that ones are the unit of measure, we can say that it is 235 ones.What if we consider that ten is the unit of measure, then we have 23 tens and 5/10 tens or 23 5/10 tens. If we consider that hundred is the unit of measure, then we have 2 hundreds and 35/100 of a hundred or 2 35/100 hundreds.

Activity # 2A: Physical models to Whole Numbers and Vice Versa See Activity # 2A Worksheet for practice on writing different numerical names for physical models by changing the unit of measure. It also asks students to show the physical model of different numerical representations, such as 30, 42, 178, 250, etc. Also, ask them to represent the numerical representation using different sized shapes for the units of measure (ones, tens, and hundreds). This is a conceptual leap for students that will help them later to conceptually understand decimals. Practice this until they are comfortable.

When we split up the one square into 10 strips, each strip is 1/10 of the one square.Use the base ten blocks, but rename them. The large square will be worth one. The strips will be worth 1/10.

Have the students model 3/10, 7/10, 1 and 2/10, 3&5/10, etc. Practice going back and forth between the physical model and the fraction or mixed number (with denominators of 10) so that the students are comfortable with translating between both representations.

Now add the small square and ask students what value it will have as a fraction.


1One

1/10one tenth

Practice modeling different mixed numbers and fractions with one hundred as the denominator and ten as the denominator so that students are comfortable going back and forth between the numerical and the physical representation.

Now ask them if there is more than one way that they can write the numerical representation of the following.

Using the large square as the unit, it can be expressed as 1 + 43/100. It can also be expressed as 1 + 4/10 + 3/100.

Activity # 2B: Physical Models to Fractions (Tenths and Hundredths) and Vice VersaPractice more physical models and naming them in two ways (using tenths and hundredths).

Inventing New Physical Models for Smaller Base Ten FractionsWe don’t have the physical model to represent thousandths, but ask the student what they think it would look like. Hopefully, they will guess that it will be a strip that is 1/10 the size of the small square.

Use the whiteboard to draw the following models representing one, one-tenth, one-hundredth, and one-thousandth. These will not be to scale, but will give the students the idea.


1/100one hundredth

1/10one tenth

1one

Practice examples of the pictorial representation and having the students express the numerical representation. Have the students name the number using different fractions.

Activity # 2C: Using New Physical Model to Model Base Ten Fractions to the Thousandths See Activity # 2C Worksheet.


One1

One tenth1/10

One hundredth1/100

One thousandth1/1000

The numerical representation could be:2 + 5/10 + 2/100 + 3/100025/10 + 2/100 + 3/1000252/100 + 3/10002523/1000

Lesson # 3 – Place Value and DeimalsNow it is finally time to introduce the decimal place as another way to numerically express base ten fractions such as tenths, hundredth, and thousandths.To do this, go back over the place value of whole numbers.

8692

Notice how the zeros (indicating powers of 10) decrease by one as the value of the place gets smaller. What can we do once we get to 2 ones to show smaller numbers to the right? Mathematicians decided to use a dot to indicate the location of the unit place. The dot lies directly to the right of the ones (units) place. Using the dot allows base ten fractions of the unit to be expressed as a numerical extension to the right of the whole numbers. The dot is named a “decimal point”. The word decimal is derived from the Latin word “deci” which means one-tenth. A decimal is a fraction that has a denominator of ten. When we write decimals, we don’t write them in fractional form with the numerator and denominator, we write them as a continuation of the whole number, separated by the decimal point.

3 + 4/10 can be written as 3.4

When we read the number 3.4 we say “three and 4 tenths”.

8 + 52/100 can be written as 8.52

When we read the number 8.52, we say “eight and 52 hundredths”

The decimal point is placed immediately to the right of the units or ones place and places to the right of the decimal are called tenths, hundredths, thousandths, ten thousandths, etc.

546.237


8 thousands8000

6 hundreds600

9 tens90

2 ones2

2 tenths 3 hundredths 7 thousandths

Activity # 3A: Name the NumberSee Activity # 3A Worksheet


Lesson # 4– The Decimal Point Indicates the Unit of Measure An important concept to understand is that the decimal indicates the unit.

Let’s give names to the following different sizes base ten shapes. Each shape is ten times the size of the shape to the immediate left:

Imagine that there is a piece that is ten times larger than the large square and lies to the left of the large square. It would be in the form of a strip of ten large squares. We will call it a “super strip”. To the left of the super strip would be a “super square”. To the right of the small strip, ten times smaller than the small strip, we will call a “tiny square”. To the right of the tiny square, ten times smaller than the tiny square is the “tiny strip”.

Depending on where the decimal place is put, the number above can be represented using different units.

