Integers and Exponents -Week 5




iPracticeMath. Integers. Picture [Online Image]

This week in math class it was all about Integers! Now, when I first heard the word integer I somewhat feared it. However, to my surprise integers are nothing to fear but something that we are surrounded by everyday. For example, driving uses integers, positive and negative experiences could be classified as integers, the stock market, time lines in history (AD and BC), above and below sea level, gambling and temperatures use integers. Below is a great video from last weeks Building Background Activities, it is about the use of the integers in the real world in regards to temperature changes and maintaining the proper temperature for an ice rink. This video helps to put into context the importance that math plays in our everyday lives. If students can understand that math is all around them and used everyday this will eliminate the "why do I have to learn this" type of questions.

http://www.learnalberta.ca/content/mejhm/index.html?ID1=AB.MATH.JR.NUMB&ID2=AB.MATH.JR.NUMB.INTE&lesson=html/video_interactives/integers/integersSmall.html

Definition of An Integer:

An integer IS a whole number that can be + positive, - negative or 0 zero. An integer is NOT a fractional number or a decimal. Thus: Integers ARE: 0, 1, 2, 6, 13, 120 etc. Integers ARE NOT: 1/4, .05, 25/50, 0.07, 8.9 etc.


Online Math Help and Learning Resources. Properties of Integers.
Retrieved from https://www.google.ca/search?q=integers&biw=1607&bih=792&source=lnms&tbm=isch&sa=X&sqi=2&ved=0ahUKEwjy37W22IbQAh
XMxYMKHYEOAC0Q_AUIBigB&dpr=0.85#imgrc=lBMaPOfePd9VLM%3A

Spencer, Erica. Picture [Photograph]
Retrieved From
https://www.pinterest.com/pin/297026537907027162/

In class we learned that the biggest problem involving integers is the fact that teachers who teach rules that don't apply to integers and make students memorize them. For example, a common rule that many teachers teach their students is that, "two negatives don't make a positive for adding, however, it does for multiplication." Thus, students need to forget the rules their teachers have taught them in regards to integers and learn the proper way to understand them.

Math Congress:

As well, in this past math class we learned about the idea of a Math Congress. We conducted it ourselves in regards to a problem our teacher assigned. A Math Congress can also be referred to as a gallery walk. Students work in groups to solve a problem and then write about their problem on chart paper. The students product's can be either hung up or left on the desks because each group will go and visit the various charts that other students have created. A math congress promotes creativity,sharing and thinking among students in various grade levels. It involves hard work by each group because the following work will be assessed by their peers. This is an excellent tool because it allows students to see how their peers are learning and what similarities and differences that they may come across.

 

CLIPS:

Furthermore, we learned about another key resource, CLIPS- Critical Learning Instructional Path  Supports. This is a multi-media resource in which the learning objects are focused on the key topics and identities as needed by a significant percent of students. CLIPS provides teachers with gap closing material for teachers which was originally meant for students to catch up. Students can get into small groups or work solo while other students move along. CLIPS, allows teachers to go through different lessons which tell you what you did right and wrong. Teachers pick the grade level they want to focus on, there are built in manipulative's and clues that can be clicked on for extra help.

 

Conclusion:

As a class we are still adding to our list of ideas for, What Makes a Good Problem,below is the collective list that we compiled together last class. It is important to keep these aspects in mind for when we are creating math problems and encouraging students to create their own problems.

1. Wide Base = everyone can start
2. High Ceiling = potential to create discipline
3. Soft Language
4. Relevant = to the class, lesson and students
5. Multiple Ways To Find The Solution
6. Use Different Manipulative's
7. Work Together = support & help
8. Creates Discussion
9. Challenging

Till next time,
Cheers, Courtney

Fractions Continued Divide and Conquer -Week 4

Last week's math class was a continuation from Week 3. In week 3 we learned the proper way to add and subtract fractions using various manipulative's. This past week we continued the fabulous journey of fractions and learned the appropriate way to divide them. Back in elementary and high school I was taught to invert the fractions and multiply, this was the only way we were taught. Now, being a young student who didn't like math to begin with or really understand it, this was extremely confusing to me. Fast forward to several years later to my teacher's college J/I math class. I have now learned that this is totally the WRONG way to teach kids how to divide fractions. Not only is this way wrong but it is also extremely complicated. PSA: dividing fractions does not have to be complicated, I'll show you below.

