Stacking Cups

I did Stacking Cups by Dan Meyer again.  I haven't done it in a few years and I think my intro made it more successful this time.  I let the students lead themselves to the problem.

I just held up a cup and asked them about it.  They talked about how it could hold water and food.  I shared that they were pretty dirty because many other students had touched them.  They were less eager for food and drinks then :)

Eventually they started talking about how you could stack cups and I asked for them to keep listing different ways to stack them until the stacked them inside of each other.  Then I set a small stack next to my water bottle and had them estimate how tall my water bottle was.  We discussed how even though the the cups stacked halfway up my water bottle, we would need more than twice as many.  I take a few estimates and then we calculate.  I set it up by asking what else we could measure and three (of four) classes decided to measure me :)  The other class chose a particular student and I went with that.  I actually think that was the most fun!

I also gave them some guiding questions based on standards we've been working on.  Here are the questions I asked them to answer in their groups:

1.  Identify the independent and dependent variables.  Select a letter to represent each.
2. How tall is one cup?  How tall are two cups?  Three?  How could you organize this information?
3. Write an equation to model the situation.  Check to make sure it matches the values that you stated in [2.]
4. What is the slope?  Interpret the meaning of the slope in the context of the situation.
5. What is the y-intercept?  Interpret the meaning of the y-intercept in the context of the situation.
6. What is your estimate for how many cups tall Mrs. Freed would be (closest without going over)?  Explain how you determined your estimate.

The next day we discussed as a class.  I compiled all of the equations groups had created and we talked about how they related to the cup.  We discussed how precision was important in order to calculate a correct estimate.  Some groups had calculated an estimate and then put their actual estimate down because I said closest without going over.

This is the class that decided to measure how many cups tall this student is.

After discussing we measured and then I had them complete an individual reflection on google classroom.  I asked them about an equation I had made up.  The questions I asked were very similar to the ones in their group.

What surprised me the most was how much more willing to do the math they were when they made up the question.  And they were so surprised when they saw that I knew they were going to come up with that :)

-Kathryn

Setting up Systems of Equations

As I was desperately searching (for a while) for something to boost my systems of equations unit with, I came across this post by Mimi at I Hope This Old Train Breaks Down.  It proved to be a great resource.  Please check it out!

Things I liked about it:

• There was more than one way to "solve" each puzzle
• The scaffolding was wonderful! Puzzle 1 and 2 were similar, puzzle 3 and 4 were similar, and puzzle 5 and 6 were similar
• The questioning at the end of puzzle 1 would help students more easily solve puzzle 2 (etc.)
• Shapes are much more friendly than x, y, and z.  Students were solving systems without knowing it

Because of all that I didn't want to just give my students the giant (8 page?) packet and have them work alone, so I made some formatting changes before using with my students:

• I put each puzzle 4/page
• I typed the questions up on powerpoint
• I printed each puzzle on a different color of paper

So in class it looked a little like this:

• We set up a bit of speed-dating scenario so they could rotate partners
• I passed out Puzzle 1 to all students
• If they completed it they were asked to answer the projected question on the back
• After 5-10 minutes I asked students to share out how they found the solution (I made sure to call on many students each time for various perspectives.)
• We rotated partners
• I passed out Puzzle 2 
• Etc.

Here is what it looked like as a quarter piece of paper.  After we were all finished I did have students staple all six together and keep them in their pocket for this section.  As we got into solving systems using substitution and elimination we kept coming back to this idea of the shape puzzle.

Some things I ran into as we worked:

• Students were using guess-and-check to find their solution.  Some kids have super-awesome number sense and could do this easily, but were then not being stretched to think about substitution or elimination.
• Students really struggled to explain clearly their solution method
• In Puzzle 3, some really struggled with that conceptual elimination that needed to happen.  I drew it out to help, but that wasn't enough for everyone.  Here is something similar to what I had on the board

My files:

• Puzzles 1-6 (4/page)
• Puzzle Questions

Note:  The link will open in drive.  You will want to download to see the full version in word.