If the decimal is placed at position c, then the square is the unit and the model represents 2.631 squares.

If the decimal is placed at position d, then the strip is the unit and the number is called 26.31 strips.

If the decimal is placed at position e, then the small square is the unit and the number is called 263.1 small squares.

If the decimal is placed at position f, then the small strip is the unit and the number is called 2631 small strips.


square Stripsmall square small strip

a bc d e f g

super strip tiny square tiny strip

super square

If the decimal is placed at position b, the unit is the super strip. There are zero superstrips, so the number is called 0.2631 super strips.

If the decimal is placed at position a, the unit is the super square. Since there are zero super squares and zero super strips, the number is called

0.02631 super squares If the decimal is placed at position g, the unit will be tiny square. Since there are

zero tiny squares, the number will be 26310 tiny squares.

Overhead Whole Class Activity: Naming by moving decimal point to change the unit Materials for teacher: Overhead of Base Ten Template – Squares and Strips (8 ½ x 14 included at end of unit), base ten blocks, and smiling decimalMaterials for students: Base Ten Template – Squares and Strips, base ten blocks, and smiling decimals Display a certain number of base-ten blocks on the overhead Base Ten Template – Squares and Strips and have the students display the same number at their desk. For example, 3 squares, 7 strips, and 4 tiny squares. Use the tagboard smiling decimal point and move it around. The decimal point always looks up and to the left to the “unit”. Each time it is placed, ask the students to orally say the number with the unit at the end. It should also be written down on the white board. Go through each unit.

Now ask students to write and say how many squares they have, how many strips they have, how many super strips, etc. They can determine this by moving the decimal point. Its eyes will look up to the left to the unit position.

When the decimal point is between the: The number is:squares strips 3.74 squaresstrips small squares 37.4 strips

small squares small strips 374 small squaressuper strips squares 0.374 super strips

super squares super strips 0.0374 super squaressmall strips tiny strips 3740 small strips


To the right…small strips, then tiny squares, then tiny strips…

squaresstrips

Small squares

To the left…super strips, then super squares

Activity # 4A: Moving the Decimal to Name the UnitsAfter doing a few at the board, have students fill out Activity # 4A worksheet where they will put out their own base ten blocks on the Base Ten Template – Squares and Strips, draw them, and then name them, changing the unit as the decimal moves. (Run off as many of page two as you need for them to practice.)

This activity should help them realize that the quantity remains the same, but as the decimal point moves, the unit changes.

This same idea can be illustrated using money. $245.84 can be expressed in the following ways:

245.84 dollars (which is the convential unit)2458.4 dimes (which are tenths of a dollar)24584 pennies (which are hundredths of a dollar)24.584 ten dollars2.4584 hundred dollars0.24584 thousand dollars

Using the metric system, the units from smaller to greater, growing by multiples of 10 are:Millimeter, centimeter, decimeter, meter, dekameter, hectometer, kilometer

43.85 meters can be expressed in different ways depending upon the unit:438.5 decimeters4385 centimeters43850 millimeters4.385 dekameters.4385 hectometers.04385 kilometers

Activity # 4B – Moving the Decimal Point to Rename the Unit using Familiar Units of MeasureSee Activity # 4B Worksheet.

Important Note!When students are asked to move the decimal using familiar units, they should not be taught to move 2 places to the left or 3 places to the right, etc. They should use their intuition to determine how many and which direction. For example, if a measure was 43.85 meters, there would be more centimeters, since centimeters are smaller. Also, since centimeter is two powers of ten away from meter (i.e. there are 100 centimeters in a meter), the decimal would be moved two places. Once students try to memorize the direction and number of decimal places, they lose understanding.


Other real-life examples of using units with decimals: If a person is 1.62 meters tall, this means that they are 162 centimeters tall. If Congress is spending $7.3 billion, the unit is billion dollars. Since a billion is a

thousand million, then $7.3 billion = $7300 million, which is $7300, 000, 000 or $7,300,000,000.

A city may have a population of 2.4 million people. The unit is million people, so there are 2,400,000 people.

Lesson # 5 – Making the Fraction-Decimal Connection Important Note! To translate between the two systems of fractions and decimals, students should rely on their understanding of the concept, not memorizing an algorithm. Teaching students to convert a fraction to a decimal by dividing the numerator by the denominator does not shed any light on the concept that fractions and decimals express the same value in a different format.

