Math3 5 Extras
Transcript of Math3 5 Extras
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Topics 1 - 6
1) Multiplication I dont want to add that over and over again.
2) Division Hey, I need to share these
3) Remainders How many 4-passenger cars to get all of youto The Avenue?
4) Factors, Multiples and Divisibility How many socks shouldbe in the dryer?
5) Fractions and Equivalent Fractions How much of thatchocolate bar did I get?
6) Fractions/Decimals/Percentages A dollar can do that!
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Multiplication
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Applications of multiplication
Quickly find the answer to adding thesame number repeatedly
Finding the area of a rectangle Calculate the total for a bill, where
multiples of several items are purchased
Scaling up proportionally, such asdoubling a recipe
Converting between units
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Sample Problem
What is the area of a room that is 10ftlong and 13 ft wide?
Area A= length * widthA=10ft x 13 ftA=130 ft2
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Important Characteristics
Multiplication by Zero
Multiplication by 1
Commutative Property
Associative Property
Distributive Property
N*0=0 3*0=0
X*1=X 3*1=3
A*B=B*A 3*2=2*3
A*(B*C)=(A*B)*C 3*(2*4)=(3*2)*4
F*(G+H)=(F*G)+(F*H)
5*(17)=5*(10+7)=(5*10)+(5*7)=50+35
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Special Application: Exponents
Finding the area of a squareWhen we multiply an number by itself,
we can use exponents the smallnumber tells how many time the numberis multiplied.The area of a square with side 7m is:7m*7m = 72m2 = 49m2
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Division
(no Remainder)
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Applications of Division
Instead of subtracting repeatedly, we divide
Distribute/Share/Sub-divide into equal portions
Determine how much each person gets whenthings are divided in proportion
Scaling down proportionally, such asrepresenting a country on map
Converting between units
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Division Example Using
Proportions
Abe, Ron and Ben share 4 pizzas during the superbowl. Eachpizza has 6 slices. Each pizza has Each time they fill their plates
Abe takes 1 slice, Ron takes 3 slices and Ben takes 4 slices, untilthe pizza is done. How many slices does each person take?
The pizza is consumed in proportion 1:3:4.Each time that they fill their plates they take a total of 1+3+4=8slices.There are 4*6=24 slices available.Number of times they can fill their plates: 24/8=3
Abe takes 3 *1 = 3 slices
Ron takes 3*3 = 9 slicesBen takes 3*4 = 12 slices(Check : 3+9+12 = 24 slices of pizza)
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Division
(with Remainder)
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Division with Remainders: Uses
Ignore (or recycle) the remainder if itdoesnt affect the outcome
Account for the need for an extra item, ifit does affect the outcome (fitting theremaining people in a golf cart after allthe others are full)
Finding the item in a certain place in arepeating pattern
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Division, With Remainder
Example
The swim team wears a different swim cap each practice.The order is always: red, orange, yellow, green, blue, indigoand violet. What color will the caps be on the 26 th day?
The pattern repeats every 7th practice:ROYGBIVROYGBIVROY.So find the remainder when 26 is divided by 7: 267 = 3 R 5
Count to the 5th place in the pattern and thats the answer,so the Blue cap!
(Note that if the remainder is zero, then the answer is thelast item in the pattern).
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Primes, Factors, Multiples
and Divisibility
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Primes, Factors and Multiples
A number that can only be divided by 1 anditself is called a prime number. In other words,a prime number has only 1 and itself as
factors. The first prime (and only even prime) istherefore 2. When two factors are multiplied, they produce
a multiple All multiples can be expressed as the product
of prime numbers and 1. Every whole number has 1 and itself as
factors.
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Factors and Primes - Application
There are 36 peoplecoming to the familyreunion. How many tablesshould I order?It depends. The factors of36 are 1,2,3,4,6,9,12,18,36.We probably dont want 36tables with a single personseated at each table, but alloff the other numbers mightwork. For example 2 tableswith 18 people each or 4tables with 9 each, etc.
Graphic: National Library of Virtual Manipulatives (Factor Tree) http://nlvm.usu.edu
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Divisibility Rules All multiples of 2 are even numbers. All multiples of 3 have digits that add up to a multiple of 3. All multiples of 4 have multiples of 4 in the last 2 digits (including 00). All multiples of 5 end in 0 or 5. All multiples of 6 are multiples of both 2 AND 3. All multiples of 7: take each digit from the units (ones) place and going
from right to left, multiply by 1,3,2,6,4,5 (repeat the pattern as often asnecessary). Add the products together the number is a multiple of 7,if this sum leaves no remainder when divided by 7. Long division maybe quicker.
