#SwDMathChat From Patterns to Algebra (part 5)

Inspired by San Diego City Schools Middle Level Mathematics Routine Bank, I started out my school year with Do Nows every day to help build my students' number sense.  I did the number of the day, area models for multiplication of 2 and 3 digit numbers, and mental math strategies.  Then somewhere about mid-November I switched the Do Nows to the Guess My Rule or pattern activities from Ruth Beatty and Cathy Bruce's book From Patterns to Algebra, but feeling sort of guilty thinking I was abandoning my number sense routines.  But the more I thought about it, I realized that these activities are all promoting number sense as well.  My students are always doing some heavy-duty thinking, trying to discover pattern rules and predicting what will happen next, and I see them using some of the strategies they learned earlier in the year.  For my special ed students, these activities are really pushing them to their limits and I'm pleasantly surprised at how well they are doing with all of them.  The very visual and hands-on lessons are perfect for them to make the strong connections they need to move onto the important 8th grade concept of graphing linear equations.  Even though this book wasn't written with the special ed population in mind, it is just what they need to be successful!  (If you haven't read about my earlier adventures with From Patterns to Algebra, please take a look at my 4 previous blog posts!)

This week we continued to work on lessons inspired by this great book.  We started out comparing patterns and finished with beginning our adventures in graphing patterns and rules.  Today I was observed by my supervisor and I think he was pretty impressed with the lesson.  He loved the hands-on aspect and was impressed with my students' ability to answer my challenging questions and make the necessary connections between the rules and the graphs.

We started out the week comparing these 2 patterns to predict which would grow faster and why, even though the first position looked similar.  I had them first identify the rule for each pattern, where they discovered that both patterns had a 5 and a 2.  Then they predicted that pattern B would grow faster because they could see that by position 3 there were definitely more color tiles.  We completed the tables to the right together and noticed that they started out with the same exact number of total tiles, but by position 3 there were many more total tiles in pattern B.  I asked them to see if they could figure out why B grew faster than A even though they both had a 5 and a 2 -- what makes the difference?  They came to a consensus that the reason for the bigger growth was that the multiplier of 5 made the pattern grow faster than the multiplier of 2.

I gave them a few more rules to compare, some with the same constant, and some where the constant with the smaller multiplier was much larger than the other rule's constant, like the one below.  I again asked them to predict which would grow faster, and of course they picked the "x6 + 2" because the multiplier was larger.  We started completing the tables one position at a time and they were surprised that their choice actually got off to a slower start.  We continued the table and eventually saw their choice overtake the other rule to take the lead - they cheered!!  I wanted to make the point that the constant really doesn't matter here - it's all about the multiplier.

The next thing we tackled lead us into my favorite part - we're finally getting to graph the patterns and rules!  As the book outlines, I started them off with patterns with no constants.  I asked them to identify the rule in the pattern (on the left) and think about how we could represent the pattern on a graph.  They really had no idea what I was talking about, so I showed them the graph (on the right).  I reminded them that graphs usually have labels on both axes, so what do we need to add to the y-axis as a label?  One student volunteered that we needed to number it so we could see how many tiles there were.  The biggest difficulty most of my students had with this was remembering to "stack" the tiles one on top of the other like a tower instead of placing them exactly as they were in the patterns.  We discussed why we needed to do this -- so there was a one-to-one correspondence to the numbers on the y-axis.

I love this Post-It graph paper!

Next I paired them up and gave them their supplies:  large Post-It note graph paper, color tiles, rulers and markers.  I had already drawn the axis and position numbers on their graphs for them, but they had to number the y-axis.  I first demonstrated how they had to place their color tiles on the graph lines, not in the spaces between so they lined up with the position numbers on the x-axis.  What we also discovered was the color tiles are just ever-so-slightly smaller than the boxes on the graph paper, so after stacking several tiles, they were not exactly lined up with a line on the graph paper -- we just had to slide each position's stack up slightly to match the next line above it.  Then they needed to draw a dot at the top center of each stack, remove the tiles, and connect the dots with the ruler.  (It's amazing how many students could not draw a straight line with a ruler and how many decided to draw their lines free-hand - ugh!)  Lastly, they labeled each line with its rule.  I had them do this for three different patterns so we could have a discussion about what what different about the 3 lines.  Trying to get them to come up with a word like "steepness" to describe the difference was like pulling teeth!  Then I asked them what it was in the rules that was affecting the steepness.  Most eventually concluded that the larger the multiplier, the steeper the line -- that was a perfect place to finish up that day's lesson.  I felt like they were really making the connection between the rule and the line on the graph - yay!!

