math
Topics
Addition
Tips, Notes, and Practice Activity on Important Addition Concepts: Intro to Addition, How to Add, Addition with Multiple Digits, Carrying Over with Addition.
Subtraction
Tips, Notes, and Practice Activity on Important Subtraction Concepts: Intro to Subtraction, How to Subtract, Subtraction with Multiple Digits, How to Subtract Larger Numbers with Borrowing, and Borrowing When There is Nothing Nearby to Borrow.
Time
Tips, Notes, and Practice Activity on Important Time Related Concepts: Intro to Time, Units of Time, and How to Tell Time.
Money
Tips, Notes, and Practice Activity on Important Money Related Concepts; Intro to Money, Types of Currency in the United States, and How to Count Money.
Multiplication
Tips, Notes, and Practice Activity on Important Multiplication Concepts: Intro to Multiplication, Multiplying One Digit Numbers, Multiplying Two Digit Numbers, and more.
Division
Tips, Notes, and Practice Activity on Important Division Concepts: Intro to Division, Short Division, Long Division.
Fractions
Tips and Notes for Important Fractions Concepts: Intro to Fractions, Comparing Fractions, and Operations with Fractions
*For Practice Activity, go to the AMA Learning 6-8 Pre-algebra Section to access the Fraction & Decimal Practice Activity.
Decimals
Tips and Notes for Important Decimals Concepts: Intro to Decimals, Comparing Decimals, and Operations with Decimals
*For Practice Activity, go to the AMA Learning 6-8 Pre-algebra Section to access the Fraction & Decimal Practice Activity.
Addition
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Addition: Notes
What is addition?
● When you add numbers together
● Numbers added together make a sum (answer)
● Several numbers/digits can be added together
● Equations/Number sentences are either horizontal or vertical
● 3 Digit Numbers (Hundreds, Tens, Ones)
How to do addition?
● 3 methods:
○ Mental math
- Students who choose to memorize each single digit addition equation
most commonly use this method - Example: Solve 2+3
- Use memorized table to say the answer is 5
○ Counting forward mentally
- Students count forward mentally to solve single digit addition equations
- Example: Solve 2+6
- Say to yourself, “2, 3, 4, 5, 6, 7, 8” or, “2, 4, 6, 8”
- Counting mentally to say the answer is 8
○ Counting forward using fingers
- Students use fingers by counting by ones to solve single addition digit
equations - Example: 3+4
- Count on fingers, “3, 4, 5, 6, 7”
- Using fingers to say the answer is 7
Addition with Multiple Digits:
○ When we add numbers with 2 or more digits we add digits of each place value
together, then add the numbers of the next place value.
- Example: 10+13
Or 10
+13
------
23
- Start with ones place, then tens places, then hundred, then thousands, etc.
- Add 0+3 to get 3 then 1+1 to get 2. Put the two new numbers together in place value order (tens place, then ones place)
- The answer would be 23.
○ Easier to solve addition equations with 2 or more digits vertical.
○ Remember which place value each number is in and goes to!!
Carrying Over with Addition:
● If the sum of the digits in a place value is more than 9, write the value of the ones place
and add 1 to the next place value over (Remember go to the left!)
● Since 9 is the last single digit number on the number line, we add a 1 to the next place
value in order to represent the 2nd digit of that number
○ Example: 230+299
Or
+1
230
+299
------
529
● First you would add 0+9 to get 9 and write it in the ones place
● Then you would go on to the tens place and add 9+3 to get 12
● You would write 2 in the tens place and write +1 on top of the hundreds place
○ Since 12 is not a single digit number, we have to carry its second digit (1) over to the next place value
● Finally you would add 1+2+2 to get 5
● Your final answer would be 529
● Remember which place value each number is in and goes to!!
Subtraction
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Subtraction: Notes
What is Subtraction?
