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MTE 280 Numbers, Operations, and Proportional Reasoning for K -8…
MTE 280
Numbers, Operations, and Proportional Reasoning for K -8 Teaching
Weeks 1-4 :pencil2: !
Week 1- Introduction to number systems
Story of One- BBC
documentary
A very fascinating hour-long documentary that covered the history of numbers, while exploring a variety of number systems that had contributed to society and the development of mathematics. The "binary" number system was also introduced in this video.
Here's a few things I learned
1) What was surprising to you about the number systems?
What was surprising to me about number systems, or specifically the "binary" number system, is the effective use of "1s and 0s". I didn't know how effective the binary system was outside of "code" which I'm not sure if that corresponds. I love as well how the documentary gave a meaning for the expansion or utilization of math was primarily to help the expansion and progression of ancient cities. (assess wealth, calculate profits and loss, and collect taxes)
2) Which numbers would you like to use regularly (besides the Hindu-Arabic numerals)? I like the Mayan system. But I'd like to learn more about all the main ones we went over in class.
3) What would you like to know more about?
I'm interested in learning how to break down and explain math more fully to kids. Math is a great weakness, but I've also seen how it can be learned and I'm so excited to learn more about math. I want to know it more to teach it better and more completely or effectively.
Types of number systems
Numeration system: a collection of properties and symbols agreed upon to represent numbers systematically.
Definition: A number base (or base for short) of a numeral system tells us about the unique or different symbols and notations it uses to represent a value. For example, the number base 2 tells us that there are only two unique notations 0 and 1. The most common number base is decimal, also known as base 10. (
https://brilliant.org/wiki/number-base/#:~:text=A%20number%20base%20(or%20base,also%20known%20as%20base%2010
.
A FEW TYPES OF BASE NUMBER SYSTEMS WE'VE REVIEWED
Base 10
Base 8
Binary Numbers
[Kahn Academy: Introduction to number systems and binary](
https://www.khanacademy.org/math/algebra-home/alg-intro-to-algebra/algebra-alternate-number-bases/v/number-systems-introduction
Definition: A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" and "1". The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. (Wikipedia)
Inventor: Gottfried Wilhelm Leibniz
[George Polya's Problem Solving Stategies](
https://math.berkeley.edu/~gmelvin/polya.pdf
George Polya (1887 – 1985) was one of the most famous mathematics educators of the 20th century (so famous that you probably never even heard of him). Dr. Polyastrongly believed that the skill of problem solving could and should be taught – it is not something that you are born with.
Four-4s activity
Use exactly four 4's to form every integer from 0 to 12, using only the operators +, −, ×, /, () (brackets), and . (decimal point).
Here are a few to get you started
0 = 44 − 44
1 = 44 / 44 or (4 + 4)/(4 + 4)
2 = 4/4 + 4/4
Addition Algorithms
Partial
Sums
Lattice
Column Addition
[Opposite Change Rule](
https://everydaymath.uchicago.edu/teaching-topics/computation/documents/add-opposite-change-ex-3.pdf
Standard Algorithm
Greatest Common Factor (GCF)
The greatest common factor (GCF) of a set of numbers is the largest factor that all the numbers share. For example, 12, 20, and 24 have two common factors: 2 and 4. The largest is 4, so we say that the GCF of 12, 20, and 24 is 4. (Kahn Academy)
Other examples: (6,8) 6: 1,
2
, 3, 6 8:1,
2
, 4, 8
Methods
Definition: The smallest positive number that is a multiple of two or more numbers (mathisfun).
Example: the Least Common Multiple of 3 and 5 is 15:
3: 3, 6, 9, 12,
15
, 18, 21
5: 5, 10,
15
, 20, 25, 30, 35
The LCM for (3,5) is 15. We see that it is the smallest number (multiple) of the two numbers.
Subtraction activity
The Divisibility Rules 1-6
1
Any integer (not a fraction) is divisible by 1
2
The last digit is even (0,2,4,6,8)
128 Yes
129 No
3
The sum of the digits is divisible by 3
381 (3+8+1=12, and 12÷3 = 4) Yes
217 (2+1+7=10, and 10÷3 = 3 1/3) No
This rule can be repeated when needed:
99996 (9+9+9+9+6 = 42, then 4+2=6) Yes
4
The last 2 digits are divisible by 4
1312 is (12÷4=3) Yes
7019 is not (19÷4=4 3/4) No
A quick check (useful for small numbers) is to halve the number twice and the result is still a whole number.