Have the students use their base ten blocks and agree to let the large square represent one. Use the large squares to represent ones and the strips and tinies to represent tenths and hundredths respectively. Use a blank square grid of 100 squares that is the same size as the base ten blocks to express the fractional part of the number. Have the students express different mixed numbers and fractions with denominators of 10 and 100. For example, have them express 2 and 35/100 as shown below:

Now have them show the same number in a base ten column sheet as below using the decimal point: Ones Tenths Hundredths


3/10 5/100

The reason why 2 and 35/100 is the same as 2.35 is because there are 2 wholes, 3 tenths, and 5 hundredths. It is important for the students to see this physically. The exact same materials that are used to represent 2 and 35/100 of the square can be rearranged or placed on an imaginary place value chart with a paper decimal point used to designate the units position.

The reverse of this activity is also worthwhile. Give the student a decimal number such as 1.68 and have then show it with base ten pieces on the place value chart. Then have them write it and show it as a fractional part of a square. Although these translations between decimals and base-ten fractions are rather simple, the main agenda is for students to learn from the beginning that decimals are simply fractions.

In the base ten system, we by convention use the ones as the unit. However, we can rename any number by changing the unit. For example, 4723.895 can be written in the following ways:

4.723895 thousands 47.23895 hundreds 472.3895 tens 4723.895 ones 47238.95 tenths 472389.5 hundredths

4723895 thousandths

Activity # 5a: Base-ten fractions to Decimals and Vice VersaSee Activity # 5A Worksheet. You can copy as many page twos as you need to provide extra practice.

Whole Class Activity # 5b: Use of the Calculator to Teach the Concept of DecimalsMaterials: Each student has a calculator.Have students use the calculator and press + 0.1 = =. Every time they hit = the value should increase by one tenth. Have them go to .9 and stop. Ask them what comes next? Many of them will guess that it is .10, since 10 comes after 9. Have them hit = and they see that the answer is 1.0. Ask them how this relates to counting from 1 to 10? Continue to count to 4 or 5 by tenths. Ask them how many pushes on the = sign it takes to get from a whole number to the next whole number. Try counting by .01 and .001. Ask the same questions. How many pushes on the = sign will it take to get from whole number to whole number counting by .01? .001?


Lesson # 6 – Developing Decimal Number Sense: Familiar Fractions Connected to DecimalsSo far the discussion has been around the connection of decimals to base-ten fractions. Number sense implies having an intuition about or a friendly understanding of numbers. To do this students should be able to

Connect decimals to familiar fractions Compare and order decimals readily Approximate decimals with useful familiar numbers

Just as students should develop a conceptual familiarity with simple fractions such as halves, thirds, fourths, fifth, and eighths, they should be able to translate these friendly fractions to decimals.

Activity # 6a with 10 x 10 Grid Students are given a “friendly” fraction (1/2, ¼, 2/4, ¾, 1/5, 2/5, 3/5, 4/5, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8…wait for 1/3, 2/3 to do as a class) to convert to a decimal. They first model the fraction using a 10x10 grid. Then they express the fraction as tenths, hundredths, or thousandths as a fraction and then write it as a decimal.(Note that halves, fourths, and fifths will be fairly easy because the denominator is a factor of 100. Eighths will be a little more difficult. Thirds may need to be done as a class activity.)

When showing ¼, many students may show it in this way as a 5 x 5 square:

Both ways show 25/100, but the second way also shows 2/10 + 5/100, which can be expressed as 20/100 + 5/100 or 25/100 or .25

Use the idea of the decibunny making “hops” of (1/10)

Encourage students to model friendly fractions using the base ten method.


Ask them if they can show the same amount using strips of ten (base ten)

1/100

Or

10/1000

This will lead them into doing 3/8. To find 3/8, they can first show 1/8, which is ½ of ¼ or 12 ½ small squares. What is ½ of a small square? Remember that in base ten, each smaller piece is broken into tenths of the next larger piece. The hundredth could be expressed as 10/1000, so half of the hundredth would be 5/1000.

Class Activity – Finding Decimal Equivalent of 1/3 and 2/3


1/812/100 + 5/1000

120/1000 + 5/1000125/1000

.125

3/837/100 + 5/1000

370/1000 + 5/1000375/1000

.3751/100 or 10/1000

5/1000

The square above represents 1 unit. Each column represents 1/10. To show 1/3 of 1,3/10 are shaded in and 1/10 is divided by 3. Dividing the 1/10 by 3 results in 3/100 shaded and 1/100 that still needs to be divided by three.

This square shows that 1/3 = 3/10 + 3/100 + 1/100 needing to be divided by 3

This 1/100 square is zoomed in on and showed to the right of the original square.

The square above represents 1/100. It needs to be divided by 3. Each column represents 1/10 of 1/100, which is 1/1000. The three shaded columns represent 3/1000. The next column is to be divided into three parts, so 3/10000 are shaded. This leaves one small square (1/10000) that still needs to be divided by 3.