● When you subtract numbers from each other
● Numbers subtracted from each other make a difference (answer)
● Several numbers/digits can be subtracted from one another
● Equations/Number sentences are either horizontal or vertical
● 3 Digit Numbers (Hundreds, Tens, Ones)
How to do Subtraction?
● 3 methods:
○ Mental math
- Students who choose to memorize each single digit subtraction
equation most commonly use this method - Example: Solve 3-2
- Use memorized table to say the answer is 1
○ Counting backwards mentally
- Students count down mentally to solve single digit subtraction equations
- Example: Solve 8-4
- Say to yourself, “8, 7, 6, 5, 4” or, “8, 6, 4”
- Counting down mentally to say the answer is 4
○ Counting backwards using fingers
- Students use fingers by counting down by ones to solve single
subtraction digit equations - Example: 5-3
- Count on fingers, “5, 4, 3, 2”
- Using fingers to say the answer is 2
How to subtract larger numbers:
○ When we subtract numbers with 2 or more digits we subtract digits of each
place value together, then subtract the numbers of the next place value.
- Example: 24-13 =
Or
24
-13
----
?
- Start with ones place, then tens places, then hundred, then thousands, etc.
- Subtract 4-3 to get 1 then 2-1 to get 1. Put the two new numbers together in place value order (tens place, then ones place)
- The answer would be 11.
○ It is easier to solve subtraction equations with 2 or more digits horizontally.
How to subtract larger numbers: Borrowing
361
-115
-------
?
In the equation above, you would start subtracting with the 1 at the end of “361” and the 5 at the end of “115 .” 1-5 is possible to do, but it is not what someone would do in this situation. In this situation, a person would use a method called “borrowing.”
● In a place value, if the top digit is less than the bottom digit you would subtract a 1 from the next place value over, then you add a 1 in front of the top digit.
● Once the top digit is greater than the bottom digit, you can subtract as usual.
● The equation would now be:
11
351
-115
-------
● First, you would subtract 11-5, which would be 6
● Then you’d move to 5-1= 4
● Then 3-1=2
● The final answer would be 246
● Remember which place value each number is in and comes to!!
Borrowing when there is nothing nearby to borrow!
Say the equation is:
3005 -1008
When you cannot borrow from the next digit over, as you cannot borrow from 0, you just go to the nearest digit with value. In this example, 5-8 is not subtractable, so you would borrow 1 from the 3 in 3005 and move it in front of the 5 in 3005, making the equation work. The new equation would be:
15
200
-1008
----------
● You would subtract 15-8, which equals 7
● Then 0-0=0
● Then 0-0=0
● Finally 2-1= 1
● Remember which place value each number is in and goes to!!
Time
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Time: Notes
What is time?
● Time is the duration of all things that happened, or the moment when they happen.
● It is a system created for order in the world, and helps people determine when certain things will happen.
● Times can be in the A.M. (ante meridiem, the latin phrase for “before noon”) and the P.M. (post meridiem, the latin phrase for “after noon”).
How do you count time?
Time is counted in:
● Seconds
● Minutes
● Hours
How do you read a clock?
This clock in the upper left hand corner is showing that the time is 10:13:
● The short black hand of the clock points to the hour
● The longer black hand points to the minute
● The red hand represents seconds.
How do you tell someone what time it is?
There are different ways to tell time: If you were telling someone the time was 2:30
● You could say: “It’s two-thirty.”
● You could also say: “It’s half past two.” If the time was 4:45, you could say:
● “It’s four-fourty-five.”
● You could also say: “It’s a quarter to five.”
Military time vs regular time?
● Military and regular are two ways you can tell time.
● You can also set a digital clock to either method of showing time.
Military:
0000 0100 0200 0300 0400 0500 0600 0700 0800 0900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300
Regular:
12:00A.M. 1:00A.M. 2:00A.M. 3:00A.M. 4:00A.M. 5:00A.M. 6:00A.M. 7:00A.M. 8:00A.M. 9:00A.M. 10:00A.M. 11:00A.M. 12:00A.M. 1:00P.M. 2:00P.M. 3:00P.M. 4:00P.M. 5:00P.M. 6:00P.M. 7:00P.M. 8:00P.M. 9:00P.M. 10:00P.M. 11:00P.M.
