Teaching Elementary Mathematics

Strategies:Students must read and understand the problem. They must reiterate the information of the problem for thorough understanding. Drawing out the problem or using manipulativesTrial and Error is a popular strategy. They must act it out then simplify the problem. Students must try different strategies until they figure out the problem.

Polya's 4 Steps to Solving ProblemsUnderstand the problem.Devise a plan.Carry out the planTake a look back and reflectTaking a look at the work you did is the most important part of the plan. Not only does it confirm that the answer is correct put also that each step was completed properly

Practice ProblemsHandshake ProblemBasketball ProblemStamp Problem

Expanded Notation375= 300+70+5375= (3x100)+(7x10)+(5x1)375= (3x10^2)+(7x10^1)+(5x1^1)214= 200+10+4(2x100)+(10x1)+(4x1)(2x10^2)+(1x10^1)+(4x1^0)

1) American Standard2) Partial Sums3) Partial Sums (B) Right to Left4) Left to Right Must write place value explicitly

1)              American Standard  11       478    +394           872                                                                                               2)              Partial Sum  4 7 8    +3  9  4                 1 2            1 6             + 7                    8  7 2    3)              Partial Sum (B)       4 7 8         +3 9 4     1      2  1      6  0       +7  0  0                8 7  2    4)              Left To Right    478  +394    700     160    + 12     872   5)              Expanded Notation 100 10              400+70+8          300+90+4          800+70+2=872                                  6)              Lattice          4 7 8         + 3 9 4                              1    1         +    7  6    2              8    7    2 Subtract: 645-2791)              Subtraction Standard 5 13       6 4 15   -2 7 9      3 6 6  2)              European/Mexican     14 15 6 4 5   3  8    - 2 7 9   3)              Reverse Indian Left to Right      14 15  6 4 5 - 2 7 9    4 7     3 6 6  4)              Left to Right Place Value Emphasis                      140 15  6 4 5      -2 7 9       4 0 0       3 0 0           7 0      6 0 +         6   3 6 6 5)              Expanded Notation Place Value Explicit 500 + 130 +15   645     600 + 40 + 5    -279   - 200 + 70 +9     366 =  300+ 60 + 6       6)              Integer Subtraction 6 4 5  - 2 7 9           -4       -3 0  - 4 0 0 3 6 6

Divisibility Rules:A is divisible by B if C exists C x B=A 5x2=10 10/2=5 without a remainder24 is divisible by 2 because they're both a multiple of 12 12x2=2410 is divisible by 5, 10 is a multiple of 55 divides 105 is a divisor of 10A number is divisible by 6 if it is divisible by both 2 and 3 Example 48/6=8 48/2=24 48/3=16Numbers are divisible by 4 when the last two digits are divisible by 4 Example 72,328/4=18,082To know if a number is divisible by 8, you have to be able to divide evenly into the last three digits.To identify numbers divisible by 7 you have to double the last number then subtract it from the remaining digits Example 12,731 1x2=2 1,273-2=1,271 127-2=125To identify numbers divisible by 11 you remove the last two digits and add it to the remaining number Example 29194; 291+94= 385; 3+85=88 88 is divisible by 11*Ending of a numberIf it ends in an even number it is divisible by 2When a number ends with 0 or 5 it is divisible by 5When a number end with 0, it is divisible by 10*Sum of the digitsYou add all the digits, if the sum is divisible by 3, then the whole number is divisible by 3 Example 375; 3+7+5=15 375/3=125The same rule applies to the number nine *Composite numbers have several factors*Prime numbers only have 2 factors 1 and itself*0 and 1 are not prime numbers*When you add zero to any number, the number does not change.0: The additive element1: The multiplicative element

Draw out problems below 3/4 of the class are girls, 2/3 have dark hair. What part of the class is a girl with dark hair? 3/4X2/3= 6/12=1/2 the class are girls with dark hair2/3 x 1/2= 2/6= 1/31/4 x 1/4= 1/163/4 x 1/3= 3/12= 1/4

