Place Value, Rounding, Comparing Whole Numbers
Place Value
Example: The number 13,652,103 would look like
We’d read this in groups of three digits, so this number would be written
thirteen million six hundred fifty two thousand one hundred and three
Example: Write the value of two million, five hundred thousand, thirty six
2,500,036
Rounding
When we round to a place value, we are looking for the closest number that has zeros in the digits to the right.
Example: Round 173 to the nearest ten.
Since we are rounding to the nearest ten, we want a 0 in the ones place. The two possible values are 170 or 180. 173 is closer to 170, so we round to 170.
Example: Round 97,870 to the nearest thousand.
The nearest values are 97,000 and 98,000. The closer value is 98,000.
Example: Round 5,950 to the nearest hundred.
The nearest values are 5,900 or 6,000. 5,950 is exactly halfway between, so by convention we round up, to 6,000.
Comparing
To compare to values, we look at which has the largest value in the highest place value.
Example: Which is larger: 126 or 132?
Both numbers are the same in the hundreds place, so we look in the tens place. 132 has 3 tens, while 126 only has 2 tens, so 132 is larger. We write 126 < 132, or 132 > 126.
Example: Which is larger: 54 or 236?
Here, 54 includes no hundreds, while 236 contains two hundreds, so 236 is larger.
54 < 236, or 236 > 54
1) Write out in words: 13,904
2) Write out in words: 30,000,000
3) Write out in words: 13,000,000,000
4) Write the number: sixty million, three hundred twelve thousand, two hundred twenty five
5) Round to the nearest ten: 83,974 6) Round to the nearest hundred: 6,873
Round 8,499 to the nearest 7) ten 8) hundred 9) thousand
Determine which number is larger. Write < or > between the numbers to show this.
10) 13 21 11) 91 87
12) 136 512 13) 6,302,542 6,294,752
14) six thousand five hundred twenty three six thousand ninety five
1.2 Adding and Subtracting Whole numbers
Note: If you are happy with the way you’ve always added or subtracted whole number, by all means continue doing it the same way!
Adding by Grouping
Example: Add 352 + 179
We can break this apart:
Add the hundreds: 300+100 = 400
Add the tens: 50 + 70 = 120
Add the ones: 2 + 9 = 11
Add the resulting pieces = 531
Adding by Rearranging
The idea that 2 + 3 is the same as 3 + 2 is called the commutative property for addition.
Example: Add 17 + 15 + 23
We can rearrange the order to be 17 + 23 + 15
Since 7+3 = 10, this makes things a bit easier
17 + 23 + 15 = 40 + 15 = 55
Subtracting using Expanded form
Example: Subtract 352 - 169
We can write this as:
300 + 50 + 2
- 100 + 60 + 9
We can’t take 9 from 2, so we borrow 10 from the 50
300 + 40 + 12
- 100 + 60 + 9
Likewise we can’t take 60 from 40, so we borrow 100 from the 300
200 + 140 + 12
- 100 + 60 + 9
--------------------------
100 + 80 + 3 = 183
Subtracting by Adjusting Values
Example: Subtract 162 - 138
If we add 2 to both numbers, the difference will be the same, but easier to compute
162 + 2 = 164
– 138 + 2 = 140
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24
_______________________________
Calculate. Use whatever techniques make sense to you.
1) 513 + 268 2) 1704 + 521 3) 88 + 26 + 32 + 4
4) 12,000 + 312 5) 92 – 75 6) 1824 – 908
7) 3000 – 73 8) 903 – 170 9) 100 – 13 + 17
11) Find the perimeter of the shape shown.
12) This year I used 606 kWh of electricity in August. Last year I used 326 kWh. How much more electricity did I use this year?
13) The bar graph shows grades on a class activity. How many students scored a C or better?
1.3 Multiplication of Whole Numbers
There are three common ways of writing “5 times 3”: , , and
Multiplying as repeated addition
Example: Multiply
This is equivalent to 42 + 42 + 42 = 126
Multiplying by thinking about area
Example: Multiply
We can think of this as 3 groups of 5 objects or 5 groups of 3 objects:
This is also why we multiply to find the area of rectangle.
