Elementary Mathematics

Learned basic problem solving skills.Used Base 10 building blocks to demonstrate our answers and give explanation & reasoning.We used combinations to help solve problems.Showed pictures, and other ways of justifying our answers.

We did example problems that helped us figure out different combinations.Ex: Given all the coins, how many combinations can you make in order to get $2.45?9 quarters, 2 dimes8 quarters, 4 dimes, 5 pennies205 pennies, 4 dimes, etc.

Used Ploya's methods to work through base 10 math problems, explain, draw pictures, & show our work. We had to determine the problem and figure out what we were being asked of. How are we going to tackle the problem? What methods will we use? Blocks? Pictures? Diagrams? Use prefferred method. Reflect on our answer & explain our work.

Worked with Manipulative Blocks1 block = 1 unit.Units (1) , Longs (10) , Squares (100).Used base 10 blocks to show expanded notation.Learned how to write numbers in the different bases.Learned how to give the base-10 numeral for given numbers.

Proved incorrect statements by using base knowledge.-> "You can't have a number bigger than the base".              

Also known as the "combination/joining" propertyUsed 3 Properties of AdditionIdentity Property of AdditionCommutative Property of Addition (Order)Associative Property of Addition (Grouping)Used 6 methods of solving addition problems:Standard AmericanPartial SumsPartial Sums w/ Place ValueLeft -> RightExpanded NotationLattice Method

Subtraction|/ *Comparison* \Takeaway *Missing Adent*

Shares 3 properties with AdditionCommutative PropertyZero PropertyAssociative PropertyOnce again, we used methods of problem solvingStandard AmericanExpanded NotationPlace ValueLattice

Properties of DivisionExample:quotient -> 4divisor -> 5 / 24 <- divident -20 --------- 4 <- remainder

Addition Mental Strategies Examples:(L->R) Compensation Ex: 347 + 129 = 475 Ex: 67 + 29 = 96 300 + 100 = 400--- 400 29 +1 -> 30 + 67 + 30 = 97 40 + 20 = 60 ------- 60 -1 + = 7 + 9 = 16----------- 16 96 Compatible Numbers Breaking Up & Bridging Ex: 130 + 50 + 70 + 20 + 50 Ex: 67 + 36 \ / \ / \ 200 + 100 + 20 67 + ( 30 + 6 ) = 320 97 + 6 = 103

Subtraction Mental Strategies Examples:(L->R) CompensationEx: 47 - 32 = 15 Ex: 47 - 29 40 - 30 = 10 29 + 1 -> 30 7 - 2 = 5 47 - 30 = 17 10 + 5 = 15 +1 = 18Compatible Numbers Breaking Up & Ex: 170 - 50 - 30 - 50 = 40 Bridging | \ / Ex: 67 - 36 = 31 170 - 100 = 70 | \ 70 - 30 = 40 67 - ( 30 + 6) 67 - 30 = 37 37 - 6 = 31

Multiplication Mental Strategies Examples:(L->R) Compatible Numbers Ex: 3 x 123 Ex: 2 x 5 x 9 x 20 x 5 / \ \ / | \ / 3 x ( 100 + 20 + 3 ) 10 x 9 x 100 \ / | | 300 + 60 + 9 90 x 100 = = 369 9,000

Division Mental Strategy Example + YouTube Video on a different way to divide using compatible numbersCompatible NumbersEx: 105 / 3 | 90 + 15 | 90 / 3 = 30 + 15 / 3 = 5 = 35

A number "a," is divisible by a second number "b," if there is a third number "c," that meets this requirement: c x b = aEx: 10 / 5 = 2 | 2 x 5 = 10 (2 & 5 are factors of 10)Important Terminology:10 is divisible by 5 or 5 divides 105 is a divisor of 105 is a factor of 1010 is a multiple of 5We know a number is divisible when we use the 5 divisibility rules listed. Ending Sum of Digits Last Digits Special Numbers Other

We know that the sum of digits of number 543 is divisible by 3 because 12 divided by 3 is 4

We know that the sum of digits of number 543 is NOT divisible by 9 because 12 divided by 9 is not possible in this instance

We have our original number of 826, we separate the last digit which is 6. We then double the last digit, making it 12, and then subtract the 12 by 82, and we get 70. 70. is now the number we check for divisibility. So we ask ourselves if 70 is divisible by 7, the answer is YES so we know the number 826 is divisible by 7.