Overall, I LOVED this as an introduction to systems of equations.  I'm not sure that I will change much about it when I do it next year.  And I have seen it benefiting students as we continue to work on solving systems of equations.

-Kathryn

Sequences

We wrapped up second semester with a unit on sequences. I like doing sequences after studying linear and exponential functions because I feel it gives an opportunity to compare and contrast linear and exponential situation.  And when we first distinguish between arithmetic and geometric sequences, students are fairly quick to make the connection to linear and exponential functions. (yay!)

Before we even do anything officially sequence related, I usually give my students four sequences (one adding, one subtracting, one multiplying, and one dividing).  I give the first four terms and ask them to find the pattern, the starting point, and the next three terms.  This gives them a little time to transition to new material that appears easy.

This time we tried a table of contents.  (Which I got from Sarah at math=love.  You can find it here.)  I liked it a lot because I used it to makes students think (just a little) about what was on each page.

Table of Contents

Definitions of sequence, arithmetic, and geometric.  I wanted to be good about including vocabulary, but I still don't think this was enough.  I needed to also show how the words were related to each other, perhaps a graphic organizer was in order...

Definitions

I displayed a bunch of sequences using my projector and had students sort them on the right side of their notebook to practice their understanding of the definitions.  Also to show that it isn't always easy to tell...sometimes they will need to subtract and divide to check.

Sort these

Some students wanted to label this page arithmetic, geometric, and sequence...that was a little concerning and shows that I did indeed need to more emphasize the relationship between the words I was teaching.

Next we discussed writing formulas for the sequences.  Since I had asked about starting point and pattern as we went, I thought recursive was a natural follow.  Instead of using all the nasty notation (a_n, a_1, etc.), I talked with my area education association and someone recommended this now = /next = notation.  

Note:  I liked it a lot, but students wanted to just put the "now" in for the "now" in the second equation so that there would only be one equation.  I think it might have been better to use start = /next = now/.  That might help.

Notes on recursive formulas

To practice I displayed a bunch of sequences and asked students to write the recursive formula for 8 of the 12.

Practice Problems to be displayed

My notebook had the answers :)

Next we looked at writing formulas explicitly.  I tied this into the equations for linear and exponential functions.  We did a mini-breakdown of the equations in class and came up with:

• Arithmetic:  a = d*n + start
• Geometric:  a = start*r^n

Those would be very familiar to them and still used a little bit of "sequence notation".

Notes on explicit formulas

We practiced writing explicit formulas the same way we did recursive (even with the same exact sequences).

My notebook :)

My last learning target for this unit is "I can explain why a sequence is a function."  It is perhaps a not-so-great learning target because it is somewhat difficult to teach and assess in a way that requires student thinking and learning.  However, it is very closely aligned to a standard (F.IF.3...?) and I struggle with the battle between whether I should be assessing standards or learning targets.  Anyway those thoughts are best left for a different post.

Here is what we did.  It was an investigation of sorts, where students were to choose a sequence and determine if the table that represented it was a function and if the graph that represented it was a function.  However we had difficulties because my students didn't do super well with determining if something is a function.  

The goal was that students would choose a wide-variety of functions and we could whiteboard individual results, do a gallery walk, and come to the conclusion that all sequences were functions.  However it didn't go down quite like that.  I perhaps needed another day of class, but it was crunch time for semester tests, so I didn't have any wiggle room.  

I ended up having a class discussion, but not assessing that standard.  It is what it is I suppose. 

Sequence = Function (p.1)

Sequence = Function (p. 2)

Notes:

• I wish I had done more vocabulary up front (ie. term, first term, second term, common difference, common ratio). It is hard for me to remember what words are new to students.
• Also a graphic organizer to show the relationship between sequence, arithmetic, and geometric.
• As I said, with recursive formula, I would change the first equation from now = ___ to start = ___.
• I've really been thinking about how I need to spiral some review in for students.  (More on this later...probably not until this summer.)  I believe this would have helped as we looked at sequences as functions.

