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Instructions:

1. Download the Frayer_Model_Workbook_Algebra_Blank

2. Choose one vocabulary word. Click + button below the vocabulary word.

3. Complete the Frayer model fields for: Definition, Characteristics, Examples, and Non-examples.

Definition (clear and simple to understand)

Characteristics (facts and properties)

Examples (with equations, sentences, or images)

Non-examples (to avoid confusion)

4. Publish or Save as Word or PDF.

5. Upload your file under the correct column in Padlet.

# data[].attributes.custom_properties.Hgnw2DZI
line- y=4x+5,"4" isn't the intercept but the slope

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Intercept is a point,it won't change in a certain formula

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- usually denoted by letter like x or t
- researched by mathematicians
- can be a number, matrix, etc.

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velocity of a moving object, v
random matrix, A

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the point where a graph intersects a coordinate axis  in a coordinate system.

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the linear function  y = 2x + 3 : The point (1, 5)

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it represents the point where a line, curve, or surface crosses (intersects) a coordinate axis in a graph, directly indicating the value of one variable when the other key variable equals zero.

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the number where a line crosses the axis

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can be positive or negative 
a straight line can't have more than one y-intercept 

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line y=2x+3 has y-intercept 3
line y=x has y-intercept 0

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1.Intercept is always a point on the axis(x or y)
2.one of the coordinates must be 0

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啊

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The values and quantities that can be changed during a specific process

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it's a necessary part in a function.it can lead to other thing's change
it is usually writen as x y z or t k b .

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The current through an electrical circuit is a fundamental concept. Its behavior depends on the components in the circuit and the power source.

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The current through an electrical circuit is a fundamental concept. Its behavior depends on the components in the circuit and the power source.

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Current (denoted as I) is the flow of electric charge. 

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If you connect a 10Ω resistor to a 5V battery, the current will be:
I = V / R = 5V / 10Ω = 0.5 A

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How fast one thing changes compared to another

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Describe an overall trend or instantaneous change

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A rate that describe the pace of one quantity changeing.

# data[].attributes.custom_properties.Hgnw2DZI
Y =2X+1,"1" is not  the rate of the change formula.

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Y =2X+1,"2" is the rate of the change formula.

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a changeble item which meets the regulation of domain and causes the result changing

# data[].attributes.custom_properties.aD7SWzR8
represented by x
can be defined
equals to what you input in
has a range in some equations

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y=3x+2 has variable x
f(t)＝4t－5 has variable t

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A symbol (like x) or name that represents a value that can change or vary.

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-A non -example of a variable is any symbol or term that represents a fixed, specific, and unchanging value, primarily called constants.

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-representation by symbol
-value can vary
-represents the unknown
-placeholder nature
-enables generalization

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-Equation: 2x+5=11. Here x is a variable.
-Function: y=f(x)=x^2. Here x is the independent variable.

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Rate of change,a basis of algebra,is just like an emperor of algebra,who empires the change rate of the data

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It is the rate of the change of something.

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A dot usually doesn't have the rate of change.

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It determines the rate that something may change with another thing

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1.y=x,2.y=5x As usual,we say that the formula 1 has a bigger rate of change comparing with the formula 2

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Slope is the measure of steepness and direction.

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The slope m of line through two points (x1,y1) (x2,y2) is m =y1-y2/x1-x2

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Slope describes the steepness and direction of a line, indicating how much the vertical value changes for a given horizontal change. A positive slope rises, a negative slope falls, and a zero slope is flat.

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In the linear equation y=2x+1,the slope is 2.

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An intercept is a point where a graph crosses an axis (the x-axis or y-axis) of the coordinate plane.

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Line y=2x+4 has an x-intercept -2 and a y-intercept 4

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A symbol that represents a value whose number is unknown 

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Its number can change or vary. It’s not fixed

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In mathematics, the term "intercept" refers to the point where a line or curve crosses a coordinate axis in a graph.‌

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tell the difference between slope and intercept carefully

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This concept is widely used in algebra and statistics to describe key characteristics of graphs or datasets.

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y=3x+4 has y-intercept 4 and x-intercept -4/3 (and it can be zero of course)

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A variable is a symbol,usually a letter (like x,y,z)that represents a quantity that can change 

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to describe a relationship,function or equation that follows a line with a fixed slope.

