Solve Me Mobile Equations

 I love using Solve Me Mobiles with my students especially as we work on solving equations.  This year I decided to dig a little deeper into the mobiles to see how I could facilitate more discussion around the thinking used to solve mobiles (which students always engage in) and connecting it to the algebraic work required in class (which some students don't understand or engage in).  

The Open Up Resources curriculum has some great lessons that use the hanger model. If you haven't already you should check them out. Along with that I found the book Making Sense of Algebra on my shelf.  Probably something I bought pre-Covid thinking I would have time.  It is a great read.  It focuses on Habits of Mind, much like math practices, that should be developed so students are successful in Algebra.  One of the chapters, Solving and Building Puzzles, discusses the Solve Me Mobiles.  The book explains that puzzles help build stamina because of the little wins that students get along the way.  That explains why students are more likely to engage in the Mobiles than in an algebra problem. The mobile is easily accessible to all students and those with low math confidence don't look at it and immediately shut down like they might be inclined to do with an algebraic equation.  The mobile allows us to transition the students to the algebraic notation that is presented in the mobile.  What really resonated with me when reading was the statement that we need to make the logic explicit.  Students are doing the thinking, but how can we take a step back and really help them think about what they are doing and how that thinking is algebraic in nature.  

So these were my 2 goals:

1. transition to algebraic notation

2. draw attention to the logic and reasoning being used

After using the 6th grade Open Up Curriculum with hanger models, I pulled this problem from the 7th grade curriculum.

I like how it gets students thinking about the relationships to come up with true and false statements.  There is nothing that needs to be solved so students get to take it all in and notice relationships.  This was my entry point for what I wanted to accomplish.

I pulled some of the mobiles and removed any numbers at the top or in the solution so that students would not need to solve the mobile. Then we focused on making statements that must be true.  

My original plan was to slowly transition to algebraic notation.  I wrote one statement and I was ready to start short-handing it.  The students caught on very quick to how what they said translated into an equation.  This mobile for example, we know that 3 moons and a hexagon weigh the same as two triangles and a hexagon, 3m + h = 2t + h.  The simplicity of this mobile allowed us to push our thinking forward.  That is really the only statement that came up at first.  Then we noticed (or I nudged them to think about) that there is a hexagon on each side of the mobile.  For some students pulled out my balance scale and removed counters from each side so they could think about how removing items affects the scale.  We were able to come to the conclusion that 3m = 2t.  Our algebraic work from our discussion looked like this:

Each image of a mobile added another layer to our algebraic thinking and work. 

This mobile allowed us to remove drops from both sides, but also think about what balances with just one square.  If you do this you will want some way to draw all over your mobile, either on a Smartboard or with an annotating extension.  We marked up the mobiles a lot in our discussions.  

By this point I was officially asking for equations, although we were still speaking in shapes rather than variables.  This next mobile added some fun discussions.  

There are 2 parts to the mobile and many shapes that repeat.  There are also two equations with 3t on one side.  This allowed us to use our equations to create other equations (shown in green), which is actually used in solving systems of equations with substitution.   We had some discussions with whole number equations to help understand this.  3+4=7 and 5+2=7, so we know that 3+4=5+2.

This was such a good conversation that I added another mobile for my groups tomorrow. 

Here are the notes I've written for myself to help guide the discussion.

I was so impressed by my students ability to think about the number relationships in this way.  By focusing on making the logic explicit and connecting it to the algebraic notation, I was able to have some great conversations with students who have not solved equations before.  I am hoping when we get to solving equations more formally they have a deeper understanding of the relationships so that they don't feel like they are simply following rules.  

Here is the presentation I used.  It has some Open Up lessons as well as what I plan to do after this activity.  Depending on when you read this it may still be a working document.  I will blog about the rest in a separate post if it goes well.  

I would love to know your thoughts or if you try this activity.  Or if there is another way you connect the mobiles to algebraic reasoning.  Leave a comment below. 

