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5.3: Decimal Operations (Part 1) - Mathematics


Skills to Develop

  • Add and subtract decimals
  • Multiply decimals
  • Divide decimals
  • Use decimals in money applications

be prepared!

Before you get started, take this readiness quiz.

  1. Simplify (dfrac{70}{100}). If you missed this problem, review Example 4.3.1.
  2. Multiply (dfrac{3}{10} cdot dfrac{9}{10}). If you missed this problem, review Example 4.3.7.
  3. Divide −36 ÷ (−9). If you missed this problem, review Example 3.7.3.

Add and Subtract Decimals

Let’s take one more look at the lunch order from the start of Decimals, this time noticing how the numbers were added together.

[egin{split} & $3.45 quad Sandwich & $1.25 quad Water + & $0.33 quad Tax hline & $5.03 quad Total end{split}]

All three items (sandwich, water, tax) were priced in dollars and cents, so we lined up the dollars under the dollars and the cents under the cents, with the decimal points lined up between them. Then we just added each column, as if we were adding whole numbers. By lining up decimals this way, we can add or subtract the corresponding place values just as we did with whole numbers.

HOW TO: ADD OR SUBTRACT DECIMALS

Step 1. Write the numbers vertically so the decimal points line up.

Step 2. Use zeros as place holders, as needed.

Step 3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.

Example (PageIndex{1}):

Add: 3.7 + 12.4.

Solution

Write the numbers vertically so the decimal points line up.$$egin{split} 3.&7 + 12.&4 hline end{split}$$
Place holders are not needed since both numbers have the same number of decimal places.
Add the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.$$egin{split} stackrel{1}{3}.&7 + 12.&4 hline 16.&1 end{split}$$

Exercise (PageIndex{1}):

Add: 5.7 + 11.9.

Answer

(17.6)

Exercise (PageIndex{2}):

Add: 18.32 + 14.79.

Answer

(13.11)

Example (PageIndex{2}):

Add: 23.5 + 41.38.

Solution

Write the numbers vertically so the decimal points line up.$$egin{split} 23.&5 + 41.&38 hline end{split}$$
Place 0 as a place holder after the 5 in 23.5, so that both numbers have two decimal places.$$egin{split} 23.&5 extcolor{red}{0} + 41.&38 hline end{split}$$
Add the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.$$egin{split} 23.&50 + 41.&38 hline 64.&88 end{split}$$

Exercise (PageIndex{3}):

Add: 4.8 + 11.69.

Answer

(16.49)

Exercise (PageIndex{4}):

Add: 5.123 + 18.47.

Answer

(23.593)

How much change would you get if you handed the cashier a $20 bill for a $14.65 purchase? We will show the steps to calculate this in the next example.

Example (PageIndex{3}):

Subtract: 20 − 14.65.

Solution

Write the numbers vertically so the decimal points line up. Remember 20 is a whole number, so place the decimal point after the 0.$$egin{split} 20.& - 14.&65 hline end{split}$$
Place two zeros after the decimal point in 20, as place holders so that both numbers have two decimal places.

[egin{split} 20.& extcolor{red}{00} - 14.&65 hline end{split}]

Subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.$$egin{split} stackrel{1}{cancel{2}} stackrel{stackrel{9}{cancel{10}}}{cancel{0}} &.stackrel{stackrel{9}{cancel{10}}}{cancel{0}} stackrel{stackrel{9}{cancel{10}}}{cancel{0}} - 1; ; 4; ; &.; 6; ; 5 hline 5; ; &.; 3; ; 5end{split}$$

Exercise (PageIndex{5}):

Subtract: 10 − 9.58.

Answer

(0.42)

Exercise (PageIndex{6}):

Subtract: 50 − 37.42.

Answer

(12.58)

Example (PageIndex{4}):

Subtract: 2.51 − 7.4.

Solution

If we subtract 7.4 from 2.51, the answer will be negative since 7.4 > 2.51. To subtract easily, we can subtract 2.51 from 7.4. Then we will place the negative sign in the result.

Write the numbers vertically so the decimal points line up.$$egin{split} 7.&4 - 2.&51 hline end{split}$$
Place zero after the 4 in 7.4 as a place holder, so that both numbers have two decimal places.$$egin{split} 7.&4 extcolor{red}{0} - 2.&51 hline end{split}$$
Subtract and place the decimal in the answer.$$egin{split} 7.&40 - 2.&51 hline 4.&89 end{split}$$
Remember that we are really subtracting 2.51 − 7.4 so the answer is negative.2.51 − 7.4 = − 4.89

Exercise (PageIndex{7}):

Subtract: 4.77 − 6.3.

Answer

(-1.53)

Exercise (PageIndex{8}):

Subtract: 8.12 − 11.7.

Answer

(-3.58)

Multiply Decimals

Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions.

Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators. So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side in Table 5.22. Look for a pattern.

Table (PageIndex{1})
AB
(0.3)(0.7)(0.2)(0.46)
Convert to fractions.$$left(dfrac{3}{10} ight) left(dfrac{7}{10} ight)$$$$left(dfrac{2}{10} ight) left(dfrac{46}{100} ight)$$
Multiply.$$dfrac{21}{100}$$$$dfrac{92}{1000}$$
Convert back to decimals0.210.092

There is a pattern that we can use. In A, we multiplied two numbers that each had one decimal place, and the product had two decimal places. In B, we multiplied a number with one decimal place by a number with two decimal places, and the product had three decimal places.

How many decimal places would you expect for the product of (0.01)(0.004)? If you said “five”, you recognized the pattern. When we multiply two numbers with decimals, we count all the decimal places in the factors—in this case two plus three—to get the number of decimal places in the product—in this case five.

Once we know how to determine the number of digits after the decimal point, we can multiply decimal numbers without converting them to fractions first. The number of decimal places in the product is the sum of the number of decimal places in the factors.

The rules for multiplying positive and negative numbers apply to decimals, too, of course.

Definition: Multiplying Two Numbers

When multiplying two numbers,

  • if their signs are the same, the product is positive.
  • if their signs are different, the product is negative.

When you multiply signed decimals, first determine the sign of the product and then multiply as if the numbers were both positive. Finally, write the product with the appropriate sign.

HOW TO: MULTIPLY DECIMAL NUMBERS

Step 1. Determine the sign of the product.

Step 2. Write the numbers in vertical format, lining up the numbers on the right.

Step 3. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.

Step 4. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.

Step 5. Write the product with the appropriate sign.

Example (PageIndex{5}):

Multiply: (3.9)(4.075).

Solution

Determine the sign of the product. The signs are the same.The product will be positive.
Write the numbers in vertical format, lining up the numbers on the right.$$egin{split} 4.07&5 imes 3.&9 hline end{split}$$
Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.$$egin{split} 4.07&5 imes 3.&9 hline 3667&5 12225&; hline 15892&5 end{split}$$
Place the decimal point. Add the number of decimal places in the factors (1 + 3). Place the decimal point 4 places from the right.$$egin{split} 4.07&5 quad extcolor{blue}{3; places} imes 3.&9 quad extcolor{blue}{1; place} hline 3667&5 12225&; hline 15892&5 quad extcolor{blue}{4; places} end{split}$$
The product is positive.(3.9)(4.075) = 15.8925

Exercise (PageIndex{9}):

Multiply: 4.5(6.107).

Answer

(27.4815)

Exercise (PageIndex{10}):

Multiply: 10.79(8.12).

Answer

(87.6148)

Example (PageIndex{6}):

Multiply: (−8.2)(5.19).

Solution

The signs are different.The product will be negative.
Write in vertical format, lining up the numbers on the right.$$egin{split} 5.&19 imes 8.&2 hline end{split}$$
Multiply.$$egin{split} 5.&19 imes 8.&2 hline 10&38 415&2; hline 425&58 end{split}$$
$$egin{split} 5.&19 imes 8.&2 hline 10&38 415&2; hline 42.5&58 end{split}$$
The product is negative.(−8.2)(5.19) = −42.558

Exercise (PageIndex{11}):

Multiply: (4.63)(−2.9).

Answer

(-13.427)

Exercise (PageIndex{12}):

Multiply: (−7.78)(4.9).

Answer

(-38.122)

In the next example, we’ll need to add several placeholder zeros to properly place the decimal point.

Example (PageIndex{7}):

Multiply: (0.03)(0.045).

Solution

The product is positive.(0.03)(0.045)
Write in vertical format, lining up the numbers on the right.$$egin{split} 0.04&5 imes 0.0&3 hline end{split}$$
Multiply.$$egin{split} 0.04&5 imes 0.0&3 hline 13&5 end{split}$$

Add zeros as needed to get the 5 places.

The product is positive.(0.03)(0.045) = 0.00135

Exercise (PageIndex{13}):

Multiply: (0.04)(0.087).

Answer

(0.00348)

Exercise (PageIndex{14}):

Multiply: (0.09)(0.067).

Answer

(0.00603)

Multiply by Powers of 10

In many fields, especially in the sciences, it is common to multiply decimals by powers of 10. Let’s see what happens when we multiply 1.9436 by some powers of 10.

Look at the results without the final zeros. Do you notice a pattern?

[egin{split} 1.9436(10) & = 19.436 1.9436(100) & = 194.36 1.9436(1000) & = 1943.6 end{split}]

The number of places that the decimal point moved is the same as the number of zeros in the power of ten. Table 5.26 summarizes the results.

Table (PageIndex{2})
Multiply byNumber of zerosNumber of places decimal point moves
1011 place to the right
10022 places to the right
1,00033 places to the right
10,00044 places to the right

We can use this pattern as a shortcut to multiply by powers of ten instead of multiplying using the vertical format. We can count the zeros in the power of 10 and then move the decimal point that same of places to the right. So, for example, to multiply 45.86 by 100, move the decimal point 2 places to the right.

Sometimes when we need to move the decimal point, there are not enough decimal places. In that case, we use zeros as placeholders. For example, let’s multiply 2.4 by 100. We need to move the decimal point 2 places to the right. Since there is only one digit to the right of the decimal point, we must write a 0 in the hundredths place.

HOW TO: MULTIPLY A DECIMAL BY A POWER OF 10

Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.

Step 2. Write zeros at the end of the number as placeholders if needed.

Example (PageIndex{8}):

Multiply 5.63 by factors of (a) 10 (b) 100 (c) 1000.

Solution

By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.

(a) 5.63(10)

There is 1 zero in 10, so move the decimal point 1 place to the right.
56.3

(b) 5.63(100)

There are 2 zeros in 100, so move the decimal point 2 places to the right.
563

(c) 5.63(1000)

There are 3 zeros in 1000, so move the decimal point 3 places to the right.
A zero must be added at the end.5,630

Exercise (PageIndex{15}):

Multiply 2.58 by factors of (a) 10 (b) 100 (c) 1000.