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Always used to describe a constant rate of change

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A equation that represent a straight line.

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The total price of goods is calculated by multiplying the unit price by the purchase quantity.

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One variable has a straight and proportional change in other.Typically in the form y=mx+c.The variable lies in simple equation.

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One quantity is directly proportional to another quantity

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Every straight line have one or two intercepts.

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A straight line passes through the y-axis and leaves the point

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On the y-axis and the horizontal axis is zero

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Delta Y Over Delta X is useful in the expression of math relationship.

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The change on x axis over the change on y axis

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It can describe the amount of relation of two variable

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y=x+1 between x axis 0 and 1 ,the delta y over delta x is 1

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The rate of change is a mathematical concept that describes how one quantity changes in relation to another quantity. 

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Static measurements, single amounts or totals

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1.  provided a numerical value 
2. can be Positive, Negative, or Zero
3. can be constant or variable
4. represented by a straight line on a graph

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1. A car travels 150 kilometers in 2 hours. Average Speed = (Change in Distance) / (Change in Time) = 150 km / 2 h = 75 km/h.
2. A company's stock price falls from $100 to $85 over a week. Average Rate = (Change in Price) / (Change in Time) = ($85 - $100) / 1 week = -$15 per week.
3. A road rises 30 meters over a horizontal distance of 100 meters. Slope = (Change in Elevation) / (Change in Horizontal Distance) = 30 m / 100 m = 0.3 (or 30%).

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A group of symbols that explain how to calculate the rate of change (the ratio of the dependent variable to the independent variable), simply with letters represent variables, operational characters and equal sign

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Delta X : variable but not a formula  delta y/delta x : imcomplete formula(without the equal sign)

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categorize as average rate of change and instantaneous rate of change 
Formula :delta y/delta x

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Acceleration: the rate of change of velocity with respect to time   a=v/t

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Key Concepts of Slope
 
- Calculation Formula: For two points (x_1, y_1) and (x_2, y_2) on a line, the slope (m) is calculated as:
m = \frac{y_2 - y_1}{x_2 - x_1}
This is often remembered as "rise over run" (vertical change divided by horizontal change).
- Direction Meanings:
- A positive slope means the line rises from left to right.
- A negative slope means the line falls from left to right.
- A zero slope means the line is horizontal (no vertical change).
- An undefined slope means the line is vertical (no horizontal change, as division by zero is undefined).

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In mathematics, especially in coordinate geometry, slope is a measure that describes the steepness, direction, and incline of a straight line on a plane. It quantifies how much the line rises (or falls) vertically for a given horizontal change.
 

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- Practical Example: If a line goes from point (1, 2) to (3, 6), its slope is \frac{6 - 2}{3 - 1} = 2. This means for every 1 unit you move right (horizonal), you move up 2 units.

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A relationship or function that satisfies additivity and homogeneity of degree 1, and its geometric representation is usually a straight line on a coordinate system with a constant rate of change 

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 If input is multiplied by a number (e.g., 3x), output is also multiplied by that number . Its graph is a straight line with a constant slope.If input is "a + b", output is "output of a + output of b".

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Where a line crosses the axis

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The point where a graph crosses or "intercepts" an axis (x-axis or y-axis).

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· The slope of the line (steepness, not a point)
· Any point that is not on an axis, like  (2, 5) 
· The variable  x  or  y  in an equation

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· The y-intercept is where  x = 0 , often represented by  b  in  y = mx + b 
· The x-intercept is where  y = 0 , it is the solution or root of the equation
· A line has exactly one y-intercept and can have zero, one, or infinitely many x-intercepts

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· In  y = 2x + 3 , the y-intercept is the point  (0, 3) 
· The graph of  y = x - 4  has an x-intercept at  (4, 0) 
· The point where a road crosses a main street on a map

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The ratio of the change in the output value of a function to the change in the input value.

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Can be constant (linear ) or variable (non-linear);can be positive ,negative,or zero,indicating increase,decrease,or no change.

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An angle α forms between a straight line and the positive direction of the x-axis, then tanα is the slope.

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The rate of change measures how one quantity changes relative to another over a specific interval. For two points (x_1, y_1) and (x_2, y_2) on a function, the formula is:
Rate of Change = \frac{y_2 - y_1}{x_2 - x_1}
It represents the "slope" between two points in linear relationships, or the average rate of change in non-linear functions.

