Arithmetic
Arithmetic begins with Unit Operations. Students may select: Add, Subtract, Multiply or Divide, Choose to Include Negative Numbers, Freeze the Second Digit, and generate 1 5 or 10 questions at a time. There are 104 Unique operating modes. Scores track Attempts and Correct Answers for Each. An Activity Log matching the Score Page, may be printed from the Tools Help. It itemizes each operation and facilitates tracking and assessing student progress. A Number Line and Multiplication Table are examples intended to aid teaching and be a study reference while learning: Add, Subtract, Multiply and Divide.
Multi-digit Operations contains 4 tools and 4 examples. Examples animate the same way every time to aid teaching. Tools generate practice problems with detailed solutions each has an easier and harder operating mode.
Exponents Squares and Roots begins with a Graphic defining Base and Power and showing how to write it. Two Examples at the top illustrate Exponential Growth both Visually and Numerically. Tools at the Bottom require Estimating Radical Values and Simplification of Perfect Squares and Radicals with a Tutorial showing Prime Factorization.
The Expressions Section begins with an O-O-O Graphic, followed by 4 examples to be simplified by clicking through. The Make Expressions Activity has students combine digits with operations to make Target Numbers. Click a Target Number to display an Expression that uses Each Digit Once and Only Once.
Arithmetic Section Video
Properties and Operations
Simplify 4 Expressions
Below is a Numerical Expression meaning it simplifies to a single number Click on it to simplify
(4 + (8 • 2²)) / ((2³ - 2) • 6)
(4 + (8 • 4)) / ((8 - 2) • 6)
(4 + 32) / (6 • 6)
36 / 36
1
Reset
Make Expressions Activity
Expressions Overview Video
Properties and Operations
Commutative
Distributive
Associative
Order of Operations
Addition accumulates Addends to produce a Sum
Addends are Commutative meaning they can be in any order and values can commute from one term to another
6 + 4 = 4 + 6 or 6 + 4 = 7 + 3
Subtraction takes away a Subtrahend from a Minuend producing a Difference
Subtraction is NOT Commutative meaning order matters
Multiplication: Multipliers produce a Product
Multipliers are Associative meaning terms can be in any order
6*4 = 4*6 or A*B = B*A
Division: Divides a Dividend by a Divisor producing a Quotient and possibly a Remainder
Division is NOT Associative meaning order matters
Distribution: is different for Addition and Multiplication
Click Expressions for Answers
2(2+3+5)=20
2(2•3•5)=60
9(1+1+1)=27
9(1•1•1)=9
5(2+0+5)=35
5(2•0•5)=0
3(2+3+5)=30
3(2•3•5)=90
reset
Order of Operations
First: ( ) ParenthesisFirst letter of operations in order are
PEMDAS
A common phrase to remember this order is
Please Excuse My Dear Aunt Sally
Reset
(4 + (8 • 2²)) / ((2³ - 2) • 6)
(4 + (8 • 4)) / ((8 - 2) • 6)
(4 + 32) / (6 • 6)
36 / 36
1
2³ • -3² / (16 + (1 - 3²))
8 • 9 / (16 + (1 - 9))
72 / (16 - 8)
72 / 8
9
10 + 5 • -2 + 20 - 4 • 5
10 - 10 + 20 - 20
0
10³ / 10² - 25
1000 / 100 - 25
10 - 25
-15
Addition accumulates Addends to produce a Sum
Addends are Commutative meaning they can be in any order and values can commute from one term to another
6 + 4 = 4 + 6 or 6 + 4 = 7 + 3
Subtraction takes away a Subtrahend from a Minuend producing a Difference
Subtraction is NOT Commutative meaning order matters
Multiplication: Multipliers produce a Product
Multipliers are Associative meaning terms can be in any order
6*4 = 4*6 or A*B = B*A
Division: Divides a Dividend by a Divisor producing a Quotient and possibly a Remainder
Division is NOT Associative meaning order matters
Addition
Subtraction
Multiplication
Long Division
Example
Example Example
Example
Online Tools: generate fresh content for practice, study and assessment. They keep track of attempts and correct answers in each mode.
Interactive Examples: show detailed operations and more, they are useful for teaching and study, replaying exactly the same every time.
Exponents have a Base and a Power
BP
BP = B • B • B • B • B... P times
103 = 10 • 10 • 10 = 1000
3 EXPONENTIAL EXAMPLES
Power of 10s Powers Cube 2N Activity
Estimate Radical Value
Simplify Square Roots
Squares and Square Roots are Inverses
√Num² = Num
√25 = √5² = 5
Review Perfect Squares thru 12
To Simplify Radicals Factor the Radicand
into Primes and Promote the Squares
√75 = √5•5•3 = √5²•3 = 5√3
Prime Factorization
Perfect Squares: Radicands are randomly generated from:
4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144,
or any of the numbers above times 100.