Answer a

(25.8)

Answer b

(258)

Answer c

(2,580)

Exercise (PageIndex{16}):

Multiply 14.2 by factors of (a) 10 (b) 100 (c) 1000.

Answer a

(142)

Answer b

(1,420)

Answer c

(14,200)


Word Problem With Multiple Decimal Operations: Problem type 1 Online Quiz

Following quiz provides Multiple Choice Questions (MCQs) related to Word Problem With Multiple Decimal Operations: Problem type 1. You will have to read all the given answers and click over the correct answer. If you are not sure about the answer then you can check the answer using Show Answer button. You can use Next Quiz button to check new set of questions in the quiz.

Q 1 - John bought 3 notebooks for .87 apiece and two boxes of pencils for $2.78 apiece. How much does he need to pay?

Answer : B

Explanation

Cost of 3 notebooks @ 0.87 apiece = 3 × 0.87 = $2.61

Cost of 2 box pencils @ 2.78 apiece = 2 × 2.78 = $5.56

Total amount = $2.61 + $5.56 = $8.17

Q 2 - Cathy earns $8.40 per hour at her part-time job and $6.50 per hour at her baby-sitting job. On Saturday she worked 3.5 hours at part-time job and spent 2.5 hours babysitting. What were her total earnings for the day?

Answer : A

Explanation

Earnings from part-time job = 3.50 × $8.40 = $29.40

Earnings from baby-sitting = 2.5 × $6.50 = $16.25

Total earnings = $29.40 + $16.25 = $45.65

Q 3 - At the malt shop a large chocolate shake takes 0.8 of a pint of milk and the medium shake takes 0.7 of a pint of milk. How many pints of milk are required for 4 large chocolate shakes and 3 medium shakes?

Answer : C

Explanation

Milk in large chocolate shake = 0.8 × 4 = 3.2 pints

Milk in medium chocolate shake = 0.7 × 3 = 2.1 pints

Total amount of milk used = 3.2 + 2.1 = 5.3 pints

Q 4 - Jamie ordered 8 pizzas and 7 burgers. Each pizza costs $12.95 and each burger costs $11.95. How much does she need to pay?

Answer : D

Explanation

Cost of pizzas = 8 × 12.95 = $103.60

Cost of burgers = 7 × 11.95 = $83.65

Total amount to be paid = $103.60 + $83.65 = $187.25

Q 5 - For first two days it snowed 2.4 centimeters each day. The next three days, it snowed 3.3 centimeters each day. How much did it snow on those five days?

Answer : B

Explanation

Amount of snowfall on first two days = 2 × 2.4 = 4.8 cm

Amount of snowfall on next three days = 3 × 3.3 = 9.9 cm

Total snowfall in the five days = 4.8 + 9.9 = 14.7 cm

Q 6 - Peter had 3 buckets that were each 0.6 full of apples and 5 buckets that were each 0.8 full of oranges. How many buckets of fruits did Peter have?

Answer : D

Explanation

Number of apples = 3 × 0.6 = 1.8 buckets

Number of oranges = 5 × 0.8 = 4.0 buckets

Total number of fruits = 1.8 + 4.0 = 5.8 buckets

Q 7 - A washing machine used 1.6 liters of water per full load of clothes to wash and 2.4 liters of water per full load of curtains to wash. If Jack washed 4.2 loads of clothes, and 3.2 loads of curtains how many liters of water did he use?

Answer : A

Explanation

Water used to wash clothes = 1.6 × 4.2 = 6.72 liters

Water used to wash curtains = 2.4 × 3.2 = 7.68 liters

Total amount of water used = 6.72 + 7.68 = 14.4 liters

Q 8 - A bottle of soda and a bottle of cola drink had 1.5 and 3.6 of the daily recommended sugar respectively. If you were to drink 0.8 of the soda bottle and 0.9 of cola bottle, how much of the daily recommend sugar would you have drunk?

Answer : C

Explanation

Sugar in soda = 1.5 × 0.8 = 1.2

Sugar in cola drink = 3.6 × 0.9 = 3.24

Total sugar in soda and cola drink = 4.44 times recommended sugar

Q 9 - Each day a company used 0.36 of a box of paper and 0.68 box of pens. How many boxes of paper and pens would they have used after 4 days?

Answer : B

Explanation

Number of boxes of paper used in 4 days = 4 × 0.36 = 1.44

Number of boxes of pens used in 4 days = 4 × 0.68 = 2.72

Total number of boxes used in 4 days = 1.44 + 2.72 = 4.16

Q 10 - At the zoo the polar bears are fed 0.2 bucket of fish a day for 3 days and 0.3 bucket of meat every day for 5 days. How many buckets of fish and meat are the polar bears fed in total?


Decimals

Up to now our “Dots & Boxes” model has consisted of a row of boxes extending infinitely far to the left. Why not have boxes extending to the right as well?

Let’s work specifically with a 1←10 rule and see what boxes to the right could mean.

Notation

It has become convention to separate boxes to the right of the ones place with a decimal point. (At least, this is what the point is called in the base ten world… “dec” means “ten” after all!)

What is the value of the first box to the right of the decimal point? If we denote its value as />, we have that ten />’s is equivalent to 1. (Remember, we are using a 1 ← 10 rule.)

From we get that .

Call the value of the next box to the right .

From we get .

If we keep doing this, we see that the boxes to the right of the decimal point represent the reciprocals of the powers of ten.

Example: 0.3

The decimal is represented by the picture:

It represents three groups of , that is:

Example: 0.007

The decimal is represented by the picture:

It represents seven groups of .

Of course, some decimals represent fractions that can simplify further. For example:

Similarly, if a fraction can be rewritten to have a denominator that is a power of ten, then it is easy to convert it to a decimal. For example, is equivalent to , and so we have:

Example: 12 3/4

Can you write as a decimal? Well,

We can write the denominator as a power of ten using the key fraction rule:

Think / Pair / Share

  • Draw a “Dots & Boxes” picture for each of the following decimals. Then say what fraction each decimal represents:

Example: 0.31

Here is a more interesting question: What fraction is represented by the decimal ?

There are two ways to think about this.

From the picture of the “Dots & Boxes” model we see:

We can add these fractions by finding a common denominator:

Let’s unexplode the three dots in the position to produce an additional 30 dots in the position.

So we can see right away that

On Your Own

Work on the following exercises on your own or with a partner.

1. Brian is having difficulty seeing that represents the fraction . Describe the two approaches you could use to explain this to him.

2. A teacher asked his students to each draw a “Dots & Boxes” picture of the fraction .

The teacher marked both students as correct.

  • Are each of these solutions correct? Explain your thinking.
  • Jin said he could get Sonia’s solution by performing some explosions. What did he mean by this? Is he right?

3. Choose the best answer and justify your choice. The decimal equals:

4. Choose the best answer and justify your choice. The decimal equals:

5. Choose the best answer and justify your choice. The decimal equals:

6. Choose the best answer and justify your choice. The decimal equals:

7. What fraction is represented by each of the following decimals?

8. Write each of the following fractions as decimals. Don’t use a calculator!

9. Write each of the following fractions as decimals. Don’t use a calculator!

10. Write each of the following as a fraction (or mixed number).

11. Write each of the following numbers in decimal notation.

Think Pair Share

Do and represent the same number or different numbers?

Here are two dots and boxes pictures for the decimal .

And here are two dots and boxes picture for the decimal .

  • Explain how one “unexplosion” establishes that the first picture of is equivalent to the second picture of.
  • Explain how several unexplosions establishes that the first picture of is equivalent to the second picture of .
  • Use explosions and unexplosions to show that all four pictures are equivalent to each other.
  • So … does represent the same number as ?

Accuplacer Arithmetic Exercises

This page has free Accuplacer arithmetic exercises. The answers and solutions are in the next section of the page.

You will find more help with Accuplacer arithmetic in the last part of the page.

Instructions:

On scratch paper, find the solutions to the 10 following Accuplacer arithmetic exercises. [You won’t be allowed to use a calculator on the Accuplacer arithmetic test.]

Check your answers, provided below, and study the solutions.

1) Mary bought a used car that was on sale for $900. The original price of the car was $1200. What was the percentage of the discount on the sale?

6) Estimate the product of: 14.9 × 10.2

7) Express 33% as a fraction.

9) Which of the following is the least?

10) A $10 shirt was for sale at a 10% discount. What was the sales price of the shirt?

Accuplacer Arithmetic – Answers

Accuplacer Arithmetic – Solutions

Solution 1:

Practical Accuplacer arithmetic problems are usually expressed in words., rather than as mathematical equations.

In order to calculate a discount, you must first determine how much the item was marked down.

Markdown = Original Price – Sales Price

Then divide the mark down by the original price and convert to a percentage.

Percentage Discount = Markdown ÷ Original Price

Solution 2:

For division problems on the arithmetic part of the exam, you must do long division until you have no remainder.

Our problem was: What is 6 ÷ 32 ?

Solution 3:

The Accuplacer arithmetic section includes long multiplication problems.

You won’t be allowed to use a calculator on the Accuplacer arithmetic test.

So, be sure your know how to do multiplication by hand, as shown below.

Solution 4:

For fraction problems, you have to find the lowest common denominator. The denominator is the number on the bottom of the fraction.

Before you subtract the fractions, you have to change them so that the bottom numbers for each fraction are the same.

You do this my multiplying the numerator [top number] by the same number you use on the denominator for each fraction:

Our question was: What is 2 /3 – 1 /6 ?

Finding the LCD

The lowest common denominator for each fraction above is 6.

In order to get the lowest common denominator, we have to convert the first fraction as follows:

When you have got both fractions in the same denominator, you subtract them.

Solution 5:

Our problem was: 6 1 /2 − 3 1 /4 = ?

In order to solve this problem, you can subtract the whole numbers separately: 6 − 3 = 3

Then combine the two results: 3 + 1 /4 = 3 1 /4

Solution 6:

Our problem was: Estimate the product of: 14.9 × 10.2

In order to solve the problem, you simply need to round each number up or down and then multiply.

10.2 is rounded down to 10

After rounding, perform the multiplication:

Solution 7:

Percentages are expressed as the number over one hundred.

Solution 8:

Remember to line all of the decimals up in a column.

Our problem was: 3.75 + .004 + .179 = ?

Line up the decimal points as shown when you add up.

Solution 9:

The number 0.0602 is the least.