1. If 1/3 cup sugar is needed to make two loaves of bread, how many cups of sugar are needed for three loaves? ½ cup of sugar 1/3 x 1/2= 1/6  1/6 x 3/1= 3/6=1/2First, I multiplied 1/3 by ½ to determine how much sugar is needed for a single loaf instead of 2 loaves. I got 1/6 I then multiplied that 1/6 by 3 to determine how much sugar is needed for three loaves and I got 3/6 which is ½ cup of sugar. 2.  When the least common denominator is used in adding or subtracting fractions, is the result always a fraction in its simplest form? Explain by giving examples. You will not always have a simplified answer even when using the least common denominator. When adding fractions, the denominators must be the same, sometimes the least common denominator will result in multiplying both denominators so you may have to simplify. It also depends on what the numerator is as well. Examples: 5/6 – 1/3= 5/6 – 1/3(2/2)= 5/6- 2/6= 3/6= 1/2 1/3 + 1/12=1/3(4/4) + 1/12=4/12 + 5/12 =9/12 =3/4

With decimals, you need to round number 5 or higher to the next whole.To divide decimals you move the decimal of the divisor to the far right to make it a whole number, do the same to the dividend. 1 A student takes a test with 45 questions and gets 37 questions right. What is their percent on the test?37/45= 0.822= 82.2%= 82%2 A factory makes sandals. If they produce 820 sandals in a day and 32% of them are blue. How many are blue?820X0.32= 262.40= 262 sandals are blueThere are three types of problems with %s8 is what % of 22? = 8=n X 22= 8/22=n=0.363= 36%8% of 22 is what #?= 22x0.08=1.768% of what # is 22= 0.08 X n=22= n= 22/0.08=275The easiest way to solve these problems is to turn them into an equation, remembering that:"is" means ="of" means X"what" means variable n 11% of 45 is what number?0.11x42=4.959% of what number is 17?0.09 X n= 17 n= 17/0.09= 18817 is what percent of 25? 25/17=0.68= 68%

1 In the United States, 13 out of every 20 cans are recycled. What percentage of cans are recycled?13/20= 13*5/20*5=65/100=0.65= 65%2 2% of Hamilton Middle School students have red hair. If there are 700 students how many have red hair.700*0.02=14 also 2% of 100 is 2 for every 100 students, that means for 700 its 14 students.There are 25 students in Ms. Johnson's class. In the class election 4 people voted for Ben, 12 voted for Sahil, 9 voted for Maria. What percentage voted for Maria?Because there are 25 students you can just multiply the number of votes for each student by 4 and get the percentage.Ben 4x4=16%Sahil 4x12= 48%Maria4x9=36%Sophie eats a 50 gram chocolate bar which contains 30% cocoa. How many grams of cocoa are in the bar.50*0.30=15 grams of cocoaWhen a grizzly hibernates its heartbeat drops to 10 beats per minute which is 20% of the normal rate. How many times does a grizzly bears hear beat at a normal rate?10/20*5=50/100 50 beats per minuteOf the 50 US states, 4 start with the letter W. What percentage of state start with the letter W?4/50=8/100=0.08=8%While waiting for a video game to download, you notice, 30% of the 32,000 kilobytes have been downloaded so far, how many kilobytes have been downloaded so far?32,000*0.30= 9,600.00 downloaded kilobytes

American StandardOne to Ten RelationshipDigits Used 0-9 in Base 10Other countries use different bases. Or example indigenous Australians use Base 5

In base 10 we operate to the tenth power and use digits 0,1,2,3,4,5,6,7,8,91s, 10s, 100s, 1,000s, 10,000, 100,000, 1,000,000.....The same principle applies to all other bases. Base 30,1,21s, 3s, 9s, 27s, 81s.....Base 40,1,2,31s, 4s, 16s, 64s, 256s...Base 5 0,1,2,3,41s, 5s, 25s, 125s, 625s...

