Math = Love

Thursday, January 18, 2018

North East South West Puzzle from Puzzle Box, Volume 1

Just a quick post today because it's already past my bedtime! This has been a crazy week because I've been spending almost all of my spare time prepping for a 2.5 hour workshop I'm giving this Saturday on interactive notebooks. The participants will actually be designing their own notebook pages during the workshop, so I've had to put a lot of time and thought into how to make our time productive and make sure people leave with an INB that they are proud of. I'm looking forward to sharing my presentation slides and documents in a couple of weeks once I've had time to tweak them and perfect them.

Today, though, I want to share about this week's puzzle table happenings. It's a short, 4-day week due to Martin Luther King, Jr. Day. And, I was so frazzled on Tuesday that I never got around to putting out this week's puzzle. So, yesterday was the first day for this puzzle on the puzzle table. With only two days of student attention so far, it has been attempted by several and solved by one pair of students I don't even have in class this year. Two girls stopped by to discuss something related to student council. The website I needed to access to answer their question was having technical issues, so they were forced to wait. They naturally gravitated toward the puzzle table and solved the puzzle. It was awesome to listen to their reasoning.


Instead of just jumping in the puzzle and placing letters in random places, they really reasoned through which letters were used the most and which letters were used the least. Then, they used this information to decide where to put these key and not so key letters in the puzzle. Once they did this, they had the correct solution in what seemed like no time at all. My other students have not been so lucky yet.

This week's puzzle, which I have named "North East South West" comes from Puzzle Box, Volume 1 (affiliate link) from Dover Publications. This is the first book in a series of three puzzle books that are edited by the Peter and Serhiy Grabarchuk. This specific puzzle is by Donald Knuth.


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Each volume has 300 puzzles, and I have found over a hundred puzzles between the three volumes that I would like to adapt to use in my classroom some day. If you love puzzles or if you are looking for resources to teach your students to reason logically, Puzzle Box, Volumes 1-3 (affiliate link) are the books for you!


This week, I decided to give my students a chance to try a word-based puzzle instead of my normal shape-based or number-based puzzles that I tend to gravitate towards. Hello, I'm a math teacher!  


Puzzlers must take the 10 letter cards and arrange them in such a way that NORTH, EAST, SOUTH, and WEST can be traced out by moving one square at a time, horizontally, vertically, or diagonally. Donald Knuth's original wording of the question mentioned moving as a chess piece. I figured that would confused my students, so I chose to reword it.


I have uploaded the files for this activity here. The puzzle board is designed to print on 11 x 17 paper or cardstock (affiliate link). The puzzle pieces are designed to print on letter sized paper.

You can print it on letter sized paper by changing the scale percentage. I had to scale it to 65% to make it fit. Be sure you remember what percent you chose because you will need to scale the puzzle pieces to the SAME percent so that the puzzle pieces fit on the puzzle board properly.

I'm off to bed. Hope you and your students have fun with this puzzle! If you haven't already, I definitely recommend that you check out the Puzzle Box books. You can get a great taste of what types of puzzles they have to offer you and your students by looking at the free Amazon Preview! Just click the "Look Inside" button for each book. If you're logged into Amazon, you can click "Surprise Me!" on the left side of the page. This will let you see quite a few of the puzzles inside the book for free. I typed up my first Puzzle Box puzzle from the free preview. Then, I did some more looking around and knew I had to order it!


Full Disclosure: I purchased Puzzle Box, Volume 1 with my own money. I was sent a free copy of Puzzle Box, Volume 2 and Volume 3 by the Grabarchuk Family.

Tuesday, January 16, 2018

Algebra 1 Unit 3 - Relations and Functions Interactive Notebook Pages

Before Christmas, we finished one of my favorite units of the year in Algebra 1 - Relations and Functions. With regards to pacing, we are WAY behind where I would like to be, but there's not much I can do about that. Last year, my pacing was off in the other direction. We spent a full two months of the school year on data and probability because we got through the rest of the standards too quickly. I do know that this year's students have a much better handle on solving equations and inequalities than last year's students, so I'm hoping this more solid foundation will allow us to progress more quickly during the second semester.

Here are our notebook pages for Relations and Functions. There are some old favorites which I reuse every single year and some new pages as well.

Each unit begins with a table of contents divider. I've blogged about these before here. The first side contains a section titled "Top Ten Things to Remember." Students complete this either as we work through the unit or right after we finish the unit. I encourage my students to flip back through their notes and decide what the ten most important things we learned were.


The other side of the divider contains a list of our SBG skills for the unit. Students are required to record their initial score for each quiz and any updated scores as a result of retaking quizzes.


We started out the unit by refreshing our memories of the various ways to represent relations. For my Algebra 1 students, they were already familiar with ordered pairs, input/output tables, and the coordinate plane. Mapping diagrams were completely new to them. Each representation got its own flap on this foldable which is one of my favorite foldables of the year.  