When dividing fractions we can remember that dividing is the same as multiplying. We must remember to just divide across the fractions as we would multiply across the fractions. We do not need the reciprocal anymore, there is no need to invert the second fraction and multiple!

MathChat. How to Divide Fractions From Annoying to Fun! (November 19 2008). Math, [Online Image]. Retrieved From https://mathchat.me/2008/11/19/dividing-fractions-from-annoying-to-fun/

For example: 6/12 divided by 2/3
Solution: divide the numerators by each other: 6 divided by 2 and divide the denominators by each other: 12 divided by 3.
Answer: 3/4

Division Problem Example: Kate divided a fraction less than one by another fraction less than one and came to an answer of 3/4. What might her fractions have been?
9/16 divided by 3/4 = 3/4
Thus, you simply divided the numerators by one another and the denominators by one another.

HOW SIMPLE WAS THAT!

Furthermore, another key aspect that I was taught in elementary and high school was that you are not allowed to create a common denominator when dividing fractions. Why? Because my teacher said so. Now, in teacher's college my teacher is stating that you CAN use common denominators to divide fractions.

Heitin, Liana. Education Week. With Fractions, Common-Core Training Goes Beyond 'Invert and Multiply' (August 12 2014). Math, [Online Image]. Retrieved From http://blogs.edweek.org/edweek/curriculum/2014/08/fractions-common-core-invert-and-multiply.html

For example: 5/6 divided by 1/3 -find the common denominator: 18 is a common denominator.
HOW did we get that you might ask? Well both 6 and 3 can be multiplied by each other to get 18. Thus, whatever you do to the denominators you must do to the numerators.
So, 5/6: 5x3=15 and 6x3=18. Now our fraction becomes 15/18.
So, 1/3: 1x6=6 and 3x6=18. Now our fraction becomes 6/18.
So, 15/18 divided by 6/18= 15/6/1 a three part fraction which = 15/6/1 because both denominators being 18 can be divided by each other, equalling 1. We drop the 1 and this fraction, 15/6 can be simplified by dividing both the numerator and denominator by 3. Thus, giving us 5/2 as the simplified answer.

One more example using the common denominator method:
4 divided by 2/3... How can we solve this? - We must find a common denominator. I choose 3.
Therefore, whatever we do to the bottom we must do to the top. So I multiply the numerator 4, by 3 = 12. Thus, my equation becomes: 12/3 divided by 2/3 = 6/1. This can be simplified to 6.

Wiki How. How to Divide Fractions by a Whole Number. Math, [Online Image] Retrieved From http://www.wikihow.com/Divide-Fractions-by-a-Whole-Number

HALLELUJAH!!! IT'S A MIRACLE

I find doing numerous examples helps me to remember the process for solving problems. Therefore, making me more successful when answering math questions. For me it's all about repetition.

Here is a fun link that I learned about from my math lesson, the object of the game is to answer the questions using your fraction skills to make several smoothie recipes.

http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.NUMB&ID2=AB.MATH.JR.NUMB.FRA&lesson=html/object_interactives/fractions/use_it.html

Rules to take away from this lesson:
-Yes you can divide fractions without inverting them. Just simply divide across.
-Yes you can use a common denominator to divide fractions- why the hell not?
-When creating a problem make sure it has a wide base= scenario where everyone can get started
-When creating a problem make sure it has a high ceiling= lots of choice in between, students can add onto the problem.