This square shows that 1/100 divided by 3 = 3/1000 + 3/10000 + 1/10000 needing to divided by 3.

This 1/10000 square is zoomed in on and is shown to the right of the 1/100 square.

3/10

3/100

3/1000

3/10000

3/1000000

3/100000

1

1/100

1/10000

1/3 means “1 divided by 3”.

As you can see, dividing by three never ends. There will always be 1 small square that still needs to be divided by 3.

From the models above, we can see that:

1/3 = 3/10 + 3/100 +

3/1000 + 3/10000 +

3/100000 + 3/1000000

1/3 = 3/10 + 3/100 + 3/1000 + 3/10000 + 3/100000 + 3/1000000 and on and on…. As a decimal, this would be written 1/3 = .3 + .03 + .003 + .0003 + .00003 + .000003 and on and on…

Adding these decimals together, we get 1/3 = .333333 and on and on…

The mathematical notation for something that goes on and on

Activity # 6b: Hundredths Disk – Estimate, then VerifyWith the blank side of the disk facing them, have students adjust the disk to show a “friendly” fraction, such as ¾. Next, turn over the disk and record how many hundredths were in the section they estimated (note the color reverses when the disk is turned over). Finally, they should make an argument for the correct number of hundredths and the corresponding decimal equivalent.


1/100 needing to be divided by 3

1/10000 needing to be divided by 3

Lesson # 7 - Repeating Decimals: Finding Decimal Equivalent for 1/3 and 2/3 as a ClassExplain to students that there are many decimals that do not come to an end, but that repeat. There are also many decimals that go on forever that do not have a repeating pattern. These will be studied in middle school. Of the “friendly” decimals that we are responsible for in 5th grade (halves, fourths, eights, thirds, sixths, twelfths, fifths, tenths, hundredths, thousandths), all of them end in the tenth, hundredth or thousandth place except for thirds and sixths. Since thirds and sixths are used frequently, we will explore their decimal equivalent as a class.

Shade in 1/3 of the 10 x 10 grid.


1/3 of a hundredth

Ones Tenths Hundredths Thousandths

33 1/3 hundredths

One

1/3 of a thousandth

Recall that we can move the decimal point to change the name of the unit.

If we put the decimal point at ten-thousandths, we have 3333 1/3 ten thousandths and on and on infinitely.

In decimal format, the number above can be expressed in the following ways:3333 1/3 ten thousandths

333.3 1/3 thousandths 33.33 1/3 hundredths

a. 1/3 tenths0.3333 1/3 ones

Each time we take it out one more decimal place, we get another 3 1/3.We can call this a “repeating 3” and take off the 1/3, knowing that it means that 3 will continue to repeat.

So 1/3 = 0.33333333333333… We usually express it as .333 or .33 as an estimation.

2/3 will therefore be two times 1/3 or 2 x .333… = 0.666

Since .6 is more than ½ (or .5), we usually round off .666 to .67 and estimate 2/3 = 0.67


Ones Tenths Hundredths Thousandths

333 1/3 thousandths

Lesson # 8: Decimals and Fractions on a Number Line Students should become familiar with placing both fractions and decimals on a number line. Using a number line helps students to visualize both fractions and decimals as real numbers and also helps them to compare them (greater than, less than, equal to).

Activity # 8a: Decimals on a Friendly Fraction Line Place the following decimals on the number lines below: 1.5, 2.125, 3.4, 3.75, 2.66, 1.33


0 1 2 3 4

0 1 2 3 4 5

0 1 2 3 4 5

1 2 3 4

Lesson # 9 - Approximation with a Nice Fraction It is important for students to be able to approximate decimals to a nice fraction. For example, .52 is close to what nice fraction? .50 or .5 or ½. The first benchmarks that you should work with are 0, ½, and 1. For example is 7.396 closer to 7 or 8? Is it closer to 7 or 7 ½ ? Using the other benchmark fractions of thirds, fourths, fifths, and eights, students should be able to find a nice benchmark near 7.396. For example, 7.396 is close to 7.4, which is 7 2/5.

Activity # 9a: Close to a Friendly Fraction Make a list of about five decimals that are close to but not exactly equal to a nice or friendly fraction equivalent. For example, use 24.8025, 6.59, .9003, 124.356, and 7.7.

The students’ task is to decide on a decimal number that is close to each of these decimals and that also has a friendly fraction equivalent that they know. For example, 6.59 is close to 6.6, which is 6 3/5. Another student may say that 6.59 is close to 6.66, which is 6 2/3.