Here's one of their patterns built on the graph paper, and the final product on the graph paper (we did not graph the 0 position or discuss the y-intercept this day):

 

Today, as I mentioned earlier, was the lesson that was observed by my supervisor.  We built on yesterday's work by adding the constants and graphing the 0 position.  We discussed where the constant and multiplier tiles should be placed on the graph and decided that the constant should be at the bottom of the stack/tower so you can more easily see that it is the same at every position.  Then we discussed what part of the rule told us where the trend line would start on the y-axis and they discovered that the constant is what determines that.

After a few examples together on the SmartBoard, I turned them loose with their partners to create their graphs.  Again they had 3 lines to graph, but unlike yesterday where I displayed a pattern on the board which they recreated as stacks on their graph paper, today I just gave them 3 rules (no visuals).  They had to decide what the constant and what the multiplier were and show them with the color tiles.  Then they proceeded to draw and label the trend line for each rule.  Each of the 3 rules had the same constant, so they noticed that all 3 lines started at the same place on the y-axis.  Connections, connections...!!

Here's some of their work from today:

      

        

#SwDMathChat From Patterns to Algebra (part 4)

This is part 4 of my experiences with using the amazing lessons from the book From Patterns to Algebra.  (If you missed the first 3 parts, you can read part 1 here, part 2 here, and part 3 here.)  These hands-on, visual lessons are so beneficial for my special ed students!  They have been really engaged in these activities every day, and I love seeing them challenged, thinking and trying so hard to understand all the connections.  I was doing these activities for a Do Now every day for the past few weeks, but this past week and for the next week or so, we will be working on these for the entire class period.  I'm gearing up to start my linear equations unit, so I want to make sure my students are prepared with the background knowledge and skills they need to be successful with linear equations.

This week, since it had been a while since we worked on the patterns lessons due to my week off before break coupled with the 10-day Christmas break, I started out with just reviewing by using the Guess My Rule, first having them come up with as many rules as they could given just one pair of input and output numbers (2 and 10 in the example below), then giving them a second pair of input-output numbers (6 and 26) to have them decide which is the only rule that works for both pairs of input-output numbers.  Once they determined the correct rule (x4+2), I first gave them another input number (7) and they had to calculate the output, then I switched it up and gave them an output number (18) and they had to figure out what the input number was.  This is always more challenging for them and even though I demonstrated a few weeks ago how to write a two-step equation to find the missing input number, they still mainly rely on guess-and-check.  I tried to point out to them that they can use the input-output pairs they already know to narrow down what numbers to use when they guess-and-check.  For instance, if the output number is 18, that falls between the output numbers of 10 and 26 that we already know, so the input number must also fall between the 2 and 6.  Some students got this strategy, while others didn't.

The next day we did the "secret pattern" lesson from the book.  I gave out slips of paper to each student with a secret rule and they had to create their own patterns with the color tiles and 2-sided chips to go along with the rules they were assigned.  Here are some of their patterns:

(One thing I realized after the fact is that I should cut the position number strips apart so they could spread them out more and line up the position numbers under the actual patterns for each position.)

Once I had checked everyone's patterns to ensure they had set them up correctly, I had them rotate to someone else's pattern to first try to write the rule for the pattern, then build the 4th position in the pattern.

Most of my students have gotten pretty good at finding the rules.  As a class, they have decided that the part of the pattern that is the same in every position is the constant and should be written as the part that is added.  They're really good at drawing the 0 position because they are good at finding that constant.  They find the multiplier by looking at the first position's tiles that are not the constant.  I've demonstrated that once they find the multiplier in the first position (for example, 1 set of 5 red tiles as in the last pattern shown above) sometimes it's easier to see that the second position is 2 sets of 5 red tiles is they make a slight separation between each set of 5 red tiles.  I also try to emphasize that they can use the multiplier and the position number to find out how many red tiles there will be at any other position (and that "total tiles = position number x multiplier + constant").  We've practiced describing what position 12 or 15 or 100 would look like - how many yellow circles and how many red tiles will there be.  They're now good at knowing that there will always be 2 yellow circles at any position and that they really just need to calculate how many red tiles by multiplying the position number I give them by the multiplier they discovered in the first position.

Something that I pointed out after they were all done with writing the rule and building the 4th position is (as the book suggests) that even though some of their rules were the same, their patterns didn't look anything alike.  And we also observed how rules that were opposite, such as x5+3 and x3+5 look different even though they have the same numbers.

Next, going off script a little from the book, I had my students relating the patterns to the tables.  First I gave them a pattern to determine the rule.  I had them write out the equations to find the total tiles in each position ("total tiles = position number x multiplier + constant").  I explained that using an organized list would make it easier, so I reminded them to start off with position 1, then 2, etc.  Next, I gave them a table and had them go through a Guess My Rule exercise.  Finally, I split the screen on the SmartBoard and had them look at the numbers from both activities to see if they saw anything in common.  Some of them spotted the fact that position 1 had a total of 8 tiles and that if the input was 1 the output was 8, so as a class we decided that the input numbers were the same as the position number in the pattern and that the output numbers were the same as the total tiles in the pattern.  Now we have a connection between the patterns and the tables in Guess My Rule.



