MATH for elementary teachers

You have 100 pencils and 70 notebooks. What is the largest number of students you can give the pencils and notebooks to so that every student gets the same number of pencils and every student gets the same number of notebooks?

#Students must be a common factor of 100 & 70 the largest such # is GCF (100,70) =10 10. 100. 70 10. 7 The largest possible # of students is 10.

Whole number

0,1,2,3,4...

counting numbers

1,2,3,4,..

Convert 11/6 to a decimal.

11/6=1.8333

Use the slide method to find the LCM and GCF of 360 and 1344.

6 360 1344 4 60 224 15 56 GCF=6x4=24 LCM=6x4x15x56=20160

Convert 8.3452452 to a fraction

8 3449/9990

State the Fundamental Theorem of Arithmetic

Every counting # greater than 1 can be factored into primes in a unique way.

Integers

...,-3,-2,-1,0,1,2,3,...

Give an example of an irrational number.

0.1010010001 0.102003000

Shannon has 56 Star Wars figures. In how many different ways can she place the figures into groups so that all of the groups are the same size and so that there are no figures left over? Explain.

1 group of 56 2 groups of 28 4 groups of 14 7 groups of 8 8 groups 7 14 groups of 4 28 groups of 2 56 groups 1

How many different rectangles are there whose side lengths in inches are counting numbers and whose area is 56 square inches? Explain.

1 in by 56 in 2 in by 28 in 4 in by 14 in 7 in by 8 in 4 rectangles 7 in by 8 in rectangle is also an 8 in by 7 in rectangle, so I not get more like 5 rectangles

Bob says that 3 is a multiple of 6 because 3 cookies can be placed into 6 groups, each containing half of a cookie (or because 6 ×1/2=3). What is Bob missing?

1/2 is not a whole number. we have to put a whole number of cookies in each group

Use the slide method to find the LCM and GCF of 2880 and 2400.

10 2880 2400 8 288 240 6 36 30 6 5 GCF= 10x8x6=480 LCM=10x8x6x6x5=14400

Find the LCM of 18 and 16 by listing multiples.

16,32,48, 64,80, 96,112,128,144,160,176,192 18,36,54,72,90,108,126, 144 L C M =144

Find all of the factors of 900.

1x900=900 2x450=900 3x300= 900 4x225=900 5x180=900 6x150=900 9x100=900 10x90=900 12x75=900 15x60=900 18X50=900 20x45=900 25x36=900 30x30-900 Answer: 1,2,3,4,5,6, 9,10,12,15,18,20,25,30,36,45,50,60,75,90,100,150,180, 225,300,450,900

Test this number for divisibility by 2, 3, 4, 5, 6, 7, 8, and 9: 767,862,340,984

2 / yes/ ends in 4 3 / no/ sum of digits = 64, which is not a multiple of 3 4 / yes/ last 2 digits form a number (84) divisible by 4 5 / no/ does not end in 0 or 5 6 / no/ not a multiple of 3 8 / yes/ last 3 digits form a number (984) which is a multiple of 8 9 / no/ sum of digits =64 , which is not a multiple of 9

Use the Sieve of Eratosthenes to find the prime numbers less than 100.

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

Use trial division to see if 287 is prime.

287÷2 = 143.5 287÷3= 95.666 287÷4=71.75 287÷5=57.4 287÷6 =47.8333 287÷7=41 → not prime!

Convert 3/7 to a decimal

3/7 = 0.428571428571

Find the prime factorizations of 360, 1344, their LCM, and their GCF.

360 10 36 2 5 6 6 2 3 2 3 360= 2^3 x 3^2 x 5 1344= 2^6 x 3 x 7 1344 8 168 4 2 8 21 2 2 4 2 3 7 GCF = 2^3 x 3 LCM= 2^6 x 3^2 x 5 x 7

5 x 4 = 20

5 factor of 20 4 factor of 20 20 multiple of 4 20 multiple of 5

Use a factor tree to find the prime factorization of 560

560=. 2^4 x 5x7 10 x 56 2x5 7x8 2x4 2x2

A large gear turns a small gear. The large gear has 60 teeth and makes 12 revolutions per minute. The smaller gear has 40 teeth. How many revolutions per minute does the smaller gear make?