If you have difficulties with this type of exercise, remove the decimals to see the answer more clearly.

Solution 10:

The shirt was normally $10, but was on sale at a 10% discount.

Sales Price = Original Price – (Original price × Discount %)

Get More Accuplacer Arithmetic Help

Our free online practice test provides in-depth practice with all of the arithmetic skills on the actual exam.

Arithmetic Skills & Question Types

Accuplacer arithmetic problems generally fall into six categories.

1) Basic Arithmetic Operations

Basic operations include multiplication, division, addition, and subtraction of whole numbers, fractions, and mixed numbers.

A problem in this skill set might look like problem 5 above.

2) Recognizing Equivalents

Recognizing equivalent fractions and decimals is also assessed on the Accuplacer arithmetic section.

These types of questions appear frequently on this part of the test.

3) Fractions, Decimals & Percentages

Be sure you know how to do basic operations with fractions and decimals.

Question 7 above is an example problem for this skill set.

You will also need to know how to do multiplication, division, addition, and subtraction of decimals and percentages.

See problem 8 above for an example.

4) Comparisons

You may be asked to make comparisons of decimals or fractions in order to find the lowest or highest number.

See problem 9 above for an example.

5) Practical Problems

Problem-solving is very common on the Next Generation Arithmetic Exam.

It includes problems on dealing with measurement, distribution, and discounts.

Problem solving questions are expressed in words, like a story or narrative, rather than as arithmetic equations.

In problem solving questions, you must work out what equation you need, and then use it to solve the problem.

See problem 10 above for an example.

Arithemetic Test Format

The Accuplacer arithmetic section includes question on:

  • Whole number operations
  • Fractions
  • Decimals
  • Percentages
  • Number comparisons and equivalents
  • Get Our Online Accuplacer Tests

Get Our Accuplacer Arithmetic Tests

You take our practice tests online.

You will your score and explanations after each question.

The arithmetic practice tests are in the same exact format as the actual exam.

Each practice test has 20 questions, covering the skills above.

Try the free sample: Accuplacer Online Test Sample

More Accuplacer Samples

If you feel confident with the skills covered on this page, then you can go ahead to the other math problems for algebra, geometry, and trigonometry.

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Note: Accuplacer is a registered trademark of the College Board, which is neither affiliated with nor endorses this website.


For positive bases $a$, you have the general rule $a^b = exp(bln(a)) = e^.$

This follows from the fact that exponentials and logarithms are inverses of each other, and that the logarithm has the property that $ln(x^r) = rln(x).$

In fact, this is formula can be taken as the definition of $a^b$ for $agt 0$ and arbitrary exponent $b$ (that is, not an integer, not a rational).

As to computing $e^<2.14ln(2.14)>$, there are reasonably good methods for approximating numbers like $ln(2.14)$, and numbers like $e^r$ (e.g., Taylor polynomials or other methods).

A decimal power can be seen as a fraction:

Of course you cannot write every number as a fraction, but you can at least approximate every number by a fraction.

I really like your question. So many students are content with learning (and so many instructors content with teaching) just the calculator key sequences that will give the correct answer. But to know math (and almost everything else in this world) you’ve got to get under the hood and “see” what’s actually going on.

Let’s start with your example, 2.14^2.14. When you look at the exponent, more than likely you intuitively get the feeling that one portion of the answer is due to the integer part, “2”, with the balance attributable to the decimal “0.14”. And you’re right.

So, let’s raise 2.14 to our integer power (which you can do by hand— though there’s nothing wrong with employing a calculator when you understand the manipulations you’re carrying out):
2.14 ^ 2 = (2.14 * 2.14) = 4.5796.

Actually, let’s back up a little and use our calculator to get the answer to our example 2.14 ^ 2.14 = 5.09431.

Now that we have ‘the answer’ and the portion attributable to the integer component of our exponent, let’s determine the increase contributed by our decimal component (5.09431/4.5796) = 1.112392. Ok, but other than the ratio, (5.09431/4.5796), just what is “1.112392”?

Fasten your seat belt— It is simply 2.14 ^ 0.14 power = 1.112392.
(Yes, use your calculator for this intermediate step)

So, 2.14 ^ 2.14 = (2.14 ^ 2 * 2.14 ^ 0.14) = (4.5796 * 1.112392) = 5.09431

Let’s try 5.27 ^ 4.34 = 1357.244436

Hope this is what you were looking for. Have fun! JE Magee

You use $exp(2.14 ln 2.14)$ or any base for logarithms you choose. But if you want pen and paper, you can help with the properties of exponents. $2.14^<2.14>=2.14^2cdot2.14^<.14>=2.14^2exp(.14(ln 2 + ln1.07))$ will converge more quickly, especially if you are willing to look up $ln 2$.

we can find $2.14 ^<2.14>$ using basic arithmetic operations +,-,/,*.

Use binomial theorem for rational number $n and $-1

note that in left hand side the power n is a fractional number but in the right hand side the powers are integers. that is, in the right hand side, each term can be calculated using basic operations +,-,*,/.

using binomial theorem two times (5 decimal places) and multiplying we get the answer

Newton's approximation for $r = sqrt$ gives the iteration $r_ = r_n - frac<^2-c><2r_n>$
$sqrt <2.14>approx 1.5 ightarrow 1.46 ightarrow 1.4628 ightarrow 1.462874 ext< (6sf)>$
Using that $10$ times gives $2.14 ightarrow 1.462874 ightarrow 1.209493 ightarrow 1.099769 ightarrow 1.048698 ightarrow 1.024059$
$ ightarrow 1.011958 ightarrow 1.005961 ightarrow 1.002976 ightarrow 1.001486 ightarrow 1.000743 ext< (6sf)>$
Thus $ln 2.14 = 2^ <10>ln 2.14^<2^<-10>> approx 2^ <10>ln 1.000743 approx 2^ <10> imes 0.000743 approx 0.7608 ext< (3sf)>$
$2.14^ <2.14>= e^ < 2.14 ln 2.14 >approx e^ < 2.14 imes 0.7608 >approx e^ <1.628> ext< (3sf)>$

The geometric series or binomial expansion gives the approximate
$2^ <-10>= (1000+24)^ <-1>approx 1/1000 - 24/1000^2 + 576/1000^3$
Thus $e^ <1.628>= (e^<1.628 imes 2^<-10>>)^<2^<10>> approx (e^<0.001590>)^<2^<10>> ext< (3sf)>$
$approx (1+0.001590+0.001590^2/2)^<2^<10>> approx 1.001591^<2^<10>> ext< (6sf)>$
Squaring $10$ times gives $1.001591 ightarrow 1.003185 ightarrow 1.006380 ightarrow 1.012801 ightarrow 1.025766 ightarrow 1.052196$
$ ightarrow 1.107116 ightarrow 1.225706 ightarrow 1.502355 ightarrow 2.257071 ightarrow 5.094369 approx 5.09 ext< (3sf)>$

which is $2.14^<2.14>$ to $3$ significant figures. I am lazy so I used a calculator for nine of the repetitions of square-root and squaring, but the above computation is clearly feasible by hand as only $O(n^3)$ operations are needed for $n$ bits of precision. It is amusing that so much work went in to produce only 3 decimal digits but I do not know any better way that can be easily extended to arbitrary precision.


Video Tutorial

Videos can also be accessed from our Full Stack Playlist 3 on YouTube.

Python math operators and PEMDAS order of operations | Python for Beginners (4:50)

Code Examples and Video Script

Welcome. Today's question: What are the rules for Math in Python?

I'm Paul, and many of us learn the rules of Math, first on paper, then in a calculator, followed by a spreadsheet, and all of these were much easier than in a programming language. At least, that's what I found.

So here I hope to make your journey less difficult than mine, by highlighting Python's seven, out of the box, basic operations.

At high school in California, we use PEMDAS to memorize the order of operations for math, and we'll see if Python conforms.

We will talk about two numerical data types: integers and floats, and see how we keep them straight with practice in Python.

Next we will cover another subject in Math: relational operators.

Step 1 - Basic Math Using Python Operators

In Project 3 (Python for Beginners) so far we installed python3 and now we'll dive into, what to me, is the most exciting part, hands on learning in Python.

Heading to the Terminal, let's review PEMDAS, which stands for parentheses, exponent, multiplication, division, addition and subtraction.

It details the order of operations, and also note, in Python we have three types of division, regular division, floor division and finding the remainder, using what's called modulo.

Step 2 - Python PEMDAS Order of Operation

Let's head to the the python3 Interpreter and cover the rest of this.

The first way to interact with Python, is like this, one line at a time. Second is a script, or text file, which is the focus of a future Project.

So the three greater-than symbols >>> are a Python signature, like the command line in Linux, meaning it's waiting for us.

$ python3 Python 3.4.2 (default, Oct 8 2014, 10:45:20) [GCC 4.9.1] on linux Type "help", "copyright", "credits" or "license" for more information. >>> #PEMDAS backwards .

The hash symbol # , or comment, too is similar, meaning the rest of the line is ignored.

Three dots . is a continuation prompt, meaning Python doesn't have enough to act on.

Python Subtraction

So let's start with the last letter of PEMDAS, subtraction, give it a simple 1 - 2 .

Python Addition

Next, A for addition. We could enter 1+1 like this, and it works.

Or 1 +1 like this, and it too works.

And it also works, but the preferred form is to put a space between each.

Step 3 - Integers & floats

Python division

Next, D, division, and here I should mention the data types: integers and floats. We've been playing with integers so far. Let's try a floating point number like 6.0 divided by the integer 2 .

Python will automatically Interpret the format for the resulting number based on how you input numbers. So input of any float results in a float.

Also, with division, in Python 3, at least, something like integer 6 divided by integer 2 , equals float 3.0 .

Floor division rounds down, so 7 floor, divided by 3 is two and a third ( 2.3333333333333335 ).

Python floor division

And 7 , floor divided by 3 is 2 , which ignores the remainder.

Python modulo division

To see the remainder only, we use the modulo, or percent sign % .

So 7 % 3 is 1 because 3 goes into 7 two times with one left over.

Modulo can help us determine if a number is positive or negative.

Python multiplication

Next, M, multiplication is pretty straightforward, so two integer 5 s equals 25 .

And one integer 5 and one float 5.0 equals float 25.0 .

Python exponent or powers

Next, E, for exponent, is entered with two stars ** , so 2 to the second power is 4 .

And to the third power is 8 .

What would be 9 to the power of .5 , or one half power?

Remember it (a number to the 1/2 power) is the square root.

Python parentheses

Next, P, for parentheses. So after following the rules of operations Python will work from left to right.