Percents with Cuisenaire Rods

I was prepared for today.  I had an intervention with a 7th grader.  We have been building his units coordination and multiplicative reasoning through fraction work.  I had it all planned out based on some success we had seen the session before.  Then he came in worried and wanted to work on an assignment from class.  Knowing that I could work some of our fraction conversations into the assignment I pushed my plans aside and we looked at his homework.  

The assignment was on finding percents using a double number line to help think about the number relationships in the problem.  A double number line is an especially nice representation for this student because it allows him to coordinate and keep track of the units (percent and value, parts and wholes) in a visible way.  

I don't remember the exact problem, but it was about basketball free throws.  12 were made and that was 75% accuracy.  It was asking how many shots were attempted.  

Here is our work as we were discussing.  

The assignment gave an open number line.  He was able to correctly place 75%.  Then wanted to add 50% as 6, because 6 is half of 12.  

As we worked on this problem together and I thought about what he knows and can do with fractions, his confusion did not surprise me.  In fact, I had a huge revelation about percent problems like this, their connection to fraction understanding, and the role of units coordination.  

I am going to frame this in terms of fractions and then you can re-read it replacing the percents into it and think about it through that lens.  

So the student is given the fact 3/4 is equal to 12.  The student needs the understand that 3/4 is made up of 3 - 1/4 pieces (which to a middle schooler is a lot clearer than knowing 75% is made up of 3 - 25% pieces).  

So the 3/4 piece needs to be considered 1 unit that needs to be partitioned into 3 pieces.  So the 12 is divided by 3 to get 4.  

Then the 1/4 piece needs to be considered 1 unit that is iterated 4 times to make up the whole.  

Throughout all of this one whole is a unit that has existed to help us know the size of each piece (our denominator or percent).  

If you kept track through that, this problem has 3 levels of units.  The whole, the unit fraction, and the non-unit fraction.  That doesn't even include the units on the top of the number line, which also has 3 levels of units being worked with in tandem with the percents.

So to do this problem the student needs to be able to:

*Understand fractions and/or percents as measurements.

*Be able to partition and iterate fractions and/or percents.

*Coordinate 3 levels of units.  

Now that I was seeing a clearer connection between fraction development and percents my brain immediately went to Cuisenaire rods.  I have used rods a lot in fraction work and they are a great concrete way help students with all 3 of those bullet points above.  

Mark Chubb (inspired by Dan Meyer) has talked about how important it is to move students up and down the ladder between concrete, representational, and abstract.  Cuisenaire rods seem like a great way to take a few steps down the ladder to make this problem more concrete.  

Because I was also thinking about the levels of units I started to think about the different layers in this problem.  For example: If the problem had given 25% (or 1/4), then there would only be 2 units to coordinate, the unit fraction and the whole.  

It also made me wonder if it is easier for a student to be given 25% and find 100% or be given 100% and find 25%.  I suppose that depends on where the student is in their understanding of partitioning and iterating.  

I decided to pull out my Cuisenaire rods and start thinking about it.  

Here is the document I came up with.

First I stripped away the top part of our double number line.  I wanted to focus on the percent and not the value attached to it.  This is much like the fraction work I have done with the rods.  Only this time students will be partitioning and iterating percents, and hopefully getting a better understanding of percents as measurement.  Many students tend to have a better part-whole understanding of fractions, so it makes sense that they have a better part-whole understanding of percents.  I am not sure I ever really thought about percents having both a part-whole and a measurement model.  (Seems so obvious, right?  Usually the big aha's are. How did I not see this before?)

The first section starts with 100% and has the student find a part.  I separated it into finding unit fractions (are unit percents a thing?) and non-unit fractions because it is the difference between coordinating 2 levels of units and 3.  The left column should be easier for students to think through as they develop their understanding.  