12/2 = 6, 6/2 = 3, 3 is a whole number. Yes
30/2 = 15, 15/2 = 7.5 which is not a whole number. No
5
The last digit is 0 or 5
175 Yes
809 No
6
Is even and is divisible by 3 (it passes both the 2 rule and 3 rule above)
114 (it is even, and 1+1+4=6 and 6÷3 = 2) Yes
308 (it is even, but 3+0+8=11 and 11÷3 = 3 2/3) No
Divisibility rules 7-12
7
Double the last digit and subtract it from a number made by the other digits. The result must be divisible by 7. (We can apply this rule to that answer again)
672 (Double 2 is 4, 67−4=63, and 63÷7=9) Yes
105 (Double 5 is 10, 10−10=0, and 0 is divisible by 7) Yes
905 (Double 5 is 10, 90−10=80, and 80÷7=11 3/7) No
8
The last three digits are divisible by 8
109816 (816÷8=102) Yes
216302 (302÷8=37 3/4) No
A quick check is to halve three times and the result is still a whole number:
816/2 = 408, 408/2 = 204, 204/2 = 102 Yes
302/2 = 151, 151/2 = 75.5 No
9
The sum of the digits is divisible by 9
(Note: This rule can be repeated when needed)
1629 (1+6+2+9=18, and again, 1+8=9) Yes
2013 (2+0+1+3=6) No
10
The number ends in 0
220 Yes
221 No
11
Add and subtract digits in an alternating pattern (add digit, subtract next digit, add next digit, etc). Then check if that answer is divisible by 11.
1364 (+1−3+6−4 = 0) Yes
913 (+9−1+3 = 11) Yes
3729 (+3−7+2−9 = −11) Yes
987 (+9−8+7 = 8) No
Overview:
Multiplication and Division
the meaning of multiplication for fractions;
the procedures for multiplying fractions;
mixed number answers to whole number division problems;
using division to convert improper fractions to mixed numbers;
interpreting division for fractions, the “invert and multiply” procedure for division;
contexts;
representations; computational fluency;
use of properties;
mental computation;
estimating results of computations; how to round; counting
Divisibility rule-12
12
The number is divisible by both 3 and 4 (it passes both the 3 rule and 4 rule above)
648
(By 3? 6+4+8=18 and 18÷3=6 Yes)
(By 4? 48÷4=12 Yes)
Both pass, so Yes
524
(By 3? 5+2+4=11, 11÷3= 3 2/3 No)
(Don't need to check by 4) No
Overview
Adding and Subtracting Fractions
Like and unlike denominators; contexts;
representations;
computational fluency;
use of properties;
mental computation;
estimating results of computations;
how to round; counting
In class we were asked: "What is the pattern? Do you recognize it?"
142,857 ✕ 2 = 285,714
142,857 ✕ 3 = 428,571
142,857 ✕ 4 = 571,428
142,857 ✕ 5 = 714,285
142,857 ✕ 6 = 857,142
142,857 ✕ 7 = 999,999
Week 5-8
Week 7- Integers cont.
Real numbers
Rational numbers (Z): a "ratio" of two numbers. 1/7, 14/2, -4/2
Irrational numbers (Q): a number that never "terminates". It goes on forever. Or, a number that cannot be written as a fraction.
Pi= 3.14159265359...
e= 2.71828182846...
Square root of 2= 1.41421356237...
Golden ration= 1.61803399
Whole numbers: A number that isnt a fraction, a positive number and zero. 0, 1, 2, 3...
Integers: -3, -2, -1, 0, 1, 2, 3...
Natural numbers: Numbers we use as we're counting, excluding decimals and fractions. 1, 2, 3...
Imaginary numbers: numbers when squared it gives the negative result
Order of operations: What is it? a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression
Example of problems:
6x4/12+72/8-9
=24/12+72/8-9
=2+72/8-9
=2+9-9
=11-9
=2
When solving an order of operations problems its important to remember "PEMDAS"
Parenthesis, exponent, multiplication, division, addition, and subtraction.
If this order is followed the student will achieve the desired outcome.
Week 8: Decimals
Week 6: Integers
Percentage change
A percent change is an increase or decrease given as a percent of the original amount.
A percent increase describes an amount that has grown.
A percent decrease describes an amount that has grown.
Percent change = amount of increase or decrease/original amount.
This is expressed as a percentage.
Ex. Finding percentage change:
1.) find the difference of the increase or decrease.
