1. Whole Numbers (Week 4)
1.1. Multiplication
1.1.1. Using repeated-addition model is one way to interpret products of whole numbers. If we have 4 groups of 3 which the equation would be 4x3=12 for the method it would be 3+3+3+3=12.
1.1.2. We can also use an array which is a number line. Multiplication also has algorithms, the Standard one, Expanded notation, Place value, and you can also use the Lattice method.
1.2. Division
1.2.1. We use three models for division; repeated-subtraction model, set(partition) model, and missing-factor model
1.2.1.1. Example: 11-3=8 8-3=5 5-3=2 so when we divide 11 by 3 we get a 3 with a R2. 11 is not devisable by 3 so we have to look more into it
1.2.2. For figuring out division we can also use the division sign, 3/5 fraction and the vinculum
2. Divisibility (Week 5)
2.1. a is divisible by b, if there is a number c that meets the requirements: bxc=a
2.1.1. Example: 10 is divisible by 5 5x2=10
2.2. These are some rules to divisibility
2.2.1. Ending: by 2; 0,2,4,6,8
2.2.2. Sum of digits: by 3 & by 9.... example: we have 543 which would be 5+4+3=12
2.2.3. Other: by 6; 2,3.... example: 3,702 which would be 3+7+2=12 we wouldn't add the 0 because its just 0.
2.2.4. Last digits: by 4: last 2 digits... example 3,728 if it was by 8 it would be the last 3 digits.
2.2.5. Special numbers: by 7... example: 826, we have to double the last digits... 6x2=12 then take 82-12=70....by 11 we "chop-off" take the last 2 digits off then add to the remaining number.
3. Factors & Prime Numbers (Week 6)
3.1. For factors we use a factoring tree and for that we find the common numbers that are divisible for that number. Prime factors can also be involved: any of the prime numbers that can be multiplied to give the original number.
3.1.1. Example: 28; (1,2,4,7,14,28) these are numbers that are able to make 28. 1x28=28 , 2x14=28 then 4x7=28 these numbers multiply into 28.
3.1.1.1. Most of these numbers are prime numbers.
3.2. A prime number are whole numbers greater than 1, that have only two factors – 1 and the number itself.
3.2.1. Prime numbers; 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59
4. Fractions (Week 8)
4.1. A fraction is part of a whole, it can also be a ratio. It's a symbol that represents the relationship between the part and the whole.
4.1.1. There are some models that we can use to figure out fractions. We can use surface area (region) by using a picture and dividing it up into what the fraction is.
4.1.2. We can also use things in the classroom like using a piece of paper by folding it or even using a number line (line segment).
4.1.3. Lastly we can use sets of things around us. We can use groups and use the students as well.
4.2. Fractions are whole numbers that can have the same denominator, or equavalent fractions, and can also not always be in simplest form.
4.2.1. Examples: 3/12 + 2/12 = 5/12 1/6 + 3/6 = 4/6
4.2.2. Examples: 5/12 - 1/12 = 4/12 4/6 - 2/6 = 2/6
4.2.3. Examples: 1/2 + 1/6 for this problem we have to find the LCM which would be 6 so the problem would be 3/6 + 1/6 = 4/6
5. Converting & Percentages (Week 10)
5.1. is: =, of: x (multiplication sign), what: n (variable), and change percent to a decimal.
5.1.1. Ex: 33% = 33/100 = 0.33, 1/10 = 0.1=0.10 = 10% 3/10=0.30 = 30%
5.2. The % symbol is crucial in identifying the meaning of a number. Converting percents to decimals can be done by writing the percent as a fraction in the form n/100 and then converting the fraction to a decimal.
5.3. A terminating decimal is when the numbers don't repeat. A repeating decimal is when the numbers continue on. For the repeating decimal you would have to put a bar on top of the number to show it's repeating.
5.3.1. Terminating example: 1/8 = 0.125 = 12.5 ( we would round up in this case if necessary) - 13% Repeating example: 5/6 = 0.83 = 83% ( for this we would put the bar on top of the three to show it is repeating.
6. Problem Solving (Week 1)
6.1. 1. Understanding the problem
6.1.1. When trying to understand the problem, we have to figure out what is to be found or what is needed in the problem. With that, we are able to find the information needed for the problem.
6.1.1.1. Natural numbers are mainly used like, 1,2,3,4,..... which is 1+2+3+4+....+10
6.2. 2. Devising a plan
6.2.1. For devising a plan, we can look for many things to help us with our end result. We can look for patterns, make a table or list, identify a subgoal, guess and check and in the end write an equation for our problem.