So we know 1 + 2 * 4 is going to give us a different answer than (1 + 2) * 4 . Right?

Step 4 - Practice with a Python Homework Problem

I'll ask you to do the last one for homework.

Do it by hand on paper before using Python, and there are a couple little tricks in there that will reinforce your understanding. Feel free to pause now, or rewind it.

And to leave the Python Interpreter type exit() followed by Enter .

Step 5 - Next: Python Relational Operators

You are welcome to join our journey to Data Science as we take one step at a time and build out our Full Stack.

  • Client : HTML, CSS, JavaScript
  • Software : Python Scientific Stack
  • Data : PostgreSQL, MySQL / MariaDB
  • OS : Linux (command line), Debian

And our next step is to talk about Relational Operators.


How to find in which number base the operation was done by looking at the corresponding operation in decimal system?

as the digit in $ a^1

Rational Expressions

A complex numbers are of the form , a+bi where a is called the real part and bi is called the imaginary part. This text will show you how to perform four basic operations (Addition, Subtraction, Multiplication and Division):

Addition:

For addition, add up the real parts and add up the imaginary parts.

Example: let the first number be 2 - 5i and the second be -3 + 8i. The sum is:

(2 - 5i ) + (- 3 + 8i ) = = ( 2 - 3 ) + ( -5 + 8 ) i = - 1 + 3 i

Subtraction:

Subtract the real parts and subtract the imaginary parts.

Example: let the first number be -3 + 7i and the second be 6 - 9i. The sum is:

(- 3 + 7i ) - (6 - 9i ) = = ( - 3 - 6 ) + ( 7 - ( -9 ) ) i = - 9 + 16 i

Multiplication

To multiply complex method use the FOIL method( First, Outside, Inside, and Last. )

Example: multiply 3 + 4i and 2 - 6i

Outside terms: 3 * (- 6i) = -18i

Inside terms: 4i * 2 = 8i

Last terms: 4i * (-6i) = -24 * i 2 = -24 (- 1) = 24

Now, combine everything together

Division

Multiply both the denominator and the numerator by the conjugate of the denominator


Go Math Grade 8 Chapter 1 Real Numbers Answer Key

Download HMH Go Math Grade 8 Answer Key Chapter 1 Real Numbers PDF for free. Learn more about the tricks and tips to practice maths with the help of the Go Math Grade 8 Chapter 1 Real Numbers Answer Key. Find a better way to make your learning simple.

Lesson 1: Rational and Irrational Numbers

Lesson 2: Sets of real Numbers

Lesson 3: Ordering Real Numbers

Mixed Review

Guided Practice – Rational and Irrational Numbers – Page No. 12

Write each fraction or mixed number as a decimal.

Write each decimal as a fraction or mixed number in simplest form

Explanation:
Let x = 0.(overline <26>)
Now, 100x = 26.(overline<26>)
100x – x = 26.(overline<26>) – 0.(overline <26>)
99x = 26
x = (frac<26><99>)

Explanation:
Let x = 0.(overline <325>)
Now, 1000x = 325.(overline<325>)
1000x – x = 325.(overline<325>) – 0.(overline <325>)
999x = 325
x = (frac<325><999>)

Solve each equation for x

Question 13.
x 2 = 144
± ______

Explanation:
x 2 = 144
Taking square roots on both the sides
√ x 2 = ± √ 144
x = ± 12

Explanation:
x 2 = (frac<25><289>)
Taking square roots on both the sides
√ x 2 = ± √(frac<25><289>)
x = ± (frac<5><17>)

Question 15.
x 3 = 216
______

Explanation:
x 3 = 216
Taking cube roots on both the sides
3 √ x 3 = 3 √216
x = 6

Approximate each irrational number to two decimal places without a calculator.

Question 16.
(sqrt < 5 >) ≈ ______

Explanation:
x = (sqrt < 5 >)
The 5 is in between 4 and 6
Take square root of each year
√4 < √5 < √6
2 < √5 < 3
√5 = 2.2
(2.2)² = 4.84
(2.25)² = 5.06
(2.5)³ = 5.29
A good estimate for √5 is 2.25

Question 17.
(sqrt < 3 >) ≈ ______

Explanation:
(sqrt < 3 >)
1 < 3 < 4
√1 < √3 < √4
1 < √3 < 2
√3 = 1.6
(1.65)² = 2.72
(1.7)² = 2.89
(1.75)² = 3.06
A good estimate for √3 is 1.75

Question 18.
(sqrt < 10 >) ≈ ______

Explanation:
(sqrt < 10 >)
9 < 10 < 16
√9 < √10 < √16
3 < √10 < 4
√10 = 3.1
(3.1)² = 9.61
(3.15)² = 9.92
(3.2)² = 10.24
A good estimate for √10 is 3.15

Question 19.
What is the difference between rational and irrational numbers?
Type below:
_____________

Rational number can be expressed as a ration of two integers such as 5/2
Irrational number cannot be expressed as a ratio of two integers such as √13

Explanation:
A rational number is a number that can be express as the ratio of two integers. A number that cannot be expressed that way is irrational.

1.1 Independent Practice – Rational and Irrational Numbers – Page No. 13

Question 20.
A (frac<7><16>)-inch-long bolt is used in a machine. What is the length of the bolt written as a decimal?
______ -inch-long

Explanation:
The length of the bolt is (frac<7><16>)-inch
Let, x = (frac<7><16>)
Multiplying by 125 on both nominator and denominator
x = (frac<7×125><16×125>) = (frac<875><2000>) =(frac<437.5><1000>) = 0.4375

Question 21.
The weight of an object on the moon is (frac<1><6>) its weight on Earth. Write (frac<1><6>) as a decimal.
______

Explanation:
The weight of the object on the moon is (frac<1><6>)
Let, x = (frac<1><6>)
Multiplying by 100 on both nominator and denominator
x = (frac<1×100><6×100>) = (frac<16.6><100>) =0.166

Question 22.
The distance to the nearest gas station is 2 (frac<4><5>) kilometers. What is this distance written as a decimal?
______

Explanation:
The distance of the nearest gas station is 2 (frac<4><5>)
Let, x = 2 (frac<4><5>)
Multiplying by 100 on both nominator and denominator
x = 2 (frac<4×100><5×100>) = (frac<80><100>) =0.8

Question 23.
A baseball pitcher has pitched 98 (frac<2><3>) innings. What is the number of innings written as a decimal?
______

Explanation:
A baseball pitcher has pitched 98 (frac<2><3>) innings.
98 (frac<2><3>) = 98 + 2/3
= (294/3) + (2/3)
296/3
98.6

Question 24.
A heartbeat takes 0.8 second. How many seconds is this written as a fraction?
(frac<□><□>)

Explanation:
A heartbeat takes 0.8 seconds.
0.8
There are 8 tenths.
8/10 = 4/5

Question 25.
There are 26.2 miles in a marathon. Write the number of miles using a fraction.
(frac<□><□>)

Explanation:
There are 26.2 miles in a marathon.
26.2 miles
262/10
131/5
26 1/5 miles

Question 26.
The average score on a biology test was 72.(ar<1>). Write the average score using a fraction.
(frac<□><□>)

Explanation:
The average score on a biology test was 72.(ar<1>).
72.(ar<1>)
Let x = 72.(ar<1>)
10x = 10(72.(ar<1>))
10x = 721.1
-x = -0.1
9x = 721
x = 721/9
x = 80 1/9

Question 27.
The metal in a penny is worth about 0.505 cent. How many cents is this written as a fraction?
(frac<□><□>)

Explanation:
The metal in a penny is worth about 0.505 cent.
0.505 cent
505 thousandths
505/1000
101/200 cents

Question 28.
Multistep An artist wants to frame a square painting with an area of 400 square inches. She wants to know the length of the wood trim that is needed to go around the painting.

a. If x is the length of one side of the painting, what equation can you set up to find the length of a side?
x 2 = ______

Explanation:
The area of a square is the square of its equal side, x
x² = 400

Question 28.
b. Solve the equation you wrote in part a. How many solutions does the equation have?
x = ± ______

Explanation:
Take the square root on both sides. Solve
x = ± 20

Question 28.
c. Do all of the solutions that you found in part b make sense in the context of the problem? Explain.
Type below:
_____________

Answer:
No. Both values of x do not make sense.

Explanation:
The length cannot be negative, hence negative value does not make sense.
No. Both values of x do not make sense.

Question 28.
d. What is the length of the wood trim needed to go around the painting?
P = ______ inches

Rational and Irrational Numbers – Page No. 14

Question 29.
Analyze Relationships To find (sqrt < 15 >), Beau found 3 2 = 9 and 4 2 = 16. He said that since 15 is between 9 and 16, (sqrt < 15 >) must be between 3 and 4. He thinks a good estimate for (sqrt < 15 >) is (frac < 3+4 > < 2 >) = 3.5. Is Beau’s estimate high, low, or correct? Explain.
_____________

Explanation:
15 is closer to 16
√15 is closer to √16
Beau’s estimate is low.
(3.8)² = 14.44
(3.85)² = 14.82
(3.9)² = 15.21
√15 is 3.85

Question 30.
Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95? How did you come up with your estimate?
x ≈ ______

Explanation:
3 √x = 95
x = 3 √95
64 < 95 < 125
Take the cube root of each number
3 √64 < 3 √95 < 3 √125
4 < 3 √95 < 5
3 √95 = 4.6
(4.5)³ = 91.125
(4.55)³ = 94.20
(4.6)³ = 97.336
3 √95 = 4.55

Question 31.
The volume of a sphere is 36π ft 3 . What is the radius of the sphere? Use the formula V = (frac < 4 > < 3 >)πr 3 to find your answer.

r = ______

Explanation:
V = 4/3 πr³
36π = 4/3 πr³
r³ = 36π/π . 3/4
r³ = 27
r = 3 √27
r = 3

FOCUS ON HIGHER ORDER THINKING

Question 32.
Draw Conclusions Can you find the cube root of a negative number? If so, is it positive or negative? Explain your reasoning.
_____________

Explanation:
Yes. The cube root of a negative number would be negative. Because the product of three negative signs is always negative.