Activity # 9b: Best Match On the board, list a scattered arrangement of five familiar fractions and at least five decimals that are close to the fractions but not exact. Students are to pair each fraction with the decimal that best matches it. The difficulty is determined by how close the various fractions are to one another. For example, use the following:11/5, 2 1/3, 2 7/8, 2 5/82.804, 2.41, 2.6271, 2.211

In Activities #9a and #9b, student will have a variety of reasons for their answers. Sharing their thinking with the class provides a valuable opportunity of all to learn. Do not focus on the answers, but on the rationales.


Lesson # 10 – Ordering Decimal Numbers When students are asked to order decimals from least to greatest, for example 0.36, 0.058, 0.375, and .4, the most common error that students make is to select the number with more digits as the larger number. Another common error, once students realize that numbers to the far right represent very small numbers, is to name the numbers with the most digits as the smaller number. Both of these are misconceptions about how decimal numbers are constructed. The following activities will help with the conceptualization of the relative size of decimal numbers.

Activity # 10a: Line ‘Em Up Prepare a list of four or five decimals numbers that students might have difficulty putting in order. They should all be between the same two consecutive whole numbers. Have students first predict the order of the numbers, from least to most. Next have them place each number on a number line with 100 subdivisions. As an alternative, have them shad in the fractional part of each number on a 10 x 10 grid using estimates for the thousandths and the ten-thousandths. In either case, it quickly becomes obvious which digits contribute most to the size of a decimal.

Activity # 10b: Close “Nice” Numbers Write a four-digit decimal on the board – 3.0917, for example. Start with the whole numbers: “Is it closer to 3 or 4?” Then go to the tenths: “Is it closer to 3.0 or 3.1?” Repeat with hundredths and thousandths. At each answer, challenge students to defend their choices with the use of a model or other conceptual explanation. A large number line, divided into ten sets of tenths, without numerals, is useful.

You can make a large number line for the board by cutting four strips of poster board measuring 6 x 28 inches. Tape end to end. Mark off 10 large sections. Then mark off ten sections within the larger sections. You will have 100 small sections. Place the number line on the chalk tray on the board and use the board to write the numbers above the number line with an arrow to the correct decimal place on the number line. For example for 3.42, you would put 3 on the farthest left mark on the number line and 4 on the farthest right. 3.42 would be placed 42 small marks to the right of 3.

Too often, rounding is taught as an algorithm (if it’s about 5 round to the next place). This does not give the child much number sense about decimals. Students should instead be taught to substitute a given number for a “nice” number near it. For example, 6.72 is close the nice numbers 6.75 and 6.66, so the student would know that it is about 6 ¾ or 6 2/3. These nice numbers have more meaning for student than 6.7.


Lesson # 11 – Introducing Percents According to Grade 5 Vermont Math GE M5:1and M5:2, students are expected to demonstrate conceptual understanding of and be able to order and compare the benchmark percents of 10%, 25%, 50%, 75%, or 100%.

Percents should be included in discussions of fractions and decimals. They are simply another way to represent a fraction. The denominator of a percent is always 100. The name “per cent” means “per hundred”, and “per” means to “divide”, so 50 per cent is 50 divided by 100 which is equivalent to the fraction ½.

Students do not need to know how to solve problems with percents as operators until Grade 6. Using them as an operator means, for example, to be able to find 42% of 500.

According to Vermont Math GE M5:1, Grade 5 students are responsible for using area, set, and linear models for benchmark percents (10%, 25%, 50%, 75%, or 100%) where the number of parts in the whole is:

equal to 100 a multiple of 100 (such as 200, 300, 400, etc.) a factor of 100 (such as 1, 2, 4, 5, 10, 20, 25, 50)

Set Model

Area Model

Linear Model

Example using set model


Shade in 50 % of the circles.Put an X through 25% of the circles.(The number of parts in the whole is 10, a factor of 100)

What percent of the area below is shaded? (The number of parts in the whole is 4, a factor or 100)

On the number line, show the location of 10%, 50% and 75% of 1. (The number of parts in the whole is 1, a factor of 100.)

0 1KEY:

represents 30 units

The number of parts in the whole is 600, which is a multiple of 100.

The 100-disk and a 10 x 10 grid can also be used to model percents using the area model. In those cases, the number of parts in the whole would be equal to 100.


How many units are there in the whole set? Answer: There are 20 shapes worth 30 units each, so there are 600 units.How many shaded units are there? Answer: There are 5 shaded shapes, which represents 150 units.The shaded units represent what percent of the whole set?Answer: There are 20 shapes in the whole set in the whole set. If 5 are shaded, then means that 5/20 or ¼ or 25% are shadedFill in the blank below, describing the units:

_______ is ________% of __________Answer: 150 is 25% of 600

Lesson # 12 – Realistic Percent Problems and Nice Numbers Although students must have some experience with the non-contextual examples using the set model, area model, and linear model shown in Lesson # 11, it is better to have students experience percents in real contexts. They hear percents used in daily life on television, radio, and newspapers. Percents are also used in stores to indicate sales and on weather reports to determine the chance of snow. The class should discuss percents used in everyday life.