The next lessons coming up will help get them more prepared for our linear equations unit.  We will be observing how changing the multiplier and keeping the same constant, or changing the constant and keeping the same multiplier effect how our patterns act.  Stay tuned for more updates...

#SwDMathChat From Patterns to Algebra (part 3)

I'm continuing to blog about my experiences with using the great ideas from the book From Patterns to Algebra.  In case you missed my first 2 blog posts, you can read the first one here and part 2 here.

So we worked on lesson 4: pattern building with composite rules this week.  The first pattern I showed them was the one on the right below with the yellow diamonds.  They were struggling a bit so I showed them the very first pattern we had worked with a few weeks ago (the one on the left without the yellow diamonds) and asked them what they noticed.  I started to hear some "ah-ha's".

My students are starting to get more flexible with this concept.  I can ask them different questions and they can apply what they know about the pattern rule to answer my questions.  For example, for the first few days, all I asked of them was to come up with the rule and what position 4 would look like.  Then I started asking them to show me position 0, or position 10, or how many tiles will there be in position 8, etc.

A few of my students can draw position 4, but are having a difficult time writing the rule independently; I have to prompt them with questions like "what's staying the same?" or "what's changing in each position?" for every pattern.  I have 8 students in this class and about 5 or 6 can come up with the correct rule and patterns for the positions I ask for fairly easily each day.  My big challenge going forward is to figure out how to help the last 2 or 3 make the connections needed to do this independently.

The last few days I really challenged them - I did not use 2 different colors, making it more difficult  for them to determine the constant.  About half of my students were able to identify it after a few minutes of productive struggle.  I love watching them all really thinking and talking about these patterns!

Here are the patterns we worked on this week:

I'm looking forward to the next lesson in which we work in reverse and I give each student a secret pattern rule and they must build what their positions 1, 2, and 3 look like.  Stay tuned...

#SwDMathChat From Patterns to Algebra (part 2)

(In case you missed my last post on using the book From Patterns to Algebra, I highly suggest you check it out before you read this post.)

The next 2 lessons in the book were similar to the first 2, but instead of just multiplying, now my students needed to think in 2 steps (can you say "nice connection to two-step equations"?) by first multiplying then adding.  We started with the "Guess My Rule" game as before.  The first day I used the rule  x2+2  as suggested in the book.  I had my students volunteer the random numbers for the input column and I gave them the output numbers one at a time.  After 3 sets of input/output numbers I asked them to try to think of the rule.  Many were stuck.  One student (the same student I mentioned in part 1 as a matter of fact) said "This is hard!" (again). But once she figured it out, she again changed her tune and said "Oh, this is easy!"  Most of my students just needed scaffolding to help them come up with the rule.  I asked them to start by looking at just one pair of numbers - "What can you multiply the input number by so that you get close to but don't go over the output number?  OK, then what do you have to add to get to the output number?  Now see if that rule works for another pair of numbers in the table."  Basically trial and error.  Eventually most of them were able to figure this out.

The next day I started by giving them the first pair of input/output numbers and told them "I can think of 6 different rules for this, can you?"  Loving a challenge, they all set out to work. That same girl I mentioned several times already had 4 correct rules and said she couldn't think of any more. That's when I broke into my rant about perseverance and productive struggle.  With my prompting she went back to work and easily came up the last 2 rules.  After we had all 6 rules written on the board, I added a new pair of inout/output numbers. I asked them "Now which rule is the only rule that works for both sets of numbers?"  A few minutes later most had written down the correct rule on their whiteboards.


The third day we started out the same way with one pair of input/output numbers and me challenging them to find 4 rules.  Once they had figured out the 4 rules, I gave them another input number and asked them to write down the output number using our rule.  This was easy for them now that they had the rule, but then I gave them an output number and asked them to figure out what the input number was.  Wow!  You would have thought I'd asked them to explain Einstein's theory of relativity!  We're in the middle of a unit on two step equations so I hinted at the connection.  Love this!  I love the way they're all engaged and challenged by this and they're starting to see the connections.  At first, one student said "But there's no x!" so I had to explain to her that x represents the unknown number, which in this case is our input number.  I saw the light bulb go on and she quickly wrote and solved the equation.

Next, I think I'll let them take turns being the rule makers as the book suggests.  I didn't do that with the first lesson, but when I let them build their own patterns for each other to solve they really loved that.  It gives them some ownership and, as I firmly believe, if you can teach something it really shows that you understand it.  I plan on using this lesson for another day or two and then moving on to patterns with two steps.  Stay tuned!

#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!