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Total value, not relative change: "A store sold 50 shirts in a day" (this is a total count, not a ratio of two changing quantities)

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(Linear Function): For y = 2x + 3, find the rate of change between x=1 (y=5) and x=4 (y=11).
Calculation: \frac{11 - 5}{4 - 1} = \frac{6}{3} = 2. (This matches the slope of the line, so linear p
functions have a constant rate of change.)

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The average change of a function

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The average rate of change of a function f(x) over the interval [a, b] is defined as:

\text{Average Change} = \frac{f(b) - f(a)}{b - a}

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1. Instantaneous Rate of Change:
  The derivative of a function at a point (e.g., f'(x)) is not the average change. It represents the rate of change at a single point, not over an interval. For example, the instantaneous speed of a car at a specific moment is not the average speed over a trip

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Slope of the Secant Line
Interval-Dependent: 

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Linear Function:
   Consider f(x) = 3x + 2 over the interval [1, 4].
   f(1) = 3(1) + 2 = 5, f(4) = 3(4) + 2 = 14.
   Average change = (14 - 5) / (4 - 1) = 9 / 3 = 3.
   This is constant for any interval, as the function is linear.

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A quantity that can change or that may take on different values. Variable also refers to a letter or symbol representing such a quantity.

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A basic concept in math and we can use the function to describe it 

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As time goes by, the speed of an object in free fall is a variable

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Always used to describe a constant rate of change 

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Variable is special quantities that can be changed through time.

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We often use X or Y as variable quantities
It can be a lot of things like letter

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y=2x+1，in this sentence，both y and x are variable.Y changes as x changes.

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It involving a series of connected events, ideas etc, that move or progress from one stage to the next.

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when the spring be broken it will not be linear

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A line that is straight and have fixed slope.

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relationship between the length of the spring and the force fit F=kx

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linear describe a relationship where the output is
directly proportional to the input 

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always used to describe the relationship between y and x 

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Variable is usually an unknown character.

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Variable is usually an unknown character.

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It is one part of a formular.Usually,a variable can be given many different numbers.

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6x+5=17,x is a variable and in this formular x is 2. 

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y=ax³＋bx²＋cx＋d，d is a number,not the rate of change.

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Describe one thing' s tendency of developing so that we can predict something easily.Likely use a,b,c,m and so on.

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y=ax ³＋bx²＋cx＋d,a b c define the rate of change

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Given two coordinates（x1,y1）,(x2,y2)，the slope is
（y2-y1）/(x2-x1)(x2-x1 can't be 0) 

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Slope is often used to describe the speed of change and the relationship between variable and dependent variable，determining whether they are positively or negatively correlated.

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（1，2），（3，5），so the slope of the straight line is 1.5.Or given a straight line y=3x+5，3 is the slope.

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y＝7   x＝9
the valua of x and y are fixed.
They are not variable.

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It is the foundation for studying functions.

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:A number which indicates the steepness of a line.
 It shows whether the linegoes uphill  or downhill 
It shows the rate of change

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Calculated as rise over run
Can be positive, negative, zero, or
undefined(a vertical line)
often be represented by "m"

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Specific figures aren't variable.And letters which represent certain figures aren't variable.

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Variable is represented by mostly x or y,while other letters are also available.

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Variable

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A quantity that may assume any one of a set of values

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A really important element in mathematics.
It usually appears in the form of letters like x,y and so on.
The value of it was uncertain.

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The rate of function helps us finding the hiden rules of algebra 

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The rate of function helps us finding the hiden rules of algebra 

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Σ( ° △ °|||)︴

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It shows a trend that how will the function change. 

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It is a number that describe the speed of the change of a line .

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It can be a number,it can also not exist

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The rate of change of a function describes how quickly the dependent variable (y) changes with respect to the independent variable (x), and is represented mathematically by the derivative (dy/dx).

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When it doesn't satisfy the definition of a function.

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Reflects the trend of variation locally or globally.

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if an object moves 5 meters in 2 seconds, its speed is 2.5 meters per second. ‌

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The rate of change of a function over an interval.

