4 - Number Sense and Basic Algebra Flashcards
This deck focuses on key math skills: computing with rational numbers, estimation, and solving problems with percent, ratios, and proportions. It emphasizes word problems, algebraic expressions, equivalent number forms (including graphs), and place value for ordering and grouping.
Define:
decimal numbers
Represent parts of a whole number, including a whole number and a part.
Examples: 0.74, 4.7 and 100.001
Explain:
What is the place value to the left of the decimal point?
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.
Identify:
The place values starting at the decimal point.
- Tenths
- Hundredths
- Thousandths
- Ten-thousandths
Name these decimal numbers:
- 2.3
- 15.78
- 0.009
- 30.0475
- 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
Explain:
How do you compare decimal numbers?
- Write the decimals with the decimal points lined up.
- Compare from the leftmost place value to the right.
Use the larger digit to determine which decimal is greater.
Explain:
The first step in ordering decimals.
Write the numbers with the decimal points lined up.
This ensures that place values correspond correctly.
Explain:
How do you determine which decimal is bigger?
- Write the decimals with decimal points lined up.
- 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.
Explain:
The process for ordering decimal numbers from least to greatest.
- Write the numbers with decimal points lined up.
- Compare from the leftmost place value to the right.
- Identify and disregard numbers already determined as larger.
Continue until all numbers are ordered.
Explain:
What is the process for ordering decimals from greatest to least?
- Write the numbers with decimal points lined up.
- 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.
Explain:
The significance of adding zeroes when comparing decimals.
It helps align the place values with different numbers of digits.
This does not change the value of the numbers.
Define:
Base ten blocks
Concrete manipulatives that replicate the base ten model of counting.
They help visualize the size of numbers by representing each place value.
Describe:
The base ten model of counting.
A system where amounts are represented by digits 0 through 9.
After 9, numbers increase in place value with each new digit.
Identify:
The different types of base ten blocks.
- Unit blocks (ones)
- Rods (tens)
- Flats (hundreds)
- Cubes (thousands)
Each type represents a specific place value.
Identify:
What does a unit block represent?
One.
It is a single cube, about 1 cubic centimeter in volume.
Identify:
What is a rod made of base ten blocks?
10 unit blocks stacked together.
It represents 1 ten and is usually 10 centimeters long.
Identify:
What does a flat represents in base ten blocks?
100 unit blocks or 10 rods.
It is 100 times the size of a unit block.
Identify:
The value represented by a cube in base ten blocks.
1,000 unit blocks, 100 rods, or 10 flats.
It represents the digit in the thousands place.
Explain:
How do base ten blocks demonstrate place values?
- Unit blocks for ones
- Rods for tens
- Flats for hundreds
- Cubes for thousands
No blocks are used for digits with a value of zero.
Explain:
How would you represent 3,427 using base ten blocks?
3 cubes, 4 flats, 2 rods, and 7 unit blocks.
This shows the values of thousands, hundreds, tens, and ones, respectively.
Explain:
How would you represent the number 105 with base ten blocks?
1 flat, 0 rods and 5 unit blocks.
The flat represents the hundreds place, while zeros indicate no blocks.
Identify:
The largest base ten block.
The cube.
It represents the digit in the thousands place.
Identify:
The purpose of using base ten blocks.
To represent numbers and understand place value.
They also help model operations like addition, subtraction, multiplication, and division.
Define:
A number line
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.
Identify:
6 steps to draw a number line.
- Label a point with zero.
- Draw a straight line to the right with an arrow.
- Draw a straight line to the left with an arrow.
- Mark points equally separated on either side of zero.
- Points to the right are positive integers.
- Points to the left are negative integers.
The ends of the number line have arrows to indicate infinite numbers.
Identify:
The types of numbers that can be represented on a number line.
Rational numbers, which include:
- Integers
- Fractions
- Finite or repeating decimals
Pi is an example of a non-rational number as it does not stop or repeat.
Explain:
How can rational numbers be graphed on a number line?
- Regular number line: Use a point to label the rational number.
- Zooming in: Use smaller lines to represent fractions or decimals.
Example: For -1/3, grids can be used to separate the rational numbers by 1/3.
Explain:
How do you represent 3/4 on a number line?
Divide the space between 0 and 1 into increments of 1/4 and mark the 3/4 position with a dot.
Make sure the integers are separated enough to clearly distinguish the fractions.
Define:
rational number
Any positive or negative number that can be expressed as a fraction.