We have our original number of 29,194. Using the chop off method requires us to chop off the last two digits giving us 291|94 we then add these two numbers together 291+94=385, we now separate the last two digits again giving us 3|85 we then again add these two numbers together 3+85=88. So we ask ourselves if 88 is divisible by 11, the answer is YES so we know the number 29,194 is divisible by 11.It is also important to note that you can repeat the process for a second time if you still need to.

We now go back to the 2 & 3 divisibility rules, and determine whether the number 3,702 is divisible by 6. We know if its divisible by BOTH 2 & 3 it is divisible by 6. When proving it by 2, we see that 3,702 ends in a 2 which is one of the ways to prove it is divisible by 2. Now we need to see if it is divisible by 3, we do so by adding the digits and dividing by 3. So we would do 3+7+0+2 = 12 and we can definitely prove that 12 is divisible by 3, so we now can confirm that 3,702 is divisible by BOTH 2 & 3, making it divisible by 6.

The list method requires you to list the factors of the number provided

When we list the factors for 24 & 36, we see that the biggest number they have in common is 12, making the GCF of 24 & 36 = 12.

When we count by 24 and 36, we see that the number they have in common was 72, making that the LCM for 24 & 36.

The Prime Factorization Method requires you to make a tree diagram with listing the factors of the number all the way down until you get prime numbers.Ex: 2424= 1 x 2424= 2 x 1224= 3 x 824= 4 x 624 < 3 x 8 OR 24 < 2 x 12 | | 4 x 2 2 x 6 | | 2 x 2 2 x 324 = 3 x 2 x 2 x 2 24 = 2^3 x 3 OR 24 = 2 x 2 x 2 x 3Ex: 3636= 1 x 3636= 2 x 1836= 3 x 1236= 6 x 636 < 2 x 18 OR 36 < 6 x 6 | | | 9 x 2 2 x 3 2 x 3 | 3 x 3 36 = 3 x 3 x 2 x 2 36 = 2 x 3 x 2 x 3Finding GCF & LCM:24 = 3 x 2 x 2 x 2 36 = 3 x 3 x 2 x 2GCF: 2 x 2 x 3 = 12LCM: GCF x 2 x 3 = 72

Meanings of Fractions: Division: Part of a Whole:Ratio: when a ratio shows a relationship between a part and a whole, it is considered a fraction. Ex 1: 20 students are in a class, of those, 12 are girls and 8 are boys. Ex. of a fraction: 8/20 = boys/whole number of students = part/whole = It is a fraction Ex of a ratio: 8/12 = boys/girls = part/part = Is NOT a fraction Ex 2: There are 35 animals, of those, 30 are cats, and 5 are dogs. Ex. of a fraction: 30/35 = cats/whole number of animals = part/whole = It is a fraction Ex. of a ratio: 5/30 = dogs/cats = part/part = It is NOT a fraction Models:Region (Surface Area) Length (Timeline) Sets (Groups of things) Pie Activity:We know that when a numerator and denominator are the same number, it equals a whole.We also know that the more pieces of a pie there are, the smaller the pieces will be.When different number of pieces equal the same surface area, they are equivalent fractions. Ex: We have 3 different pies, one divided in 1/2, one divided in 1/6, and one divided into 1/4. 1/2 = 3/6 = 2/4 *Although these fractions are different, they are equivalent fractions.

Ex 1: 1/4 + 1/6 = 5/12 (GCF) 4: 4, 8, 12 6: 6, 12We know that 12 is the GCF of 4 & 6. We know that to get from 4 to 12, we multiply 4 by 3 & to get from 6 to 12, we multiply 6 by 2, and whatever we do to the numerator we must do to the denominator. So this leaves us to multiply... 1/4 x 3 = 3/12 1/6 x 2 = 2/123/12 + 2/12 = 5/12 Ex 2: 1/2 + 1/6 = 8/12 (LCM) 2: 2, 4, 6, 8, 10, 12 6: 6, 12We know that 12 is the LCM of 2 & 6.We know that to get from 2 to 12, we multiply 2 by 6 & to get from 6 to 12, we multiple 6 by 2, and whatever we do to the numerator we must do to the denominator. So this leaves us to multiply... 1/2 x 3 = 6/121/6 x 2 = 2/126/12 + 2/12 = 8/12Ex 3: 3/12 + 1/4 = 1/2 (Simplify)We know that we can simply 3/12 down to 1/4 by dividing both the numerator and denominator of 3/12 by 3 since it the numerator 3 can only be reduced down to 1 by dividing it by 3. We know that we can no longer simplify 1/4 as it is already reduced, so now we have.. 1/4 + 1/4 = 2/4 which can be reduced to 1/2