In either case the answer is a perfect square.
Radicals: Generates a radicand from 5 prime numbers
either 2's, 3's, 5's, and possibly a 7 or an 11. Students
factor the number into primes removing perfect squares.
Attempt / Correct
Sq:0/0 Rad:0/0
resethide
Perfect Squares
Radicals
NewAnswer
How to Simplify
Easy mode: generates a target number made from 3 prime numbers either 2s, 3s or 5s.
Hard mode: generates a target number made from 5 prime numbers either 2s, 3s or 5s and may include either a 7 or an 11.
Enter the prime factors in any order
Prime Factorization for
96 and 97
Attempt / Correct
E:0/0 H:0/0
resethide
EasyHard
NewAnswer
Finding Factors of 2 3 5
Factorization of 96 97
Divisibility Rules for 2's 3's and 5's
If the number is even; 2 is a factor
If the number ends in 0 or 5; 5 is a factor
If the sum of digits is divisible by 3; 3 is a factor
This tool generates numbers up to 10000. Students click the line-segment where the square root lies
Students learn the Square and Root relationship between:
10 20 30 40 50 60 70 80 90 100
100 400 900 1600 2500 3600 4900 6400 8100 and 10000.
Powers of Ten
Graph : Y = 2X for 1 Month
The X-axis is simple, each square is 1 and we need 30 squares.
The Y-axis is not so simple. We need a Logarithmic Scale where the bottom squares increase by 1 to be able to show the first few values and the top squares have units increasing by Millions.
We do not show such graphs in this program.
N is #Day : 2N = Pennies
1 = 21 = 1¢
2 = 22 = 4¢
3 = 23 = 8¢
4 = 24 = 16¢
5 = 25 = 32¢
6 = 26 = 64¢
7 = 27 = 128¢
8 = 28 = 256¢
9 = 29 = 512¢
10 = 210= 1024¢
11 = 211 = 2048¢
12 = 212 = 4096¢
13 = 213 = 8192¢
14 = 214 = 16384¢
15 = 215 = 32768¢
16 = 216 = 65536¢
17 = 217 = 131072¢
18 = 218 = 262144¢
19 = 219 = 524288¢
20 = 220 = 1048576¢
21 = 221 = 2097152¢
22 = 222 = 4194304¢
23 = 223 = 8388608¢
24 = 224 = 16777216¢
25 = 225 = 33554432¢
26 = 226 = 67108864¢
27 = 227 = 134217728¢
28 = 228 = 268435456¢
29 = 229 = 536870912¢
30 = 230 = 1073741824¢
Discuss How to Graph this Function
Save 1 Penny on Day 1, then 2 Pennies on Day 2, then keep doubling Pennies each day for one month?
#Pennies = 2n
Where n is the Day
21 = 2 ¢
22 = 4 ¢
23 = 8 ¢
24 = 16 ¢
25 = 32 ¢
- - - - - -
How many Pennies will you save on Day 30?
230 = 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 = 1073741824 Pennies = $10,737,418.24 → Over 10 Million Dollars.
Show Details
Attempt / Correct
0 / 0
reset
hide
New Number
Example estimate: √3711
√3711 is between √3600=60 and √4900=70 and
closer to √3600, click Line-Segment 60 to 65
Click the line-segment containing the value
2-Digit mode generates two 2-digit numbers to be added
4-Digit mode generates three 4-digit numbers to be added
NOTE: the answer field is different - the digits are entered in reverse order. See the Help Video.
Attempt / Correct
2Dgt:0/0 4Dgt:0/0
hidereset
2-Digit 4-Digit
NewAnswer
NOTE: the answer field tabs right to left so students may enter Sum as calculated. See an example in the Help Video.
Clear Input
Solution
Easy mode: Divisors are 2 or 10
Hard mode: Divisors are 3 through 9
Division is the Inverse of Multiplication. If you have learned Multiplication Facts, Division Facts are the same Numbers re-arranged. See examples below:
3 * 5 = 15 ←→ 15 ÷ 3 = 5 ←→ 15 ÷ 5 = 3
6 * 7 = 42 ←→ 42 ÷ 7 = 6 ←→ 42 ÷ 6 = 7
12 * 8 = 96 ←→ 96 ÷ 8 = 12 ←→ 96 ÷ 12 = 8
Long Division Example
Step-by-Step Solution
Full Solution
Show Solution
Answer: 140.1111111... repeating
Written as: 140.1
Attempt/Correct
E:0/0 H:0/0
hide
reset
divisors
2,10 3 - 9
NEWANSWER
Use pencil and paper to solve
then enter answer
remainder
Multiply: 3 * 7 = 21, write the 1, carry 2
Multiply: 3 * 2 = 6, add 2, write 8
Multiply: 3 * 3 = 9, write 9
Multiply 1 by 327, shift partial product using a "0"
First, shift partial product 2 digits using a "00"
Multiply: 5 * 7 = 35, write 5, carry 3
Multiply: 5 * 2 = 10, add 3, write 3, carry 1
Multiply: 5 * 3 = 15, add 1, write 16
Add Partial Products to get Final Answer
Easier mode: generates a 2-digit number and a 1-digit number to be multiplied.
