1

Math 141: Math for Elementary Teachers Unit 4 Notes

The word decimal comes from the Latin, decem, meaning ten. Technically, any number written in base-ten positional numeration can be called a decimal. However, decimal most often refers only to numbers such as 17.38 and 0.45, which are expressed with decimal points. The number of digits to the right of the decimal point is called the number of decimal places. The positions of the digits to the left of the decimal point represent place values that are increasing powers of 10 (1, 10, 102, 103, …). The positions to the right of the decimal point represent place values that are decreasing powers of 10

(10-1, 10-2, 10-3, …) or reciprocals of powers of 10 (1

10,

1

102,

1

103, … ).

Like whole numbers, decimals can be written in expanded form to show the powers of 10. For example,

473.2865 can be expanded to 4(102) + 7(10) + 3(1) + 2 (1

10) + 8 (

1

102) + 6 (1

103) + 5 (1

104)

Write 1854.273 in expanded form. Express the value of the underlined digit as a fraction whose denominator is a power of 10.

1) 47.35 2) 6.089 3) 14.27 Write the decimal equivalent of each expression.

1) 8

1+

7

10+

3

100+

9

10,000 2)

2

1+

3

100+

1

1000+

7

10,000

Reading and Writing Decimals The digits to the left of the decimal point are read as a whole number, and the decimal point is read as and. The digits to the right of the point are also read as a whole number, after which we say the name of the place value of the last digit. Consider the decimal 16,854.273. The place values are listed below. Notice the similarity in pairs of names to the right and left of the units digit. 1 6, 8 5 4 . 2 7 3

1 t

en-t

ho

usa

nd

s

6 t

ho

usa

nd

s

8 h

un

dre

ds

5 t

ens

4 u

nit

s

2 t

enth

s

7 h

un

dre

dth

s

3 t

ho

usa

nd

ths

16,854.273 is read “sixteen thousand, eight hundred fifty-four and two hundred seventy-three thousandths”

Decimals and Rational Numbers 6.1

2 Write the name of each decimal.

1) 3.472

2) 16.14

3) 0.3775 Models for Decimals Models are important for providing conceptual understanding and insight into the use of decimals. Decimal Squares - The Decimal Squares model illustrated the part-to-whole concept of decimals and place value. Unit squares are divided into 10, 100, and 1000 equal parts, and the decimal tells us what part of the square is shaded.

In the tenths square above, represent 0.3. In the hundredths square above, represent 0.64. And in the thousandths square above represent 0.161. Describe the square that would represent each decimal.

1) 0.32

2) 0.7

3) 0.256

3 Number Line - The number line is a common model for illustrating decimals. On the number line below, mark the appropriate location of each decimal: 0.4 0.75 -0.8 -0.15

-1 0 1

Equality of Decimals Equality of Decimals can be illustrated visually by comparing the shaded amounts in their decimal squares. In the tenths square below, represent 0.4. And in the hundredths square, represent 0.40.

You should notice that 4 parts out of 10 and 40 parts out of 100 are represented by the same amount of shading. In each decimal square, four columns are shaded. This illustrates that 0.4 = 0.40 Inequality of Decimals Research indicates that students from elementary school through college often have difficulties determining inequalities for decimals. One source of confusion is thinking of the digits in the decimals as whole numbers. For example, a student might think that 0.789 is larger than 0.82 because 789 is larger than 82. Consider the decimals 0.47 and 0.6. Represent these decimals in the decimal squares below.

Write an inequality relating the two decimals above.

4 Place Value Test for Inequality of Decimals: The greater of two postitive decimals that are both less than 1 will be the decimal with the greater digit in the tenths place. If these digits are equal, this test is applied to the hundredths digits, etc. Place the following decimals in order from least to greatest.

0.26 0.1 0.698 0.240 0.8 0.3154 Place the following decimals in order from least to greatest.

0.316 0.31 0.032 0.036 0.361 0.3106 Rational Numbers

Up to this point, numbers written in the form 𝑎

𝑏, where 𝑎 and 𝑏 are integers with 𝑏 ≠ 0, have been called

fractions. This is the terminology commonly used in elementary and middle schools. However, the word fraction has a more general meaning and includes the quotient of any two numbers, integers or not, as long as the denominator is not zero.