Question 33.
Make a Conjecture Evaluate and compare the following expressions.
(sqrt < frac < 4 > < 25 >> ) and (frac < sqrt < 4 >> < sqrt < 25 >> ) (sqrt < frac < 16 > < 81 >> ) and (frac < sqrt < 16 >> < sqrt < 81 >> ) (sqrt < frac < 36 > < 49 >> ) and(frac < sqrt < 36 >> < sqrt < 49 >> )
Use your results to make a conjecture about a division rule for square roots. Since division is multiplication by the reciprocal, make a conjecture about a multiplication rule for square roots.
Expressions are: _____________

Answer:
Evaluating and comparing
√4/25 = 2/5
√16/81 = 4/9
√36/49 = 6/7
Conjecture about a division rule for square roots
√a/√b = √(a/b)
Conjecture about a multiplication rule for square roots
√a × √b

Question 34.
Persevere in Problem Solving
The difference between the solutions to the equation x 2 = a is 30. What is a? Show that your answer is correct.
_____

Explanation:
x 2 = a
x = ±√a
√a – (-√a) = 30
√a + √a = 30
2√a = 30
√a = 15
a = 225
x 2 = 225
x = ±225
x = ±15
15 – (-15) = 15 + 15 = 30

Guided Practice – Sets of real Numbers – Page No. 18

Write all names that apply to each number.

Question 1.
(frac<7><8>)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 2.
(sqrt < 36 >)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers

Question 3.
(sqrt < 24 >)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
e. Irrational Numbers

Question 4.
0.75
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 5.
0
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers

Question 6.
−(sqrt < 100 >)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers

Question 7.
5.(overline < 45 >)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 8.
−(frac<18><6>)
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integers

Tell whether the given statement is true or false. Explain your choice.

Question 9.
All whole numbers are rational numbers.
i. True
ii. False

Explanation:
All whole numbers are rational numbers.
Whole numbers are a subset of the set of rational numbers and can be written as ratio of the whole number to 1.

Question 10.
No irrational numbers are whole numbers.
i. True
ii. False

Explanation:
True. Whole numbers are ration numbers.

Identify the set of numbers that best describes each situation. Explain your choice.

Question 11.
the change in the value of an account when given to the nearest dollar
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Explanation:
The change can be a whole dollar amount and can be positive, negative or zero.

Question 12.
the markings on a standard ruler

Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
b. Rational Numbers

Explanation:
The ruler is marked every 1/16t inch.

ESSENTIAL QUESTION CHECK-IN

Question 13.
What are some ways to describe the relationships between sets of numbers?

Answer:
There are two ways that we have been using until now to describe the relationships between sets of numbers

  • Using a scheme or a diagram as the one on page 15.
  • Verbal description, for example, “All irrational numbers are real numbers.”

1.2 Independent Practice – Sets of real Numbers – Page No. 19

Write all names that apply to each number. Then place the numbers in the correct location on the Venn diagram.

Question 14.
(sqrt < 9 >)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers

Question 15.
257
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers

Question 16.
(sqrt < 50 >)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
e. Irrational Numbers

Question 17.
8 (frac<1><2>)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 18.
16.6
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers

Question 19.
(sqrt < 16 >)
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Answer:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers

Identify the set of numbers that best describes each situation. Explain your choice.

Question 20.
the height of an airplane as it descends to an airport runway
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Explanation:
Whole. The height of an airplane as it descents to an airport runway is a whole number greater than 0

Question 21.
the score with respect to par of several golfers: 2, – 3, 5, 0, – 1
Options:
a. Real Numbers
b. Rational Numbers
c. Integer Numbers
d. Whole Numbers
e. Irrational Numbers

Explanation:
Integers. The scores are counting numbers, their opposites, and zero.

Question 22.
Critique Reasoning Ronald states that the number (frac<1><11>) is not rational because, when converted into a decimal, it does not terminate. Nathaniel says it is rational because it is a fraction. Which boy is correct? Explain.
i. Ronald
ii. Nathaniel

Explanation:
Nathaniel is correct.
A fraction is a rational real number, even if it is not a terminating decimal.

Sets of real Numbers – Page No. 20

Question 23.
Critique Reasoning The circumference of a circular region is shown. What type of number best describes the diameter of the circle? Explain your answer.

Options:
a. Real Numbers
b. Rational Numbers
c. Irrational Numbers
d. Integers
e. Whole Numbers

Explanation:
Circumference of the circle
A = 2πr
π = 2πr
Diameter is twice the radius
2r = 1
Whole

Question 24.
Critical Thinking A number is not an integer. What type of number can it be?
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
b. Rational Numbers
e. Irrational Numbers

Question 25.
A grocery store has a shelf with half-gallon containers of milk. What type of number best represents the total number of gallons?
Options:
a. Real Numbers
b. Rational Numbers
c. Integers
d. Whole Numbers
e. Irrational Numbers

Answer:
b. Rational Numbers

FOCUS ON HIGHER ORDER THINKING

Question 26.
Explain the Error Katie said, “Negative numbers are integers.” What was her error?
Type below:
_______________

Answer:
Her error is that she stated that all negative numbers are integers. Some negative numbers are integers such as -4 but some are not such an -0.8

Question 27.
Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational? Explain.
Type below:
_______________

Explanation:
Not always.
If the calculator shows a terminating decimal, the number is rational but otherwise, it is not possible as you can only see a few digits.

Question 28.
Draw Conclusions The decimal 0.(ar<3>) represents (frac<1><3>). What type of number best describes 0.(ar<9>) , which is 3 × 0.(ar<3>)? Explain.
Type below:
_______________

Explanation:
let x = 0.9999999
10x = 9.99999999
10x = 9 + 0.999999999
10x = 9 + x
9x = 9
x=1.

Question 29.
Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form. Why is this?

Answer:
Because irrational numbers are nonrepeating, otherwise they could be represented as a fraction. Although a potential counter-example to this claim is that some irrational numbers can only be represented in decimal form, for example, 0.1234567891011121314151617…, 0.24681012141618202224…, 0.101101110111101111101111110… are all irrational numbers.

Guided Practice – Ordering Real Numbers – Page No. 24

Compare. Write <, >, or =.

Question 1.
(sqrt < 3 >) + 2 ________ (sqrt < 3 >) + 3

Explanation:
(sqrt < 3 >) is between 1 and 2
(sqrt < 3 >) + 2 is between 3 and 4
(sqrt < 3 >) + 3 is between 4 and 5
(sqrt < 3 >) + 2 < (sqrt < 3 >) + 3

Question 2.
(sqrt < 11 >) + 15 _______ (sqrt < 8 >) + 15

Answer:
(sqrt < 11 >) + 15 > (sqrt < 8 >) + 15

Explanation:
(sqrt < 11 >) is between 3 and 4
(sqrt < 8 >) is between 2 and 3
(sqrt < 11 >) + 15 is between 18 and 19
(sqrt < 8 >) + 15 is between 17 and 18
(sqrt < 11 >) + 15 > (sqrt < 8 >) + 15

Question 3.
(sqrt < 6 >) + 5 _______ 6 + (sqrt < 5 >)

Explanation:
(sqrt < 6 >) is between 2 and 3
(sqrt < 5 >) is between 2 and 3
(sqrt < 6 >) is between 7 and 8
(sqrt < 5 >) is between 8 and 9
(sqrt < 6 >) + 5 < 6 + (sqrt < 5 >)

Question 4.
(sqrt < 9 >) + 3 _______ 9 + (sqrt < 3 >)

Explanation:
(sqrt < 9 >) + 3
9 + (sqrt < 3 >)
(sqrt < 3 >) is between 1 and 2
(sqrt < 9 >) + 3 = 3 + 3 = 6
9 + (sqrt < 3 >) is between 10 and 11
(sqrt < 9 >) + 3 < 9 + (sqrt < 3 >)

Question 5.
(sqrt < 17 >) – 3 _______ -2 + (sqrt < 5 >)

Explanation:
(sqrt < 17 >) is between 4 and 5
(sqrt < 5 >) is between 2 and 3
(sqrt < 17 >) – 3 is between 1 and 2
-2 + (sqrt < 5 >) is between 0 and 1
(sqrt < 17 >) – 3 > -2 + (sqrt < 5 >)

Question 6.
10 – (sqrt < 8 >) _______ 12 – (sqrt < 2 >)

Explanation:
(sqrt < 8 >) is between 2 and 3
(sqrt < 2 >) is between 1 and 2
10 – (sqrt < 8 >) is between 8 and 7
12 – (sqrt < 2 >) is between 11 and 10
10 – (sqrt < 8 >) < 12 – (sqrt < 2 >)

Question 7.
(sqrt < 7 >) + 2 _______ (sqrt < 10 >) – 1

Explanation:
(sqrt < 7 >) is between 2 and 3
(sqrt < 10 >) is between 3 and 4
(sqrt < 7 >) + 2 is between 4 and 5
(sqrt < 10 >) – 1 is between 2 and 3
(sqrt < 7 >) + 2 > (sqrt < 10 >) – 1

Question 8.
(sqrt < 17 >) + 3 _______ 3 + (sqrt < 11 >)

Explanation:
(sqrt < 17 >) is between 4 and 5
(sqrt < 11 >) is between 3 and 4
(sqrt < 17 >) + 3 is between 7 and 8
3 + (sqrt < 11 >) is between 6 and 7
(sqrt < 17 >) + 3 > 3 + (sqrt < 11 >)

Question 9.
Order (sqrt < 3 >), 2 π, and 1.5 from least to greatest. Then graph them on the number line.
(sqrt < 3 >) is between _________ and _____________ , so (sqrt < 3 >) ≈ ____________.
π ≈ 3.14, so 2 π ≈ _______________.

From least to greatest, the numbers are ______________, _____________________ ,_________________.
Type below:
___________

Explanation:
(sqrt < 3 >) is between 1.7 and 1.75
π = 3.14 2 π = 6.28

1.5, (sqrt < 3 >), 2 π

Question 10.
Four people have found the perimeter of a forest using different methods. Their results are given in the table. Order their calculations from greatest to least.

Type below:
___________

Explanation:
(sqrt < 17 >) – 2
(sqrt < 17 >) is between 4 and 5
Since, 17 is closer to 16, the estimated value is 4.1
1+ π/2
1 + (3.14/2) = 2.57
12/5 = 2.4
2.5
(sqrt < 17 >) – 2, 1+ π/2, 2.5, 12/5

ESSENTIAL QUESTION CHECK-IN

Question 11.
Explain how to order a set of real numbers.
Type below:
___________

Answer:
Evaluate the given numbers and write in decimal form. Plot on number line and arrange the numbers accordingly.

Independent Practice – Ordering Real Numbers – Page No. 25

Order the numbers from least to greatest.