When multiplying fractions, we know that the "x" sign means "of"Ex 1: 1/2 x 1/2 aka 1/2 of 1/2 = 1/4 Since we know that "x" means "of" we know that the problem is really asking us what 1/2 of 1/2 is, when we solve it we get 1/4.Ex 2: 2/3 x 1/4 aka 2/3 of 1/4 = 2/12 Since we know that "x" means "of" we know that the problem is really asking us what 2/3 of 1/4 is, when we solve it we get 2/12.Ex 3: 3/4 x 1/3 aka 3/4 of 1/3 = 3/12 Since we know that "x" means "of" we know that the problem is really asking us what 3/4 of 1/3 is, when we solve it we get 3/12.Ex 4: 3/4 x 2/3 aka 3/4 of 2/3 = 6/12 Since we know that "x" means "of" we know that the problem is really asking us what 3/4 of 2/3 is, when we solve it we get 6/12.*When you multiply, the product gets smaller & the denominator gets bigger*

Ex 1: 2/3 / 1/5 = 2/3 x 5/1When dividing fractions one way to solve the problem is by using the reciprocal of the second fraction given. The reciprocal is the opposite of the fraction given. For example the reciprocal of 1/5 is 5/1. Ex 2: 1 / 1/2 Another way to divide fractions is by asking yourself how many times can "b" go into "a". For example, in this problem 1="a" and 1/2="b". So how many times does 1/2 go into 1? 2 times. So now we divide 1/2 by 2. Which gives us our answer of 1/4.

Practice with FractionsEx 1: 3/4 of a class are girls, 2/3 girls have black fair, what fraction are girls that have black hair? For this problem we are only focusing on the girls, which make up 3/4 of the class, so for this problem we can ignore the boys which make up 1/4 of the class. So now we are left with 3/3 being girls, but how many of those girls have black hair? We know from the information given, that 2/3 of the class are girls with black hair. So now we can bring back the 1/4 of the boys because they are apart of the class total. So we now know that 1/2 of the class are girls who have black hair. Ex 2: Jim and his friends went trick-or-treating. Jim took 1/4 of all the chocolate bars. His friend Ken took 1/3 of the remaining chocolate bars. Len took 1/3 of that remaining, and we know that Max ends up with 8 chocolate bars in the end. How many chocolate bars were there? and how many did each of them get? So we know that Jim took 1/4 of all the chocolate bars so we can safely divide the total bars by 4, and we know that Jim took 1 of those 4 sections so we can put his aside, which leaves us with 3/4 of the total bars. We know that Ken took 1/3 of the 3/4 left. We know that since Jim took 1 of the 4 sections, there are 3 left. Ken took one of those 1/3 sections, leaving us with 2/3. We know that the 2/3 sections we have left are divided into 3 pieces each because of our denominator of 3, so we know in total there are 6 pieces left now between the 2/3 sections. Len takes 1/3 of those, so he takes 1/3 of the 6 pieces, which is 2 sections. And since Max took 8 bars, we know that every 1/4 in the beginning is work 6 chocolate bars. Max gets 8 pieces, Len gets 4 pieces, and Jim & Ken get 6 each.

Ex 1: If **** = 2/7 of a whole, what does the whole look like?So if we know that **** = 2/7 we know that ** = 1/7** ** ** ** ** ** **2/7 + 2/7 + 2/7 + 1/7So in total we have 14 * and that is what it would look like.

When working with decimals we know...It works on a base 10 systemThere's a one - to - ten relationshipPlace Value is emphasizedEx: 375.372 3 7 5 . 3 7 2 | \ \ | | | hundreds tens ones tenths hundredths thousandthsEx 1: I ran 25.3 miles | tenths place *Knowing that the .3 is in the tenths place, that means that .3 is 3/10ths of 1 mile*Ex 2: I slept 6.4 hours | tenths place *Knowing that the .4 is in the tenths place, that means that .4 is 4/10ths of 1 hour*