2.) divide by old or original number (difference/original number)
3.) multiply by 100 to get end %
10 to 22
22-10=12
12/10=1.2
1.2x100%=120
120% increase
200 to 110
200-110=90
90/200=.45
.45x100=45
-45% decrease
25 to 30
30-25=5
5/25=.2
.2x100=20
20% increase
80 to 115
115-80=35
35/80=.4375
.4375x100=43.75
43.75% increase
Vocabulary
Markup
A markup is an amount by which an original pricedis increased. Markup is a % of the original price.
Final price = original price + markup.
Caleb buys a necklace at wholesale cost at $48 each. He then marks up the price by 75% and sells them. What is the amount of the markup? What is the selling price?
48x.75=$36
48+36=$84
or
100%+75%=175%
1.75%x48= $84
84-48=36
Discount
A discount is an amount by which an original priced is reduced. Discount is a % of the original
price. Final price = original price – discount.
Admission to the museum is $8. Students receive a 15% discount. How much is the discount? How much do students pay?
8x.15= $1.20 (discount)
8-1.20= $6.80 (pay)
This resembles a "percent decrease".
Another method for the same problem (subtract from 100%):
100%-15%=85%
0.85(8)= $6.80
8-6.8=$1.20
Christo used a coupon and paid $7.30 for a pizza that normally costs $10.50. Find the percent discount.
10.50-7.30=3.20
3.2/10.5=0.3047...
x100=30%
The following are the four main properties of real numbers:
Commutative property:
Formula: a+b=b+a
Basically adding in any way and we'll get the same answer.
"The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example a+b=b+a and\,(a)(b)=(b)(a). A fraction is a part of a whole." (libretexts)
Associative property:
"The associative property is a math rule that says that the way in which factors are grouped in a multiplication problem does not change the product.." (kahnacademy)
Ex. 3/5 + (2/5+1/5)=(3/5 + 2/5) + 1/5
Distributive property:
"The distributive property allows us to multiply one number or term with a set of terms in parentheses. All you do is multiply the term outside the parenthesis by each term inside the parentheses. Fractions follow the same rules as any other kind of term in algebra" (study.com)
Ex. 1/3(x+6)
Identity property:
"states that if we add a fraction to zero, it will give the same fraction as a result." (splashlearn)
Anything times 1 is going to be equal to that number
In-class multiplication: Introducing 3 new multiplication methods:
Method 1: Build up strategyLinks to an external [site.](
https://www.youtube.com/watch?v=NPC1mMKOl5I
Method 2: Multiplication with base 10 blocks (
https://www.youtube.com/watch?v=G_QPPzRWATI
)
Method 3: Multiplication arrays
GCF&LCM
Prime factorization-"finding which prime numbers multiply together to make the original number" (mathisfun) Here's a good video:
https://www.youtube.com/watch?v=XBnUWjo3TgM
Prime factorization includes "composite" and "prime" numbers. A prime number is a number that can be divided by 1 and itself. When you have a composite number you cant divide it anymore than it is. A prime number however can continue to be divided until it reaches a "prime" number.
Greatest common factor-"the largest number that is a factor of two or more numbers" (mometrix)
The GCF of 16 and 24 is 4.
The LCM is the least number of two different numbers multiples. For instance: The LCM of 10 and 6:
6,12,18,24,30,36,42
10,20,30,40,50
We see that 30 is the LCM of both numbers.
Here's a video that goes into more detail:
https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/cc-6th-lcm/v/least-common-multiple-exercise
Week 9-11
Week 9: Decimals
Weeks 12-14
Week 12:
Rational Number (fraction) Operations
Fractions represented by words, diagrams, and symbols;
Modeling fractions as parts of a whole or as a count of a subset;
Placing fractions on a number line; equivalent fractions; common denominators;
Simplest form; mixed numbers and improper fractions;
Day 1: Equivalent fractions with Cuisenaire models & Comparing fractions
Showing equivelent fractions:
1 whole" 8/8, 1/1, 6/6.
1/2: 4/8, 5/10, 10/20.
3/4=6/8=12/16/24/32
1/2/8=1/1/4
Fractions using area models
2/5 Pie+1/5 pie=3/5 pie
We had to draw this out
Additionally, showing what "3/5" looks like on a number line. The number line starts at zero and as we move to the right we have 1/5=5/5. You can demonstrate 2/5+1/5 above the number line so we can see the "size" of 3/5 up against the fifths number line. This also adds a demension to allow the learner to see and process mathematical information in a different way.