In correction, any number that can be written in the form 𝑎

𝑏, where 𝑎 and 𝑏 are integers with 𝑏 ≠ 0, is called a

rational number. Rational numbers can be expressed by many different number symbols or numerals. When

the denominator of a rational number equals 1, the rational number is an integer. For example, 6

1= 6 and

−4

1= −4. Therefore, integers are also rational numbers. In addition,

3

10= 0.3 is a rational number. All rational

numbers 𝑎

𝑏 can be written as decimals.

Convert each fraction to a decimal without using a calculator.

1) 64

100

2) 7283

1000 3)

54

10,000

Sometimes when the denominator is not a power of 10, the fraction can be replaced by an equal fraction whole denominator is a power of ten. Convert each fraction to a decimal without using a calculator.

4) 1

4 5)

3

5 6)

27

20

5 In the previous examples, all the decimals have a finite number of digits. Such decimals are called terminating decimals. However, there are decimals that are nonterminating. Here is the rule: If a nonzero rational number 𝑎

𝑏 is in simplest form. It can be written as a terminating decimal if and only if 𝑏 has only 2s and/or 5s in its

prime factorization. Which of the following rational numbers can be written as a terminating decimal? (no calculator)

1) 5

6 2)

3

70

3) 11

40 4)

9

15

When a decimal does not terminate and contains a repeating pattern of digits, it is called a repeating decimal. The block of digits that it repeated over and over is called the repetend. It is customary to place a bar above the repetend to indicate the repeating pattern. The division algorithm can be used to obtain a repeating decimal (or terminating decimal, for that matter) for any rational number. Write the decimal for each rational number using your calculator. Use a bar to show the repetend.

1) 5

6 2)

3

11 3)

5

12

Terminating decimals can be written as fraction whose denominators are powers of ten. If you can correctly read the decimal, you can write it as a fration. For example, 0.378 is read “three hundred seventy-eight

thousandths.” Therefore, 0.378 written as a fraction is 378

1000. This can be reduced to

189

500.

Write each terminating decimal as a fraction without using a calculator. You do not need to reduce.

1) 0.4 2) 0.67 3) 0.34

4) 0.142 5) 3.92 6) 0.0283

6 Repeating Decimals A repeating decimal can also be written as a fraction whose numerator and denominator are integers. Let’s

first look at the repeating decimals for the fractions 1

9,

1

99, and

1

999 .

We can use these 3 repeating decimals to write other repeating decimals as fractions. For example, consider the repeating decimal 0.888…

0.888 … = 8 × 0.111 … = 8 ×1

9=

8

9

A similar process is used when the repetend has two digits. Consider the repeating decimal 0.373737…

0.3737 … = 37 × 0.0101 … = 37 ×1

99=

37

99

Any digits in the decimal that precede the retetend can be separated by multiplying powers of 10. For

exmaple, consider 0.2373737… We want to start by writing this decimal as 1

10× 2.373737 …

1

10× 2.373737 … =

1

10× 2

37

99=

1

10×

235

99=

235

990

Replace each repeating decimal with a quotient of two integers. You do not need to write the fractions in lowest terms.

1) 0. 17̅̅̅̅ 2) 0. 7̅ 3) 0. 238̅̅ ̅̅ ̅

4) 0.34̅ 5) 0.1029̅̅ ̅̅ ̅ 6) 0.3015̅̅̅̅

7 Density of Rational Numbers Last unit, we saw that rational numbers, when written as fraction, are dense. That is, between any two such numbers there is always another. To find decimals between two given decimals, we can express both decimals with a greater number of decimal places. Find three decimals between each pair of decimals.

1) 0.124 and 0.125 2) 1.1 and 1.2 3) 0.47 and 0.621

Estimation Rounding to a given place value is the most common method of obtaining decimal estimations.

Locate the place value to which the number is to be rounded, and check the digit to its right.

If the digit to the right is 5 or greater, then all digits to the right are dropped and the digit with the given place value is increased by 1.

If the digit to the right is 4 or less, then all digits to the right of the digit with the given place value are dropped.

Round each decimal to the given place value.

1) 5.487 (hundredths) 2) 32.149 (tenths) 3) 0.45952 (thousandths) Teaching Question Suppose an elementary school students asked you why writing zeros to the right of a whole number increases the value of the number, but when zeros are written to the right of a decimal point, the value of the number does not increase. Write a response that would make sense to this student.

