4 - Number Sense and Basic Algebra

Question 1

Define:

decimal numbers

Answer 1

Represent parts of a whole number, including a whole number and a part.

Examples: 0.74, 4.7 and 100.001

Question 2

Explain:

What is the place value to the left of the decimal point?

Answer 2

The place values to the left of the decimal point are represented as powers of ten:

• ones
• tens
• hundreds

These increase as you move left from the decimal point.

Question 3

Identify:

The place values starting at the decimal point.

Answer 3

• Tenths
• Hundredths
• Thousandths
• Ten-thousandths

Question 4

Name these decimal numbers:

• 2.3
• 15.78
• 0.009
• 30.0475

Answer 4

• 2.3 → Two and three tenths
• 15.78 → Fifteen and seventy-eight hundredths
• 0.009 → Nine thousandths
• 30.0475 → Thirty and four hundred seventy-five ten-thousandths

Question 5

Explain:

How do you compare decimal numbers?

Answer 5

1. Write the decimals with the decimal points lined up.
2. Compare from the leftmost place value to the right.

Use the larger digit to determine which decimal is greater.

Question 6

Explain:

The first step in ordering decimals.

Answer 6

Write the numbers with the decimal points lined up.

This ensures that place values correspond correctly.

Question 7

Explain:

How do you determine which decimal is bigger?

Answer 7

1. Write the decimals with decimal points lined up.
2. Compare each place value starting from the left until you find a difference.

The number with the larger digit at the first point of difference is greater.

Question 8

Explain:

The process for ordering decimal numbers from least to greatest.

Answer 8

1. Write the numbers with decimal points lined up.
2. Compare from the leftmost place value to the right.
3. Identify and disregard numbers already determined as larger.

Continue until all numbers are ordered.

Question 9

Explain:

What is the process for ordering decimals from greatest to least?

Answer 9

1. Write the numbers with decimal points lined up.
2. Compare from the leftmost place value to the right.

The process is similar to ordering from least to greatest.

Identify and disregard numbers already determined as smaller.

Question 10

Explain:

The significance of adding zeroes when comparing decimals.

Answer 10

It helps align the place values with different numbers of digits.

This does not change the value of the numbers.

Question 11

Define:

Base ten blocks

Answer 11

Concrete manipulatives that replicate the base ten model of counting.

They help visualize the size of numbers by representing each place value.

Question 12

Describe:

The base ten model of counting.

Answer 12

A system where amounts are represented by digits 0 through 9.

After 9, numbers increase in place value with each new digit.

Question 13

Identify:

The different types of base ten blocks.

Answer 13

• Unit blocks (ones)
• Rods (tens)
• Flats (hundreds)
• Cubes (thousands)

Each type represents a specific place value.

Question 14

Identify:

What does a unit block represent?

Answer 14

One.

It is a single cube, about 1 cubic centimeter in volume.

Question 15

Identify:

What is a rod made of base ten blocks?

Answer 15

10 unit blocks stacked together.

It represents 1 ten and is usually 10 centimeters long.

Question 16

Identify:

What does a flat represents in base ten blocks?

Answer 16

100 unit blocks or 10 rods.

It is 100 times the size of a unit block.

Question 17

Identify:

The value represented by a cube in base ten blocks.

Answer 17

1,000 unit blocks, 100 rods, or 10 flats.

It represents the digit in the thousands place.

Question 18

Explain:

How do base ten blocks demonstrate place values?

Answer 18

• Unit blocks for ones
• Rods for tens
• Flats for hundreds
• Cubes for thousands

No blocks are used for digits with a value of zero.

Question 19

Explain:

How would you represent 3,427 using base ten blocks?

Answer 19

3 cubes, 4 flats, 2 rods, and 7 unit blocks.

This shows the values of thousands, hundreds, tens, and ones, respectively.

Question 20

Explain:

How would you represent the number 105 with base ten blocks?

Answer 20

1 flat, 0 rods and 5 unit blocks.

The flat represents the hundreds place, while zeros indicate no blocks.

Question 21

Identify:

The largest base ten block.

Answer 21

The cube.

It represents the digit in the thousands place.

Question 22

Identify:

The purpose of using base ten blocks.

Answer 22

To represent numbers and understand place value.

They also help model operations like addition, subtraction, multiplication, and division.

Question 23

Define:

A number line

Answer 23

A straight line labeled by equally spaced integers, extended indefinitely from either side of zero.

It includes both negative and positive numbers, known as integers.

Question 24

Identify:

6 steps to draw a number line.