Define:
fraction
Mathematical representation of dividing two integers.
Most commonly used when conducting rational number calculations.
Define:
denominator
The bottom integer in a fraction.
For example, in the fraction 1/4, 4 is the denominator.
Define:
numerator
The integer on the top of a fraction.
Define:
least common denominator
(LCD)
The smallest number that all denominators can go into.
It should be used to find a common denominator.
Describe:
What is required to add or subtract rational numbers in fraction form?
Identify a common denominator.
A common denominator is a multiple of both denominators.
The denominators must be identical for the operation.
Identify:
The first step in adding rational numbers.
Find the least common denominator (LCD).
This ensures the fractions can be added together.
Explain:
How do you find the LCD of two fractions?
- Find the prime factors of each denominator.
- List the prime factors including the greatest number of times each appears.
- Multiply the list of factors together.
Example: The LCD of 24 and 16 is 48.
Explain:
The process to add fractions once the LCD is found.
Convert each fraction to an equivalent fraction using the LCD, then add the numerators.
Retain the LCD as the denominator of the final answer.
Describe:
Procedure for subtracting rational numbers.
Ensure the denominators are the same, convert to equivalent fractions if necessary, then subtract the numerators.
The common denominator remains unchanged.
Identify:
1/2 + 1/4 = ____
3/4
The LCD is 4, so 1/2 becomes 2/4.
Explain:
How do you reduce a rational number to its simplest form?
Divide both the numerator and denominator by their greatest common factor.
Example: 10/20 reduces to 1/2.
Explain:
How do rational numbers differ from fractions?
- Rational numbers can have integers as their numerator and denominator.
- Fractions can have whole numbers as their numerator and denominator.
All fractions are rational numbers, but not all rational numbers are fractions.
Define:
greatest common factor
(GCF)
The greatest factor that two numbers are both divisible by.
It is used to simplify rational numbers.
Explain:
How to multiply two rational numbers.
Multiply the numerators together for the new numerator and the denominators together for the new denominator.
For rational numbers ab and cd, ab×cd = a×c / b×d.
Explain:
The process of simplifying a rational number.
Divide both the numerator and the denominator by their GCF.
Example: To simplify 16/20, divide both by 4 to get 4/5.
Define:
The reciprocal of a rational number.
It is the result of flipping the numerator and denominator.
For the rational number ab, the reciprocal is ba.
Explain:
How to divide two rational numbers
Multiply the first rational number by the reciprocal of the second rational number.
For rational numbers ab and cd, ab÷cd = ab×dc.
Explain:
Why does the order matter when dividing rational numbers?
Because ab÷cd ≠ cd÷ab.
This can lead to different results.
Identify:
The formula for dividing rational numbers.
ab÷cd = a×d / b×c.
This involves keeping the first number and flipping the second.
Define:
Estimate
A quick guess that is close enough to work with but not necessarily accurate.
Estimation is useful when strict accuracy is not required.
Identify:
The most common way to estimate.
Rounding numbers.
Rounding helps simplify calculations by reducing complexity.
Identify:
The magic number for rounding.
5
If the digit to the right of the desired rounded digit is 5 or higher, round up; otherwise, round down.
Identify:
The steps to round a number.
- Look at the digit to the right of the desired digit.
- If 0-4, round down.
- If 5-9, round up.
Following these steps ensures accurate rounding.
Explain:
The purpose of estimating sums.
To simplify addition problems and quickly gauge totals.
Estimating sums can help in budgeting and checking calculations.
Explain:
How should you estimate when dealing with money?
Round all values up to the next higher whole dollar.
This prevents underestimating total costs.
Describe:
What is estimating totals with multiplication used for?
To estimate counts in a population based on a sample.
This method is useful when counting an entire population is impractical.
Define:
Equivalent
When two or more quantities (or things) are equal in value.
Identify:
What forms can equivalent quantities take?
- Number
- Fraction
- Decimal
- Percentage
Define:
Equivalent fractions
Fractions that represent the same ratio or value.
Describe:
The role of a numerator in a fraction.
How many parts are being used.
Describe:
The role of a denominator in a fraction.
Number of parts into which a whole is divided.
Explain:
How can you verify if two fractions are equivalent?
By cross-multiplication.
Explain:
Cross-multiplication
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.
Describe:
What does an equivalent decimal represent?
A portion of a full value, with a whole number part and a decimal part.
Describe:
How to create an equivalent decimal.