When adding and subtracting decimals, it's as easy making sure you line up the decimals correctly.Adding Decimals Subtracting Decimals Ex 1: Ex 2: Ex 1: Ex 2: 0.02 0.005 59.5 7.690+ 1.10. +21.5 - 0.41 - 0.1---------- ------------- ------------ ---------- 1.12 21.505 59.11 7.590

When multiplying decimals, there is emphasis on place value.Ex: 2 . 3 x 3 . 2 ---------- 1 4 6+ 6 9 0------------- 7 5 6 ---> So where do we place the decimal? | In the problem we have .3 and .2, these are equivalent to 3/10ths and 2/10ths

Percent means "per 100"When dealing with money, there are 100 cents in a dollar Real Life Ex: There is a $300 dress that is on sale for 30% off. How much is the dress after applying the discount? *We know we can break down the 300 into 3 groups of 100* $300 / | \100 100 100 *We know the dress is on sale for 30% off so we can subtract 30 from each 100 shown* $300 / | \100 100 100 -30 -30 -30------- ------ --------70 70 70*So now we get an answer of $70 as the final price of the dress after applying the discount*KEY: (Whenever we see these 3 key words we can assume they mean the following...)is -> =of -> x what -> n (variable)*Knowing about the key above, we can now turn these word problems into equations*Ex 1: 8 is what percent of 22? ----> 8 = n x 22 ----> n = 8/22 ----> n = .36 ----> .36 x 100 = 36% 8 is 36% of 22Ex 2: 8% of 22 is what number? ----> 0.08 x 22 = n ----> n = 1.761.76 is 8% of 22Ex 3: 8% of what number is 22? ----> 0.008 x n = 22 ----> n = 22/0.08 ----> n = 2758% of 275 is 22

When dealing with integers there are two methods that are mainly used:Chip MethodTimelineChip Method Key:Red Side of chip = NegativeYellow Side of chip = Positive**1 yellow side (positive) + 1 red side (negative) = zero pair**Zero pair: consists of 1 red chip and 1 yellow chip that cancel each other out making them equal to zero -> zero pair

Ex 1: +3 + -2 = +1*We start our problem by drawing 3 positives & 2 negatives*+ + + - - *Since we now have prior knowledge of zero pairs, we can start grouping them up*(+-) (+-) +*In this picture above, we can see we have 2 zero pairs, we know a zero pair is = to 0, so we can get rid of those 2 zero pairs since they are =0. We are now left with our answer of +1.Ex 2: -6 + -2 = -8*We start our problem by drawing 6 negatives & 2 negatives*- - - - - - - - *Since they are both negatives, we can simply grouping them together and add*- - - - - - - - *In this picture above, we can see that there are 8 negatives, and we are now left with our answer of -8.Ex 3: +3 + -4 = -1*We start our problem by drawing 3 positives & 4 negatives*+ + + - - - - *Since we now have prior knowledge of zero pairs, we can start grouping them up*(+-) (+-) (+-) -*In this picture above, we can see we have 3 zero pairs, we know a zero pair is = to 0, so we can get rid of those 3 zero pairs since they are =0. We are now left with our answer of -1.Ex 4: -2 + -1 = -3*We start our problem by drawing 2 negatives & 1 negative*- - - *Since they are both negatives, we can simply grouping them together and add*- - - *In this picture above, we can see that there are 3 negatives, and we are now left with our answer of -3.

Ex 1: +5 - +1 = +4*We start our problem by drawing 5 positives & 1 positive*+ + + + + + *Since we know we are taking away 1 from +5 we are now left with 4 positives*+ + + +*In this picture above, we can see that we are now left with our answer of +4.Ex 2: -5 - -1 = -4*We start our problem by drawing 5 negatives & 1 negative*- - - - - -*Since we know we are taking away 1 from -5 we are now left with 4 negatives*- - - - *In this picture above, we can see that we are now left with our answer of -4.Ex 3: -5 - +1 = -6*We start our problem by drawing 5 negatives & 1 positive*- - - - - +***We are asked to take away 1 positive, but we don't have any, we only have 5 negatives, so we know we need to add a zero pair***- - - - - (+-)**Now we do have +1 to take away from the -5, and we are left with 6 negatives as our answer of -6.

We can solve integer division problems by looking for relationshipsEx: +6 / +2 = +3 /+3 x +2 = +6 \ +6 / +3 = +2 -6 / -2 = +3 /+3 x -2 = -6 \ -6 / -3 = -2 +6 / -2 = -3 /-3 x -2 = +6 \ +6 / -3 = -2