Answer 24

1. Label a point with zero.
2. Draw a straight line to the right with an arrow.
3. Draw a straight line to the left with an arrow.
4. Mark points equally separated on either side of zero.
5. Points to the right are positive integers.
6. Points to the left are negative integers.

The ends of the number line have arrows to indicate infinite numbers.

Question 25

# Identify: The **types** of **numbers** that can be represented on a **number line**.

Answer 25

**Rational numbers**, which include: * Integers * Fractions * Finite or repeating decimals ## Footnote Pi is an example of a non-rational number as it does not stop or repeat.

Question 26

# Explain: How can **rational numbers** be **graphed** on a **number line**?

Answer 26

1. **Regular number line**: Use a point to label the rational number. 2. **Zooming in**: Use smaller lines to represent fractions or decimals. ## Footnote Example: For -1/3, grids can be used to separate the rational numbers by 1/3.

Question 27

# Explain: How do you **represent** **3/4** on a **number line**?

Answer 27

Divide the space between 0 and 1 into increments of 1/4 and mark the 3/4 position with a dot. ## Footnote Make sure the integers are separated enough to clearly distinguish the fractions.

Question 28

# Define: rational number

Answer 28

Any **positive** or **negative** **number** that can be expressed as a fraction.

Question 29

# Define: fraction

Answer 29

Mathematical **representation** of **dividing** two **integers**. ## Footnote Most commonly used when conducting rational number calculations.

Question 30

# Define: denominator

Answer 30

The **bottom** integer in a **fraction**. ## Footnote For example, in the fraction 1/4, 4 is the denominator.

Question 31

# Define: numerator

Answer 31

The integer on the **top** of a **fraction**.

Question 32

# Define: least common denominator | (**LCD**)

Answer 32

The **smallest number** that all denominators can go into. ## Footnote It should be **used to find a common denominator**.

Question 33

# Describe: What is **required** to **add** or **subtract** rational numbers in **fraction form**?

Answer 33

Identify a **common** **denominator**. | A common denominator is a multiple of both denominators. ## Footnote The denominators must be identical for the operation.

Question 34

# Identify: The **first step** in **adding** **rational numbers**.

Answer 34

Find the least common denominator (**LCD**). ## Footnote This ensures the fractions can be added together.

Question 35

# Explain: How do you **find** the **LCD** of **two fractions**?

Answer 35

1. **Find** the prime factors of each denominator. 2. **List** the prime factors including the greatest number of times each appears. 3. **Multiply** the list of factors together. ## Footnote Example: The LCD of 24 and 16 is 48.

Question 36

# Explain: The **process** to **add fractions** once the LCD is found.

Answer 36

Convert each fraction to an equivalent fraction using the **LCD**, then **add** the **numerators**. ## Footnote Retain the LCD as the denominator of the final answer.

Question 37

# Describe: Procedure for **subtracting rational numbers**.

Answer 37

Ensure the **denominators** are the **same**, convert to equivalent fractions if necessary, then **subtract** the **numerators**. ## Footnote The common denominator remains unchanged.

Question 38

# Identify: 1/2 + 1/4 = \_\_\_\_

Answer 38

3/4 ## Footnote The LCD is 4, so 1/2 becomes 2/4.

Question 39

# Explain: How do you **reduce** a **rational number** to its **simplest form**?

Answer 39

**Divide** both the **numerator** and **denominator** by their **greatest common factor**. ## Footnote Example: 10/20 reduces to 1/2.

Question 40

# Explain: How do **rational numbers** **differ** from **fractions**?

Answer 40

* **Rational numbers** can have **integers** as their numerator and denominator. * **Fractions** can have **whole numbers** as their numerator and denominator. ## Footnote All fractions are rational numbers, but not all rational numbers are fractions.

Question 41

# Define: greatest common factor | (**GCF**)

Answer 41

The **greatest factor** that **two numbers** are both **divisible by**. ## Footnote It is used to simplify rational numbers.

Question 42

# Explain: How to **multiply** two **rational numbers**.

Answer 42

**Multiply** the **numerators together** for the new numerator and the **denominators together** for the new denominator. ## Footnote For rational numbers ab and cd, ab×cd = a×c / b×d.

Question 43

# Explain: The process of **simplifying** a **rational number**.

Answer 43

**Divide** **both** the **numerator** and the **denominator** by their **GCF**. ## Footnote Example: To simplify 16/20, divide both by 4 to get 4/5.

Question 44

# Define: The **reciprocal** of a **rational number**.

Answer 44

It is the result of **flipping** the **numerator** and **denominator**. ## Footnote For the rational number **ab**, the reciprocal is **ba**.

