# Qtr 2: Relationships and Multiple Representations

That is our Big Idea for Algebra 1 class this quarter.

In the past couple of weeks we have been focusing our counting circle on time (minutes).  Counting up/down by 15 mins and starting whatever I chose (most likely multiples of 60).

After we practiced counting a  few times (couple days) I started asking questions to have students convert total minutes into hours and minutes.  I then asked the question:

“Do you know what else people say for 2:15pm instead of just ‘two, fifteen’?”

Students: “Quarter after?”  (as anticipated)

Me: “Why do they say that?  QUARTER AFTER? what does that mean? I thought a quarter was \$0.25?”

Students: “Well but its different…” (somewhat confused faces)

Couple of students:  “Well yeah but \$0.25 is with money, this is time.”

Me: “Can anyone tell me why its different? What is the base number for time?”

S: “60, that is why it is different because 60 mins is 1 hr”

Me: “What is the base number for \$0.25?”

S: **crickets** (many students don’t know our number system is base 10)

S: “\$1.00?”

Me: “nope not \$1.00, the numbers we use everyday (0, 1, 2, 3…) what are they all based on?”

Students: “100, 5, 2, …” (randomly yelling out answers so of course 10 comes out)

Me: “YES!  10 base 10 that is what our number system is based on”

Now continues a little history lesson (~3 mins long) about sexagesimal (my favorite word to teach teenagers) and decimal number systems.  Including a little mention about Babylonians and why I love to teach them this word.  We closed up that discussion with 1/4  not necessarily equal to \$0.25 but means different things based on the number we are working with.  “A quarter of what?”

Following this “Math Exchange” we start this week’s succession of counting circles (we only spend ~15 mins each period on these):

Monday: counting up by 15 mins, starting at 65 mins (not divisible by 60 but 60+5).

Tuesday: No school.

Wednesday: counting up by \$0.25, starting at \$1.05 (I hope you see where I am going with this).  After our counting circle on this day, I asked them if they think there is a relationship between Monday’s counting circle and today’s counting circle?  They were very familiar with counting  by \$0.25 because we did that for the most part of quarter 1.

So we discussed the “quarter” idea that we have been mulling over since last week.  We also discussed how \$1.05 is kind of like 1 hr and 5 minutes (65 mins).  Before they went to their seats to start our math lesson for the day I told them “You need to think about these relationships because tomorrow I am going to give you an assessment that asks you to describe the relationship between two situations like these.  They probably won’t be the same situations but they will be mins and \$ so be prepared for tomorrow”

THAT SAID………

Thursday: Assessment (see picture below) I gave them 2 different counting circle scenarios with a number line like we would normally come up with if we did the counting circle.  Then I asked them those 3 questions.  Notice how the questions successively get harder as you go down.  I really love how only Q#1 has one answer but the other 2 questions have multiple answers and ways to prove their answers.  I love how most students would be able to answer Q#1 (getting them started) and then move their way down the ladder.  I also love how Q#3 basically wraps Q#1 &Q#2 up in a beautifully wrapped math present for them.

I will admit this assessment wasn’t a piece of cake.  But we have been working on these connections and counting circle numbers for quite some time now.  I discussed it with them yesterday!!  I EVEN TOLD THEM WHAT WOULD BE ON THEIR ASSESSMENT!!

Did they struggle with it?  ABSOLUTELY!

Would I change anything? NO WAYS!

Do I think these discussions are happening too late in their math career?  WHOLEHEARTEDLY YES!!

When do I think they should be discussing these relationships between number systems and fractions??  Somewhere around Grades 3-5 and continue them throughout their years in math class.

Are they happening in my school’s Grade 3-5  classrooms?  HECK NO!

Will I stop asking my students to look for and point out these relationships?  HECK NO!

Do I love my job? HECK YES!!