By adding zeros to the end of the decimal.
Describe:
The method for converting decimals into fractions.
Determine the denominator based on the last digit’s place value.
Explain:
How whole numbers can be represented as fractions.
By writing them as the number over 1.
e.g., 5 as 5/1
Explain:
How to convert percentages into fractions.
Write the percentage as the numerator and 100 as the denominator.
Identify:
The process of converting a percentage to a decimal.
Move the decimal point 2 spaces to the left.
Define:
percent
Per one hundred.
A percent represents an amount divided into 100 units.
Describe:
What is a method for solving percent problems?
When you see a percent of something, translate ‘of’ into multiplication.
Example: 75 percent of 20 translates to 75% x 20.
Explain:
What does “a number out of another number” translate to in mathematical terms.
Division
Example: 65 out of 80 translates to 65 / 80.
Explain:
If a shoe costs $218 and is discounted by 55%, what is the amount paid after the discount?
$98.10
Calculate 45% of $218 after determining the remaining percentage from the discount.
Define:
ratio
A means to show relative size between two or more values.
Ratios compare values rather than giving actual amounts.
Identify:
What are 4 different ways ratios can be written?
- With a colon (e.g., 2:1)
- With a forward slash (e.g., 2/1)
- As a decimal (e.g., 2.0)
- As a percentage (e.g., 200%)
The simplest form is often used for clarity.
Identify:
What is the simplest form of the ratio 4:20?
1:5
For example, this could mean there are 5 birds for every tree.
Define:
proportion
A statement that two ratios are equal to each other.
For example, 4:2 = 2:1.
Identify:
What is the ratio of pink to blue to orange if 10 people like blue, 12 like pink, and 7 like orange?
12:10:7
This ratio is already in its simplest form.
Identify:
Represent this recipe as a ratio: for every 1 cup of sugar, 4 cups of water are needed.
1:4 (sugar to water).
Identify:
In a town with 400 commuters, 120 drive to work while 280 use bicycles. What is the ratio of drivers to cyclists?
120:280
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?
16
This is found by setting up the proportion 35:2 = 280:g.
Identify:
What operation is primarily used to solve proportions?
multiplication
This can be done vertically, horizontally or diagonally (cross-multiplication).
Explain:
When is the vertical method best used while solving proportions?
When there is a multiple between the quantities in the same ratio.
Example: 39=5x, where 3 is the multiple.
Explain:
What is the horizontal method used for?
To find the missing part of equivalent fractions.
It is best used when there is a common multiple between the two numerators or denominators.
Describe:
The diagonal method in solving proportions.
Also known as cross-multiplication; it involves taking the product of opposite numerators and denominators.
This method is useful when vertical or horizontal multiples are not easily recognizable.
Identify:
The formula for a proportional relationship.
a/b = c/d
It can also be expressed as a:b::c:d.
Define:
word problem
A math problem expressed in English rather than symbols.
Word problems often require translation into algebraic equations.
Identify:
The three methods to solve a proportion.
- Vertical
- Horizontal
- Diagonal (cross-multiplication)
Each method is suited for different types of proportions based on the relationships between the quantities.
Identify:
The three steps in solving word problems.
- Visualize the problem.
- Write your equations.
- Solve the equations.
Define:
one-step algebra equation
An equation that requires only one operation to solve.
The operations can be addition, subtraction, multiplication, or division.
Identify:
Common operations used in one-step algebra equations.
- Addition
- Subtraction
- Multiplication
- Division
These operations constitute the single step needed to solve the equation.
Describe:
The first step in translating a word problem.
Identify and highlight important words and phrases.
Look for numbers and operation indicators.
Explain:
What does “less than” imply in a word problem?
Indicates subtraction and can reverse the order of numbers.
For example, ‘four less than a number’ translates to ‘number - 4’.
Identify:
The correct translation of “four less than a number equals eight”.
x - 4 = 8
The correct order is crucial; the larger number comes first.
Explain:
How do you isolate the variable in the equation x - 4 = 8?
Add 4 to both sides of the equation.
This results in x = 12.
Identify the equation:
Charles has twice the amount of dollars that Josh has.
C = 2 * J
C represents the dollars Charles has, and J represents the amount of dollars Josh has.
Identify:
The four steps for solving algebra word problems.
- Analyze the problem.
- Gather information.
- Translate into an equation.
- Solve the equation.
These steps provide a structured approach to solving word problems.