Harder mode: generates two 2-digit numbers to be multiplied.
Use Pencil and Paper to Copy and Solve Each Operation
☺ Practice Often ☺
Attempt / Correct
E:0/0 H:0/0
hide
reset
Easier
Harder
NewAnswer
Solve Using Pencil and Paper
No Calculators Please
=
Easier mode: generates two 2-digit numbers to subtract.
Harder mode: generates two 4-digit numbers to subtract.
3 Examples
Reset
Answer
ones
tens-hundreds
Answer
ones
tens-hundreds
Answer
ones
tens-hundreds
reset
Attempt / Correct
2Dgt:0/0 4Dgt:0/0
hidereset
2-Digits
4-Digits
NewAnswer
The Answer Field is entered right to left ← ← so students may enter the answer as they calculate it. The Help Video shows an example.
Show Solution
You may:
• Select: Add, Subtract, Multiply or Divide
• Choose to include Negative Numbers
• Freeze Second Digits
• Generate 1, 5 or 10 Questions at a time
• View Scores Until the Window closes
• Use Number Line to explain, Add and Subtract
• Use Multiplication Table, to explain Patterns
Attempts/Corrects
NewAnswer
1510
#questions
Add
Sub
Mult
Divide
positive only
include negatives
Freeze Second Digit
no
12
34
56
7
8 9 10 11 12hide
How To
Add and SubtractMultiply and Divide
Multiplication Table
Number Line
Use numbers on each die to make a Target Number
Click a Target Number to display a solution.
(!) Factorial and more
4 : (1 - 1) * 2 * 3 + 4
14 : 4 * 3 + 2 + 1 - 1
24 : 2 * 3 * 4 + 1 - 1
34 : 2³ * 4 + 1 + 1
44 : 4(3² + 1 + 1)
54 : 3²(4 + 1 + 1)
64 : 2 * 4(1 + 1)³
74 : 34 - (2 + 1)! - 1
84 : 34 + 2 + 1 * 1
94 : (4 + 1)! - (3 + 1)! - 2
Hide Expressions
New Roll
In this activity instructors may allow students to use the numbers more than once or not at all directly effecting the difficulty in making expressions. The expressions shown here, use each number exactly 1 time, to accomplish this for some target numbers a Factorial (!) Operation is used.
Factorial means multiplying down to 1, for Example:
3! = 3 * 2 * 1 = 6
4! = 4 * 3 * 2 * 1 = 24
5! = 5 * 4 * 3 * 2 * 1 = 120
Factorial is used in statistics to calculate probability and more. We do not discuss it further in this program.
Please extend this activity if you like! Roll your own dice and choose your own target numbers.
Use numbers on each die to make a Target Number
Click a Target Number to display a solution.
(!) Factorial and more
7 : 5 + 2 + 2(3 - 3)
17 : 5 * 2 + 3! + (3 - 2)
27 : 5² + 2 + 3 - 3
37 : 2³ * 5 - (3! ÷ 2)
47 : 5 * 2(2 + 3) - 3
57 : (3! + 5)(2 + 3) + 2
67 : 5² * 3 - 2³
77 : (5 + 2)(2³ + 3)
87 : 3² * 5 * 2 -3
97 : 5²(3! - 2) - 3
Hide Expressions
New Roll
Show Groups Made
You made 0 groups
New Numbers
☺☺☺
make-10
count groups made
Multiplication thru 12x12
Any Number Times 0 = 0
Any Number Times 1 is the Same Number
Multiplying by 10 Adds a Zero to the Number
1•10=10; 2•10=20 3•10=30 4•10=40 . . .
Multiples of 5 End in 0 or 5, they are Half
of their Corresponding Multiple of 10
Multiplying by 2 Doubles the Number, Use
Halves and Doubles in Counting Section for Practice
Multiplying by 9 is like Multiplying by 10
then Subtracting the Number
Multiplying by 11 is like Multiplying by 10
then Adding the Number
Multiplying by 12 is like Multiplying by 10
then Adding the Number Doubled
A Number Multiplied by Itself it is called Squared
1•1=1² 2•2=2² 3•3=3² . . . X•X=X²
Multiplication is Associative meaning is is the same forward of backwards. Half the table is the same!
Say them! Write them! Learn them!
Number Line
Unit Operations
-5 to 5-50 to 50-500 to 500
Multiplication Table
Unit Operations