Day 2: Area models for addition and subtraction.
How can we compare these proper fractions?
3/4 & 1/6
The first step would be to ask your self "is there a common denominator?" If not thats our first step. In this case the LCM of 4 and 6 is 12. So we have to do the necessary multiplication with each fraction to turn the denominator into 12.
For 3/4 we would multiply top and bottom by 3 getting 9/12. 1/6 would be multiplied by 2. We would get 2/12. We are analyzing which is the larger fraction.
In this case 9/12>2/12
Finding the result of a percentage increase or decrease
Find the result when 30 in increase by 20%:
0.20 (30)=6
30+6=36
Find the result when 80 is increased by 65%
0.80(65) or 80(.65) = 52
65-52= 13
Writing decimals
13,400,000
expanded form: Thirteen million four-hundred thousand
scientific notation: 1.34x10^7
3,000,000
400,000
0
0
0
0
0
10,000,000
Write the following numbers in expanded form and scientific notation.
1/2 is equivilent to 3/6
1/3 is equivelent to 2/6
1/2+1/3=5/6
Example: 3/4 x 1/3 so filling the two yellow hexagons with a red half would equal ".25/100".
If two hexagons are a whole.
• use pattern blocks to evaluate (on the slides of mathigon see link in comments)
• Record your solutions with a pictorial model (see previous slide)
How many 1/2's are in 3 wholes?
1/2/3=6
I remember gaining knowledge and insight on a variety of new number systems that I had never heard of before.
Week 2- Bases
Week 3-Number Sense- Order of operations
Week 4
Divisibility, Prime and Composite Numbers, Fundamental Theorem of Arithmetic, GCD/GCF, LCM (Odd and even numbers; factors and multiples; divisibility tests; contexts; definitions; computational fluency)
Least Common Factor (LCM)
[Greatest common factor explained | Factors and multiples | Pre-Algebra | Khan Academy(
https://www.youtube.com/watch?v=jFd-6EPfnec&t=61s
[Least common multiple exercise: 3 numbers | Factors and multiples | Pre-Algebra | Khan Academy(
https://www.youtube.com/watch?v=D6yHKOYJiso
https://youtu.be/sD-mvb9A0RE
Use of blocks
https://youtu.be/VkCd5SQoRe4
Using black manipulatives to me seems to be the beginning stages of developing those rational and spatial thinking patterns that will allow the child to essentially build, create, and perceive what is in front of them. To see patterns or connections through the blocks could be fundamental to doing the same as the child grows older. I like how this strategy had a structure with the "two-land" and "three-land" and encompassing the different sizes of blocks. It was also noted that the kids can "think logically regardless of the arrangement of the blocks," which I think gives kids the autonomy for a creative learning experience.
Integers can consist of positive and negative numbers
Absolute value: the distance a number is from "0" Ex: |-20| = 20 because its 20 spaces from 0.
[Absolute value](
https://www.youtube.com/watch?v=LnfhdtjcpVY
This week it negative and positive integers were described pictorally using number lines, different objects like animals and colred chips. Descibing integers pictorially and concretely
[Kahn Academy Introduction to rational and irrational numbers](
https://www.youtube.com/watch?v=cLP7INqs3JM
Types of Integers
Video: Types of Integers
Video: What are the Types of Numbers? Real vs. Imaginary, Rational vs. Irrational
Video: Base 10 system
Video: Base 8 system
For any base number system that we use there's going to be some significant patterns. The way we can solve for an answer is fairly simple.
For the common base 10 number system that we all have used, we know that in one direction (going left) the numbers increase (since they're multiplied by 10): 10 (10^1), 100 (10^2), 1000 (10^3), 10,000 (10^4), 100,000 (10^5) in this fashion. If you were to go "right" in this array, the numbers would decrease: 1/10, 1/100, 1/1000, 1/10,000. It's important note that the exponent becomes "negative" as the number decreases.
Base 8 is quite similar although you are working with "8" instead of "10". Going left, the numbers increase by multiples of 8: 8, 64, 512, 4,096, 32,768, and so on.
A question that can be asked is what is "513" in base 8 form? You could draw this out. I would post a picture but for some reason on here the picture comes out blurry so I'll do my best to describe with words.
Your base 8 numbers would coorespond with 513. If you were to write you base 8 numbers down you could do: 512, 64, 8, 1. To get the answer to this question you ask yourself :what and how many base 8 numbers do I need to equal 513? You need one 512, zero 64's, zero 8's, and one "1". So essentially "1001" base 8 would equal 512 base 10. 1001 would be your answer.