Important time lengths?
● 60 seconds in a minute
● 60 minutes in an hour
● 24 hours in a day
● 365 days in a year (366 in leap years, years that have a Feb 29)
● 10 years in a decade
Money
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Money: Notes
What is money?
● Way to exchange items (Pay money to buy an item)
● Means of payment
● Way to measure value of an object
● Bills or Coins
● United States: Dollars ($) or Cents (¢)
How do you count money?
● Using Coins or Dollar Bills
● Coins have a heads side and tails side
● 100¢ in $1
● Most Common Coins and Dollar Bills:
○ Penny:
- Worth 1 Cent (¢)
- Has picture of Abraham Lincoln on heads and Lincoln Memorial on tails
○ Nickel:
- Worth 5¢
- Has picture of Thomas Jefferson on heads and Monticello on tails
○ Dime:
- Worth 10¢
- Has picture of Franklin Roosevelt on heads and olive branch on tails
○ Quarter:
- Worth 25¢
- Has picture of George Washington on heads and bald eagle on tails
○ $1 Bill:
- Worth 1 dollar ($: Always before the value)
- Has picture of George Washington
○ $5 Bill:
- Worth $5
- Has picture of Abraham Lincoln
○ $10 Bill:
- Worth $10
- Has picture of Alexander Hamilton
○ $20 Bill:
- Worth $20
- Has picture of Andrew Jackson
When do you use money?
● You want to purchase a good or service
- ○ Buy a house
- ○ Pay a handyman to fix a broken roof
Multiplication
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Multiplication: Notes
What is multiplication?
● Important order of operation
● Adding a number to itself a specific number of times
● Multiplication Sentence: Multiplicand (#) x Multiplicand (#) = Product (Answer)
● Can be performed with any number, no matter how many digits it has
● Can multiply horizontally or vertically
● Symbols used to multiply: x, *, ᐧ
How to multiply?
● You are adding a number to itself a said number of times
○ Example: Multiply 6 x 3
Or
6
x 3
-----
○ Add 6 to itself 3 times
○ You would think in your head, “6+6+6”
○ The answer would be 18
● The best way to perform multiplication in to memorize the multiplication tables (SEE CHART IN PDF FILE ABOVE)
○ If you memorize the multiplication tables, when you get a question like 7 x 7, you won’t need to think, “What is 7+7+7+7+7+7+7 ?”
Multiplying Numbers With More Than One Digit:
● You can also multiply numbers that don’t have the same number of digits.
- Place the number with more digits above the number with less number of digits.
- Multiply the numbers in the ones place together, and write the answer under the
bottom number’s ones place. - Multiply the bottom number’s ones place with the top number’s tens place, and
write that answer to the left of your answer to Step 2 to get your final answer.
a. Always remember to solve from right to left.
● Example: Multiply 21 x 4
Or
21
x 4
------ (Write the equation like this!)
- You would first multiply 1 x 4 to get 4, and write that under the 4
21
x 4
-----
4
- Then, you would multiply 4 x 2 to get 8, and write that next to the 4
21
x 4
------
84 (Final Answer)
● If the top number has 2 or more digits, you can use this strategy to solve those equations
as well!
Carrying Over:
● In multiplication, sometimes you have to carry over a number because the product is
more than 9.
● Similarly to addition you have to carry over the tens digit to the next step of solving
the equation and add it to the next product.
● Follow the same steps as before, but if the product of a step is greater than 9, then add the
tens digit to the next product in the line
● Example: Multiply 35 x 8
Or
35
x 8
------ (Write the equation like this!)