Question 12.
(sqrt < 7 >), 2, (frac < sqrt < 8 >> < 2 >)
Type below:
____________

Explanation:
(sqrt < 7 >), 2, (frac < sqrt < 8 >> < 2 >)
(sqrt < 7 >) is between 2 and 3
Since 7 is closer to 9, (2.65)² = 7.02, hence the estimated value is 2.65
(frac < sqrt < 8 >> < 2 >)
(sqrt < 8 >) is between 2 and 3
Since 8 is closer to 9, (2.85)² = 8.12, hence the estimated value is 2.85
2.85/2 = 1.43

(frac < sqrt < 8 >> < 2 >), 2, (sqrt < 7 >)

Question 13.
(sqrt < 10 >), π, 3.5
Type below:
____________

Explanation:
(sqrt < 10 >), π, 3.5
(sqrt < 10 >) is between 3 and 4
Since, 10 is closer to 9, (3.15)² = 9.92, hence the estimated value is 3.15
π = 3.14
3.5

π, (sqrt < 10 >), 3.5

Question 14.
(sqrt < 220 >), −10, (sqrt < 100 >), 11.5
Type below:
____________

Explanation:
(sqrt < 220 >), −10, (sqrt < 100 >), 11.5
196 < 220 < 225
√196 < √220 < √225
14 < √220 < 15
√220 = 14.5
√100 = 10

-10, √100, 11.5, √220

Question 15.
(sqrt < 8 >), −3.75, 3, (frac<9><4>)
Type below:
____________

Explanation:
(sqrt < 8 >), −3.75, 3, (frac<9><4>)
(sqrt < 8 >) is between 2 and 3
Since, 8 is closer to 9, (2.85)² = 8.12, hence the estimated value is 2.85
-3.75 = 3
9/4 = 2.25

−3.75, (frac<9><4>), (sqrt < 8 >)

Question 16.
Your sister is considering two different shapes for her garden. One is a square with side lengths of 3.5 meters, and the other is a circle with a diameter of 4 meters.
a. Find the area of the square.
_______ m 2

Explanation:
Area of the square = x²
Area = (3.5)² = 12.25

Question 16.
b. Find the area of the circle.
_______ m 2

Explanation:
Area of the circle = πr² where r = d/2 = 4/2 = 2
Area = π(2)² = 12.56

Question 16.
c. Compare your answers from parts a and b. Which garden would give your sister the most space to plant?
___________

Answer:
12.25 < 12.56
The circle will give more space

Question 17.
Winnie measured the length of her father’s ranch four times and got four different distances. Her measurements are shown in the table.
a. To estimate the actual length, Winnie first approximated each distance to the nearest hundredth. Then she averaged the four numbers. Using a calculator, find Winnie’s estimate.

______

Explanation:
(sqrt < 60 >) = 7.75
58/8 = 7.25
7.3333
7 3/5 = 7.60
Average = (7.75 + 7.25 + 7.33 + 7.60)/4 = 7.4815

Question 17.
b. Winnie’s father estimated the distance across his ranch to be (sqrt < 56 >) km. How does this distance compare to Winnie’s estimate?
____________

Answer:
They are nearly identical

Explanation:
(sqrt < 56 >) = 7.4833
They are nearly identical

Give an example of each type of number.

Question 18.
a real number between (sqrt < 13 >) and (sqrt < 14 >)
Type below:
____________

Answer:
A real number between (sqrt < 13 >) and (sqrt < 14 >)
Example: 3.7

Explanation:
(sqrt < 13 >) = 3.61
(sqrt < 13 >) = 3.74
A real number between (sqrt < 13 >) and (sqrt < 14 >)
Example: 3.7

Question 19.
an irrational number between 5 and 7
Type below:
____________

Answer:
An irrational number between 5 and 7
Example: (sqrt < 29 >)

Explanation:
5² = 25 and 7² = 49
An irrational number between 5 and 7
Example: (sqrt < 29 >)

Ordering Real Numbers – Page No. 26

Question 20.
A teacher asks his students to write the numbers shown in order from least to greatest. Paul thinks the numbers are already in order. Sandra thinks the order should be reversed. Who is right?

_____________

Answer:
Neither are correct

Explanation:
(sqrt < 115 >), 115/11, 10.5624
(sqrt < 115 >) is between 10 and 11
Since, 115 is closer to 121, (10.7)² = 114.5, hence the estimated value is 10.7
115/11 = 10.4545
10.5624
Neither are correct

Question 21.
Math History
There is a famous irrational number called Euler’s number, symbolized with an e. Like π, its decimal form never ends or repeats. The first few digits of e are 2.7182818284.
a. Between which two square roots of integers could you find this number?
Type below:
_____________

Answer:
The square of e lies between 7 and 8
2.718281828
(2.72)² = 7.3984
Hence, it lies between (sqrt < 7 >) = 2.65 and (sqrt < 8 >) = 2.82

Question 21.
b. Between which two square roots of integers can you find π?
Type below:
_____________

Answer:
3.142
(3.14)² = 9.8596
Hence. it lies between (sqrt < 9 >) = 3 and (sqrt < 10 >) = 3.16

FOCUS ON HIGHER ORDER THINKING

Question 22.
Analyze Relationships
There are several approximations used for π, including 3.14 and (frac<22><7>). π is approximately 3.14159265358979 . . .
a. Label π and the two approximations on the number line.

Type below:
_____________

Answer:

Question 22.
b. Which of the two approximations is a better estimate for π? Explain.
Type below:
_____________

Answer:
As we can see from the number line, 22/7 is closer to π, so we can conclude that 22/7 is a better estimation for π.

Question 22.
c. Find a whole number x so that the ratio (frac<113>) is a better estimate for π than the two given approximations.
Type below:
_____________

Answer:
355/113 is a better estimation for π, because 355/113 = 3.14159292035 = 3.14159265358979 = π

Question 23.
Communicate Mathematical Ideas
What is the fewest number of distinct points that must be graphed on a number line, in order to represent natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers? Explain.
_______ points

Explanation:
There need to be plotting of at least 2 points because a rational number can never be equal to an irrational number. So let’s say 5 points are the same among six but the 6th will be different as there both rational numbers and irrational numbers included.

Question 24.
Critique Reasoning
Jill says that 12.(ar<6>) is less than 12.63. Explain her error.
Type below:
_____________

1.1 Rational and Irrational Numbers – Model Quiz – Page No. 27

Write each fraction as a decimal or each decimal as a fraction.

Explanation:
1.(overline < 27>)
x = 1.(overline < 27>)
100x = 100(1.(overline < 27>))
100x = 127((overline < 27>))
x = .(overline < 27>)
99x = 127
x = 127/99
x = 1 28/99

Explanation:
1 (frac<7><8>)
1 + 7/8
8/8 + 7/8
15/8 = 1.875

Solve each equation for x.

Question 4.
x 2 = 81
± ______

Explanation:
x 2 = 81
x = ± 81
x = ± 9

Question 5.
x 3 = 343
______

Explanation:
x 3 = 343
x = 7

Question 7.
A square patio has an area of 200 square feet. How long is each side of the patio to the nearest 0.05?
______ feet

Explanation:
The area of a square is found by multiplying the side of the square by itself. Therefore, to find the side of the square, we have to take the square root of the area.
Let’s denote with A the area of the patio and with s each side of the square.
We have:
A = 200
A = s.s
s = (sqrt < A >) = (sqrt < 200 >)
Following the steps as in “Explore Activity” on page 9, we can make an estimation for the irrational number:
196 < 200 < 225
(sqrt < 196 >) < (sqrt < 200 >) < (sqrt < 225 >)
14 < (sqrt < 200 >) < 15
We see that 200 is much closer to 196 than to 225, therefore the square root of it should be between 14 and 14.5. To make a better estimation, we pick some numbers between 14 and 14.5 and calculate their squares:
(14.1)² = 198.81
(14.2)² = 201.64
14.1 < (sqrt < 200 >) < 14.2
(sqrt < 200 >) = 14.15
We see that 200 is much closer to 14.1 than to 14.2, therefore the square root of it should be between 14.1 and 14.15. If we round to the nearest 0.05, we have:
s = 14.15

1.2 Sets of Real Numbers

Write all names that apply to each number.

Question 8.
(frac < 121 > < sqrt < 121 >>)
Type below:
___________

Answer:
Rational, whole, integer, real numbers

Question 9.
(frac<π><2>)
Type below:
___________

Answer:
Irrational, real numbers

Question 10.
Tell whether the statement “All integers are rational numbers” is true or false. Explain your choice.
___________

Explanation:
“All integers are rational numbers” is true, because every integer can be expressed as a fraction with a denominator equal to 1. The set of integer A a subset of rational numbers.

1.3 Ordering Real Numbers

Compare. Write <, >, or =.

Question 11.
(sqrt < 8 >) + 3 _______ 8 + (sqrt < 3 >)

Explanation:
4 < 8 < 9
(sqrt < 4 >) < (sqrt < 8 >) < (sqrt < 9 >)
2 < (sqrt < 8 >) < 3
1 < 3 < 4
(sqrt < 1 >) < (sqrt < 3 >) < (sqrt < 4 >)
1 < (sqrt < 3 >) < 2
(sqrt < 8 >) + 3 is between 5 and 6
8 + (sqrt < 3 >) is between 9 and 10
(sqrt < 8 >) + 3 < 8 + (sqrt < 3 >)

Question 12.
(sqrt < 5 >) + 11 _______ 5 + (sqrt < 11 >)

Explanation:
(sqrt < 5 >) lies in between 2 and 3
(sqrt < 11 >) lies in between 3 and 4
(sqrt < 5 >) + 11 lies in between 13 and 14
5 + (sqrt < 11 >) lies in between 8 and 9
(sqrt < 5 >) + 11 > 5 + (sqrt < 11 >)

Order the numbers from least to greatest.

Question 13.
(sqrt < 99 >), π 2 , 9.(ar < 8 >)
Type below:
_______________

Explanation:
(sqrt < 99 >), π 2 , 9.(ar < 8 >)
99 lies between 9² and 10²
99 is closer to 100, hence (sqrt < 99 >) is closer to 10
(9.9)² = 98.01
(9.95)² = 99.0025
(10)² = 100
(sqrt < 99 >) = 9.95
π² = 9.86
9.88888 = 9.89

π 2 , 9.(ar < 8 >), (sqrt < 99 >)

Question 14.
(sqrt < frac < 1 > < 25 >> ), (frac<1><4>), 0.(ar < 2 >)
Type below:
____________

Explanation:
(sqrt < frac < 1 > < 25 >> ), (frac<1><4>), 0.(ar < 2 >)
(sqrt < frac < 1 > < 25 >> ) = 1/5 = 0.2
1/4 = 0.25
0.(ar < 2 >) = 0.222 = 0.22

(sqrt < frac < 1 > < 25 >> ), 0.(ar < 2 >), (frac<1><4>)

Essential Question

Question 15.
How are real numbers used to describe real-world situations?
Type below:
_______________

Answer:
In real-world situations, we use real numbers to count or make measurements. They can be seen as a convention for us to quantify things around, for example, the distance, the temperature, the height, etc.