8

Addition The concept of addition of decimals is the same as the concept of addition of whole numbers, integers, and fraction; it involves putting together, or combining two amounts. Represent the sum 0.2 + 0.6 using the decimal squares below.

+

=

Represent the sum 0.47 + 0.36 using the decimal squares below.

+

=

Represent the sum 0.5 + 0.24 using the decimal squares below.

+

=

Operations with Decimals 6.2

9 Pencil-and-Paper Algorithm for Addition In one pencil-and-paper algorithm for addition of decimals, the digits are aligned, tenths under tenths, hundredths under hundredths, etc. When the sum of the digits in any column is 10 or greater, regrouping is necessary. Compute each sum using the pencil-and-paper algorithm described above.

1) 62.47 + 114.86 2) 4.039 + 17.18 3) 0.267 + 0.5163

Subtraction Subtraction of decimals, like subtraction of whole numbers and fractions, can be illustrated with the take-away concept. In the decimal squares below, represent the given differences.

1) 0.48 – 0.12 2) 0.89 – 0.2

Pencil-and-Paper Algorithm for Subtraction In one pencil-and-paper algorithm for subtraction of decimals, the digits are aligned as they are for addition of decimals. Subtraction then takes place from right to left, with thousandths subtracted from thousandths, hundredths from hundredths, etc. When regrouping (borrowing) is necessary, it is done just as it is in subtracting whole numbers. Compute each difference using the pencil-and-paper algorithm described above.

1) 46.32 – 18.47 2) 0.4074 – 0.356 3) 15.02 – 2.743

10 Multiplication The product of a whole number and a decimal can be illustrated by repeated addition. Represent the product 2 × 0.7 using the decimal squares below.

+

=

To multiply a decimal by a decimal, you can think of these decimals as fractions or as an area model. Represent the product 0.5 × 0.3 in the decimal square below.

Decimals with smaller place values are more difficult to illustrate, therefore we will not discuss it. Pencil-and-Paper Algorithms for Multiplication Traditional: To compute products involving decimals, we can multiply the numbers as though they were whole numbers then locate the decimal point in the product. The number of decimal places in the answer is the total number of decimal places in the original two numbers. Compute each product using the traditional algorithm for multiplication.

1) 3.7 × 2.5 2) 4.6 × 0.35 3) 1.8 × 0.473

11 Lattice: For lattice multiplication, you can use what you already know about multiplying whole numbers to multiply decimals. To place the decimal point, find the intersection of the decimal points along the horizontal and vertical lines; then slide it along the diagonal. Estimating the product helps to place the decimal correctly. Compute each product using lattice multiplication for decimals.

1) 6.2 × 1.8 2) 2.3 × 0.54 3) 3.1 × 0.627 Division The two concepts of division, the measurement (subtractive) concept and the sharing (partitive) concept, are both useful for illustrating division of decimals. To illustrate the division of a decimal by a whole number, we can use the sharing concept. In this case, the divisor is the number of equal parts into which a set or region is divided. Represent the quotients in the decimal squares provided.

1) 0.8 ÷ 4 2) 0.4 ÷ 5

12 To illustrate the division of a decimal by a decimal, we can use the measurement concept. The measurement concept involves repeatedly measuring off or subtracting one amount from another. The number of times the smaller amount can be subtracted from the larger amount is the quotient. Represent the quotients in the decimal squares provided.

1) 0.84 ÷ 0.12 2) 0.9 ÷ 0.15

Pencil-and-Paper Algorithm for Division To divide any decimal by a whole number, first divide using the long division algorithm for whole numbers. Then place the decimal points in the quotient directly above its location in the dividend. In the long division algorithm for dividing with decimals, we never actually divide by a decimal. Before we divide, an adjustment is made so that the divisor is always a whole number. The rule for dividing by a decimal is to count the number of decimal places in the divisor and then move the decimal points in the divisor and the dividend that many places to the right. Thus, division of a decimal by a decimal can always be carried out by dividing a decimal (or whole number) by a whole number. Compute each quotient using the pencil-and-paper algorithm described above.

1) 106.82 ÷ 7 2) 0.498 ÷ 0.6 3) 34.44 ÷ 1.4

13 Multiplying and Dividing by Powers of 10 To multiply a decimal by a power of 10, move the decimal point one place to the right for each power of 10. To divide a decimal by a power of 10, move the decimal point one place to the left for each power of 10. Compute each product of quotient mentally.

