Elementary Mathematics
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Elementary Mathematics  Annabelle Hawkins
Mindmap going over what we learned through weeks 715.
Mindmap going over what we learned through weeks 715.
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Elementary Mathematics
Week 9
Integers: Build, Sub, Multiply
1. Building Subtraction:
eg. 83=?
*** Students should say the problem as "8 take away 3".
Step 2: Place 8 postive counters on the table, then take away 3 of them.
Final answer= 5
For negative numbers:
eg. 5 (3)=?
Step 1: Add five negative tiles to the board.
    
Step2: take away three of the negative tiles.
 
Final answer= 2
When you can't take away enough tiles:
eg. 47=?
Step 1: Add four positive tiles to the board.
++++
Step 2: create a zero bank that is big enough for us to take away 7.
++++ +++++
    
Step 3: Now take away 7
++
    
The two positives and negatives cancel out, leaving us with 3.
Build multiplication
Build multiplication:
eg. 4(3)
*** say the problem as "4 groups of three positives".
Students should put four groups of three yellow tiles onto the table.
For negative numbers:
eg. 4(3)
Step1: Start with the number 0, change problem to 04(3).
Step 2: Create a zero bank that is big enough to take away four groups of postive three.
Step 3: Left over numbers cancel out, leaving you with 12.
Integers: Solving sub and Division
Integers: Solve Subtraction
1. If two answers come up as the same, that means they are equivalant.
2. Subtraction is the same thing as adding the opposite.
3. "Keep change change" Keep the first number the same, change the symbol, and change the sign of the second number.
eg. 38 (41)=?
K C C
38+(+41)=?
=79
Integers: Multip/ Div rules
1. neg x neg= positive
2. Neg x pos= negative
3. neg x pos= negative
4. pos x pos= positive
***Never use the word "and" when talking about these rules..
Simpler rules:
1. Same signs= positive
2. Opposite signs= negative
Week 10
Intro to Fractions
1. Context is important when talking about fractions. 6 cars will be bigger than 10 plants.
2. Know how big each number is.
3.The numerator is the number of things we have, the denominator is the total amount possible.
4. The denominator will tell you the size of the piece.
5. The bigger the denominator, the smaller the piece (numerator).
Solving fractions: add/subtract
Solving fractions with understanding:
1. Always add and subtract whole numbers first.
2. Instead of "common denominators, say "same size piece".
3. When subtracting a fraction from a whole number, you can give the number a "common denominator" by taking away one of the pieces from the whole number and writing it as a fraction with the same denominator.
eg. 8 4/7=?
7 7/7 4/7
= 7 3/7
Adding subtracting fractions with different denominators:
eg. 7/18 1/6=?
Step 1: We need the denominators to be the same, so we need to figure out what their greatest common factor is.
Step 2. 18 can be broken down into 3 groups of 6, so we multiply 1/6 by 3 (both the top and bottom). This gives us 7/18 6/18.
Final answer= 1/18
Comparing Fractions
Determining which fraction is bigger:
1.Multiply both numerators by 2 to see which one makes it closer to 1/2 (anchor fraction). Whichever is closest is the bigger number.
2. If the numerator is the same, the one with the smaller denominator is the bigger number.
3. If they are both missing the same amount, the one with the bigger denominator is the biggest because it is only missing one small piece.
4. If they have whole numbers, the one with the bigger whole number will always be the biggest.
5. If the numbers have common factors, you can multiply them to make the denominators the same. This will make it easier to see which is bigger.
Week 8
Divisibility Rules
Divisibility Rules:
2: All even numbers, end in 0, 2, 4, 6, 8
3: The sum of digits is divisible by 3
4: If the last two digits are divisible by 4, then yes.
5: Last digit is a 5 or 0.
6. If 2 and 3 both work, then 6 works.
8: If the last digit is divisible by 8, then yes.
9: the sum of the digits is divisible by 9.
10. if the last digit is 0, then yes.
Intro to integers: Build and Show
1. Build integers using two color counters:
*** Red= negative, yellow= positive. Negative numbers go on the bottom, positive numbers go on the top.
eg. Build 3 using 5 tiles.
Step 1: Place three positive (yellow) color counters on the table.
+++
Step 2: Create a "zero bank" using positive and negative counters. You can add one positive and one negative counter, giving you 5 counters in totoal. The positive and negative will cancel out, giving you 3.
+++ +