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Taking the distance function of uniform linear motion as an example: Let the distance be s(t) = 2t,within the interval [1, 3], the rate of change =s(3)- s(1)/3-1= 6-2/2=2 kilometers per hour, which means moving a fixed 2 kilometers per hour and corresponds to the slope of the straight line in the function graph.

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The change in y divided by the change in x.

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linear is always usd to describe something straight and no intercept.

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linear is always usd to describe something straight and no intercept.

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The two variables have a proportional characteristic relationship

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In some wrong equation variable  is meaningless ,like zero plus x equal six ,x is meaningless

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It changes with place where it appear 
It can represent one number and  enormous number

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Equation x +5=6  x=1  x can only represent one number
Equation 6x+y=11 x can represent enormous number

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Rate of change is a measure of how quickly one quantity changes relative to another.

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Think of instantaneous velocity as average velocity.

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It is essentially a measure of "how fast things are changing" and it is widely used.

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1:Instantaneous velocity: The reading of the car's speedometer at a certain moment is the instantaneous rate of change at that moment.
2: The slope of a curve at a point

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describle the speed of the number's change

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you get one yuan,one yuan is the change,isn't the rate of change.

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in one Minute,you get One yuan,one yuan per minute is the rate of change.

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If a linear function pass through two points (a,b),(c,d),the slope is b-d/a-c

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In the function y=4x+8,the number “8”is not the slope

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Slope describe how steep a linear function is and the direction of it like from top-left to bottom-right or vise versa

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In the function y=3x+7,the number “3” is the slope

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Slope-Intercept Equation of a Line

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A form of linear equations."k"is the slope,"b"is the  intercept.

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These are other forms of linear equations: y-y1=k(x-x1), Ax+By+C=0,k=y2-y1/x2-x1,x/a+y/b=1

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The form is "y=kx+b",it can describe the shape of a linear.

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y=2x+1,"2" is the slope,"1"is the intercept

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AAA

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a value that is used to describe a line's ratio of it's delta y and delta x.

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it can help us describe a line's property 

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it's the basic of calculus

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how fast one thing changes compared to another 

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use f'(x) by the definition of average rate of change instead of the instantaneous rate of change

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can be described as the words like quick or slow

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a car speeds up from 10m/s to 20/s in 1s,so the average rate of change of the car's velocity is 10m/s＾2

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The ratio of the variation to the reference quantity

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‌Rate of change (ROC) measures how a quantity changes relative to another, often expressed as the ratio of output change to input change 

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linear

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Its core meaning is relating to, resembling, or capable of being represented by a straight line.

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"The movie Pulp Fiction is famous for its non-linear plot, where scenes are presented out of sequence."

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Proportional, non-curved, with one input quantity corresponding to one output quantity.

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Equation: The equation y = 2x + 1 is a linear equation because its graph is a straight line.
· Relationship: "The spring exhibits a linear relationship between force and extension; if you double the force, the extension doubles." (This is Hooke's Law).
· Function: A linear function is one that satisfies the property f(a + b) = f(a) + f(b).

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The **rate of change** is a mathematical concept that describes how one quantity changes in relation to another quantity. It measures the speed at which a variable quantity changes. In calculus, the instantaneous rate of change at a specific point is defined as the derivative of a function. The average rate of change is calculated over an interval.

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Temperature:** A specific temperature reading, like 25°C, is a state, not a rate. The rate of change would be "the temperature is rising at 2°C per hour."
Distance Traveled:** This is a total amount or a position, not the rate at which it changes. For instance, "100 kilometers" is a distance, while "50 kilometers per hour" is the rate of change of that distance (speed).

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*Velocity:** This is the rate of change of an object's position with respect to time. For example, a car moving at 60 miles per hour has a position that changes by 60 miles for every hour that passes.
Acceleration:** This is the rate of change of velocity with respect to time. If a car's velocity increases from 0 to 60 mph in 10 seconds, its acceleration is 6 mph per second.

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it is the rate of y-axis changes dividing x-axis changes

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A symbol used to represent a quantity that can be assigned a value or can vary.

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Represent unknown and variable quantities with symbols.

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A straight line that extends infinitely, without any width or thickness.

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Rate of change is useful#

# data[].attributes.custom_properties.DYR1H3xk
The rate of change refers to how fast a physical quantity changes over time.