Even though the Vermont GEs do not require solving problems with percents as operators until Grade 6, Grade 5 students can be introduced to problems using the benchmark percents on numbers in which the number of parts in the whole are 100, a factor of 100, or a multiple of 100.

John VandeWalle, the author of the book that this unit is derived from, suggests the following maxims for teaching a unit on percents:

Limit the percents to familiar fractions (in our case tenths, fourths, halves) and use numbers compatible with those fractions. (In our case, use number of parts in the whole as 100, a factor of 100, or a multiple of 100).

Do not suggest any rules or procedures for different types of problems. Do not categorize or label problems. Once you do this, children memorize and the conceptual learning stops.

Use the terms part, whole, and percent (or fraction). Fraction and percent are interchangeable. Help students see percent models above as the same types of models they used for fractions.

Require students to use models or drawings to explain their solutions. It is better to assign three problems requiring a drawing and an explanation than to give 15 problems requiring only computation and anwers. Remember that the purpose is the exploration of relationships, not computational skill.

Encourage mental computation.

Activity # 12a The following sample problems meet these criteria for easy fractions and numbers. Try working each problem, identifying each number as a part, a whole, or a fraction. Draw linear, area, or set models to explain or work through your thought process.

1. The PTA reported that 75% of the total number of families were represented at the meeting last night. If children from 200 families go to the school, how many were represented at the meeting?

2. The baseball team won 25% of the 24 games it played this year. How many games were lost?

3. In the Grade 5 class, 20 students, or 25%, were on the honor roll. How many students are in the Grade 5 class?

4. George bought his new gameboy at 10% discount. He paid $180. How many dollars did he save by buying it at a discount?


5. If Joyce has read 60 of the 240 pages in her library book, what percent of the book has she read so far?

6. The hardware store bought widgets at 90 cents each and sold them for 99 cents each. What percent did the store mark up the price of each widget?

Below are some examples of percent problems using models to solve them. Work the above problems using similar models.

A store has $80 coats marked 25% off. How much will a coat cost on sale?

This year, 20 more students rode the bus than last year. If that is a 10% increase, how many rode the bus last year?

The highway department is responsible for 600 miles of two lane roads and 200 miles of four-lane roads. What percent of the roads are two-lane?


25% of 80 is 20. If the entire bar at left is the original cost of the coat, then each square is worth $16.

20 x 3 = 60

therefore, the coat costs $60 on sale.

75%$16 $20each

20 20 20 20 20 20 20 20 20 20

10%

20 is 10%, which is 1/10, so if we put down ten 20s, we have 10 x 20 = 200 students who rode this bus last year. That means that 220 ride the bus this year.

6002-lane

2004-lane

200 200 200 200

The total number of miles is 800. 200 is 25% of 800 and 600 is 75% of 800. Therefore 75% of the roads are two-lane.

25%

Name _________________________________________Date___________

Activity # 1A Worksheet

Circular Disk Model:Take the Circular Disk apart and look at the orange or blue colored slice.

1. How many large divisions (slices) are there?

2. What fraction of the whole disk is does one slice represent?

3. Each large slice is divided into how many parts?

4. What fraction of the whole disk does the space between each small mark represent?

5. In each small section, one small mark is longer than the others. Why?

Put the two parts of the circular disk back together. Following the example below, answer the questions about each fraction.

Example: Show the fraction 65/100 in orange using the two disks. a. Is the fraction more or less than ½? (answer: more)b. Is the fraction more or less than 3/4? (answer: less)c. What are two ways to write this fraction using numbers or a sum of numbers?

(Show the different combinations of hops from one end of the orange slice to the other.)(Answers: 65/100 or 6/10 + 5/100)

d What fraction(s) are represented with blue? (Answers: 35/100 or 3/10 + 5/100)1. 34/100

a.

b.

c.

d.


2. 87/100

a.

b.

c.

d.

3. 46/100

a.

b.

c.

d.

4. 12/100

a.

b.

c.

d.


Activity # 1B Worksheet

Meter Stick Model:On a meter stick the whole stick represents one unit, and the unit is a meter. Notice that there are three sizes of division marks dividing the meter into smaller units. We will call these divisions large, medium, and small. Using a meter stick, answer the following questions

1. There are ten large divisions on the meter stick, each called a decimeter. What fraction of the whole meter stick does one decimeter represent?