A great website to practice the value of decimals using blocks
One thing we did during this week was demonstrated decimals through blocks. Namely "flats", "longs", and "units".
Flat = 100
long = 10
unit = 1
Ex. 3.21 expressed using blocks could be 3 flats, 2 longs, and 1 unit.
You could list these blocks as anything.
4.4 could equal 4 longs and the units could equal tenths.
Video: Multiplying with blocks
Another concept we covered this week was "multiplying with blocks. This allowed me to gain an additional perspective of multiplication in relation to "place value". Though challenging it could allow learners to grasp multiplication and decimals a little more.
Dividing a decimal by a decimal: Ex. 5.76/0.3 Is the divisor (outside number) a "whole number"? If not multiply by 10 or 100 depending on how many places you need to go. You multiply both numbers by 10 or hundred. In this case we would multiple 0.3 and 5.76 by 10 to get 19.2.
[Division of decimals](
https://www.youtube.com/watch?v=Val4TmjHXRY
Week 5: Divisibility
Week 9: Multiplying decimals
Video:Multiplying a decimal by a decimal
This is hard to see but an exercise where we had to do 1x0.7 and 0.5 and 0.7 and fill in a "flat" sized grid.
For a) Since the answer would be "0.7" you would fill in 7/10 "longs". B) The answer would equal .35 so you would fill in 35/100 "unit" sized spaces within your "flat" to represent the decimal.
It frustrates me that I can't make these pictures LARGER without upgrading but they are examples of what multiplying decimals would look like using a grid or base 10 block system.
Using different colors could help differentiate the problem and highlight the answer.
Here's a great video that shows multiplying decimals using blocks
This week gave us experience as groups to review Horace's chapter homework on decimals. We went through it and made corrections and analyzed specific problems.
I liked the practical experience this illustrated.
Scientific Notation
This was our week overview: GCD/GCF, LCM (Factors and multiples; contexts; definitions; computational fluency), Rational Numbers (Fractions
represented by words, diagrams, symbols; modeling fractions as parts of a whole or as a count of a subset; placing fractions on a number line; equivalent fractions; comparing fractions; interpreting fractions as division; common denominators; simplest form; mixed numbers and improper fractions)
Here's a great video that shows some sweet divisability techniques
Divisibility
Factoring
Kahn Academy: Finding factors of a number
Math Antics: Factoring
Simply stated, factoring is basically "un-multiplying". If you want to find the factors of "120" you start from 1 and go to 120. Here's an example"
120=1x120
120=2x60
120=3x40
120=4x40
120=5x24
120=6x20
120=7x doesnt work
120=8x20
120=9xdoesnt work
120=10x12
120=11xdoesnt work
You basically start with one and as you go through each lower factor you essentially find out what the larger factors are and eventually meet in the middle. Once you've "met in the middle you know your done.
Long division is a methodf that can be used to know if a certain numer is or is not a factor.
mathisfun.com
Negative integers are the opposite of positive integers: -3 is the opposite of 3
4 is the opposite of -4
|50|=-50
|-3|=3
|555|=-555
|0|=0
Here's a good video for this object lesson using chips to represent pos and neg integers
https://www.youtube.com/watch?v=KQwvjE7eypE
Scientific notation comes in handy especially when you're representing a REALLY large number in a small way. For instance 10x10=100 could be shortened to 10^2 and 10x10x10=10^3.
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-numbers-operations/cc-8th-scientific-notation/v/scientific-notation
"Grids can be a useful tool for exploring the structure and value of a decimal number."
[Modeling multiplying decimals]
https://www.youtube.com/watch?v=8B2CpiJO-uI
What happens when you multiply a tenth by a tenth?
0.1x0.1=0.01
1/10x1/10=1/100
10^-1x10^-1=10^-2
Learning decimals through the 10 base system using blocks allowed me to really see things at work.
What happens when you 10 by 10 or a tenth by a tenth puts into perspective that happens on a mathematical level either increasing or decreasing on the number line.
The base 10 method regarding multiplying decimals allowed me to see how a decimal could become smaller (tenths turn into hundreths) through visual/colorful methods.