Selected Response – Mixed Review – Page No. 28

Question 1.
The square root of a number is 9. What is the other square root?
Options:
a. -9
b. -3
c. 3
d. 81

Explanation:
We know that every positive number has two square roots, one positive and one negative. We are given the principal square root (9), so the other square root would be its negative (-9). To prove that, we square both numbers and we compare the results:
9 • 9 = 81
(-9). (-9)= 81

Question 2.
A square acre of land is 4,840 square yards. Between which two integers is the length of one side?
Options:
a. between 24 and 25 yards
b. between 69 and 70 yards
c. between 242 and 243 yards
d. between 695 and 696 yards

Answer:
b. between 69 and 70 yards

Explanation:
The area of a square is found by multiplying the side of the square by itself. Therefore, to Bud the side of the square, we have to take the square root of the area.
Let’s denote with A the area of the land and with each side of the square. We have:
A = 4840
A = s . s
A = s²
s = √A = √4840
Following the steps as in °Explore Activity on page 9, we can make an estimation for the irrational number:
4761 < 4840 < 4900
(sqrt < 4761 >) < (sqrt < 4840 >) < (sqrt < 4900 >)
69 < (sqrt < 4840 >) < 70
Each side of the land is between 69 and 70 yards.

Question 3.
Which of the following is an integer but not a whole number?
Options:
a. -9.6
b. -4
c. 0
d. 3.7

Explanation:
Whole numbers are not negative
-4 is an integer but not a whole number

Question 4.
Which statement is false?
Options:
a. No integers are irrational numbers.
b. All whole numbers are integers.
c. No real numbers are irrational numbers.
d. All integers greater than 0 are whole numbers.

Answer:
c. No real numbers are irrational numbers.

Explanation:
Rational and irrational numbers are real numbers.

Question 5.
Which set of numbers best describes the displayed weights on a digital scale that shows each weight to the nearest half pound?
Options:
a. whole numbers
b. rational numbers
c. real numbers
d. integers

Answer:
b. rational numbers

Explanation:
The scale weighs nearest to 1/2 pound.

Question 6.
Which of the following is not true?
Options:
a. π 2 < 2π + 4
b. 3π > 9
c. (sqrt < 27 >) + 3 > 172
d. 5 – (sqrt < 24 >) < 1

Explanation:
a. π 2 < 2π + 4
(3.14)² < 2(3.14) + 4
9.86 < 10.28
True
b. 3π > 9
9.42 > 9
True
c. (sqrt < 27 >) + 3 > 172
5.2 + 3 > 8.5
8.2 > 8.5
False
d. 5 – (sqrt < 24 >) < 1
5 – 4.90 < 1
0.1 < 1
True

Question 7.
Which number is between (sqrt < 21 >) and (frac<3π><2>) ?
Options:
a. (frac<14><3>)
b. 2 (sqrt < 6 >)
c. 5
d. π + 1

Explanation:
a. (sqrt < 21 >) and (frac<3π><2>)
(sqrt < 21 >) = 4.58
(frac<3π><2>) = 4.71
14/3 = 4.67
b. 2(sqrt < 6 >) = 4.90
c. 5
d. π + 1 = 3.14 + 1 = 4.14

Question 8.
What number is shown on the graph?

Options:
a. π+3
b. (sqrt < 4 >) + 2.5
c. (sqrt < 20 >) + 2
d. 6.(overline < 14 >)

Explanation:
6.48
a. π+3 = 3.14 + 3 = 6.14
b. (sqrt < 4 >) + 2.5 = 2 + 2.5 = 4.5
c. (sqrt < 20 >) + 2 = 4.47 + 2 = 6.47
d. 6.(overline < 14 >) = 6.1414

Question 9.
Which is in order from least to greatest?
Options:
a. 3.3, (frac<10><3>), π, (frac<11><4>)
b. (frac<10><3>), 3.3, (frac<11><4>), π
c. π, (frac<10><3>), (frac<11><4>), 3.3
d. (frac<11><4>), π, 3.3, (frac<10><3>)

Explanation:
10/3 = 3.3333333
11/4 = 2.75

Question 10.
The volume of a cube is given by V = x 3 , where x is the length of an edge of the cube. The area of a square is given by A = x 2 , where x is the length of a side of the square. A given cube has a volume of 1728 cubic inches.
a. Find the length of an edge.
______ inches

Explanation:
V = x 3
A = x 2
1728 = x 3
x = 12
The length of an edge = 12 in

Question 10.
b. Find the area of one side of the cube.
______ in 2

Explanation:
A = (12)² = 144
Area of the side of the cube = 144 in 2

Question 10.
c. Find the surface area of the cube.
______ in 2

Explanation:
SA = 6 (12)² = 864
The surface area of the cube = 864 in 2

Question 10.
d. What is the surface area in square feet?
______ ft 2

Explanation:
SA = 864/144 = 6
The surface area of the cube = 6 ft 2

Conclusion:

If you are looking for the Grade 8 maths notes and textbook, then refer to Go Math Grade 8 Answer Key Chapter 1 Real Numbers. It is the best source for students to learn maths and get a good score in the exam.


Parent Primer: Math

Mathematics is an extremely important subject that provides learners with the tools to: think critically, use inductive and deductive reasoning, problem solve, apply logic, make connections, select the right tools, represent, interpret and analyze information. Children learn math by investigating, solving problems, gathering information to organize, graphing and analyzing, explaining, proving and representing their thinking.

Be positive and encourage risk taking in mathematics. Make mathematics a daily, authentic component in your child’s life. Encourage problem solving and persistence and ask your child to explain their thinking when working through math problems. And most importantly, don't worry! Just by refreshing yourself with one to three of the concepts below, you'll be understanding your child's math's lessons that much more.

K?8 Mathematics is typically structured around the following topics:

  • Math Terms
  • Geometry
  • Number and Operations
  • Patterns and Algebra
  • Fractions & Decimals

These larger topics are then divided into sub topics such as fractions, place value, properties of two and
three dimensional shapes and figures. The parent primer in mathematics is intended to help parents
understand and support the concepts taught in mathematics. The parent primer is organized around the
key topics and provides information about what children need to know.

Symbol Definition
+ Plus add and
- Minus less subtract take away
* Times multiply
/ or ÷ Divided by
= Equals is equivalent to
Does not equal not equal to
Approximately about equal to roughly
< Less than
> Greater than
Less than OR equal to
Greater than OR equal to
% Percent per hundred
° Degree/s
Square root of
! Factorial
|| Absolute value
Pi
Infinity

  • Absolute Value - Technically, a number's distance from zero on a number line. An easier way to think of it is the positive value of any number. So the absolute value of -5 is 5. ( |-5|=5.)
  • Cardinal Numbers - A fancy name for numbers such as 4, 67, etc.
  • Decimal - A fraction whose denominator is a power of ten (10, 100, 1000, etc.) and written by putting the numerator of that fraction to the right of a decimal point. So, 22/100 is 0.22 .056 is equivalent to 56/1000.
  • Denominator - The bottom number in a fraction. In 1/2, 2 is the denominator.
  • Factor - One number is a factor of another number if it can divide into it exactly, i.e. 3 is a factor of 9, or 5 and 11 are factors of 55.
  • Fraction - A number that represents some part of a whole and written a/b. So 1/2 means 1 of 2 parts, or one-half of something.
  • Improper fraction - A fraction bigger than 1, such as 3/2 or 9/4.
  • Integers - All the whole numbers plus all their negative counterparts (-1,-2,-3). Does not include fractions or mixed numbers.
  • Mixed Number - A number that contains both an integer and a fraction, such as 2 1/2 or 3 1/3.
  • Multiple - The product of a given whole number and another whole number. On times tables, each number will list its multiples beneath it - i.e. the multiples of 6 are 12, 18, 24, 30, 36, 42, etc.
  • Negative Numbers - Any number less than zero.
  • Numerals - A fancy word for numbers.
  • Numerator - The top number in a fraction. In 1/2, 1 is the numerator.
  • Ordinal Numbers - A number that indicates order or position, such as 1st, 3rd, 27th, etc.

  • Percent - Means per hundred and shows the ratio of a number to 100.
  • Place Value - Where a single number is placed in a larger figure tells you it's value: whether that number stands for the number of tens, hundreds, thousands, etc.

So, in the number 3,245,093.2.

You can see there are 3 millions, 2 hundred thousands, and so on. The 5 is in the thousands place.

  • Prime Number - Any number that can only be divided by 1 and itself.
  • Whole Numbers - All the positive numbers and zero - the counting numbers (0,1,2,3. ). Does not include fractions or mixed numbers.

Related to Operations

  • Dividend - The number to be divided in a division operation. In the problem, 60ུ, 60 is the dividend.
  • Divisor - The number that's doing the dividing in a division problem. In the operation, 60ུ, 4 is the divisor.
  • Equation - A mathematical statement that says two amounts or expressions have the same value Any number sentence with an 'equals' sign (2+3=6-1).
  • Exponent - In the number 4³, the 3 on top is called an exponent. It indicates that 4 is being raised to the power of 3, or multiplied by itself 3 times.
  • Expression - Each part of any number sentence that combines numbers and operation signs (+,-,*, /) is an expression a number sentence without an equals sign.
  • Factorial - Any number factorial (written 3! Or 15!) means that you multiply that number by all the whole numbers less than that number. So 6! means 6*5*4*3*2*1.
  • Inequality - a mathematical statement that says that two quantities are not equal. A number sentence with >,<, or .
  • Inverse - Related but opposite operations or numbers are inverses of one another. Addition and subtraction are inverse operations. 3 is the inverse of 1/3.
  • Operation - Adding, subtracting, multiplying, or dividing two or more numbers.
  • Parentheses - Used to show which operation to perform first. For example, in (2+3)*4, you would first add 2+3 and then multiply that sum by 4.
  • Power - If you multiply a number by itself, the number of times you multiply it is called a power. For example, 4*4*4 is 4 raised to the third power. It is written 4³.
  • Product - The result of multiplying two numbers together. Instead of asking, "What is 3 times 4?" The question might be phrased, "What is the product of 3 and 4?" (The answer is 12 for both.)
  • Quotient - The number that results from a division problem, not including the remainder. In the problem, 60ུ, the quotient is 15.
  • Remainder - A number 'left over' from a long division problem. In the problem, 61ུ, the answer is 15 with a remainder of 1, or 1r.