Please leave any thoughts or suggestions in the comments, or tweet me (@kathrynfreed).

-Kathryn

Solving One-Variable Linear Equations

Confession:  I taught an entire semester of Algebra 1 without spending time solving linear equations. 

Why?  Because I knew my students had seen a lot of it in 8th grade, and I also knew that they were all at different places and ready for different challenges.  I also knew that they had forgotten some of what they had learned and a little reminder might take them a lot further.  All of this is hard to address in a class of 20+ students.

However, I knew it was something I needed to address at the start of this semester so that we could solve systems and eventually some quadratics.  I gave a pretest, but it was evident that not a lot was immediately recall-able for them.  So I spent a few days focusing on solving 2-step equations.  Here are a few reasons why I chose to start with 2-step equations:

• I can say they can all be solved in 2-steps, which helps the students process what they need to do
• I can address issues like "Ah, there's a fraction!" (but it really just means division) and I can even throw in some parenthesis
• I can challenge all students with things like -t + 10.2 = -23.1 (the negative variable is really tricky the first few times they see it)
• It is not out of the reach of most students (I do have a few students who still struggle to solve one-step...but those are students who did not take 8th grade math in my district last year.)
• It is enough to bring back a lot of what they learned about solving equations

I have one Algebra class that is a little bit quicker than the others, so I got to move away from 2-step equations with them on Friday.  I couldn't decide exactly how to do it, because there are SO MANY ways to solve different equations.  I really just want to ensure they are aware of the various options they have and give them some practice at choosing what to use in what scenarios.  I don't want to say:  "always distribute" or "put the variable terms on the left and the constants on the right" or anything that shows that there is only one way to solve.  I want problem solvers, not procedure followers.

Anyway, here is what I ended up doing.  I just gave them 4 "challenge" equations to solve.  And for the most part, they worked HARD for 30 minutes to work out solutions.  Here are the equations:

• 2(-1x + 6) = 22
• 2x + 2 = 32 + 5x
• 3 - 2x + 6x = 15
• 5x - 7 = 2(x + 1)

I put the four equations on the board and said "I challenge you to use the resources available in this room to find the solutions (and perhaps a solution method) to these equations today."  We discussed what resources were available and then they got to work.  The review we had done with 2-steps was enough for several students to remember even how to move variable terms to the other side.

There were a few students who "tried" and then gave up and didn't accomplish much at all, but I would say at least 85% of my students worked hard during the time I gave them.  I was SO PROUD of them.  I just wanted to brag a little about my students, because it was everything I hoped it would be.  

We will continue to work with advanced linear equations on Monday and Tuesday to get them more fluent at being aware of the strategies that are available and choosing an appropriate one.  Perhaps with some whiteboarding...

-Kathryn

Linear Functions (Part 2)

See my first post on linear functions here.  This is what the rest of the unit looked like:

Above was supposed to be a matching game, but the students didn't enjoy it very well.  I was frustrated, but maybe we should have played bingo like @mpershan (see his post it's trig bingo).  Anyway here is my template.  (To view file open in google docs and download original in word.)  I had each student graph 6 equations before starting, and then pair up with someone who had six different equations and play memory.

Then we wrote linear equations.  Kids practiced a few and then came up with their own.  It is hard to figure out how to think about something that is happening over and over again, so it was a struggle.  Also I feel this is something that is very difficult to differentiate for and provide interventions for....

That was the end of our unit on linear functions.  We moved onto exponential functions!

-Kathryn

Linear Functions

We jumped into linear functions this week.  Here are some of the things we did (and by some I mean pretty much my entire week of lessons).