60 teeth & 12rpm 40 teeth& ?rpm 60x12=40x? ? = 60x12/40 =18 The small gear makes 18 revolutions per minute

Use trial division to see if 8303 is prime.

8303 ÷2=4151.5 ÷3=2767.6 ÷ 5=1660.6 ÷7=1186.1 ÷ 11 =754.8 ÷13 =638.6 ÷17=488.4 ÷19=437 → not prime!

Convert 0. 878787 to a fraction.

87/99 X=0.3452452 1000x =345.2452 x=. 0.3452 999x=344.9 X=344.9 / 999 X = 3449 / 9990 8.3452452 =8 3449/ 9990

What does it mean for a counting number to be even?

A counting number N is even if N beans can be placed into pairs with none left over

Draw a Venn diagram showing the counting numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

All real numbers Irrational. Rationals integers Whole # Counting #

Shannon is trying to find all of the factors of 40 by dividing 40 by 1, 2, 3,... Here is her work so far: 1 ×40=40 2 ×20=40 4 ×10=40 5 ×8 =40 8 ×5 =40 Explain why Shannon can stop looking.

Any new factor on the left would be larger than 8 and would have already appeared on the right. Any new factor on the right would be less than 8 and would have already appeared on the left.

State the Fundamental Theorem of Arithmetic

Every counting number greater than 1 can be factored into primes in a unique way.

Find the GCF of 224 and 336 by listing factors.

GCF =112

Explain why we exclude 1 from the definitions of prime and composite.

If we include 1, we lose "uniqueness" in the fundamental theorem of arithmetic

Joey thinks that 5 is even. His reasoning is this: A number 𝐴 is even if there is a number 𝑘 so that 𝐴 = 2𝑘. Well, 5 = 2 × 2.5, so 5 is even. What is Joey missing?

K must be an integer, but 2.5 is not an integer

A large gear turns a small gear. The large gear makes 300 revolutions per minute. The smaller gear makes 1536 revolutions per minute. What is the smallest number of teeth each gear could have?

LCM(300,1536) =38400 3. 300 1536 4. 100 512 25 128 3 x 4x25x128 =38400 Large 38400/300 =128 teeth Small 38400/300 =25 teeth

To test a counting number for divisibility by 5, you need only look at the ones digit. Use base 10 bundles to explain clearly why this is true.

Suppose A is a counting number. place A beans into base 10 bundles. divide each large bundle into bundle into bundles of 10. now our beans are in bundles of 10 along w/ our pile of ones. divide each bundle of 10 into 2 piles of 5. All of our beans are now in piles of 5 except the pile of ones. we can finish placing the beans into piles of 5 exactly if we place the ones into piles of 5.

Pencils come in packs of 24. Erasers come in packs of 10. What is the smallest number of pencils and erasers you can buy so that each pencil is paired with exactly one eraser?

The # needs to be a multiple of 24 and of 1 0 . The smallest such # is LCM ( 24,10) =120 2. 24. 10 12. 5 LCM =2x12x5=120 The smallest #of pencils & erasers is 120

Explain what it means for a counting number to be composite.

The number is not 1 and is not prime. OR: the number has a factor other then 1 and itself

Explain what it means for a counting number to be prime.

The number is not 1, and its only factors are 1 and itself

What does it mean for a counting number A to be a factor of a counting number B?

There is a counting number (or integer, whole # ) C so that axc=b

What does it mean for a counting number A to be a multiple of a counting number B?

There is a counting number (or integer, whole#) C so that a=bxc

State what it means for a real number X to be rational.

They are integers A and B w/ X=a/b and B does not equal 0

State what it means for a real number X to be irrational.

X is not rational

What are the rational numbers?

fractions of integers

Use algebra to explain why an odd counting number plus an odd counting number is even.

suppose A and B are odd counting numbers. there are integers K and also that A=2K+1 and B=2L+1. Then A+B=2k+1+2l+1=2(k+l+1) thus A+B is even

Use pairing to explain why an odd counting number plus and odd counting number is even

suppose that A and B are odd and that we have a pile of A beans and a pile of B beans. Place the A beans and pears with one left over. Then place the B beans in pairs w/ one left over. Now place the two leftovers in a pair. We have put A+B beans into pairs w/ none left over, so A+B is even.