Related to Graphs, Charts, & Statistics

  • Data - A set of information. For example, all the answers collected for a survey would be the data for that survey.
  • Impossible Number - A number that while it's the correct result of an average, it has an impossible real-world value, i.e. the average family has 2.5 children, but you can't have .5 of a person.
  • Mean - The average of a group of two or more amounts. To get the mean or average, add the numbers, then divide the result by the number of amounts you summed. Simply, the mean of 3, 15, and 21 is 3+15+21 divided by 3, which is 13.
  • Median - The number in the exact middle of a set of numbers. So, in the set: 1,3,4,6,13,15,21 - 6 is the median of the set.
  • Pictograph - A graph displaying information through pictures.

  • Area - The amount in square units contained in a two-dimensional shape or surface.
  • Circumference - The distance around a circle.
  • Closed Curve Shapes - A plane shape made with curved lines, like a circle or oval.
  • Congruent figures - Figures that are the same shape and the same size. They can be rotated or flipped and still be congruent.
  • Diameter - Any line that passes through the center of a circle that joins two points on the circle Twice the radius.
  • Geometry - Math related to shapes and figures such as area, size, volume, and length.
  • Line - A straight set of points that extends infinitely in both directions.
  • Line Segment - A part of a line with a beginning point and an end point.
  • Perimeter - The distance around a polygon the sum of the lengths of the sides of a two-dimensional figure.
  • Plane shape - A two-dimensional or flat shape.
  • Polygon - A plane shape with 3 or more straight sides (line segments), like a triangle, hexagon, or rectangle.
  • Radius - The distance from the center of the circle to any point on it Half the diameter.
  • Ray - A part of a line that has one end point and continues infinitely in one direction.
  • Solid Shape - A three-dimensional shape, such as a cube, sphere, cone, or pyramid.
  • Tessellations - Using a single shape repeatedly to make a larger pattern or mosaic.
  • understand that polygons are two?dimensional shapes with straight sides that intersect at vertices and the number of vertices is equal to the number of sides
  • name the regular polygons (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, hendecagon, dodecagon)
  • use properties (vertices, sides, shapes )to classify polygons. For example, a triangle has 3 vertices and 3 sides
  • classify triangle types (scalene, isosceles, equilateral and angle types (right, acute and obtuse)
  • understand the properties of and irregular shapes and solids 2 and 3?Dimensional Shapes and Objects
  • sort and classify 2?dimensional shapes (triangles, quadrilaterals, pentagons…) and 3?dimensional solids (cubes, cylinders, cones, spheres, pyramids… using attributes like edges, vertices, and sides
  • construct nets and determine the 3-dimensional object based on the net
  • understand that reflections transform the object by flipping it across a line making it look like a mirror object
  • identify reflective or rotational symmetry (reflective symmetry looks the same when reflected on either side of the line and rotational symmetry looks the same when rotated, a heart would have reflective symmetry, a square would have both reflective and rotational symmetry
  • 2 shapes are considered to be congruent if one of the shapes can be transformed into the other shape by flipping, sliding or turning it
  • understand the concept of slides, flips and turns
  • tell time with digital and analog clocks and solve problems involving lapses of time, conversions of minutes to hours and days and time as it relates to the future
  • measure in both standard (inches, feet, gallons, miles) and non-standard units (paces, finger widths)
  • use customary units of measure (feet, inches, gallons, pints, miles, pounds…)to solve a variety of measurement problems and compute measurement conversions
  • calculate area, mass, perimeter, volume and capacity
  • understand that a line extends forever in both directions, a line segment has two defined endpoints and a ray has one defined endpoint and the other end extends forever
  • use the Pythagorean relationship

An angle is the distance between two rays or two line segments. Three kinds of angles you will encounter are:

Acute Angle Right Angle Obtuse Angle
A positive angle measuring less than 90 degrees An angle of exactly 90 degrees. Indicated in pictures as a square An angle whose measure is greater than 90 degrees

Three general types of triangles:

Scalene Triangle Isoceles Triangle Equilateral Triangle
A triangle with no sides of the same length A triangle with two sides of the same length A triangle where all the sides (and angles) are the same length

And one special one, a Right Triangle, which is any triangle containing a right angle:

Parallelogram Rhombus Trapezoid
A 4-sided plane shape with 2 pairs of parallel sides of equal length A parallelogram with 4 sides of the same length A 4-sided plane shape with 1 pair of parallel sides

Area of a Triangle, Parrallelogram, Square and Rectangle

You can always form a rectangle, square or a parallelogram from two triangles which is why a triangle has an area of half of the rectangle, square or parallelogram. This makes remembering the area of a triangle easy!

The area of a rectangle: A = base x height

The area of a triangle: A = 1/2 (base x height)

Numbers refer to counting, comparing and ordering of whole numbers, decimals, integers and fractions. Operations refer to computations of numbers which include addition, subtraction, multiplication and division.

  • count forwards and backwards using a variety of starting points
  • skip count by 2’s, 3’s, 4’s, 5’s starting at different numbers, sometimes using a hundred’s chart
  • answer questions like: what is 2 more than, 3 less than, 10 more than….
  • understand that cardinal numbers indicate how many are in a set (9 cats, 12 books) and ordinal numbers refer to position (fifth, seventh, ninth…)
  • add, subtract, multiply and divide (the four operations) and solve word problems involving the four operations with whole numbers, integers, fractions and decimals
  • add, subtract, multiply ad divide when regrouping is involved (carrying, borrowing from the next column)
  • use strategies to discover prime (numbers that can only be divided by one and itself: 3, 7, 11…) and composite numbers. (numbers that can be divided by more than one and itself: 2, 4,6, 8, 9…)
  • find multiples and lowest common multiples of a pair of numbers
  • find factors and greatest common factors
  • understand the value in placement of numbers (2348.76: 2 is in the thousands place, 2 is in the hundred’s place, 4 in in the ten’s place, 8 is in the one’s place, 7 is in the tenth’s place and 6 in the hundredth’s place

Students begin working with patterns in kindergarten. As they move into the fifth and sixth grades through to high school, algebra begins to replace patterns. Patterns are the foundation of algebra.

  • create, extend, describe, and compare growing, shrinking, recursive and repeating patterns using various attributes: shape, color, number, sounds etc.
  • determine pattern rules
  • use T-charts/function tables to organize information
  • predict and prove what comes next in a pattern

For Example: Jen learned that for each year old a person is, a dog is about seven years old. Determine the rule to write the people years for any
number of years for a dog.

People Dogs
1 7
2 14
3 21
4 28
5 35

A recursive pattern provides the start item or number of a pattern shows how the pattern continues.

For example, a recursive rule for the pattern 5, 8, 11, 14, … is start with 5 and add 3. A shrinking or growing pattern shrinks or grows, the start item(s) or number(s) show how the pattern continues:

(95, 85, 75, 65 ___ ___ ___ or A BB AA BBB AAA BBBB _____ _____ _____ )

A repeating pattern repeats: * * X X X * * X X X ** X X X

Functions and Relationships

A function is a relationship which will often refer to the output when the input is known. For example, if the function is to triple the number, the input would be the number and the output would be what the tripled number would be, the relationship is what you do to the input number to get the output number. Function is finding the rule.

Variables and Equations

?work with variables which can be objects, shapes or letters that represent unknown values or quantities (5 + x = 12)

?solve problems by working backwards, guessing and checking or by maintaining a balance

?solve inequalities by finding the values and whole number solutions of the inequality (15 — X < 8 how many whole number solutions are there?)

Data and probability concepts begin in kindergarten. Data concepts involve collecting and analyzing data and constructing graphs and charts to display the information. Probability is the likelihood for events to happen, it is determining if an event is impossible or possible and
measures of likelihood are expressed qualitatively or quantitatively.

  • use tally marks, line plots, charts and lists to organize information
  • read, construct and analyze pictographs and bar graphs
  • construct and analyze stem and leaf plots
  • work with the mean, median and the mode
  • use the algorithm for finding the mean
  • describe the likelihood of events happening using likely, unlikely, certain or possible or impossible
  • identify the possible outcomes of probability experiments
  • understand the concepts of more likely and less likely with spinners and number cubes
  • use a tree diagram to analyze probability experiments

First things first. Fractions are a lot easier to work with if you first reduce them to their simplest form, which means that 1 is the only number that can divide evenly (meaning without a remainder) into the numeration and denominator.

For example: 330/550
We can see right away that both numbers can be divided by 10, reducing the fraction to
33/55
Both top and bottom can also be divided by 11, giving us the equivalent fraction
3/5
Neither of these numbers has any common factors, so 3/5 is the simplest form.

Another way to do this would be to find the greatest common multiple of 330 and 550, which is 110. This allows you to reduce the fraction in one step.

To add or subtract fractions with the same denominator, simply add or subtract the numerators of those fractions.

If the fractions have different denominators, you must convert them to fractions with the same denominator. To do this, find the smallest common multiple of both numbers and convert both fractions to ones with that multiple as the denominator:

3/5 + 3/4 The smallest multiple of 5 and 4 is 20

12/20 + 15/20 = 37/20 or 1 17/20

To multiply fractions, simply multiply the numerators and denominators:

To divide by a fraction, such as:

Simply flip the fraction (which is the divisor) and multiply by that number like so:

Another way to think of it is, "How many ¼'s would fit into 4?" Or how many quarters make up 4 dollars. Any way you look at it, the answer is 16.

To convert a fraction into a decimal, simply divide the denominator into the numerator. So ¼ = 1 ÷ 4 = 0.25

Adding and subtracting decimals is no different than with whole numbers. You just line up the decimal points and perform the operation. It only gets tricky when you multiply or divide.

When you multiply decimals, line up the numbers on the right (not with the decimal points) and multiply as you normally would:

then place the decimal point in the product by adding up how many numbers in the original are to the right of the decimal point.

23.21 (2 numbers two the right of the decimal or 2 decimal places)
* 4.2 (1 number two the right of the decimal or 1 decimal place)
---------
97.482 (3 numbers to the right of the decimal or 3 decimal places)

When you divide decimals.
_____
2.4 /48.96

If the divisor contains a decimal, get rid of it! Move the decimal place over an equal amount in both numbers until the divisor is a whole number (multiply both the divisor and the dividend by whatever power of ten is necessary to get rid of the decimal).
_____
24 / 489.6

Then perform the division as you normally would. The decimal point moves up into the answer at exactly the same point as it is in the dividend:


Watch the video: Multiplying Decimals Concrete (November 2021).