As a future teacher it is important to remember the, 6 Fail Safe Ways for Opening Problems:
1. Begin with the answer.
2. Ask for similarities and differences- How is multiplying fractions like dividing fractions? How are they different?
3. Leave certain information out of the problem. Ie. omit number's.
4. Provide several numbers and math words so the student can create a sentence using all of them.
5. Use soft language, Ie. Two fractions are almost but not quite equal, what could they be? Example: 3/4 and 8/9.
6. Ask to prove if an idea is true or false.

This past week we learned a lot in our math class. We learned the PROPER way to divide fractions by dividing across the fractions as well as, finding common denominators to divide the fractions. These are the appropriate methods that I will use when teaching my students to divide fractions. It is important to remember the take home rules and the 6 Fail Safe Ways for Opening Problems for your students.

Till next time,
Cheers, Courtney

Fun With Fractions! - Week 3

http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.NUMB&ID2=AB.MATH.JR.NUMB.FRA&lesson=html/video_interactives/fractions/fractionsSmall.html

This week's math class was all about FRACTIONS! Above is an introductory link to Fractions which was provided in our week 4 folder on Sakai. The video is short and sweet! It discusses how fractions are used in our everyday world such as in hotels and restaurants. Fractions are used through multiplication, division, subtraction and addition in order to keep a kitchen running smoothly. It is always important to show our students how math is used in the real world. This will avoid statements such as: "I'll never use this in my job." & "Why do I need to know this, it doesn't apply in the real world."

Our class this week started off with each student choosing a fraction that they found interesting. I chose 1/4 because this fraction can be related to money and measurements for baking. We were to model our fractions in as many ways as possible, state why this is a fraction, and what's important about it. Even though every student came up with different examples of a fraction, we all had similar definitions. For instance, a fraction has a numerator-the top number and a denominator- the bottom number. Both of these numbers are separated by a line, creating a fraction.

Furthermore, we learned that there are various types of fractions. There are proper fractions which are part of a fraction, unit fractions which are fractions with a 1 as the numerator, mixed fractions which have a whole number and then the fraction, ex. 1 1/2 and lastly, there are improper fractions in which the numerator is more than the denominator, ex. 25/5. We also learned that fractions can be added together and subtracted from one another. However, the fractions must have a common denominator. To find a common denominator we must find a number that both denominators can be multiplied to. Then we multiply the numerators by what the denominators were multiplied by.

For example: 2/3 + 5/8
Common denominator =24- How did we get that? We multiplied the first denominator, 3 by 8 and the second denominator, 8 by 3= 24. Therefore, what we do to the bottom has to be done to the top. So, the first numerator 2, would be multiplied by what it's denominator was multiplied by =8 and the same for the second fraction. Thus, 5 would by multiplied by 3.
Answer: 2/3 + 5/8
= 2/24 + 5/24 -denominators done
= 16/24 + 15/24 -numerators done
=31/24

Also, we can simplify fractions by finding a common number that can be divide by both the numerator and the denominator.
For example: 18/24 can be simplified by dividing both numbers by 6
Answer: 18 divided by 6 = 3 and 24 divided by 6= 4
Simplification: 3/4

Furthermore, every week we are encouraged to provide context and real-world applications to our teachings and problems in order to keep the students interested. This week, Pat taught part of her Fraction's lesson using The Hershey's Milk Chocolate Fractions Book. This book teaches parts of whole fractions to students. Pat provided each group with a 16 piece Hershey's chocolate bar. As she read the story we separated the chocolate bar into the various fractions stated in the book. This exercise was EXCELLENT! Every group had to work together in order to solve the problems, thus promoting collective learning. This activity is both fun and informative, it engages the students and makes them want to learn math! This is an excellent and tasty idea for a lesson that teaches Junior/Intermediate students the basics of Fractions.


Helt, Courtney. 2016

As well, Pat stressed the importance of manipulative's again this week. Manipulative's are extremely useful for a lesson involving fractions.

