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MAT011
TOPIC: WHOLE NUMBERS
1.1
STANDARD NOTATION
I. 2, 20, and 200 are very different numbers because the “digit” 2 has different “place values.”
II. In the place-value chart below, notice how the hundreds-tens-ones repeat and that the words
“tens” and “ones” are not usually mentioned when we say numbers!
A. TRILLIONS
BILLIONS
MILLIONS
THOUS’DS
(ONES)
HUN’S/(TENS/ONES)
HUN’S/(TENS/ONES)
HUN’S/(TENS/ONES)
HUN’S/(TENS/ONES)
HUN’S/(TENS/ONES)
B. Commas (but not the word “and”
) are inserted to create three-digit groups in “word
names” and “standard notation.” (Often the comma is omitted if less than 10,000.)
EX forty-two thousand, twelve EX 42,012
C. In “expanded notation,” each digit is next to its place value and pluses link it up.
EX 7200 = 7 thousands + 2 hundreds + 0 tens + 0 ones.
1.2
ADDITION
I. When adding, stack numbers “flush right” and “carry” (little guy!) as necessary.
1
EX
II.
1.3
28
15
“Addends” usually are stacked largest to smallest.
4
47
“the sum”
“Perimeter” means “measure around” so add up the lengths of the sides.
SUBTRACTION
I. For each “addition sentence” there are two “related subtraction sentences” and vice versa.
(Remember that a sentence must have a verb and in math that’s usually the “=”.)
EX 2+3=5 implies 5-3=2 and 5-2=3
EX 10-1=9 implies 9+1=10 and 1+9=10
(Notice how the related sentences always involve the number that’s alone on one side of the
equal sign with one of the other two. They’re good for “checking.”)
II. Just as in adding, stack the numbers flush right and now “borrow” as necessary.
A. No borrowing:
EX
29
-14
15
2 1
B. When the digit below is larger, you borrow:
1
EX
32
-17
15
C. When you need to borrow and only find a zero above, go west until you find a non-zero.
Borrow and then go east distributing the goods and borrowing as necessary:
9 9
3 1 1 1
EX
1.4
4003
-1436
2567
ROUNDING AND ESTIMATING; ORDER
I. “Rounding” instructions must tell you where to round to!
STEP 1: Underline the digit they ask you to round to.
STEP 2: Look to the right of it.
If 0-4, replace it and all to its right with zeroes.
If 5-9, add one to the underlined digit and follow that with zeroes.
EX Round 6429 to the nearest hundred.
Do this: 6429≈6400
(Notice those wiggly equal signs. They are used when rounding because 6429 does not
really equal 6400, does it?)
II. “Estimating” often means rounding first as instructed and then calculating.
III. “Order” is about what’s left and right on the number line.
A. “Inequality symbols” include these two:
1. “<” is read “is less than.”
2. “>” is read “is greater than.”
B. Trick!
Some people think of “Pac-man” mouths always hunting the bigger number.
EX 2<5 or 5>2
1.5
MULTIPLICATION
I. Multiplication is repeated addition.
EX 2+2+2=2 three times=2x3=(2)(3)=2∙3=6
(Notice how many ways you can write one idea!)
II. Just as in addition and subtraction, stack flush right.
A. Put numbers with more digits on top.
B. If same number of digits, place number with zeroes on right below.
EX 212
x400
C. You must line up your calculations with the lower digit you’re working with or insert zeroes.
(Some people use squared paper.)
D. Again, you may need to carry.
EX 211
x 43
1 633
844
Here, things are lined up rather than using zeroes.
9073
III. Estimating again means rounding first as instructed.
IV. “Area” is the number of little squares within, so memorize formulas in back cover, or just count!
2
1.6
DIVISION
I. For each “multiplication sentence” there are two related division sentences and vice versa.
EX 4 x 5 = 20 implies 20 4 = 5 and 20 ÷ 5 = 4
EX 12 ÷ 3 = 4 implies 3 ∙ 4 = 12 and 4 ∙ 3 = 12
(Just as with subtraction, these are good for checking.)
II. Just as with multiplication, division can be written a number of ways!
26 ÷ 2 or
EX
or 2 26
= 13
(Please be careful here: First is top is inside number. Second is bottom is outside number.
