Notes from Place Value and Forms of Numbers
- Magnitude-how big or small something is
- Place-the position a digit is in a number
- Value-how much a digit is worth within its given place
• Standard Form- a common way of writing a number with commas separating groups of three digits starting from the right.
•Ex. 3,458
•Word Form-a way to write numbers using words
•Ex. Three thousand, four hundred fifty eight
•Expanded Form-a way to write a number that shows the value of each digit
•Ex. 3000 + 400 + 50 + 8
Notes for Comparing, Ordering, and Rounding Presentation
Comparing Made Easy…..
Here are the steps!
Example: Compare 18,066,000 and 18,131,000
1.Line up the numbers by the place value.
2.Start from the left. Compare the digits until they are different.
Step 1:18, 066, 000 Step 2: 18, 066, 000
18, 131, 000 18, 131, 000
The hundred thousands digits are different. 1 is greater than 0, so 18,131,000 > 18,066,000.
Ordering Numbers
•We can use the same method when we are ordering three or more numbers.
ex: Put these numbers in order from greatest to least: 16,640,000; 26,444,000; 17,755,000
1.Line the numbers by place value.
2.Start at the left. Compare the digits. Continue comparing until all numbers are used.
16,640,00016,640,000
26,444,000 2>117,755,000 7>6
17,755,00017,755,000 > 16,640,000
26,444,000 is the greatest number.Therefore 26,444,000>17,755,000>16,640,000
The fabulous World of Rounding!!!
•Just as with ordering and comparing, there are steps to rounding that will guarantee an accurate response every time!
•Ex: Round the population of Tokyo to the nearest million. 26,444,000
•Steps:
•1. Identify the place you want to round. I normally underline this number.
26,444,000
2. Circle the digit to the right.
26,444,000
3. If the digit to the right is 5 or greater increase the rounding place by 1. If the digit is less than 5, do not change the rounding place digit. Then replace all digits with zeros.
26,444,000 4<5, therefore this rounds to 26,000,000.
Estimating Solutions
When instructed to round to a particular place value,
numbers 5 through 9 round up and numbers 0 through 4 round down.
When not instructed how to round, students might use a variety of estimation techniques.
Estimate Solutions - Decimals
13.25tens / ones / decimal / tenths / hundredths
1 / 3 / . / 2 / 5
Rounding to a Particular Place Value
1.Look at the digit directly to the right of the place value being rounded to.
2.If the digit to the right is 5 or greater, add 1 to the digit in the place value.If it is less than 5, the value of the digit will not change.
3.If the place value being rounded to is to the left of the decimal place, fill in all places to the right with zeros back to the decimal place.
Estimate Solutions - Fractions
To estimate solutions withfractionsthat have the same denominator, add or subtract the numerators and then compare the numerator with the denominator.
Example 1:
Which of the following would be the best estimate of the following sum?
A.B.C.D.
Solution:
Since the fractions have a common denominator, add the numerators.
When estimating, compare the numerator with the denominator.
Notice that the numerator, 26, is about one-half of the denominator, 50.
26 × 2 = 5250
So, the best estimate of the sum is.
Example 2:
Which of the following would be the best estimate of the following difference?
A.B.C.D.
Solution:
Since the fractions have a common denominator, subtract the numerators.
When estimating, compare the numerator with the denominator.
Notice that the numerator, 9, is about one-third of the denominator, 30.
9 × 3 = 2730
So, the best estimate of the difference is.
Using a common fraction is a way to estimate a calculation that involves fractions.For example,can be approximated asbecauseis close to, which is equivalent to.
Example 3:
Dorise needscups of water to make cookies andcups of water to make a pitcher of lemonade.About how many cups of water does she need in all?
Solution:
Add together the two amounts of water.Use common fractions to simplify the calculation.
Dorise needs about4cups of water.
Fractions
Addition of Fractions
Adding Fractions with Common Denominators
When adding fractions with a common denominator,
add the numerators and keep the common denominator.
Adding Fractions with Uncommon Denominators
When adding fractions with different denominators, theleast common
denominatorneeds to be found before the fractions can be added.
After rewriting the fractions using the least common denominator,
add the numerators and keep the common denominator.
Subtracting Fractions with Common Denominators
When subtracting fractions with a common denominator,
subtract the numerators and keep the common denominator.
Subtracting Fractions with Uncommon Denominators
When subtracting fractions with different denominators, theleast common
denominatorneeds to be found before the fractions can be subtracted.
After rewriting the fractions using the least common denominator,
subtract the numerators and keep the common denominator.
Tips for Adding and Subtracting Mixed Numbers
1.Before adding and subtracting mixed numbers, make sure that the fractions havecommon denominators.If the fractions do not have a common denominator, rewrite the fractions using the least common denominator first.
2.Whenaddingmixed numbers, first add the fractions.If the sum of the fractions is greater than one, regrouping may be necessary.Then, add the whole numbers.
3.Whensubtractingmixed numbers, first determine the fractions can be subtracted without regrouping.If regrouping is not necessary, then subtract the whole numbers.
Addition of Decimals
1.Place the numbers so that the decimal points are aligned vertically.
2.Add each column, starting on the right and working left.
3.If the sum of a column is greater than nine, carry the one to the next column on the left.
If the decimals end in different place values, such as a number with tenths added to a number with hundredths, use a zero as a place holder so that both numbers show the same place values.
Subtracting Decimals
- Place the numbers so that the decimal points are aligned vertically. (Add zeros if needed.)
2.Subtract each column, starting on the right and working left.If the number being subtracted is larger than the number it is being subtracted from, then add ten to the number and subtract one from the number in the next left column.(This is called regrouping, or borrowing.)
Geometric Patterns
When finding the missing picture in a list of pictures, figure out the pattern in the list.
First, look at the change in the number of objects in each picture in the list.Then, figure out how much bigger or how much smaller each picture is than the picture before it.
Number Patterns
A number pattern is a sequence of numbers arranged according to a rule or formula.
The rule can be used to predict the value of unknown numbers in the pattern.
Number patternscan be found in tables and graphs.
Finding the number pattern in a table can help solve problems.
A function is a mathematical relationship between two values.
A function table displays input and output values of a function.
The output values are determined by the function.
Equations
In order to simplify equations, perform operations (addition, subtraction, multiplication, or division) on equations by keeping both sidesbalanced.
After finding a solution, check the answers by substituting the value of the variable found back into the original equation.
Rates of Change
Patterns of change can be described as constant, increasing, or decreasing.