- Multiply 5 x 8 to get 40, and write 0 under the 8 and add a +4 on top of the 3
+4
35
x 8
------
- Multiply 8 x 3 to get 24 and add 4 to it to get 28. Write 28 next to the 0.
+4
35
x 8
------
280 (Final Answer)
Double & Triple Digit Multiplication:
● Always make sure that you solve double and triple digit multiplication problems vertically!
● When multiplying double digit numbers, make sure to follow these step by step rules:
1. Multiply the bottom ones place number with the top ones place number, and write the answer beneath the bottom ones place number.
2. Multiply the bottom ones place number with the top tens place number, and write the answer beneath the bottom tens place number.
3. Write a zero under the answer to Step 1. This is a placeholder: it will allow you to multiply the bottom tens number.
4. Multiply the bottom tens number to the top ones number, and write the answer under your answer to Step 1.
5. Multiply the bottom tens number to the top tens number, and write the answer to the left of your answer to Step 4. This will create a hundreds place in your answer. 6. Finally, add your answers from Step 2 and Step 5 to get a final answer.
● Note: You can use these steps to solve 3 digit multiplication problems as well! You would just need to add another line of products from the bottom hundreds place under the products of the bottom tens place.
● Example: Multiply 34 x 23
Or
34
x 23
-------- (Write the equation like this.)
- First you would multiply 4 x 3 to get 12. You would write 2 under the 3 and add a +1 on top of the 3.
+1
34
x 23
------
2
- Then, you would multiply 3 x 3 to get 9 and add 1 to it to get 10. You would write 10 next to the 2.
+1
34
x 23
------
102
- You would add 0 under the 2 as a placeholder.
+1
34
x 23
------
102
0
------
- You can choose to get rid of the +1 since you don’t need it anymore. You would multiply 2 x 4 to get 8, and write that next to the 0.
+1
34
x 23
------
102
80
------
- You would multiply 3 x 2 to get 6, and write that next to 8.
+1
34
x 23
------
102
680
------
- Finally you would add 102+680 to get 782 as a final answer!
+1
34
x 23
------
102
680
------
782 (Final Answer)
Division
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Division: Notes
What is division?
● Division splits a number into a number of equal groups
○ For example, 8 divided by 4 = 2
■ There are 4 groups of 2 in 8. You can check this with multiplication: 2*4 = 8. You can also check with addition: 2+2+2+2 = 8. There are 4 groups of 2!
● When the dividend is divided by the divisor to make the quotient (final answer)
○ In the problem above, the dividend is 8, the divisor is 4, and the quotient is 2.
Dividend
------------ = Quotient
Divisor
8
----- = 2
4
What will it look like?
● Division appears in 3 forms:
○ Dividend ÷ divisor - Most common division sign
○ Dividend / divisor - Another sign for division, looks like a fraction
○ Divisor ⟌Dividend - Long division
How to divide?
● Dividing is simply splitting a number into a number of equal groups. The dividend is split into the number of groups that the divisor is.
○ If the dividend was 9 and the divisor was 3, that would mean to solve the problem, a person has to split 9 into 3 groups. The answer, or the quotient, would be 3, because 3+3+3 = 9.
○ If the dividend was 10 and the divisor was 5, the answer would be two, because 10 = 2+2+2+2+2.
● You should try to remember your division tables just like you do your multiplication tables, because then you won’t have to think very hard when you encounter a simple division problem
How do I long divide?
● Division gets more complicated
○ Sometimes numbers don’t come out even when divided
○ Sometimes numbers are too large to be remembered as fast facts, for a fast facts chart, look below
● For larger numbers, there is something called long division.
● Long division looks like this:
○ Divisor ⟌Dividend
**TO SEE A MORE CLEAR/DESCRIPTIVE EXPLANATION OR EXAMPLE, PLEASE GO TO THE PDF VERSION**
Fractions
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Fraction: Notes
What are fractions?
