#SwDMathChat From Patterns to Algebra

I recently discovered the book From Patterns to Algebra by Dr. Ruth Beatty and Dr. Catherine D. Bruce and thought it was perfect for my students to give them the hands-on and visual connections to linear equations and support their algebraic thinking.  As they say in the introduction, "Mathematics has been called 'the science of patterns'."  The lessons presented in the book are meant to be followed in the given sequence.   The first chapter of the book, which is where I am now with my students, has them playing "Guess My Rule" and pattern building.  I chose to purchase the DVD with the book which gave me access to video vingettes and Smart notebook files for each lesson.


In the first lesson, "Guess My Rule", I asked students to give me a random number between 2 and 9 (1 and 10 make it too easy), and then apply the rule I'm thinking of and write the answer.  As they figure out what the rule is (what I'm multiplying their input numbers by), they write it down on their whiteboards.  When I think everyone has figured it out, I take one more random number and have everyone write down the output number on their whiteboards to check.  Some of my students lack multiplication fact fluency, so this is good practice for them.  As the authors advise, don't have the input numbers in order because that leads to students looking for additive rules instead of multiplicative rules.


In the next lesson, I displayed various patterns with 'position number' cards underneath each iteration of the pattern.  The students must again figure out what the rule is.   The first day we did the patterns, I heard one student say "This is hard!"  I told her that once she sees how the first one is done, she will think they're a lot easier than she originally thought.  They were all overthinking it!  Once everyone figured out the rule and wrote it on their whiteboards, I had them draw what position number 4 would look like.  One student (the same student who thought this was hard) expressed the rule additively instead of multiplicatively at first (for the first example below she said you're adding 3 each time).  I asked her to try to see what the position number was being multiplied by instead of how the pattern was changing each time.

After doing the patterns for a few days, today I had them create the first 3 positions of their own patterns with color tiles.  After I checked their patterns, I had them rotate to someone else's pattern, write down the rule on the whiteboard, then try to build the 4th position themselves.  They really liked this activity!  I loved how engaged everyone was with creating their own patterns and figuring out each other's patterns.  Here's some of the patterns they created:

The lessons coming up next deal with composite rules.  I think this will be more difficult for my students because they have to see the two steps being done (multiplication and addition) instead of just one (multiplication).  I hope I'm pleasantly surprised and they pick this up very quickly.  I'll keep you updated on our progress!

#SwDMathChat Expressions and Combining Like Terms ... "Can I Take Your Order?"

In keeping with my new school year resolution to introduce new topics with real world situations, I start out teaching combining like terms by telling my class we're going on a pretend class trip - after going to see a play, we're going to stop at McDonald's for lunch.  Instead of waiting to get there to place our orders, I'm going to call in the orders ahead of time as soon as we get back on the bus.  This is where I really grab my students' attention - I pass out 9 magnets to each student with pictures of McDonald's food and drink items (I made these magnets a few years ago with the magnetic paper that you can run through a printer).  They can choose 2-5 items that they want to order.  I have each student make their decision and then stick their magnets on the whiteboard in the front of the room, circle them with a dry erase marker, and write their name next to their circle.  Then I grab my cell phone and pretend to call in the order to McDonald's, naming EVERY.SINGLE.ITEM separately, faking being out of breath from having to say so many items!  Then I ask them if that's how I should have done it, or is there a better way?  Right away they all have an answer for me, although they don't all know exactly how to put it into words.  Basically, they tell me I should have counted up how many of each item I wanted to order, so we go through all their orders and count up how many burgers, chicken nuggets, fries, salads, sodas, chocolate milks, milkshakes, ice creams, and apple pies we all ordered.  Then I make my call again the easier way.  Then I segue into my combining like terms lesson...

I have a SmartBoard notebook file set up with infinitely cloned color tiles (in the rectangle on the left), which I drag out across the screen.  I have my students practice writing an expression to represent the color tiles by combining like terms (colors in this case).  For example, the expression we wrote for the color tiles shown here was 2R + 2B + 1Y + 3G.  We only do one together as a whole class, then I have my students come up and take turns dragging out the color tiles for the rest of the class to write an expression.

Then I move on to defining parts of a variable expression and making sure they understand that terms must have the same exact variable raised to the exact same power in order to be considered "like".  I give them several sample expressions and they practice naming the terms, coefficients, constants, and like terms.  We end this day with two games of Kahoot! to practice identifying the parts of an expression, one regular Kahoot! and then a ghost mode round of the same Kahoot! game.  They love trying to beat their ghosts!


Next I have them practice actually combining the like terms - I teach them several strategies to do this.  I demonstrate using shapes or colors to group the similar or like terms.  Each student can come up to the board to drag the shapes over the terms to show which terms are alike.  Then they have to write the final expression with the like terms combined.  I start out using only positive terms, and once they have the hang of combining these, I throw in negative terms - you should hear the moans and groans!



For more practice with identifying like terms, I created a card sort in the Desmos Activity Builder (love this!).  I used an idea from Cathy Yenca's "Becoming an "Expert" blog post of displaying the teacher dashboard on my SmartBoard in the anonymous mode until I found my first "expert".  Then I let them know which set of cards was theirs.  After they had all green cards, they moved on to the second screen which asks them to combine the terms they identified as like on the first screen.


After that, we did a Combining Like Terms QR code scavenger hunt.  As soon as they walked into the classroom and saw the QR codes hanging around the room they cheered!  They love doing these scavenger hunts!  I love watching them moving around the room so focused on their work.



The last activity I do before the study guide and quiz is centers.  One center had a (boring) worksheet on which they had to identify parts of an expression.  The second center was an iPad app called DigitWhiz which has several combining like terms activities.  I had them do the "Simplify" activity (shown at the right).  At the third center, they had Combining Like Terms Sort Cards - they had to first decide how to sort them and then combine each set of like terms.

They had their quiz on expressions and combining like terms on Friday - most of them did really well!  The biggest issue most of them have is not recognizing that a subtraction sign indicates the number following it is negative.  I remind them to change the subtraction to addition by "adding a line, changing the sign", but some forget to do this.  I'm hoping this will improve as we move into solving equations.

One topic I am skipping this year is expressions and equations that require distribution.  Teaching resource classes means I go at a slower pace than the general ed classes, so unless I leave out certain topics, I will never be able to get to every topic in the 8th grade curriculum.  It's always difficult for me to decide what to leave out and every year I feel like I leave out something different.  I chose to skip distribution this year because it's always so frustrating for my students (and me!).  Now it's on to one-step equations...