GRAPHING

Counting  Arithmetic  Fractions  Graphing  Algebra  Geometry

Points and Coordinates

Parts of a Graph

Slope and Y-Intercept

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GRAPHING

The Graphing section begins with Points and Coordinates featuring two tools. First is a Number-line where students Identify Points, Distance and Vectors, entering .5 for half units. The second tool generates 5 Shapes on a Grid. Students locate the (X,Y) coordinates, entering them next to the Shape. Easy Mode uses positive units 0 to 10, while Hard Mode uses units -10 to 10.

Parts of a graph contains a series of Examples and Activities starting with a 20x20 grid. Animations add Origin, Axis, Units and Functions. A graphic discussing Y as a Function of X, explains the dependent relationship of Y, and the independent X. The Dollar per Day Word Problem Activity, has students use Functions to calculate Y when X=1,2,3..etc, and produce a Neatly Labelled Graph.

The Slope and Y-Intercept section features a Tool where students determine slope(M) and y-intercept(B) from two points. In easy mode one point is on the Y-Axis and only the slope need be calculated and entered in lowest terms. In Hard mode one (X,Y) point is substituted into the Slope Intercept Equation Y=mX+b to solve for the Intercept (B). Detailed Solutions are generated for all lines.

An example of Slope(M) shows how to determine Rise over Run. Lines with the same slope highlight the effect of changing B. An Estimate Slope Intercept Activity ties the concepts together by having students drop a toothpick on a grid - mark dots on the ends - draw a line through the dots, and estimate the y=Mx+B equation.

Play Section Video

Screen Print Grid

-10-9-8-7-6-5-4-3-2-11234567891010987654321-1-2-3-4-5-6-7-8-9-10

ABC123MATH 2024

Start with a 20 x 20 Grid

Origin   Axis & Units   Functions

Origin (0,0) Origin in the Center Draw X,Y Axes Through the Origin Add Units -10..10 on Each Axis Functions Relate X and Y Values Y is a Function() of X Y = f(X) 3 Examples Y = -X Y = 2X Y = ½X + 5 origin(0,0) X - a x i s Y - a x i s -10-9-8-7-6-5 -4-3-2-112 345678 910 1098765 4321-1-2 -3-4-5-6-7-8 -9-10 (-8,8) (8,7) (3,-3) Line:   Y = -X • • • • XY -55 -22  00  2-2  5-5 Line: Y = 2X • • • • XY -4-8 -2-4  00  24  48 Line: Y = ½X + 5 • • • • XY -62 -24  05  26  68

Functions: Y=f(X)

$1/day Word Problems


Functions relate X and Y values. Y=F(X) is read:
Y is a Function of X. We say Y depends on X, and create a Data Table where X=1,2,3... and Y is Calculated.

Below X is Time and Y is Temperature. We may collect (X,Y) points in a Data Table and Graph them.


Time for Ice Water to Boil

Temperature C° Time in Minutes 01020304050607080901001100123456789101112131415 Data Tables(Big Pot)XY 1 2 3 4 5 6 7 8 910 111213101525304045556070758590100 (Small Pot) XY 1 2  3 4 521 426384100

Notice the Graph has a Title, X and Y Axis labelled with units, Different colors used as a key.

-10-20-30-40-500 1020304050 X - Axis

The Y-Axis is similar but vertical, up and down.

The 2 axes cross at the Origin (0,0) of the graph, the point where both X=0 and Y=0, creating a coordinate plane with infinite (X,Y) points.

-10-20-30-40-500 1020304050

<=-SUBTRACTION (-)        (+) ADDITION-=>

-20 - 20 = -40 where to start ⇒ direction to move ⇓ how many to move ⇑

Estimate Slope and Y-Intercept

and write the Equation   y = Mx + B


1. Drop a toothpick on a grid and mark a dot at each end. Draw a line through the dots and add arrows to each end of the line.