All multiples of 8 have multiples of 8 in the last 3 digits (long division
sometimes faster than this test). All multiples of 9 have digits that add up to a multiple of 9. All multiples of 10 end in 0.
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Greatest Common Factor (GCF)
The biggest factor that 2 or more numbers havein commonI have 35 roses and 20 carnations, what is thelargest number of bouquets that I can make, ifeach buoquet must have the same number ofeach kind of flower?I need to find the GCF of 35 and 20:20=4x5=22x5 and 35=5x7The largest number that 20 and 35 have incommon is 5. I can make 5 buoquets, each with7 roses and 4 carnations.
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Least Common Multiple (LCM)
The smallest number that is divisible by the numbers forwhich I must find the LCM.
A package has 12 hotdogs and each bag has 8 buns. How
many packages and bags must I buy in order to have abun for each hotdog and spend as little as possible?
I must find the LCM of 8 and 12. First I must factorize eachnumber: 8=2x2x2 and 12=2x2x3. Find the maximumnumber each prime occurs in each of the two factors, so3 twos (2x2x2) and 1 three (3). Multiply these together to
get the LCM: 2x2x2x3 = 24. I need 24 hotdogs (2packages of 12 hotdogs) and 24 buns (3 packages of 8buns).
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Fractions and
Equivalent Fractions
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Fractions and Equivalent
Fractions
Answers are typically given in the simplest or smallestfraction.
Equivalent Fractions: Whatever you do to the numerator,you must do to the denominator.
When ordering, adding, or subtracting fractions, we needthem to have the same denominator, so we must findequivalent fractions
When multiplying or dividing fractions its often easiest totry to simplify or find an equivalent fraction that makes thecalculations easier.
Use the LCM of the denominators to find the newdenominator for all the fractions. We then call this theLeast Common Denominator (LCD).
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Equivalent Fractions example
Sammy the slug traveled 3/4m, 1/3m, 5/8m, 2/9m and6/7m to find his meals today. How far did he traveltoday?
3 and 7 are prime, 4=2x2, 9=3x3, 8=2x2x2LCM=2x2x2x3x3x7 = 504
Total distance =
504
378
7332
7332
22
3
4
3
504
168
73222
73222
3
1
3
1
504
315
733
733
222
5
8
5
504
432
332*2*2
332*2*2
7
6
7
6
504
112
7222
7222
3*3
2
9
2
m5043972
504
1405
504
432
504
112
504
315
504
168
504
378
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Fractions, Decimals and
Percentages
Hint: $1.00
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Applications
Simple interest
Sales tax
Sales price markup and discount(percent increase and percent decrease)
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A Dollar - Decimals
Were used to adding money line up thedecimal point and add the numbers that fall in
the same column. Keep the decimal point inthe same place in the answer. A tip that helpswith addition and subtraction is to fill in theempty spaces with zero.
Find the sum of 103.275, 7.09 and 0.9876.
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A Dollar - Percentages A dollar has 100 pennies. When we express the number of pennies that we
have as a fraction we are actually writing a percentage. If we have 65 cents,we have 65/100 pennies or 65% of a dollar. When we express any fraction asan equivalent fraction with 100 in the denominator, we are writing a percentage.
A whole is 100% (100 cents of the dollar), 200% would be 2 wholes.
When finding a percentage ofa number, write the percentage as a fraction,then multiply the number by percentage. Simplify the fractions before findingthe final answer.
Write 1/5, 3/10, 7/40 and 230/1000 as percentages.
.
%23100
23
101000
10230
1000
230
%5.17100
5.17
5.2
5.2*
40
7
%30100
30
10
10*
10
3
%20100
20
20
20*
5
1
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A Dollar - Fractions A quarter (25cents) and a half dollar (50cents) are fractions that we
use all the time. With fractions, the keyword of tells us to multiply. Convert whole
numbers to fractions, by dividing them by 1.
When multiplying fractions, first simplify, by canceling any number inthe denominator by any number in the numerator.What is 5/6 of 30? What is 1/7 of $21.35?
.