Then I came up with options for starting with the part and determining the whole or 100%.  Again, breaking it apart into unit fractions and non-unit fractions.  If given 25%, students would iterate that piece 4 times to find 100%.  If students are given 75%, they have to determine 25% and either compose the 2 parts or iterate the 25% to get to 100%.  

Then I noticed in my fraction work that I have some where students are given a fraction and have to find another fraction.  So I wanted to create the same with percents.  Students are never specifically asked about the whole (100%), but they still need to be aware of it as they think about the number relationships.  I think that means that all of these require coordinating 3 levels of units, but maybe because it's percents it doesn't.  (UC friends, let me know if I'm wrong about that.)

Given 20% find 60%.  This requires iterating the 20% 3 times to find 60%.

Given 75% find 50%.  This requires a student to know that 75% can be partitioned into 25% pieces, 2 of which make 50%.

All of this percent work could definitely be done as part of understanding percents and fraction, decimal, percent conversions.  In the future I will do this type of activity earlier in the school year, before students have to find percents of a number.  

Then to extend this work to what my student was working on I can simply attach a value to the rod a well as a percent.  Here are some examples in each category:

I spent a lot of time by myself playing with the Cuisenaire rods as I processed all this information.  I learned so much and thought about percents in a way I hadn't before. The document I came up with is simply a cheat sheet for me.  I have found that I cannot think these things through on the fly.  If I try I end up choosing a rod as a percent that doesn't work with the other rods that are there - although I should note that the "piece" that students find does not have to be just one rod.  It opens up the possibilities if students can combine rods together to make the needed lengths.   Jo Boaler actually has a nice activity like that in her Mindset Mathematics Grade 6 curriculum.  It uses fractions so would have to be converted to a percent activity for my purposes here. 

Of course it doesn't stop there.  Mark Chubb, as a true math coach, got me thinking:

Kids totally need to do what I did today.  There was so much great relational thinking.  

• What rods can I choose?  
• Which rod won't work as 50%? 
• Which rod doesn't have a 50% rod to go with it?  
• What other percents can I make with 20%?

Obviously these are all questions I can ask students as we explore together.  

When I have done fraction activities, I have had students make up their own puzzles for us to solve with the rods. Then we share them.  This would also be a great way to engage students in the exact task I was doing today.  Since my document is just a cheat sheet I won't be using all the options with the students.  That leaves options for them to find through exploration.  

If you made it to the end of this very long post, I would love to know your thoughts.  

• What might percent increase problems look like?
• What other types of percent problems am I missing? 
• What struggles do you see getting in the way of students being successful with percents?

Charging Batteries - An exploration of Percents, Fractions, and Decimals

Today in class I used the Battery - Percents, Decimals & Fractions Desmos activity by Andrew Stadel.  Last week we had done a prompt to find 20% off and my students understood the context of percent off but did not clearly understand that it meant the amount they paid also had a percent attached to it.  For example, if you have 20% off, you pay 80%.  I wanted an activity that helped them better understand the complements that add up to 100%.  The Desmos activity uses the context of a charging battery and it was perfect.  I started with having students estimate the charge on my Chromebook.  They were able to easily tell me the charge on my battery, the percent used, and clearly understood that the two needed to add up to 100%.

From there we launched into the Desmos task.  It is great.  You really need to check it out.  Here is a quick snapshot to entice you.

The task itself is designed to help students make connections between fraction, decimal, and percent values.  We have been doing lots of work with Clothesline Math to help students understand the relationships between numbers and keep track of the different units in a problem.  We were able to use our clotheslines to help us see the connections as we discussed the batteries that were in the task. 

I liked this activity and context so much I decided I needed a few ways to extend it to future warm ups in my classroom.  One thing that I wish I had had for my clothesline were the actual pictures of the batteries.  It could be much like Fraction Talks on a clothesline.  So I used the sliders in the Desmos activity to create this printable with batteries, fractions, decimals, and percents. 