● A part of a whole
● Numerators and Denominators:
Numerators: Number of parts (Top #)
-----------------------------------------------------------------------------------
Denominators: Number of Parts the Whole is divided into (Bottom #)
What are Equivalent fractions?
● Fractions that look different but are actually the same
○ 4 2 1
--- = ---- = ---
8 4 2
● These fractions are the same: as you go to the right, you will notice that they are cut in half
● These fractions are reduced or simplified
● Reduced and Simplified Fractions are found by multiplying or dividing the numerator and denominator by the same number
How do you convert fractions to decimals?
● Simply divide the numerator into the denominator
○1
---
2. = 1 ÷ 2 = 0.5
What about percentages?
● A fraction out of one hundred
● Make the denominator fraction equal to 100 and find the missing numerator
○ Percentage of 1 x25 = 25
----------------
4 x25 = 100 25%
What are proper and improper fractions?
● Proper fractions: Numerator is less than the denominator
● Improper fraction: Numerator is greater than the denominator
What about mixed numbers?
● Simplified Improper Fractions
● Whole Number with a fraction next to it
● Mixed # to Improper Fraction: Multiply denominator to whole #, add numerator (Numerator value), denominator stays the same
7
○ 3 1⁄2 =---
2
- Multiply 3 x 2 to get 6, then add 1 to get 7 as numerator, 2 stays the same
● Improper Fraction to Mixed #: Divide numerator into denominator (Whole #), and take remainder value (numerator) and put it over the denominator which stays the same
○ 7
----
2 = 7 ÷ 2. = 3 R1 = 3 1⁄2
- Divide 7 into 2 to get 3 R1, put the 1 over the 2 to get the mixed number answer of 3 1⁄2
How to Add and Subtract Fractions?
● Need the same denominator in order to add or subtract fractions
● Use the Least Common Denominator (LCD) to make the denominators the same
● To find the LCD, you can do 2 things:
○ Multiply the 2 denominators together and find the LCD
■ 1 3 1 6 3 9
---- --- --- --- ---- ----
3 and 6 3 x 6=18 so 3 = 18 and 6 = 18
○ List the multiples of each denominator and select the first multiple that both numbers have
■ 1 3 3: 3, 6, 9 1 2 3 3
---- ---- ---- ----- ---- -----
3 and 6 6: 6, 12 8 so 3 = 6 and 6 = 6
● Once you find the LCD, MAKE SURE TO SIMPLIFY THE FRACTIONS
○ Make sure that you have all the equivalent fractions when you use the LCD
● Keep the denominators the same and add or subtract the numerators as usual
● Example: 1 3
--- + ---
3 6
- Find the LCD and simplify the fractions:
1 3 3: 3, 6, 9 1 2 3 3
--- --- ---- --- ---- ---
3 and 6 6: 6, 12 8 so 3 = 6 and 6 6
- Add the numerators regularly
2 3 5
--- + --- = ---- (Final Answer)
6 6 6
● Example: 3 1
--- - ---
6 3
- Find the LCD and simplify the fractions:
1 3 3: 3, 6, 9 1 2 3 3
--- --- ---- --- ---- ---
3 and 6 6: 6, 12 8 so 3 = 6 and 6 6
- Subtract the numerators regularly
3 2 1
--- - --- = ---- (Final Answer)
6 6 6
● This is the basic method for adding and subtracting decimals
○ You may be given more challenging problems with mixed numbers and improper fractions
■ Just always make sure that you follow this basic format all the time and simplify your fractions!
How to Multiply and Divide Fractions?
● To multiply fractions, simply multiply the numerators together and the denominators together and simplify as desired
○ Example: 3 x 1 = 3
--- --- ---- (Final Answer)
6 x 3 18
- Multiply 3 x 1 and 6 x 3 together to get 3/18 as the final answer
● To divide fractions, reverse the seconds fractions numerator and denominator and multiply as usual
● Example: 3 ÷ 1 = 3 x 3 = 9 or 3
----- ----- ---- ---- ---- ---- (Final Answer)
6 ÷ 3 = 6 x 1 = 6 or 2
- Switch 1 and 3 in the second fraction
- Multiply 3 x 3 and 6 x 1 together to get 9/6 or 3/2 as the final answer
Decimals
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Decimals: Notes
What are decimals?