1) 100 × 4.5 2) 1000 × 0.32714 3) 0.35 ÷ 10 4) 5.9 ÷ 1000 Compatible Numbers for Mental Calculations Sometimes in computing products mentally, it helps to recognize the decimal equivalents of a few simple fractions. Here are some that are useful.

. 25 =1

4 .5 =

1

2 .75 =

3

4 .2 =

1

5 .4 =

2

5 .6 =

3

5 .8 =

4

5 .125 =

1

8

Compute each product by replacing the decimal by an equivalent fraction.

1) .25 × 800 2) 0.2 × 30 3) 0.6 × 45 Estimation Rounding: Estimate each product mentally by rounding the decimals to the nearest whole number.

1) 4.6 × 8.21 2) 10.263 × 5.9 Compatible Numbers: Decimals can be replaced by compatible decimals or fractions for estimating. Estimate each computation by replacing a decimal by a more compatible decimal or fraction.

1) 27.7 – 1.8 2) 2.87 + 5.15

3) 0.19 × 45 4) 6 ÷ 0.26

14

Ratios A ratio is a pair of positive numbers that is used to compare two sets. A ratio gives the relative size of two sets, but not the actual number of objects in those sets. For example, the fact that the ratio of boys to girls in a certain classroom is 1 to 3 tells us that for every boy there are three girls, or that the number of boys is one-third the number of girls, but it does not tell us the number of boys or girls. If the ratio of boys to girls is 1 to 3,

we could write this as 1:3 or 1

3

In recent years, for every 2 female inmates in the United States, there were 11 male inmates. What is the ratio of the number of male inmates to the numbers of female inmates? Proportions An equality of ratios is called a proportion. Each ratio gives rise to many pairs of equal ratios. Proportions are useful in problem solving. Typically, three of the four numbers in a proportion are given and the forth is to be found.

1) If the ratio of teachers to students in a school is 1 to 18 and there are 360 students, how many teachers are there?

2) The Bay City Cardinals have won 5 of 8 games. At the same rate, how many games will they have to play to win 60 games?

3) If 4.8 pounds of flour costs $8.40, how much will 6 pounds cost?

Ratio, Percent, and Scientific Notation 6.3

15 Percent The word percent comes from the Latin per centum, meaning out of 100. Percent was first used in the fifteenth century for computing interest, profits, and losses.

Percents are ways of representing fractions with denominators of 100. For example, 15 percent means 15

100 and

is written 15%. Diagrams are one method of gaining an understanding of percents. A 10×10 grid with 100 equal parts (a hundredths decimal square) is a common model in elementary school texts for illustrating percents. Describe a decimal square to represent each percent.

1) 90%

2) 9%

3) 35.5% Notice the similarity between the percent symbol and the numeral 100. This is helpful in remembering how to replace a percent by a fraction or decimal. First, drop the percent symbol and write the percent as a fraction with a denominator of 100. Then, to obtain a decimal, divide the numerator by 100. (recall: dividing by 100 moves the decimal point two places to the left) Write each percent as a fraction and as a decimal without using a calculator. You do not need to write the fractions in lowest terms, but do not leave a decimal within a fraction.

1) 42%

2) 3%

3) 65.2%

4) 213

4%

16 To write a decimal as a percent, reverse the previous process. Multiply the decimal by 100 and then include a percent symbol. Write each decimal as a percent without using a calculator.

1) .07 2) .953

3) 0.008 4) 3.25

To write a fraction as a percent, first write it as a decimal. Then multiply the decimal by 100 and include a percent symbol. Alternatively, you can rewrite the fraction as one with a denominator of 100. Write each fraction as a percent without using a calculator.

1) 7

10 2)

9

20

3) 3

5 4)

1

8

Fill in the chart below without using a calculator. You do not need to write the fractions in lowest terms, but do not leave a decimal within a fraction.

Fraction Decimal Percent 1

4

0.82

4%

0.191

11

50

3.86%

17 Calculations with Percents Calculations with percents fall into three categories:

1. Given the whole and the percent, find the part. 2. Given the whole and the part, find the percent. 3. Given the percent and the part, find the whole.