2. Each decimeter is divided into how many medium divisions?

3. Each medium division is called a centimeter. What fraction of the whole meter stick does each centimeter represent?

4. Each centimeter is divided into how many small divisions?

5. Each small division is called a millimeter. What fraction of the whole meter stick does each millimeter represent?

Answer the questions below using the example as a guide:Example: Find the fraction 352/1000 on the meter stick.

a. Is the fraction more or less than ½? (answer: less)b. Is the fraction more or less than 3/4? (answer: less)c. What are three ways to write this fraction using numbers or a sum of numbers?

(Hint: think of the different combinations of hops that decibunny can make)(Possible Answers: (352/1000) or (35/100 + 2/1000) or (3/10 + 52/1000) or (3/10 + 5/100 + 2/1000)

6. 783/1000

a.

b.

c.


7. 12/100

a.

b.

c.

8. 499/1000

a.

b.

c.

9. 8/10

a.

b.

c.

Name _________________________________________ Date _______________Grade 5 Unit # 3 – Fractions, Decimals, PercentsWilliamstown Elementary School – developed by Elaine Watson, Ed.D.Primary Source: John A. VandeWalle (2004), Elementary and Middle School Mathematics: Teaching Developmentally of 65

Activity # 1C Worksheet

Number Line Model:Using a number line from 0 to 1, answer the following questions:

1. How many large divisions are there?

2. What fraction of one does one large division represent?

3. Each large division is divided into how many small divisions?

4. What fraction of one does each small division represent?

Answer the questions below using the example as a guide:Example: Find the fraction 52/100 on the number line (approximately).

a. Is the fraction more or less than ½? (answer: more)b. Is the fraction more or less than 3/4? (answer: less)c. What are two ways to write this fraction using numbers or a sum of numbers?

(Possible Answers: (52/100) or (5/10 + 2/100)) (Hint: think of the decibunny.)


0 1

52/100


0 1

0 1

0 1

5. 74/100a.

b.

c.

6. 18/100a.

b.

c.

7. 9/10

a.

b.

c.

Name ___________________________________________ Date __________________

Activity # 2A Worksheet: Physical Models to Whole numbers and Vice Versa

Example: Name the following physical models in different ways:

a. Using ones as the unit: 213 onesb. Using tens as the unit: 21 3/10 tensc. Using hundreds as the unit: 2 and 13/100 hundreds or 2 +1/10 +3/100 hundreds

1.

a. Using ones as the unit:

b. Using tens as the unit:

c. Using hundreds as the unit:


hundredten one

2.




3.





Use the following shapes to represent the numbers below (Use the fewest number of shapes possible):

5. 23 4/10 tens

6. 1 2/10 hundreds

7. 5 3/10 tens


One hundred ten one

Name ________________________________________ Date ________________

Activity # 2B Worksheet:

On this worksheet, the large square represents one. Name the strip and the smaller square as base ten fractions:

One ________________ of one _____________ of one

Check with the teacher to make sure the above two answers are correct before you complete the rest of the worksheet!

Given the pictorial representation with the large square representing one, write the numerical representation using different units.

Example: What number would the three shapes above represent:a. If one is the unit of measure?

Possible Answers: ( 1 and 2/10 and 4/100 ones) or (12/10 and 4/100 ones) or (124/100 ones) or (1 and 24/100 ones)

b. If one tenth is the unit of measure?Possible Answers: (12 and 4/10 tenths) or (124/10 tenths)

c. If one hundredth is the unit of measure? Possible Answer: (124 hundredths)


1.

What number would the above picture represent if:a. one is the unit of measure?

b. one tenth is the unit of measure?

c. one hundredth is the unit of measure?

2.

What number would the above picture represent if:d. one is the unit of measure?

e. one tenth is the unit of measure?

f. one hundredth is the unit of measure?

Use the following shapes to illustrate the numbers below (Use the fewest number of shapes possible):Grade 5 Unit # 3 – Fractions, Decimals, PercentsWilliamstown Elementary School – developed by Elaine Watson, Ed.D.Primary Source: John A. VandeWalle (2004), Elementary and Middle School Mathematics: Teaching Developmentally of 65

3. 21 tenths

4. 1 4/10 tenths

5. 35 hundredths

Name _________________________________________ Date ________________


One One tenth One hundredth

Activity # 2C Worksheet

Use the following models to represent one, one-tenth, one-hundredth, and one-thousandth.

Example: Name the various fractions or sums of fractions that represents the picture below:

1.