2. Showing integers using + and 
eg. 3 + 4= ?
*** Positives still go on top, negatives on the bottom.
++++

The positive and negative cancel out, leaving us with 3.
Solving and Adding Integers
Adding Integers Algorithm: Use + and  symbols to figure out whether we add or subtract.
eg. 25+ (12)=?
Step 1: Write two positives above the 25 and one negative above the 12. This reminds us that we would have more positive tiles since 25 is a bigger pile, and less negative tiles since 12 is a smaller pile.
Step 2: Since we have two postives and one negative, this means we subtract because we have one of each sign.
25
12
=13
*** If you have all negatives, you would add the numbers together. If you have two negatives and a positive, you would subtract.
Week 7
Subtraction Using Expanded Form
To subtract using expanded form, break the numbers into separate place values using addition and then subtract them one at a time. Then, add what you have together.
eg. 426234= ?
400+20+6
200+30+4
=100+10+2
Final answer= 112
Subtraction Using Equal Addends
If you do not want to use borrowing, the equal addends method for subtraction can be used to make the bottom number smaller than the top.
To use equal addends, add any number to the bottom that will make the last digit of the bottom number smaller than the top digit.. Then, add that same number to the top.
eg. 36 28=?
36
28
We can add 2 to 28 to make it an even 30. Then, we'll add 2 to 36 as well.
36+2
28+2 =
38
30=
8
Spiraling Content
Spiraling content is going back and revisiting previously learned content throughtout the semester. This way, it's harder to forget.
Multiplication Facts
1. Time tests are ineffective because they put students under too much pressure, leading them to do poorly on the test.
2. Progress charts that show improvement can encourage students to keep working.
3. Flash cards are a great way to memorize the multiplication table.
4. Multiplication tricks are terrible because students don't atually learn anything about multiplication and they don't work for all numbers.
Multiplication Alt algorithms
1. Expanded form: Break each number apart by place value using addition. Multiply each number by each other once, then add what you have together at the end.
eg. 35 x 23=?
30+5
x 20+3
600+100
+240+40
=840+140
Final answer=980
2. Left to right: Instead of putting numbers into expanded form, write them normally and solve the same way. Keep in mind their actual place value.
eg. 47x53
47
x 53
2000
350
21
120
=2441
3. Lattice: Draw the lattice box, making diagnol cuts for however many place values there are. Then, put one number on the top and one number on the right. Multiply across from right to left.
Division: Long,repeated subtraction. and upwards
1. Long division (or traditional division): This method is tricky for students because they have a hard time remembering which number goes on the inside. The small number does not always go on the outside. Remainders in fractions are too confusing.
eg. 218 / 4
218 is the number being split, so it would go on the inside. We say "218 divided into four groups."
2. repeated subtraction: Use the highest factor that the students know and then write it on the side. Then, subtract and keep repeating until that number doesn't work anymore. The remainder is written as a fraction, with the remainder as the numerator and the outside number as the denominator. Pros: This method is simpler than long division. Cons: The remainder can be confusing.
3.Upwards division: Write the equation vertically, or as a fraction. Make sure the top numbers are spaced apart so that you can write other numbers in between. Figure out how many times the denominator goes into each top number, then write that number in the answer spot. Multiply these two numbers so it can be subtracted from the first digit, then repeat the process again for the next numbers. The remainder will be the numerator and the denominator will stay the same.
Week 12
Multiplying Fractions Algorithm
1. Show multiplying fractions using diagrams:
ex. (3/4)(2/3)
Step 1: Draw a box, breaking it into four parts horizontally. Then, color in three pieces.
Step 2: Break the box into three parts vertically, then fill in two of the vertical pieces using a different color.
Step 3: Count how many squares overlap. You should end up with (6/12).
2. Backwards C:
To multiply fractions with whole numbers, use the backwards C to turn them into improper fractions.
ex. (3 3/4)(2 4/5)
Step 1. Multiply the whole number by the denominator and then add the numerator to the product.
3x4=12> 12+3=15> 15/4
Step2. Do the same with the other fraction.
2. 2x5=10> 10+4=14> 14/5
Step 3. Now multiply the fractions like normal.
(15/4)(14/5)= 210/20
step 4. Simplify: 10.5
3. Funky Ones:
Step 1. When multiplying fractions, break the numerators and denominators down into simplified groups.
ex. (8/10)(3/4) can be written as ((2x4)/(2x5)).
Step 2. Cross out any numerators and denominators that both of the fractions share, creating a "funky one." In this problem, the fours would cross out because of the fractions have them.
step 3. Now multiply straight across.
Dividing Fractions Algorithm
Dividing Fractions
Step 1. When dividing fractions, use the KCF method (keep, change, flip). Keep the first number the same, change the division sign into a multiplication sign, and flip the second number so that the denominator is on top and the nator is on the bottom.
ex. (8/15) / (10/30)> (8/15) X (30/10)
Step 2. Simplify the fractions using the funky ones method. Cross out numerators and denominators that are the same.