What does it mean to be linear?
I really wanted to emphasize the various representations of linearity, so that students could see how they all work together.  So I started with this foldable to define linear.  I gave them the algebraic and graphic definitions, but we used those to come up with what it means to be linear in a table.

I did this by giving them each a graph of a linear function, with four points marked on it.  (Here are the graphs I used.)  Then I had them in groups of two create a table on the board for their graph.  I was hoping this would help students with the concept that a line is made up of points as we have been struggling with that idea.  Once we were done with that we made lots and lots of observations about the tables.  This took a LONG time.  I let them notice anything, so it took a while to get away from "they all have a 0/1/-2" type of statements into noticing the various patterns that were there.

But eventually we came up with "x and y both change by constant amounts".  I was very happy because in every class students recognized that they could choose any of the tables on the board as an example.  And students were willing to try to think of non-examples.  So each student got to make that part their own.  I know that is the purpose of things like this, but we don't get there often enough.  It was nice to get that far with this.

Here is the inside.  (And here is a link to the foldable.)

I created a card sort as a way to help them practice telling if something is linear.  I really wanted to get at the fact that linear is a word that describes a group of functions.  So something must be a function before it can be linear.  So we sorted between "Not a Function", "Function-Linear", and "Function-Nonlinear".  It was a lot harder for them than I thought it would be, but eventually we began to get the hang of it.  They still struggle with messing up whether VLT is for function or linear...but one step at a time.

As you can see, instead of gluing the cards in, we recorded the answers in our notebooks.  I hope that students will try to sort the cards on their own as a way to review.

Thoughts:

• My card sort is not perfect.  Two of my tables are non-functions for the same reason, that was a typo...I want to fix that.
• Also didn't really mean to have the equation x=2 in there because we didn't discuss what makes an equation a function...but it did lead to a decent conversation.  I think I would prefer to replace the card with a linear function that is not in slope-intercept form.
• One class got into a really good discussion about whether or not y=(1/3)x was linear or not.  Even when b=0 was thrown out as an idea, one student was still adamant that it needed to be written down for it to be in slope-intercept form.  I kept the conversation going for a while and then moved on without giving up who was correct.  While I was talking a student looked to the student sitting next to him and asked, "so is it linear or not?".  Aren't I mean?  The next day I eventually showed them the graph and they all agreed that it was linear.  I also explained that mathematicians tend toward laziness and prefer not to write something if it is unnecessary.
• I went hard-core with colors in my notes, but students didn't in theirs.  That is something I need to work on being more intentional about.

Finding slope and y-intercept

To begin our discussion of slope and y-intercept, I had students write "what I know..." and "what I want to know..." on the board.  Here is one example:

We used this to jump start our graphic organizer for notes.  Starting with what they already knew, but still getting it into the notes was a win-win.  They got to feel smart for knowing it and I got them all to put it into their notes anyway, especially since not all students knew it.  Here is what we came up with.

I don't think they've seen the slope formula, and I really wanted to share that with them for the Numeric part, but they thought smarter than me and were ready with ideas by the time we got their.  They remembered finding the change in y and x when deciding if the table was linear or not, and figured that would work!  It is really quite brilliant, and I feel lame because I didn't think of it first, but by the time I made it to the end of the day I had abandoned the slope formula.  I want to work back to it for tables that don't have a constant change in x, but still might be linear...we'll see if I can manage that :)

Then we practiced finding slope and y-intercept with a boring worksheet.  But it was quick and worked as a check to see where students were at.  I used my name cards to call students for answers at the end of class.

Thoughts:

• I really wish I could have just had them create their own graphic organizer for this, but I'm not sure I could have been clear enough for them to understand the expectations
• Once again I went hard-core with color and they did not (but look how cute it is!)
• The worksheet was boring, but didn't take too much time.  I think that's OK, because I just needed enough for them to get back into things.  I would have done things slightly differently if this was the first time they had been introduced to slope-intercept form.