211 base3(2x9)+(1x3)+(1x1)(2x3^3)+(1x3^1)+(1x1^0)18+3+1=22102base3(1x9)+(0x3)+(1x2)(1x3^2)+(0x3^1)+ (1x2^0)9+0+2=111111base3(1x27)+ (1x9)+(1x3)+(1x1)(1x3^3)+(1x3^2)+(1x3^1)+(1x3^0)27+9+3+1=40

Addition:Identity Property:When you add zero to any number that number's identity does not change. Example 7+0=7Communitive Property: Order propertyWhen you have 2 numbers and you add them, the order in which you add them does not matter 4+3=7 3+4=7Associative Property: Grouping PropertyWhich means the way you ground them does not matter example (a+b)+c=d (a+c)+b=dMultiplication:Is grouping of things or numbers, example 3x2 is three groups of 2Identity property applies to multiplication example 8x1=8Communitive Property applies to multiplication, the order in which you multiply does not affect the outcomeAssociative PropertyApplies to multiplication (axb)c=(ax(bxc)zero property:Any number you multiply by zero is zeroDivision is repeated subtraction

FractionsA fraction is a symbol, a representation of the relationship between a part and a wholeAt 2:20 she explained that if the numerator and denominator are the same, the fraction equals 1 Example 8/8=1 6/6=1 5/5=1 3/3=1Quotient: 3÷5= 3/5 Ratio: 20 students, 12 girls, 8 boys Boys to students 8 to 20= 8/20 Girls to students 12 to 20= 12/20= 3/5 Boys to Girls ratio 8:12 this is NOT a fraction because these are two parts, no wholeFractional parts are equivalent partsThe more pieces the whole is cut into the smaller the pieces get Example of equivalent fractions: 1/2= 2/4= 4/8Using the least common multiple does NOT mean you will get a simplified answer *Dividing Fractions: Leave it, change it, flip itLeave the first fraction the sameChange the division symbol to a multiplication symbolFlip the numerator and denominator of the 2nd fraction and multiply across Example 2/3÷4/5= 2/3 X 5/4= 10/12= 5/63 groups of 2= ** ** **= 3x2ModelsArea; The surface area Length; Number lineSet: Groups of things

375.3273 Hundreds7 is Tens5 is Ones3 is Tenths2 is Hundredths7 is Thousandths

Janice is preparing a recipe that calls for ¾ of a cup of oil per serving. If Janice needs to prepare 2 and 2/3 servings, how many cups of oil will she need?3/4 x 8/3= 24/12= 2 cups of oilMarc opened a pizza box. Inside there was ¾ of a pizza. Marc ate ½ of what was in the box. How much pizza did Marc eat3/4 x 1/2= 3/8s of the whole pizzaIf **** represents 2/7 of the whole, draw what the whole looks like.**= 1/7****= 2/7**************= 7/7Jim, Ken, Len and Max have a bag of miniature candy bars from trick-or-treating together. Jim took ¼ of all the bars, and Ken and Len each took 1/3 of all the bars. Max got the remaining 4 bars. How many bars were in the bag originally? How many bars did Jim, Ken and Len each get?1/4 + 1/3 + 1/3= 1/4(3) + 2/3(4)= 3/12 + 8/12= 11/12Max's 4 bars= 1/12Total Bars 48Jim 1/4= 12 barsKen 1/3= 16 barsLen 1/3= 16 bars12+16+16+4=48