We can solve each of the above percent problems by setting up a proportion.

part

whole=

percent

100

The “part” is many times identified by the word “is”, and the whole is many times identified by the word “of”. Solve using a proportion. Round to the nearest tenth, if applicable.

1) What is 7% of 91? 2) 45 is 22% of what number?

3) 29 is what percent of 88? 4) 53 is 36.5% of what number?

5) A survey of football players revealed that 15% of 1180 players had knee injuries. How many players had knee injuries?

6) If $880 of a $2000 loan has been paid off, what percent of the loan has been paid off?

18 7) You just hired a new employee to work in your bakeshop. The new employee burned 250 chocolate

chip cookies. If this represented 20% of the day’s production, how many cookies were baked that day?

8) Approximately 19% of the voters voted for the independent candidate. If 212,822 people voted, what is the number of people who voted for the independent candidate?

9) If 48 of the 60 seats on a bus were occupied, what percent of the seats were not occupied?

10) After eating 25% of the jelly beans, Brett had 72 left. How many jelly beans did Brett have originally? Discounts A common occurrence of percents is found in computing discounts. In the case of discounts, we want to subtract a certain percent of the original cost. What is the cost of an item that is listed for $36.00 with a 15% discount? Sales Tax Another common occurrence of percents is found in computing sales tax. We are typically adding the cost of the object and the sales tax. What is the cost of an item that is listed for $45 after a 6% sales tax is applied?

19 Solve the following problems regarding discounts and sales tax.

1) The regular price of a tool kit is $39.99. If the tool kit is marked 20% off, what is the final cost of the tool kit? Include a 7% sales tax.

2) Yolanda bought a sweater that was marked down 25%. What was the original cost of the sweater if she saved $9.00?

3) Jerry paid $2421 for a treadmill after having been given a 10% discount. What was the presale price of the treadmill?

4) A restaurant paid $111 for a piece of equipment after having been given a 40% discount. What was the original price of the equipment?

5) Cameron purchased $125 worth of merchandise but only paid $93.75. What percent discount did he receive?

6) A department store is having a huge sale. All clearance items will be discounted an additional 30% at the register. Sam picks up a sweater on the clearance rack that is already marked down 25%. The original price of the sweater is $75. What will Sam pay for the sweater?

20 Mental Calculations with Percents Certain percents are convenient for calculations. One of these is 10% because multiplying by 0.10 is just a matter of moving the decimal point. For example, 10% of $16.50 = 0.10 × 16.50 = 1.65. Once we know 10% of a number, we can use that amount to determine other percents such as 5%, 15%, 25% and 30%. Find the following percents mentally.

1) 20% of $82

2) 15% of $30

3) 25% of $65 For some computations, it is convenient to replace a percent by a fraction.

1) 25% of 88

2) 20% of 55

3) 75% of 24

4) 331

3% of 45

A sporting goods store buys baseball mitts at a wholesale price and then marks them up 40% to sell. This week they are running a sale on the mitts, and they are begin discounted by 25%. If the wholesale price is $50, what are the mitts selling for this week?

21 Scientific Notation Very large and very small numbers can be written conveniently by using powers of 10.

Some computers can perform 400,000,000 calculations per second. Using a power of 10, we can write 400,000,000 = 4 × 108.

The average thickness of human hair, which is approximately 0.003 inch, can be written 0.003 = 3 ×10-3. Any positive number can be written as the product of a number from 1 to 10 and a power of 10. This method of writing numbers is called scientific notation. Fill in the missing numbers in the table.

Positional Numeration Scientific Notation

Years since age of dinosaurs 150,000,000

Orbital velocity of Earth (km/h) 4.129 × 104

Wavelength of gamma ray (m) .0000000000003048

Size of viruses (cm) 9.14 × 10-7

Class Discussion Questions Some students in a middle school lass were having difficulty understanding the relationship between ratios and fractions. One student asked, “Since the ratio of the girls to boys in our class is 2 to 3, why isn’t it correct to say 2/3 of the class are girls?” How would you respond to this question? Your friend’s rent went down 10% last year and rose 20% this year. She concludes that over the two years her rent went up 10%. Do you agree with your friend? Explain.

22

Earlier this chapter, terminating and repeating decimals were discussed. Terminating and repeating decimals

can be written as fraction. For example, 0.83 =83

100 and 0. 7̅ =

7

9

The purpose of this section is to introduce a new type of number whose decimals are nonrepeating. Such nonrepeating decimals are called irrational numbers. For example, the number 0.070070007… has a pattern, but there is no block of digits that is repeating. Therefore, this is an irrational number. Label each number as rational or irrational.