Name the various fractions or sums of fractions that represent the picture above:

2. Name the various fractions or sums of fractions that represent the picture below:


One1

One tenth1/10

One hundredth1/100

One thousandth1/1000

Answer: The numerical representations could be:2 + 5/10 + 2/100 + 3/100025/10 + 2/100 + 3/1000252/100 + 3/10002523/1000

3. Name the various fractions or sums of fractions that represent the picture below:

Name ________________________________________ Date _____________________


Given the following number:


74.625

1. What numeral is in the tens place? ___________

2. What numeral is in the thousandths place? ___________

3. What numeral is in the ones place? _____________

4. What numeral is in the hundredths place? ____________

5. What numeral is in the tenths place? _____________

6. Write the number as a mixed number with the base ten denominator:

Given the following number:

193.0527. What numeral is in the hundredths place? _______________

8. What numeral is in the tens place? _______________

9. What numeral is in the tenths place? __________________

10. Write the number out as it would be said (Hint: Use “and” when you see the decimal point.)

11. Write the numerical representation of “two thousand thirty-two and five thousandths”

12. Write the numerical representation of “sixty-four hundredths”. (Hint: this could also be called “zero and sixty-four hundredths”

13. Write the numerical representation of “eight and five tenths”.


Name ______________________________________ Date _________________


Place some base ten blocks on the different columns on your Base Ten Template. Sketch them in the space below. Move the decimal point and fill in the table below the picture.

SuSq.

SuStrip

Squares Strips Small Squares SmStr

TiSq

When the decimal point is between the: The number is: (use units)squares strips

strips small squares

super strips squares

super strips super squares

small squares small strips

small strips tiny squares


To the right…small strips, then tiny squares, then tiny strips…

squares

strips

Small squares

To the left…super strips, then super squares

Place some base ten blocks on the different columns on your Base Ten Template. Sketch them in the space below. Move the decimal point and fill in the table below the picture.

SuSq

SuStrip

Squares Strips Small Squares SmStr

TiSq

When the decimal point is between the: The number is: (use units)squares strips

strips small squares

super strips squares

super strips super squares

small squares small strips

small strips tiny squares


Name ______________________________________Date ________________

Activity # 4B Worksheet

$358.24thousand dollars hundred dollars ten dollars dollars dimes pennies

When the decimal is at

Point

The number is: (use units)

F

3582.4 dimes

C

A

E

B

D

G

437centimetersGrade 5 Unit # 3 – Fractions, Decimals, PercentsWilliamstown Elementary School – developed by Elaine Watson, Ed.D.Primary Source: John A. VandeWalle (2004), Elementary and Middle School Mathematics: Teaching Developmentally of 65

A B C D E F G

kilometers hectometers dekameters meters decimeters centimeters millimeters

When the decimal is at

Point

The number is: (use units)

F

C

A

H

B

D

G

E

Name ____________________________________________ Date _________________


A B C D E F G H

Activity # 5A WorksheetUse the Base Ten Template – Real Numbers and Base Ten Blocks where the blocks have the following values:

Place base ten blocks on the Base Ten Template – Real Numbers and sketch your blocks below. Move the smiley face decimal point to show different numbers. Record the numbers in the table below, making sure to include the unit at the end of the number.thousands hundreds tens Ones tenths hundredths thousandths

When the decimal point is between the: The number is: (use units)ones tenths

hundredths thousandths

hundreds tens

tenths hundredths

tens ones

thousands hundreds

Activity # 5A Worksheet (Page 2)


one tenth hundredth

Place base ten blocks on the Base Ten Template – Real Numbers and sketch your blocks below. Move the smiley face decimal point to show different numbers. Record the numbers in the table below, making sure to include the unit at the end of the number.

thousands hundreds tens Ones tenths hundredths thousandths

When the decimal point is between the: The number is: (use units)ones tenths

hundredths thousandths

hundreds tens

tenths hundredths

tens ones

thousands hundreds

.

Activity # 5A Worksheet (Page 3)


Sketch the base ten blocks (square = 1, strip = 0.1, small square = 0.01) and the decimal point in the appropriate columns given the decimal number. Fill in the blank with the correct number and decimal point placement to indicate the number of ones

1. .312 tens = ___________________________________ ones


2. 14.2 tenths = ___________________________________ ones


3. .0013 hundreds = ___________________________________ onesGrade 5 Unit # 3 – Fractions, Decimals, PercentsWilliamstown Elementary School – developed by Elaine Watson, Ed.D.Primary Source: John A. VandeWalle (2004), Elementary and Middle School Mathematics: Teaching Developmentally of 65


4. 4100 thousandths = ___________________________________ ones


Name ____________________________________________ Date _________________




Example: Shade in

Williamstown Elementary School - WatsonMath.com€¦ · Web viewThe word decimal is derived from...

Documents

Transcript of Williamstown Elementary School - WatsonMath.com€¦ · Web viewThe word decimal is derived from...