# data[].attributes.custom_properties.Hgnw2DZI
If it just means a quantity of change,it isn't the rate of change.

# data[].attributes.custom_properties.aD7SWzR8
It is a quantity which describes the change in velocity

# data[].attributes.custom_properties.eQQ066o4
The rate of change of velocity is acceleration.a=🔺v/t

# data[].attributes.custom_properties.DYR1H3xk
The rate of change measures the speed at which one number changes in relation to another.

# data[].attributes.custom_properties.aD7SWzR8
Has direction,it can be positive negitave or zero

# data[].attributes.custom_properties.DYR1H3xk
A variable is a symbol that represents a quantity that can change or vary. 

# data[].attributes.custom_properties.eQQ066o4
y=2x+3 Both the letters x and y are variable 

# data[].attributes.headline
Rate of change refers to the speed at which a variable changes within a specific time or space range, and it is usually measured by the amount of change per unit time (or per unit distance, etc.). 

# data[].attributes.subject
Rate of change refers to the speed at which a variable changes within a specific time or space range, and it is usually measured by the amount of change per unit time (or per unit distance, etc.). 

# data[].attributes.body
Rate of change usually means a velocity of change,however,in my opinion,it symbolize a kind of romance to find stability in changes

# data[].attributes.custom_properties.DYR1H3xk
Rate of change refers to the speed at which a variable changes within a specific time or space range, and it is usually measured by the amount of change per unit time (or per unit distance, etc.). 

# data[].attributes.custom_properties.DYR1H3xk
The Rate of Change is a measure of how quickly a quantity is changing over a specific period of time or in relation to another changing quantity.

# data[].attributes.custom_properties.Hgnw2DZI
also y = 2x + 1  1 in this example is constant instead of rate of change.

# data[].attributes.custom_properties.aD7SWzR8
The sign of the rate of change is its most immediate characteristic, indicating the direction of the change.

# data[].attributes.custom_properties.eQQ066o4
y = 2x + 1  the rate of change of this example is 2 

# data[].attributes.body
describe the change of two variables in a line

# data[].attributes.custom_properties.DYR1H3xk
the quantity describe the speed of change

# data[].attributes.custom_properties.aD7SWzR8
often be used to describe the trend of the change of speed 

# data[].attributes.custom_properties.DYR1H3xk
Two number which change follow an exact proportion could be a linear relationship

# data[].attributes.custom_properties.aD7SWzR8
It's a exact relationship between two number and one of them is variable.

# data[].attributes.custom_properties.DYR1H3xk
A variable refers to a symbol or name that represents a variable in mathematics, computer science, and other fields. Its core characteristic is that it has no fixed value and can change or be solved according to the context.

# data[].attributes.body
the quantity that can be change

# data[].attributes.headline
 A quantity that changes with another quantity

# data[].attributes.subject
 A quantity that changes with another quantity

# data[].attributes.body
I am SHAOSHUAI

# data[].attributes.custom_properties.DYR1H3xk
"Linear" means something that is straight or follows a direct sequence. It can also describe a relationship where change happens at a constant rate, like a straight line on a graph.

# data[].attributes.custom_properties.DYR1H3xk
Delta Y Over Delta X is a tool that can help you know the rate of y or x changing.

# data[].attributes.custom_properties.Hgnw2DZI
It is not the way that represent the rate by point

# data[].attributes.custom_properties.aD7SWzR8
Using x and y to prepesent the rate.
It displays the rate straightly.

# data[].attributes.custom_properties.DYR1H3xk
linear pertains to equations, functions, or relationships that can be represented by a straight line(a variable has a straight and priportnal change in other)

# data[].attributes.custom_properties.DYR1H3xk
Is a character where the Graph crosses the axis

# data[].attributes.custom_properties.Hgnw2DZI
don't confuse slope andintercept,for instance,2is the slope of the formula,and 6 is the intercept of it

# data[].attributes.custom_properties.aD7SWzR8
It can be positive negative or zero,it's used to describe the position of graph

# data[].attributes.custom_properties.eQQ066o4
for example,the formula“y=2x+6” 's intercept is 6

# data[].attributes.custom_properties.DYR1H3xk
The final variety of two at a extremely short time

# data[].attributes.custom_properties.aD7SWzR8
It can show the outcome about two variety over