((2x4 )/ (5x3)) / ((2x15) / (2x5))
Step 3. Multiply across.
Final answer= 3/5
2. For dividing fractions with whole numbers, use the Backwards C method to turn the fractions into improper fractions. Then go on to do KCF and multiply across.
Showing Fractions: Add, sub, multiply
1. When drawing fractions, use rectangles instead of circles because they are easier for children to understand.
2. Adding fractions by drawing diagrams:
ex. (1/3) + (1/4)
Step 1. Draw two rectangles about the same size, one with three pieces and one with four pieces. Fill in one piece for each of them, since their numerators are both 1.
Step 2: Now draw another rectangle about the same size. Draw three pieces vertically, then four pieces horizontally using a different color. Fill in one third and one fourth.
Step 3: Count how many pieces are filled in, and that's your answer.
= (7/12)
3. Subtrcating Fractions by Drawing Diagrams:
ex. (3/4)  (1/3)
Set up problem by drawing rectangles for each fraction, just like ad addition problem. Cross out the 1/5 from the 3/4, giving you (1/5).
Week 13
Showing Decimals
Showing Adding Decimals
1. If possible, use manipulatives to show decimals.
ex. .13 + .26
2. Emphasize that these numbers are different. Tens become tenTHS, hundreds become hundreTHS. The first digit, going left to right, would be the tenths place, and second digit would be the hundredths place.
3. Decimals are just fractions written differently.
4. Decimals can be shown by using base ten blocks, just like fractions.
5. The zero in front of the number means we have zero whole numbers.
6. To show the hundredths place, cut the rectangle into ten pieces horizontally and vertically. The numbers in the "tenths place" are shown by filling in whole pieces (ten squares), and numbers in the "hundredths place" are shown by filling in a single piece (one square).
Showing Subtracting Decimals
1. Show the first number by filling in the tenths place and hundredths place in your base ten block. Draw a circle around how many pieces are being taken away, showing the amount you actually have left.
2. If needed, whole pieces can be broken down into singular pieces to make subtracting from the hundredths place easier.
Showing Multiplying Decimals
1. Draw a rectangle with one hundred pieces by drawing ten pieces vertically and horizontally.
2. Fill in how many pieces you have for each number using two different colors, and the answer is the double shaded regions.
Solving Decimals: Adding/ Subtracting/ Multiplying
Adding Decimals
1. Have students estimate the answer before they actually solve the problem.
2. Students should set up their addition problems by lining up the whole numbers.
3. Students should not "line up the decimals" because knowing where to place the decimal can get too confusing.
4. Once the whole numbers are lined up, have the students solve the problem like a normal addition problem.
5. Place the decimal according to the estimation.
Subtrcating Decimals
1. Set up problem the same way by lining up the whole numbers.
2. Estimate the answer
3. Subtract, place decimal according to the estimation
Multiplying Decimals
1. Estimate what the answer will be
2. Then multiply the numbers WITHOUT decimals
3. Once students have the answer, place the decimal according to the estimation.
Week 14
Order of Operations
1. Why does order of operations matter?> If long equations are not solved in the correct order, students will not get the right answer.
2. PEMDAS is not necessarily the "correct" order; can also be done as PEDMSA
3. Problems are done in this order: Grouping, exponents, Mupltiplication/Dvision (left to right), and then Addition/Subtraction (Left to right).
Exponents
1. When squaring negative numbers, they will still end up being negative. 5^2 would equal 25, because it's the same thing as saying 05^2. The only exception would be when the exponent is outside of the parentheses
Organizing Equations into Groups
1. Draw a line anywhere you see an addition or subtraction sign, which signals that these numbers are a certain group.
2. *** Negative signs are counted as subtraction signs.
3. Solve the numbers in each group. After simplifying all of them, you should end up with only addition and subtraction.
4. ***When there are fractions involved, the fraction becomes it's own group.
Week 15
Scientific Notation
1. What is scientific notation?> A way to manipulate very small or very large numbers so that they are easier to look at.
2. The first number is always between 1 and 10. The number will also be times 10 and there will always be an exponenet above the 10.
3. ***Exponent can be positive or negative.
ex. 3.07 x 10^14
Scientific Notation Without Counting the Decimal or Moving Left/Right
1. Decide if the number is big or small. Big numbers are numbers with whole numbers before the decimal. Small numbers are numbers without whole numbers at the front. Big number= positive exponent, small number= negative exponent. Big numbers move to the right, small numbers move to the left.
2. Place the decimal between the first two nonzero numbers.
ex. 2,040,000,000> 2.04
Add the x10 after this number> 2.04 x 10
3. Exponent will be positive since it is a big number.
4. Count how many numbers are after the decimal. This will be your exponent.
=2.04 x 10^9
Putting Scientific Notation Back into Standard Form
1. Decide if the number is big or small.
ex. 2.15 x 10^5 > Small number because the exponent is negative.
2. Since the exponent is 5, add five zeros to the left of the whole number> 00000215
3. Place decimal between first two zeros > 0.0000215
Week 11
Exam 2