Helt, Courtney. 2016

As a class we experimented with the Hershey's Chocolate bar, egg cartons, various blocks, number lines and strips, clocks, as well as pie plates and cards from the student presentations. Each manipulative proved that fractions can be solved easily if we have something visual to work with.


As a future teacher I would highly encourage and recommend to my students the use of manipulative's, even if the students claim they do not need them. KIDS NEED MANIPULATIVE'S. NO CHOICE!
They provide a fun, engaging and collaborative learning environment for all students. 


Helt, Courtney. 2016

 

Consequently, this week was filled with manipulative's and that made learning about Fractions extremely fun! Teachers teach  through problem solving and visual aids help students to solve their problems. Below is a terrific link that Pat informed us about. Math Playground has numerous games and problem solving activities that students can complete for various units. I look forward to the coming weeks and what new teaching methods we will learn from Pat.

 

 

http://www.mathplayground.com/


Till next time,
Cheers, Courtney



Algorithms: Making Sense For All Students - Week 2

Last week's math class opened with the question, "Why encourage Alternative Algorithms?" When I heard this question I instantly put my head down because I did not understand the concept of an algorithm. I shyed away from the word because I thought it was something extremely complex and out of my mathematical league. However, to my surprise and embarrassment this big word, "algorithm", is simply a step by step process for how to solve a problem.

In my opinion, teachers MUST encourage alternative algorithms because not every student learns in the same way. Algorithms promote creative thinking for how to solve a problem. This thinking might be understood in numerous ways depending on the student. One way of solving a problem might make total sense to one student but another way might make even more sense to a different student. As a result, I believe in promoting a classroom that encourages my future students to learn in a way that best suits them. There is no need for one right solution because there are thousands of right solutions. It is the responsibility of each individual student to find that one, or multiple right ways that best suits their learning.

Below is an example of Skip Counting presented in last week's class. Skip Counting is an algorithm that uses a number line to solve addition and subtraction problems. When using a number line it is important to remember that the line always increases to the right and decreases to the left. Skip counting can be used for students in all grades. I love this method because it makes sense. I am an extremely visual learner and this method appeals to me because I can see the process it takes in order to get the answer. The use of a number line helps students to keep track and visualize their math problem.

Problem: (Nathan's class goal is to sell 50 tickets, they have already sold 332 in the first week, how many will they have to sell to meet their goal?) 500-332=168 The 332 is broken down into 300, 30 and 2 across the number line. Younger students could break down the 300 into (100, 100 and 100) to better visualize the skip counting.

Helt, Courtney. 2016

As well, this picture is taken from last weeks math class. Below is an example of how to solve a multiplication problem using the concept, objects in an array. This is one way for understanding multiplication. Again, being a visual learner this solution makes sense to me. The problem is separated into a rectangular chart. Students can break down the process of multiplication through sections to further their understanding.

The multiplication equation 25x 28 is broken down into 20 + 5 and 20 + 8. Each number is multiplied by the adjacent number and the totals are added together. Therefore the equation is further broken down, and expanded to provide insight into the equation.

Helt, Courtney. 2016

Furthermore, it is crucial as new teachers to understand the importance of teaching our students how to solve equations in numerous ways. It is also crucial to show our students visually what we are doing in order for them to fully understand there math problems. Providing students with open problems will encourage interest and creativity. This will keep our students wanting more from math!

Jo Boaler couldn't have put it more perfectly when describing, Brain Crossing and it's importance to learning math, "Drawing, and visualizing and using symbols together in math is the most powerful math learning." Hence,it is key as future teachers to provide students with multiple ways to solve a problem and encourage the use of visual solutions.

https://www.youtube.com/watch?v=qZBjub36Bvs
Boaler, Jo. [S Lamb] 2015, August 25. "Day 2 - Brain Crossing" Retrieved from https://youtu.be/qZBjub36Bvs

Till next time,
Cheers, Courtney