If you have a 100 unit flat a"bar" or row would equal a tenth. The individual units are 1/100. As you multiply decimals say .87x.13 you color these on the grid and they intersect leaving you with the answer, .29
Very important video to help understand concepts from this week in greater detail!
https://www.youtube.com/watch?v=IueVrMlmQ2I
What are real numbers?
https://www.youtube.com/watch?v=3YwrcJxEbZw
Types of decimals to know:
terminating: 0.475
non-terminating: 0.7777...
non-terminating/non-recurring: sq root of 2=1.414
Everytime the number "142,857" is multiplied by a sequential set of numbers a re-occuring pattern follows.
Week 10: Percentages
Percentage of change-percent increase and decrease:
https://www.youtube.com/watch?v=jAcDJDjQk2g
Scientific notation
Week 11: Exam review/exam 2
0.000465
expanded form: four hundred and sixty-five millionths
scientific notation: 465x10^-5
0.0
0.00
0.000
0.0004
0.00006
0.000005
Terminating and non-termanating decimals
Terminating: 0.375, 0.472, .437
Non-terminating:.777, .00333
How can you predict if a fraction will be a terminating or non-terminating decimal?
Just divide the numerator by the denominator . If you end up with a remainder of 0 , then you have a terminating decimal. Otherwise, the remainders will begin to repeat after some point, and you have a repeating decimal.
Show with blocks: 0.22 x 8 =
REVIEW
Multiplying base 10 blocks using decimals
https://www.youtube.com/watch?v=3hMHdGB9ILk
I was able to review the multiplication with blocks. It's important to note the "overlapping" method only works if youre multiplying decimals that are not a whole number otherwise if it is a whole number decimal x another whole number decimal you simply add on the necissary amount of verticle and horizontal blocks.
Adding decimals with base 10 blocks
https://www.youtube.com/watch?v=jjvyB4zj_lQ
Adding base 10 blocks is fairly easy. You're representing the number using flats, rods and units. You add the decimal numbers then prepare the necissary blocks to represent your final answer.
Dividing base 10 blocks using decimals
https://www.youtube.com/watch?v=NI8s9noqSGc
Dividing base 10 blocks is fun. You have a number like .48/12. You arrange your blocks as so: 4 longs, 8 units and you split these up where you will have 4 groups of 1 long and 2 units. The 4 groups representing the answer: .04
1.4/2 in base 10 blocks will be arranged in this way: 1.4- You'll have 1 flat and 4 longs.
2- You'll have 2 flats.
For this problem, since the answer is not going to be a whole number we can substitute the flats for more additional longs. You'll have 34 longs
Topics
Adding and subtracting fractions with unlike denominators
https://www.youtube.com/watch?v=XsW8HJutIgM
Proper and Improper fractions
https://www.youtube.com/watch?v=2h8XiqSnzaU
Multiplying and dividing fractions
https://www.youtube.com/watch?v=7EWPkw_R-MA
Here's a great video that explains comparing proper fractions
https://www.youtube.com/watch?v=u_QTuWj107o
Comparing fractions using bars
https://www.youtube.com/watch?v=ory05j2jgBM
I wish I could draw or post this exercise better! but during this week it was interesting how we did the activity where we:
Compared 2 fractions
Drew the fractions using "bars"
and then plotted fractions on the number line.
These exercises allowed the learner to grasp a new or additional dementions to "comparing fractions"
Week 13
Equivalent fractions (identity property
https://www.youtube.com/watch?v=qvKI1jdElOw
Giant ruler
Pattern blocks: This lesson was really fun. We paired with a partner and basically "rolled" or "spun" a thing that would tell us which shape we could choose. The goal of the game was to fill the "octagons" with the correct shapes the quickest. A great game that could be played with kids.
Fraction bars
The methods of teaching we learned this week was:
Fraction bars
Pattern blocks
and using a giant ruler to understand fractions
Multiplying fractions with "pattern blocks"
https://www.youtube.com/watch?v=xvIYmbuJ-qY
This week the one thing that stood out to me was the power of shapes when teaching/learning math.
[Mathigon](
https://mathigon.org/polypad#fraction-bars
Awesome Base 10 blocks interactive website
Week 14
Egyptian fractions
The first day consisted of multiple exercises that had to do with multiplying fractions using shapes as visuals. The shapes to the left were the ones we used.
the yellow hexagons=a whole
red trapezoid=1/2
Blue parallelagram= 1/3
green triangle= 1/6
Multiplying fractions using "pattern blocks"
Dividing fractions using pattern blocks
It was also shown how to do this with "strips". Lining up 3 whole strips with 3 whole strips below (that are halved) this shows how 3 is equivalent to 6 1/2 strips.