1) .006006006… 2) .060060006… 3) .01001

4) .73173117311173… 5) .737373… 6) .21060606… Square Roots The square root of a non-negative number is defined as a number that, when multiplied by itself, yields the original number. For example, 3 is the square root of 9, since 3 × 3 = 9. The square root of a number 𝑏 is

written as √𝑏. Calculate each square root.

1) √49 2) √20.25 3) √0.64 4) √

1

4

The square roots of perfect squares (√1, √4, √9, √16, etc. ) are whole numbers, and are therefore rational

numbers. The square roots of all other whole numbers greater than 0 (√2, √3, √5, etc. ) are irrational. These

numbers have nonrepeating decimals. Label each number as rational or irrational without using a calculator.

1) √81 2) √10 3) √30

4) 29

30 5) √

4

9

6) √0.16

Irrational and Real Numbers 6.4

23 Pythagorean Theorem Irrational numbers were first recognized by the Pythagoreans, followers of the Greek mathematician Pythagoras who lived in the fifth century B.C. It is possible that the discovery of such numbers arose in connection with the Pythagorean Theorem. This theorem concerns right triangles. The two shorter sides of such a triangle are called legs, and the longest side, which is opposite the right angle, is called the hypotenuse. The theorem states that for any right triangle with legs of lengths 𝑎 and 𝑏 and hypotenuse of length 𝑐:

𝑎2 + 𝑏2 = 𝑐2 Use the Pythagorean Thoerem to find the missing length in each triangle.

The converse of the Pythagorean Theorem also holds. If the sum of the squares of two sides of a triangle equals the square of the third side, then the triangle is a right triangle. A triangle has side lengths 7, 24, and 25. Is the triangle a right triangle? Justify your answer. The product of the square roots of two numbers is equal to the square root of the product of the two

numbers. That is, for positive numbers 𝑎 and 𝑏, √𝑎 × √𝑏 = √𝑎 × 𝑏. This theorem is useful for simplifying square roots. A square root is in simplified form if the number under the square root has no factor other than 1 that is a perfect square.

24 Write each square root in simplified form.

1) √24

2) √50

3) √80

4) √54

5) √108

For any non-negative number 𝑏, √𝑏 × √𝑏 = 𝑏. Evaluate the following expressions.

1) √14 × √14 2) √9 × √9 3) (√6)2

Use the Pythagorean Theorem to find the missing length in each right triangle.

25 Cube Roots

The cube root of a number 𝑛 is written as √𝑛3

. This is the number 𝑠 such that 𝑠 × 𝑠 × 𝑠 = 𝑛. The cube roots of perfect cubes are whole numbers, and are therefore rational. The cube roots of all other whole numbers are irrational numbers. Evaluate without a calculator.

1) √643

2) √273

3) √−1253

Evaluate using a calculator. Round your answer to the nearest tenth.

4) √203

5) √1003

Real Numbers The irrational numbers (I) and the rational numbers together form the set of real numbers. The set of rational numbers and the set of irrational numbers are disjoint, and their union is the set of real numbers (R). The rational numbers (Q) contain the integers (Z), and the integers contain the whole numbers (W).

26 Use the letters W, Z, Q, I, and R to indicate to which set(s) each number belongs.

1) −12 2) √15 3) 0.23

4) 130 5) 3

5 6) 0. 27̅̅̅̅

7) √183

8) √25 9) 12.5

Rationalizing a denominator

Quotients of real numbers, such as 2 ÷ √3 are often written as fractions, such as 2

√3. When the denominator

of a fraction contains a square root, cube root, etc., it is sometimes necessary to find an equal fraction that has a rational number for its denominator. The process of replacing a denominator that is irrational by a denominator that is rational is called “rationalizing the denominator”.

2

√3×

√3

√3=

2√3

3

Rationalize the denominator of each fraction.

1) 2

√5

2) 3

√6

3) 21

√7

27 CASA Questions: Answer the following without a calculator.

1) Between which two whole numbers is √30?

2) Between which two whole numbers is √67 − 4 ?

3) Which of the following plotted points could be 2√3 + 1 ?