Question 45

# Explain: How to **divide** two **rational numbers**

Answer 45

**Multiply** the **first rational** number **by** the **reciprocal** of the **second rational** number. ## Footnote For rational numbers ab and cd, ab÷cd = ab×dc.

Question 46

# Explain: Why does the **order** matter when **dividing rational numbers**?

Answer 46

Because **ab÷cd ≠ cd÷ab**. ## Footnote This can lead to different results.

Question 47

# Identify: The **formula** for **dividing rational numbers**.

Answer 47

ab÷cd = a×d / b×c. ## Footnote This involves keeping the first number and flipping the second.

Question 48

# Define: Estimate

Answer 48

A **quick guess** that is close enough to work with but **not necessarily accurate**. ## Footnote Estimation is useful when strict accuracy is not required.

Question 49

# Identify: The most common way to **estimate.**

Answer 49

**Rounding** numbers. ## Footnote Rounding helps simplify calculations by reducing complexity.

Question 50

# Identify: The magic **number** for **rounding**.

Answer 50

5 ## Footnote If the digit to the right of the desired rounded digit is 5 or higher, round up; otherwise, round down.

Question 51

# Identify: The **steps** to **round a number**.

Answer 51

1. Look at the digit to the right of the desired digit. 2. If 0-4, round down. 3. If 5-9, round up. ## Footnote Following these steps ensures accurate rounding.

Question 52

# Explain: The purpose of **estimating sums**.

Answer 52

To **simplify** addition problems and quickly **gauge totals**. ## Footnote Estimating sums can help in budgeting and checking calculations.

Question 53

# Explain: How should you **estimate** when dealing with **money**?

Answer 53

**Round** all values **up** to the **next higher whole dollar**. ## Footnote This prevents underestimating total costs.

Question 54

# Describe: What is **estimating totals** with **multiplication** used for?

Answer 54

To estimate **counts** in a population **based on a sample**. ## Footnote This method is useful when counting an entire population is impractical.

Question 55

# Define: Equivalent

Answer 55

When two or more quantities (or things) are **equal** in **value**.

Question 56

# Identify: What **forms** can **equivalent quantities** take?

Answer 56

* Number * Fraction * Decimal * Percentage

Question 57

# Define: Equivalent fractions

Answer 57

**Fractions** that represent the **same** **ratio** or **value**.

Question 58

# Describe: The **role** of a **numerator** in a fraction.

Answer 58

**How many parts** are being used.

Question 59

# Describe: The **role** of a **denominator** in a fraction.

Answer 59

Number of **parts** into which a **whole** is **divided**.

Question 60

# Explain: How can you **verify** if **two fractions** are **equivalent**?

Answer 60

By cross-multiplication.

Question 61

# Explain: Cross-multiplication

Answer 61

The **product** of the **numerator** of the **first** fraction and the **denominator** of the **second** **equals** the **product** of the **denominator** of the **first** and the **numerator** of the **second**.

Question 62

# Describe: What does an **equivalent decimal** represent?

Answer 62

A **portion** of a **full value**, with a **whole number** part and a **decimal** part.

Question 63

# Describe: How to create an **equivalent decimal**.

Answer 63

By **adding zeros** to the end of the decimal.

Question 64

# Describe: The method for **converting decimals** into **fractions**.

Answer 64

**Determine** the **denominator** based on the **last digit's place value**.

Question 65

# Explain: How **whole numbers** can be represented **as fractions**.

Answer 65

By writing them **as the number over 1**. ## Footnote e.g., 5 as 5/1

Question 66

# Explain: How to convert **percentages into fractions**.

Answer 66

Write the percentage as the numerator and 100 as the denominator.

Question 67

# Identify: The process of **converting a percentage** to a decimal.

Answer 67

Move the **decimal point 2 spaces** to the left.

Question 68

# Define: percent

Answer 68

Per one hundred. ## Footnote A percent represents an **amount divided into 100** units.

Question 69

# Describe: What is a method for solving **percent** problems?

Answer 69

When you see a percent of something, **translate 'of' into multiplication**. ## Footnote **Example**: 75 percent of 20 translates to 75% x 20.

Question 70

# Explain: What does "**a number out of another number**" translate to in mathematical terms.

Answer 70

Division ## Footnote **Example**: 65 out of 80 translates to 65 / 80.

Question 71

# Explain: If a shoe costs **$218** and is **discounted by 55%**, what is the **amount paid** after the discount?

Answer 71

$98.10 ## Footnote Calculate 45% of $218 after determining the remaining percentage from the discount.

Question 72

# Define: ratio

Answer 72

A means to show **relative size** between **two or more** values. ## Footnote Ratios compare values rather than giving actual amounts.