Order matters in division! )
III. The idea of division as repeated subtraction is a little confusing!
(It may be easier to see it as same-sized subgroups: EX 6 ÷ 2 = 3)
OR
*
**
**
*
*
*
*
*
IV. Zero is a perfectly good number,
but when dividing, make sure it’s not the
*
*
*
*
*
bottom/second/outside number! This is “undefined.”
or 3 ÷ 0 or 0 3
EX
(You can’t cut a pizza into 0 pieces!)
*Some people use the “no/o.k. trick”: (not defined) (just fine!)]
V. One pops up also: = 3 and = 1
VI. The trickiest part of long division is remembering that when a digit does not “go in,” you must
write a 0 at the top before bringing down the next digit.
EX
1.7
102
8 816
-8
16
-16
0
SOLVING EQUATIONS
I. “Equations” have equal signs in them and are more than “expressions.” A “solution” of an
equation is the number (or numbers) that, when substituted for the “variable” (letter), make it
true or the same on both sides.
EX x = 2 is the solution for x + 8 = 10; x = 3 is not.
II. If the variable is alone on one side of the =, then just calculate.
EX x = 2 ∙ 12 means x = 24.
(Notice that you don’t just write “24”; you write “x = 24”.)
III. If the variable is not alone, there are two ways to proceed:
A. Use related sentences to get variable alone.
EX 3 ∙ x = 24 implies 24 ÷ 3 = x
EX 7 + y = 26 implies 26 – 7 = y
EX a ÷ 4 = 16 implies 16 ∙ 4 = a
EX b – 7 = 10 implies 10 + 7 = b
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B. A more formal version is to work down the page using “opposite” operations:
addition/subtraction and multiplication/division. If you plan to go on in math, do this.
EX 3x = 24
EX 7 + y = 26 EX a ÷ 4 = 16 EX b – 7 = 10
3
3
-7
-7
x4 x4
+ 7 +7
x=8
1.8
y = 19
a = 64
b = 17
APPLICATIONS AND PROBLEM SOLVING
I. Generally, “Please Let Frankie Sing!” or Picture/Label/Formulas/Substitute
II. Vocabulary can give things away:
A. total, perimeter is probably addition
B. difference, “____-er” is probably subtraction
C. area, amounts as rates is probably multiplication
D. splitting into even pieces is probably division
III. In many word problems, students don’t know whether to multiply or divide, and, if dividing,
which number goes into which number!
It is well worth learning how to set up and solve a “proportion”!!
EX If 6 cupcakes per package, how many packages for 672 cupcakes?
Step 1: Usually two numbers are linked in a phrase and sometimes the number 1 is implied.
Write them, with units, vertically in fractional form.
Step 2: Set that fraction equal to another “empty” fraction.
Step 3: Using units to guide you, insert the remaining number and a variable for what you
want to know.
Step 4: “Cross multiply” (butterfly shape) without units.
Step 5: Solve as before; divide by number in front of variable.
=
6 x = 672
6
6
x = 112 so you’ll need 112 packages
1.9
EXPONENTIAL NOTATION AND ORDER OF OPERATIONS
I. Exponents are shorthand for repeated multiplication.
EX 2 x 2 x 2 x 2 = = 16
A. In the above example, we say “2 to the 4th power.”
2 is the “base” and 4 is the “exponent.”
B. Because 2 and 3 are so commonly used as exponents when calculating area and volume,
they have nicknames:
1.
is “five squared.”
2.
is “five cubed.”
II. When simplifying a string of numbers connected with symbols, people use tricks to remember
the order in which it should be done.
A. PEMDAS or Please Excuse My Dear Aunt Sally means this:
1. Do any calculations in side parentheses first.
2. Do any exponents second.
3. Do multiplication and division left to right third.
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4. Do addition and subtraction left to right fourth.
B. Carefully note the words “left to right.”
EX 16 ÷ 8 ∙ 2 = 4 (not 1)
C. With “nested parentheses” work from inside out: parentheses, then brackets, and then
braces.
D. Set up may help. Divide your page in half vertically with problem on the left and
calculations on the right. Underline what you’re focusing on and touch nothing else. Insert
implied multiplication symbols if you like.
EX 2{44 – [1 + 3(2 + 1)]}
= 2{44 – [1 + 3∙3]}
= 2{44 – [1 + 9]}
= 2{44 – 10}
= 2 ∙ 34
= 68
III. An average is calculated by finding the total of some numbers and then dividing by the number
of numbers. Average =
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