If you scroll down there is a section that has just the batteries.  With just the batteries I can have students place them on the clothesline, thinking about placement and spacing, and then they could come up with the fraction, decimal, percent values on their own.  

I also really liked the slides that had students sketch using certain parameters.  

It reminded of the Menu Math activities that Nat Banting shares on his website.  So I decided to make a Menu Math task using the context of the batteries.  

This was my first time creating a menu math task and I loved trying to find the right combinations so that it could not be easily done in just 1 or 2 batteries.  My daughter and I tested it out and on our first attempt we were each able to do it with 4 batteries.  Plus I loved the discussions we had about some of the wording "at least", "up to", "no more than".  It was partially a lesson on inequalities as well.  You will notice I also tried to incorporate the amount drained/used in order to continue that discussion about the complements that should add up to 100%.  

If you try these activities or have any ideas of other ways to extend this context I would love to hear about it in the comments below.  

Cuisenaire Rod Exploration Day 1

This year for my middle school interventions I have decided to focus on units coordination.  At this point I am envisioning that most of our work with be around multiplicative and fractional reasoning.  Although now that I am starting I may have to do some additive reasoning first with a few students.  

I like to start the year focusing on characteristics of mathematicians (taken from Tracy Zager's Becoming the Math Teacher You Wish You'd Had, I wanted some open activities that allowed us to notice, question, and explore.  I chose to use the Cuisenaire Rods and some activities that I learned from Simon Gregg.  

Here was the first day's lesson plan:

Making the counting numbers.

Start with 3

How many ways can we make 3?

Ask Questions: Are 2,1 and 1,2 considered different arrangements?

Prove: Have we found them all?  How do you know?

Move to 4

How many ways can we make 4?

Ask questions: How does that compare to 3?  

Move to 5

Intuition: How many combinations do you think there will be?

How many ways can we make 5?

Work together and alone: As it gets to be more combination will they each make combinations or work to find different ones to make the entire collection?

What do you notice?  What do you wonder?

• Look for patterns

• Make predictions

• Notice relationships (a 3 rod can be replaced with 3 other possibilities).

• Generalization: Is there a way to figure it out without trying to build them all?

Here are the results of some of our explorations:

I let the groups decide if order mattered.  When it did we found lots of different ways to arrange the combinations and looked for patterns so we could try to predict:

 

We discovered that if there are 3 blocks there are 3 ways to arrange them. 

If there are 4 blocks there are 4 ways to arrange them.

We didn't have time but this would be a great Always/Sometimes/Never exploration.  (I'll let you try it and see).

For the groups that decided order doesn’t matter there aren’t as many combinations but it was harder to explain/prove we had found them all (maybe because we didn’t spend as much time thinking about all the possibilities and arranging them to make sure we had them all). 

From 5-6, a student realized they could use all the combinations of 5 and then add one to each.  Then looked for additional combinations.  

We noticed a pattern, but our pattern broke.  

Reflection:

This activity on the surface seems very elementary.  I had a moment of doubt before my first group of students.  "Am I really going to ask 8th graders what numbers add up to 4?"  Thankfully I pushed that doubt aside, knowing the power of the Cuisenaire rods.  I loved every minute of this.  It was challenging to try to find all combinations.  It reminded me a lot of the work we do with probability before students know how to make probability tables or trees.  We needed an organized list, a way to organize our combinations to make sure we had them all.  It was very cool to see the different ways that students chose to arrange their collection and explain how they knew they didn't miss any.  

It was interesting to see the predictions and how they would change them once they started making their collection without any prompting.  

Even after class I continued to explore on my own because I was so fascinated by it all.  Here are some resources Simon shared with me as I explored.

Here is a blog post with a proof that confirms our doubling pattern. 

It also shows how Pascal's Triangle shows up in the pattern.

As for our broken pattern, I still am not sure about that, but it turns out that Rananujan worked on that and his work is celebrated in this beautiful children's book.