Question 73

# Identify: What are 4 different ways **ratios** can be written?

Answer 73

1. With a colon (e.g., 2:1) 1. With a forward slash (e.g., 2/1) 1. As a decimal (e.g., 2.0) 1. As a percentage (e.g., 200%) ## Footnote The simplest form is often used for clarity.

Question 74

# Identify: What is the **simplest form** of the **ratio 4:20**?

Answer 74

1:5 ## Footnote For example, this could mean there are 5 birds for every tree.

Question 75

# Define: proportion

Answer 75

A statement that **two ratios** are **equal** to **each other**. ## Footnote For example, 4:2 = 2:1.

Question 76

# Identify: What is the **ratio** of pink to blue to orange if 10 people like blue, 12 like pink, and 7 like orange?

Answer 76

12:10:7 ## Footnote This ratio is already in its simplest form.

Question 77

# Identify: **Represent** this recipe as a **ratio**: for every **1 cup** of sugar, **4 cups** of water are needed.

Answer 77

1:4 (sugar to water).

Question 78

# Identify: In a town with **400 commuters**, **120 drive** to work while **280 use bicycles**. What is the **ratio** of drivers to cyclists?

Answer 78

120:280

Question 79

# Identify: If car batteries and decorative garden gnomes sell in a ratio of **35:2**, how many gnomes are sold after selling **280 car batteries**?

Answer 79

16 ## Footnote This is found by setting up the proportion 35:2 = 280:g.

Question 80

# Identify: What **operation** is primarily used to **solve proportions**?

Answer 80

multiplication ## Footnote This can be done vertically, horizontally or diagonally (cross-multiplication).

Question 81

# Explain: When is the **vertical method** best used while **solving proportions**?

Answer 81

When there is a **multiple between the quantities** in the same ratio. ## Footnote Example: 39=5x, where 3 is the multiple.

Question 82

# Explain: What is the **horizontal method** used for?

Answer 82

To **find** the **missing part** of **equivalent fractions**. ## Footnote It is best used when there is a common multiple between the two numerators or denominators.

Question 83

# Describe: The **diagonal method** in solving proportions.

Answer 83

Also known as **cross-multiplication**; it involves taking the product of opposite numerators and denominators. ## Footnote This method is useful when vertical or horizontal multiples are not easily recognizable.

Question 84

# Identify: The **formula** for a **proportional relationship**.

Answer 84

a/b = c/d ## Footnote It can also be expressed as **a:b::c:d**.

Question 85

# Define: word problem

Answer 85

A math **problem expressed in English** rather than symbols. ## Footnote Word problems often require translation into algebraic equations.

Question 86

# Identify: The **three methods** to **solve a proportion**.

Answer 86

1. Vertical 1. Horizontal 1. Diagonal (cross-multiplication) ## Footnote Each method is suited for different types of proportions based on the relationships between the quantities.

Question 87

# Identify: The **three steps** in solving **word problems**.

Answer 87

1. **Visualize** the problem. 2. **Write** your equations. 3. **Solve** the equations.

Question 88

# Define: one-step **algebra equation**

Answer 88

An equation that requires only **one operation to solve**. ## Footnote The operations can be addition, subtraction, multiplication, or division.

Question 89

# Identify: Common **operations** used in **one-step algebra equations**.

Answer 89

* Addition * Subtraction * Multiplication * Division ## Footnote These operations constitute the single step needed to solve the equation.

Question 90

# Describe: The **first step** in **translating a word problem**.

Answer 90

**Identify** and **highlight** **important** words and phrases. ## Footnote Look for **numbers** and **operation indicators**.

Question 91

# Explain: What does "**less than**" imply in a word problem?

Answer 91

Indicates **subtraction** and can **reverse the order** of numbers. ## Footnote For example, 'four less than a number' translates to 'number - 4'.

Question 92

# Identify: The correct translation of "**four less than a number equals eight**".

Answer 92

x - 4 = 8 ## Footnote The correct order is crucial; the larger number comes first.

Question 93

# Explain: How do you **isolate** the **variable** in the equation **x - 4 = 8**?

Answer 93

**Add 4 to both sides** of the equation. ## Footnote This results in x = 12.

Question 94

# Identify the equation: Charles has **twice** the amount of dollars that Josh has.

Answer 94

C = 2 * J ## Footnote C represents the dollars Charles has, and J represents the amount of dollars Josh has.

Question 95

# Identify: The **four steps** for solving **algebra word problems**.

Answer 95

1. **Analyze** the problem. 2. **Gather** information. 3. **Translate** into an equation. 4. **Solve** the equation. ## Footnote These steps provide a structured approach to solving word problems.