Homework #6March 16, 2022Solve both problems. As always, show your work.1.    A set of marbles can be divided in equal shares among 2, 3, 4, 5, or 6 children with no marbles left over. What is the least number of marbles this set could have? 60 marbles2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,602x30=60 Each child would get 30 marbles 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,603X20=60 Each child would get 20 marbles 4,8,12,16,20,24,28,32,36,40,44,48,52,56,604X15=60 Each child would get 15 marbles 5,10,15,20,25,30,35,40,45,50,55,605X12=60 Each child would get 12 marbles 6,12,18,24,30,36,42,48,54,606X10=60 Each child would get 10 marbles 2. Mary spent two thirds (2/3) of her money. She lost two thirds (2/3) of the remaining amount and then she had $8 left. How much money did she start with? $72      $48 spent, $16 lost, and $8 remaining 

375.3273 Hundreds7 Tens5 Ones.3 tenths2 hundredths7 thousandthsDecimals separate the whole from the part. The decimal always sits to the right of a unit. In money, the dollars are the unit and the decimal are the cent parts of the dollar.Moving the decimal point to the right will show you how many dimes you need to have that specific amount.3/10= 0.3= 30/1003/100= 0.037/10= 0.7012/100= 0.120.72= 72/100Adding a zero to the decimal can help students understand the value 0.3= 0.30Students who don't fully understand place value will make the mistake of thinking 0.73 is greater than 0.8. Adding the zero can help demonstrate that 0.73<.080Same goes for digits like 0.125 and 0.9 0.900>0.125Terminating decimals means it has no remainder. 0.125 is a terminating decimal.Repeating decimals require a bar over the digits that repeat. Be sure to put the bar above the entire repeating number. π is not a decimal because it does not stop or repeat any pattern, its considered an irrational #

0.1= One tenth0.01= One hundredth0.101= One hundred one thousandths0.001= One Thousandth0.011= Eleven ThousandthsYou can turn the decimal into a fraction to determine which one is smaller and multiplying by 10 if necessary to get equivalent fractions. Be sure to line up the decimals when adding and subtracting to ensure all the digits get added to the correct place value. This needs to be emphasized to school children.To subtract $5.00-$1.23 you tell your students to exchange $1 for 10 dimes and 1 dime for 10 pennies. When they remove $1, 2 dimes and three pennies they will see that they are left with $3.77With multiplying, there is no need to line up the decimals when writing out the problem. After you complete the operation count the number of spaces after the decimal, that will tell you how many spaces over your decimal needs to be in your answer. Percent means out of 1003/100= 0.03= 3%Fraction to percentage2/3= 0.66= 66% 3 goes into 2 0.66 times5/4= 5/4*20=100/80= 1.25= 125%5/6= 0.832= 83.2% 6 goes into 5 .832 times

Make sure you put the bar over the repeating decimal in the right position.Examples 1.636363 bar goes over 631.666666 bar goes over a 6Research shows that a vertical number line helps students more than a horizontal number line.Reminder multiplication is repeated addition 3x2 is 3 groups of 2Chip method:This method helps students understand the concept of positive and negative integers, using positive and negative chips. If a problem calls for you to remove or add an integer that you don't have you can add zero pairs, basically a positive and a negative that cancel each other out.In multiplication a negative times a negative is a positive. When you see the negative integer, your answer should have the opposite sign.Also in multiplication when an equation starts with a negative integer, rearrange the equation so that the first number is positive.

Addition and Subtraction+3 + -2= +1 + + + - - two cancel out leaving +1-6 + -2= -8 - - - - - - - - add 2 negative left with -8+3 + -4= -1 + + + add - - - - you are left -1-2+ -1= -3 - - - ass 1 negative left with -3+4 - -2= +6 + + + + add zero pairs - + - + left with +6Multiplication-2 x 5= -10+5 x -2-- -- -- -- ---2 x 3= -63 x -2= -6 -- -- --What will be on the final:Prime NumbersWhat is a factor and what is a multiple2x5=10 2 and 5 are FACTORS2,4,6,8,10,12 are MULTIPLES Of 2Decimals: Comparing fractions, long division, repeating or terminating decimals bars in the correct placeChip methodPercentage of different #sProblem solving with fractionsFind least common multiple