6.2.1.1. We see that there is a pattern; 1,2,3,4,5,6,7....,10 we are adding one each time
6.3. 3. Carrying out the plan
6.3.1. How we carry out the plan, we have to be able to check each step through the process. Checking our work.
6.4. 4. Looking back
6.4.1. Looking back at the problem, we wanna check our work to make sure it works. If there are other ways for it to make sense, use those techniques for the problem as well.
7. Whole Numbers (Week 3)
7.1. Addition
7.1.1. Properties & Algorithms
7.1.1.1. Identity property; a+0=a, commutative property (order); a+b=b+a, associative property (grouping); (a+b)+c=a+(b+c).
7.1.1.2. There is 6 different algorithms for addition, first we have the Standard American, Partial Sums, Partial Sums with place values, Left-to-Right, Expanded notation and Lattice.
7.1.2. Addition is a binary operation because two numbers are combined to form another number. It is important for children to use different strategies that make sense to them.
7.2. Subtraction
7.2.1. Algorithms
7.2.1.1. For subtraction we also have 6 different algorithms. First the normal American standard, European-Mexican, Reverse Indian, Left-to-Right, Expanded notation, and Integer Subtraction
7.2.2. Subtraction is the inverse operation of addition. We sometimes use the take-away model to help students better understand.
7.2.2.1. With subtraction we can also use properties as well; commutative, associative and identity property.
8. GCF & LCM (Week) 7
8.1. We use the greatest common factor and the least common multiple for factors. GCF: The highest number that divides exactly into two or more numbers. LCM: The smallest positive number that is a multiple of two or more numbers.
8.1.1. One way which seems to be the easiest would be the list method. With the numbers we are given, we find what can be multiplied into that number. When we see that there is something common in it then we decided what the GCF is or the LCM is.
8.1.1.1. Example: 24: 1,2,3,4,6,8,12,24 then we have 36: 1,2,3,4,6,9,12,18,36 for these two numbers we see that 12 is the GCF.
8.1.1.2. Example: 24; 24,48,72,96 to understand better we are just adding 24 until we see a number the same for 36: 36,72 so the LCM would be 72
9. Decimals (Week 9)
9.1. Decimals follow the same rules when it comes to place value: one-to-ten
9.2. If we have a number like 375. we know that the hundreds place is the 3, the tens place is the 7, and the ones place is the 5.
9.2.1. For decimals we can do addition, multiplication and division.
9.2.1.1. For trying to figure out each decimal we have to line up each decimal to get the correct answer. Ex: 3.23 + 1.2 we line them up then add which gets us 4.43.
9.2.1.2. For multiplication if we have numbers like 1.9 and 2.1 we estimate each to 2 instead which will get us 4 because 2x2 is 4. In reality the actual answer would be 3.99.
9.2.1.3. For division, we have 369.36/3 we would have the bigger number in what looks like a box and the 3 would be on the outside. To figure it out we have to move the decimal up to start off. Then from there we figure out each number that divides into 3.
9.3. Decimals can be introduced with concrete materials and we can also use decimal grids. We can use whole units, tens units, and ones units.
9.4. We also can use terminating decimals which is numbers that can be written with a finite number of places to the right of the decimal point.
10. Integers (Week 11)
10.1. A way we are able to find integers is a method called the "chip" method. For negative numbers we use to red chips and for the positive numbers we use the yellow chips. A positive and a negative together makes a zero pair. The negative integers are opposites of positive integers.
10.2. Example for Addition: +5 + +1 so for this one we would add 5 positive chips and then once we have the five we add an extra because it is asking to add one positive chip. Which then it gets us +6
10.2.1. Example for Subtraction: -5 - +1, for this problem we would have to add 5 negative chips but then we need to add one positive chip to the circle. Since we are not able to do that, we have to add a zero pair. +&-. So then we cancel out the new positive chip and left with a negative chip to add to our circle of chips. Which then gets us -6.
10.2.1.1. Example for Multiplication: +3 x +2, for this problem it is asking 3 groups of +2. So we would have three circles and we have to add 2 positive chips into each circle. Which then gets us +6
10.2.1.2. Another example for Multiplication: -3 x+2 for this problem we have to use the Commutative Property. So we flip the numbers around and get +2 x -3. It is then asking 3 negative groups of +2. We have two circles and we have to add 3 negative chips to each circle. Which our answer would be -6.