2. Estimate Slope(M) and Y-Intercept (B). Then write the equation in y=Mx+B format.

Print Activity Sheet

Calculate Slope & Intercept

The easiest line to graph is Y = X

Y-Axis X-AxisY = 1X + 0M = 1     B = 0 Data TableX     Y -7-7-3-3 002½2½ 66 •••••

Slope (M)

Y-Intercept (B)

Make Lines Activity

Parallel Lines have Equal Slopes

-10-9-8-7-6-5-4-3-2-112 34567891010987654321-1-2-3-4-5-6-7-8-9-10 Hover Equations Y = 2X + 0 Y = 2X + 2 Y = 2X + 4 Y = 2X + 6 Y = 2X + 8 Y = 2X - 2 Y = 2X - 4 Y = 2X - 6 Y = 2X - 8 Y = 2X + 8Y = 2X + 6 Y = 2X + 4Y = 2X + 2 Y = 2X - 2Y = 2X - 4Y = 2X - 6 Y = 2X - 8 • -4-8 -2-4 00 24 48 • • • Data Table XY Slope = 2Y-Intercept = 0

Slope (M)

Y=Mx+B Example

Slope (M) =riserun△Y△X= ΔY = 13ΔX = 13 ΔY = -12ΔX = 12 ΔY = 20ΔX = 2 ΔY = -20ΔX = 2 ΔY = 2ΔX = 20 ΔY = -2ΔX = 20 ΔY = 0ΔX = 20 ΔY = 20ΔX = 0 M = 1M = -1M = 10M = -10M = 1/10M = -1/10 M = 0 cannot divide by 0 Undefined Slope ΔY = 13ΔX = 13 ΔY = -12ΔX = 12 ΔY = 20ΔX = 2 ↑ ΔY = -20ΔX = 2 ↓ ΔY = 2ΔX = 20 ΔY = -2ΔX = 20 ΔY = 0ΔX = 20 ΔY = 20ΔX = 0 -10-9-8-7-6-5-4-3-2-112 34567891010987654321-1-2-3-4-5-6-7-8-9-10 Hover Slopes 1 -1 10 -10 1/10 -1/10 0 undefined

Y-Intercept

y=Mx+B Example

X-Axis: DaysY-Axis: Dollars

$1/Day Word Problems Data Table ←y-axis→ ← x - axis → 01020304050607080901001100123456789101112131415 XY 1 2 3 4 5 6 7 8 9101112131413610152128364555667891104

It took 14 Days to get over $100

More Word Problems

Parts of a Graph

It is easy to change the Function(x). Instead of one dollar a day use 2 dollars a day, or make the Target Amount $200, or both. Then have students produce a data table and neatly labelled graph. Modify criteria and repeat as necessary.

If you get 1 dollar on day 1,   2 dollars on day 2,
3 dollars on day 3, and so on...   How many days until you have Over $100 dollars?

Show Solution

Print Activity Sheet

This tool generates 5 shapes on a grid. Students locate and enter (X,Y) Coordinates.

Easy: 0..10 Grid       Hard: -10..10 Grid

Attempt / Correct
Easy : 0/0     Hard : 0/0
resethide

←Select Grid→
EasyHard
NewAnswers

Enter (X,Y) Coordinates

(x,y)

(,)

(,)

(,)

(,)

(,)

-10-9-8-7-6-5 -4-3-2-112 345678 910 1098765 4321-1-2 -3-4-5-6-7-8 -9-10 0012 3456 78 910 1098 7654 321 ; ; ; ; ;


Easy Mode: X=0 for one Point, so the y-Intercept (B) may be read off the graph without any calculation.
The Slope (M) may be calculated, or counted, then entered in lowest terms

Hard Mode: requires a slope calculation same as above then the point with the smallest X value gets substituted into the Equation: y = Mx + B to solve for y-Intercept (B)

Attempt/Correct
E:0/0   H:0/0
resethide

Easy Hard
NewAnswer

M = /     B = /

Great Job! detailed solution -10-9-8-7-6-5 -4-3-2-112 345678 910 1098765 4321-1-2 -3-4-5-6-7-8 -9-10

y=Mx+B Example

In Points mode a point is on the line and students enter the coordinate.

In Distance mode two points are on the line and students count or calculate the distance between them.

In Vector mode Direction and Distance from A to B is required, so the answer may be positive or negative.

Enter .5 for Half Units

Attempt/Correct
P:0/0   D:0/0   V:0/0
hidereset

Points Distance Vector
NewAnswer

01 23456 78910 ← X - AXIS →

(X,Y) Coordinates