= 15
Multiplying using place values
The idea that is called the distributive property.
Example: Multiply
We can write this as Then
= 120 + 48 = 168
Example: Multiply
We can think of this as
This can be thought of as areas, as pictured to the right
You also write this in the more “traditional” way. By working with place values, you can avoid having to carry. It’s more writing, but less likely to
Example: Multiply
162
x 24
2000 100 x 20
400 100 x 4
1200 60 x 20
120 60 x 4
40 2 x 20
+ 8 2 x 4
3768
Multiplying large numbers
When you have a lot of trailing zeros, you can multiply without them, then tack on the extra zeros at the end.
Example: Multiply
, and there were 4 trailing zeros, so the result is: 420,000
________________________________
Calculate. Use whatever techniques make sense to you.
1) 2) 3)
4) 5) 6)
7) Estimate the value of by rounding the values first
For each problem, decide if you should add or multiply, then calculate the result
8) You and three friends each order a $3 slice of pie. How much is your total bill?
9) You ordered a burger for $9 and your friend ordered pasta for $12. How much is your total bill?
10) You earn $12 an hour. How much do you make in an 8 hours shift?
11) You need to order 120 new t-shirts for each of 6 stores. How many t-shirts do you need to order?
13) You buy two bottles of headache medicine, one with 100 pills, and the other with 75 pills. How many pills do you have total?
Worksheet – 1.3.2 Areas – Whole numbers Name: ________________________________
Find the area of each figure.
1) 2)
3) 4)
6) A bedroom measures 12 feet by 13 feet. What is the area of the room? If carpet costs $3 per square foot, how much will it cost to carpet this room?
(see the picture – the shaded part is the brick)
1.4 Dividing Whole Numbers
There are 3 common ways of writing “6 divided by 3”: , , and
You can think of division as splitting something into equal groups.
Example: You have 12 cookies and 3 kids are going to share them. How many does each kid get?
We divide the 12 cookies into 3 groups: cookies per kid
Sometimes we might have something left over; this is a remainder
Example: Find
Dividing 66 items into 7 piles, we couldn’t put 10 in each pile, since that would require items. So we can put 9 items in each pile, using up items, and we have 3 items left over.
= 9 with remainder 3
Long Division
Example: Divide
We’re trying to find a number that multiplied by 8 will give 2896
Since = 8000, we know the number is smaller than 1000.
What’s the biggest 100’s number that, when multiplied by 8, is not bigger than 2896?
, , .
Looks like 300 is the biggest hundred, so we write a 3 in the hundreds place, multiply and write the result 2400 below the 2896.
Now subtract. This is how much is still left.
Next, we’re going to look for the biggest 10’s number that, when multiplied by 8,
is not bigger than 496. , ,
Looks like 60 is the biggest ten that fits, so we write a 6 in the tens place,
multiply and write that below the 496.
Now subtract. This is how much is still left.
Lastly, we’re going to look for the biggest number that, when multiplied by 8,
is not bigger than 16. Since , we write a 2 in the ones place, multiply
and write that below the 16, and subtract. Since we have nothing left,
there is no remainder.
So
Division involving zero
If we have nothing, and we divide it into any number of piles, each pile will have nothing, so
However, dividing by zero doesn’t make sense. For example think about . That’s asking what number, when multiplied by 0, gives 5. There isn’t one! Dividing by zero is undefined.
Worksheet – 1.4 Dividing Whole numbers Name: ________________________________
Calculate.
1) 2) 3)
4) 5) 6)
For each problem, decide if you should add, subtract, multiply, or divide, then calculate the result
8) Four roommates agree to split the $1500 rent equally. How much will each pay?
9) A team for the Alzheimer’s walk has raised $375. How much more do they need to raise to reach their goal of $1000?
10) A car insurance quote is $744 for six months. How much is that a month?
