Integers, the negatives and positives

When I saw that integers was the topic for this week I gave a sigh of relief! My simple, approachable whole numbers! I welcomed them with open arms! As I was going over the readings I was feeling pretty confident. But then I realized... wait.. I'm going to need to explain this to students without falling back on "oh use the good old rule! two negatives makes a positive!" or "use that amazing number line!" I loved doing subtraction, multiplication, division and even addition of integers as a student because I knew the rules and my favourite manipulative (ye old number line) and they gave me what I needed, the right answer! Who cared about how I got to it or why it made sense? It just did!

In class this week however I realized that I didn't really understand why I was doing something.. I was just utilizing the algorithms I had been taught, I hadn't really rationalized things on a level that meant the learning was being deeply embedded. Fortunately there were many "ah-ha!" moments in class today especially when it came to working with negative integers. 

The first activity we completed in class was a comparison of Teagan's height with a Giant's height based on their hand sizes.

The first thing we did was complete the problem individually then we moved on to comparing our methods as a small group and finally as a class. Well I had completed the question with ease and quickly too, I realized I was one of the few who had not immediately recognized that I could use a ratio to solve my problem... and my fixed mindset whispered "well you clearly don't get Math".. which is ridiculous! Because I did arrive at the answer, but I did it in a different way! I unknowingly was using ratios and decimals to get my answer! Hurrah for being creative in Math! That's what made this such a great problem, it was approachable and open to different approaches.

I'm glad we do so much discussion in our groups as I can gain insight about how else I can approach a problem and really understand how I get to an answer. How great are congresses??? This was one of my highlights for class today. Pat approached Nuha and I to help lead discussion to teach our group how another class had arrived at the unit price of each cat food can in Joel's problem and which algorithm they used to get to the answer. 

                        

This was such a great exercise. It was powerful to see how many different approaches our classmates had taken, there were so many different ways of thinking of the question that I hadn't realized! It also helped me to have to explain my thinking and then expand my understanding as I heard each group member's perspective. 

The final "ah-ha!" moment occurred for me during Adam's lesson on integers. I had never given negative integers a story or a real life representation to better understand them. His comparison of negative integers to negative experiences gave them a context that was easy to relate to and remember! For example, if you have had 4 great experiences this morning then you have four levels of happiness! If I prevented two of the potentially negative experiences you would encounter that day [4 - (-2)] then I have increased your levels of happiness by two by being such a great friend! That's how I see it anyways... I'm excited to read what else my classmates learned from today! 

The aftermath of our fraction lesson

 Multiplying and dividing fractions.... *shudder.* This was the lesson I dreaded! I was having nasty flashbacks of middle school math class where I sat at my desk racking my brain trying to make sense of why the algorithms we were using actually made sense... Well they didn't! Multiplying reciprocals? Dividing like a crazy person? FINDING COMMON DENOMINATORS?

 I never seemed to understand the concepts we were being taught and I wanted to cry when I saw all of the fraction questions on Elevate my Math.. more like elevate my blood pressure am I right?

I don't want to give up though! I know that math is important and that I may very well be teaching it at some point in the future. Though I do have to admit that fractions are still a concept I am challenged by. I have realized that the context in which a problem is presented does seem to make a difference, if a fraction problem is related to baking or clocks I do experience an "ah-ha!" moment! Clearly I am a visual learner...

I love baking, and had three brothers growing up, so that meant a lot of doubling or tripling of recipes! But I always added the fractions in my mind.. I didn't multiply them! ... or did I? 1/3 x 3 is 1/3 and 3/1... right... Okay, so I guess I did multiply them. OK I can do this!

My favourite activity covered during our class was building the fraction squares.

I love puzzles! I thought this was a creative and exciting way to teach students about the concept of "parts of a whole" and to make fractions fun! I know this won't appeal to every student, but there was also a chance to create different designs and shapes from the fraction pieces.

I still am in love with how much we collaborate as a class. I can't do this on my own, so why should I expect students to? Why shouldn't they be able to work on assignments in groups and share their ideas and algorithms? I do think we learn best when we work together. I love that Pat is trying to teach us what makes an effective and valuable math problem. I do want to get there! I don't' want to make students discouraged like I was when I was younger. All are welcome to learn in a math class, and everyone can do it!

Making Math Meaningful

This week I wrote a mini lesson & activity that related fractions to musical notation. I never realized before doing this activity the potential of relating a math lesson to a music lesson. Time signatures, especially 4/4 time, are a great avenue to explore fractions in your music or math classroom!

The activity I delivered broke down a 4/4 measure of music into a fraction pie. A whole pie can cover the measure, which would represent a whole note, two blue half pies each represent half notes, and red quarter pies represent quarter notes, etc. I geared the activity to grade 4 and 5, as this is when fractions in fourth are covered in the math curriculum. In the arts curriculum students are exploring rhythmic musical notation at the same time. This provides a great connecting tool between the two classes! This is also a great way to have students visually and physically represent equivalent fractions using their pie manipulatives.

I was very excited to teach the activity because as I played around with my fraction pies I was making new connections to mathematics myself! Equivalent notes is not an easy concept for students to grasp. We typically see this chart breakdown on music worksheets.

Tango Musicology. (2016, January 16). Notes and Rests: Their Duration [Online Image]. Retrieved from http://www.tangomusicology.com/wordpres/music-rudiments/notes-and-rests-their-duration/ 

Why not provide students with a hands on activity that is relating to their other coursework? This gives them another algorithm to use in their music classroom!

As I walked around the room I had a learning experiences of my own! I had not taken into account that this is old news for some of my peers and they haven't seen a bar of music since grade 9! I wish I would have taken some more time to go over clapping out the rhythms. We do learn from our own experiences, however, and now I realize I will need to get more feedback from my classes to make sure my problems have a wide base at the start!

I have included a brief excerpt from my favourite work by Maurice Ravel. This is an adaptation of his opera L'Enfant et les Sortileges. In this section of the work, the young boy is having a nightmare about his math homework. The large extending pointed finger is what got to me the most. It's so accusatory! I've included a little translation after the video.




A brief translation: 

LITTLE OLD MAN

Water out of two pipes flows into a pool!

Two cabs leave the depot right at

Intervals of twenty minutes every hour,

Hour, hour, hour!

Once a country peasant,

Peasant, peasant, peasant,

Carried all his eggs to market!

Then an advertiser,

Iser, tiser, tiser,

Bought up sixty seconds airtime!

CHILD

freaking out

Oh, God! It’s arithmetic!  LITTLE OLD MAN
fascinated, repeating

acquiescingly

Tickle, tickle, tickle!

(He dances around the Child in a more harassing way)

PUPPETS

rising up and shrieking

Tickle, tickle, tickle!

LITTLE OLD MAN

in falsetto

Eight and eight, twenty,

Twelve and six, thirteen,

Eight and eight, twenty,

Nine from three, sixteen.

CHILD

Nine from three, sixteen?

It's clear the boy doesn't understand a lot of his math problems as his teacher in the dream is giving out the wrong answers! I think most of us can relate to that overwhelming feeling of confusion in certain lessons but thankfully math educators are making changes to ensure their students don't get left behind.

My favourite fraction manipulative we used in class this week was the fraction clocks! When we had to add 3/4 and 1/3 the answer came to us so much faster than converting the fractions and then adding! I have always been a strong visual and kinesthetic learner and knowing this I will need to incorporate strategies in my classroom to make sure that the auditory learners are not excluded.

I am excited to see the next group of math activities in our class. I love seeing how my peers approach tasks so differently. There is so much to learn from the variation of strategies and delivery of each task.