Now that we've reviewed relations, it's time to talk about a very specific type of relation - a function. To introduce the idea of a function, we completed a Frayer Model. 


Next, we practiced writing sentences to justify whether a relation is a function or is not a function. I find that students need explicit practice regarding how to properly justify something like this. 


Now that we know all about functions and non-functions, it's time for a function/not a function card sort. I blogged about this card sort and shared the file for it here



Since I started teaching, I have struggled to teach independent and dependent variables in a way I am proud of. Each year, it seems like I try something new. And, no matter what, the same 60% of kids who intuitively understand independent vs. dependent get it. And, the same 40% of kids who mix up dependent and independent EVERY SINGLE TIME still mix them up every single time. This happens no matter how well I feel like I've explained it.

Here is this year's attempt: 


I'm especially proud of the inside. I think that having students sketch a graph has helped their understanding of independent vs. dependent variables. 


This approach, however, was not the magic cure. I still had a fair number of students who switched the variables every single time.

Now, let's take a closer look at discrete vs. continuous functions. 


This discrete vs. continuous card sort still makes me so happy. I blogged about this activity here last month. 


Once again, I chose to pull out the good ol' DIXI ROYD mnemonic device for Domain and Range. 


I was feeling really uninspired when it came time to type up domain/range practice problems for our notebooks. Luckily, Math by the Mountain came to the rescue! These next two booklet foldables are her work. You can learn more about her relations and functions unit in this blog post.

Domain and Range of Discrete Relations:



Domain and Range of Continuous Relations: 



Up next: Domain and Range Restrictions. I simply edited my booklet foldable a bit from last year to update some of the examples.



I added a new graphic organizer this year for rate of change. This resulted from making the decision to NOT introduce the term "slope" during our relations and functions unit. Instead, I decided to wait until our linear graphs and inequalities unit to begin referring to rate of change as slope. 


Rate of Change Practice. We did four problems and stapled them together so they only took up one page in our notebooks. 





After looking at graphs with a constant rate of change, we shifted to graphs that have various rates of change.  I'm a bit disappointed that both graphs I chose were distance-time graphs. This is definitely an area of improvement for next year!



 It's graphing story time! This was my second time using popcorn graphs with my students. The conversations were awesome this year as well!


I was a bit short on time, so I just ended up using the same graphing stories foldable as last year. 



Here are close-ups of the two tasks inside. I found both of these tasks online. 




I created this puzzle to motivate my introduction of function notation. I really liked the concept, but my presentation could use some work. All of the values I picked as my examples could be found on the same linear graph (despite the entire graph not being linear). So many of my students thought that the function notation just meant to multiply by 6. :( It did lead to some interesting conversations when some students had interpreted the puzzle as multiply by six and others had interpreted it as look at the ordered pairs that the graph goes through.


Another big change I made this year was introducing evaluating functions by only looking at graphs and tables FIRST. Once my students were comfortable with evaluating this way, then I introduced evaluating from an equation. In the past, I did it backwards of that and started with equations. I think it was too much too soon. This year seemed to work a million times better!

Evaluating Functions From a Graph:


Evaluating Functions From a Table: 


Evaluating Functions From an Equation: 


Now, it's time to practice writing functions and using them to solve problems. My students found these problems to be a bit difficult. I think we should have spent more time practicing these than we did. 



For graphing functions, I chose to do the Win Some Cash task again from last year. Again, my students got super sucked into the scenario. 


While doing the task this year, I realized a mistake I had made last year. :( Last year, I had my students connect the dots to more easily see the shapes made by exponential, linear, and quadratic functions. This year, I highly emphasized discrete vs. continuous graphs and when it makes sense to connect the dots and when it doesn't. This is actually a discrete function, so we couldn't connect the dots this year. With this in mind, I'd like to edit the activity a bit next year to bring out the general shapes of different types of functions more clearly.

Also, note to self: make the graph bigger. I made a note about that last year, and I forgot to fix it before printing it again for this year! 


We lots of graphing practice. Students had to graph each function by making an input/output table and classify each function as linear, quadratic, absolute value, or exponential.  My students really struggled with knowing how to connect the dots. When I instructed them to connect the dots from left to right, this just confused them even more. Not sure how to remedy this for next year. Maybe I should make some sort of connect the dots puzzle for them to work out that goes left to right...



This year, I meant to make a summary page that discusses key points about each type of function, but we ran out of time.  It's definitely on the list of things to fix for next year. 


This two page spread in our notebooks just makes me smile. 


A Frayer Model for "Linear." Students had to create their own examples and non-examples. 


Our last notebook page of the unit was a linear/non-linear card sort.

You can download the files for this unit here