11) Your friend with a flock of chickens wants to give you 65 eggs. How many egg cartons (the kind that holds a dozen eggs) should you take with you to carry the eggs?
12) If you make $2,240 a month, how much do you make each year?
13) If you make $2,240 a month, how much do make each week (roughly – assume 4 weeks in a month)?
1.5 Exponents, Roots, and Order of Operations
Exponents and Roots
If we have repeated multiplication, like we can write this more simply using exponents:
Example: Write using exponents
Since we are multiplying 3 times itself 5 times, the base is 3, and the exponent is 5:
Example: Evaluate
Undoing squaring a number is finding the square root, which uses the symbol . It’s like asking “what number times itself will give me this value?” So since
Example: Find
because
Order of Operations
When we combine multiple operations, we need to agree on an order to follow, so that if two people calculate they will get the same answer. To remember the order, some people use the mnemonic PEMDAS:
IMPORTANT!! Notice that multiplication and division have the SAME precedence, as do addition and subtraction. When you have multiple operations of the same level, you work left to right.
Example: Simplify
Example: Simplify
______________________________
Evaluate.
1) 2) 3) 4)
5) 6) 7) 8)
9) 10) 11)
12) For a rectangle, the formula Perimeter = 2L+2W is often used, where L is the length and W width. Use this formula to find the perimeter of a rectangle 10 feet long and 4 feet wide.
Write out the mathematical expression that would calculate the answer to each question:
13) A family of four goes out to a buffet, and pays $10 each for food, and $2 each for drinks. How much do they pay altogether?
14) Don bought a car for $1200, spent $300 on repairs, and sold it for $2300. How much profit did he make?
1.6 Mean, Median, Mode
The mean (sometimes called average) of a set of values is
Example: Marci’s exam scores for her last math class were: 79, 86, 82, 93. The mean of these values would be: (79 + 86 + 82 + 93) divided by 4:
Example: On three trips to the store, Bill spent $120, $160, and $35. The mean of these values would be
It would be most correct for us to report that “The mean amount Bill spent was $105 per trip,” but it is not uncommon to see the more casual word “average” used in place of “mean”.
Median
With some types of data, like incomes or home values, a few very large values can make the mean compute to something much larger than is really "typical". For this reason, another measure, called the median is used.
To find the median, begin by listing the data in order from smallest to largest. If the number of data values is odd, then the median is the middle data value. If the number of data values is even, there is no one middle value, so we find the mean of the two middle values.
Example: Suppose Katie went out to lunch every day this week, and spent $12, $8, $72, $6, and $10 (the third day she took the whole office out). To find the median, we'd put the data in order first: $6, $8, $10, $12, $72. Since there are 5 pieces of data, an odd number, the median is the middle value: $10.
Example: Find the median of these quiz scores: 5 10 9 8 6 4 8 2 5 8
We start by listing the data in order: 2 4 5 5 6 8 8 8 9 10
Since there are 10 data values, an even number, there is no one middle number, so we find the mean of the two middle numbers, 6 and 8: . So the median quiz score was 8.
Mode
The mode of a set of data is the value that appears the most often. If not value appears more then once, there is no mode. If more than one value occurs the most often, there can be more than one mode. Because of this, mode is most useful when looking at a very large set of data.
Example: The number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season are shown below.
37 33 33 32 29 28 28 23 22 22 22 21 21 21 20
20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6
Looking at these values, the value 18 occurs the most often, appearing 4 times in the list, so 18 is the mode.
________________________________
Find the mean, median, and mode of each data set
1) 3, 4, 2, 6, 1, 2
2) 2, 4, 1, 5, 28
3) A small business has five employees, including the owner. Their salaries are $32,000, $40,000, $28,000, $65,000, and $140,000. Find the mean and median salary.
4) The graph shown shows the number of cars sold at a dealership each week this month. Find the mean and median sales per week.
1.7 Areas and Perimeters of Quadrilaterals
Rectangles
Perimeter:
Area:
Parallelogram
Perimeter: Sum of the sides
Area:
Trapezoid
Perimeter: Sum of the sides
Area:
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