● Decimals are parts of a whole; they are fractions of a whole number
● They can represent the same numbers as fractions
● They can exist on their own (with a zero on the whole number side of the dot,
Ex: 0.23426 or with a whole number on the other side of the dot, Ex: 8.45634)
How to convert decimals to fractions?
1. Take the decimal and put in a fraction-like form over 1:
0.25
------
1
2. Multiply the top and the bottom of the fraction x100:
0.25 x100 25
-------------- = ------
1 x100 100
3. Simplify the fraction:
25 ÷25 1
------------ = -----
100 ÷25 4
4. Smile because your decimal is now a fraction and most math teachers prefer fractions!
What about percentages?
● To turn a decimal into a percentage, just move the decimal point two places to the right,
or this way: →
○ Then, remove the decimal point and add a % at the end!
● Decimal:
○ 0.02 → 002.
■ Take away the zeros in front and the period and you’ll see that 0.02 is equivalent to 2%
How do you round decimals?
● Most times when working with decimals throughout your math class career,
the worksheet will ask you to round to a certain decimal, or you will find it
convenient to do so yourself.
● The problem will most likely ask you to round to the nearest tenth or hundredth of a decimal.
○ The nearest tenth decimal place is right here: 0.1
○ The nearest hundredth place is right here: 0.11
○ The nearest thousandth is here: 0.111
● When rounding to any decimal place, you need to look at the digit in the place you know will be the end of your new rounded number and the digit right next to it on the right (→) side. If the digit on the right side of your last digit is 4 or below, you will cut off the rest of the numbers including your right side digit, and that will be your new number:
○ Example: You have 0.344 and you need to round to the nearest hundredth. You will look at the 4 in the middle, the number you need as your last number, and the 4 at the end, the number that determines what happens to your last number. 4 and under means you keep the rest the same, so your rounded answer would be:
■ 0.34
● If the digit on the right side of your last digit is 5 or above, you will cut off the rest
of the numbers including your right side digit, and you will raise your last digit up one whole number digit:
○ Example: Say you have the decimal: 3.7846534
○ Your teacher tells you to round to the nearest thousandth, so you look at the
3rd and 4th decimals from the decimal point, which in this case are 4 and 6. You would see that 6 fits into the category of 5 or above, which would mean you would raise the value of the number in the thousandth place one whole digit. Your rounded answer would be
■ 3.785
What are repeating/recurring decimals?
● Repeating or Recurring decimals are decimals are decimals that go on forever and never stop. They repeat the same pattern of numbers again and again!
○ Pi is NOT a recurring/repeating decimal! It is a non-terminating, non-repeating decimal. It never repeats the same pattern of numbers
● Example: 1 ÷ 3 or the fraction 1⁄3 is a recurring decimal. It looks something like 0.333333333... and it goes on forever.
○ This can also be written: __
○ With a line over the repeating number, sort of like this: 0.33
How to add and subtract decimals?
● To add and subtract decimals, you do the same thing as regular addition and subtraction, but you have to remember to keep the decimal points in place!
How to multiply and divide decimals?
● To multiply decimals, you do so in the same way you multiply other numbers, but you just have to remember to keep the decimal point in place!
● To divide decimals, you should divide them with long division
- You take the divisor and make it have a whole number if it doesn’t already:
0.34 to 3.4 - You take the dividend and move it the same amount of points as you did the
divisor: 16.45 to 164.5 - Line up the decimal point in the quotient with the decimal point in the
dividend: ___.__
3.4⟌164.5 - Divide as usual!
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