Mixture Problems in Algebra One

This intiative by #MTBoS has been great - I've been better about blogging and visually documenting what I do in class.  And it's been great fun to read other blog posts on the weekly initiatives. This latest directive is to blog about a lesson. This is mostly what my blog is about - blogging on lessons I do and the interactive student notebook materials I use/create. 

Teaching mixture problems using systems of equations can be challenging because it can be difficulty to visualize what they are about. I'm all about visualizing and understanding concepts (for myself and for my students). This lesson sets the stage for building understanding about the concepts underlying mixture problems.

The class is Algebra One, honors level, ninth grade. We've been working with systems of equations for about a week and a half now. Students learned about systems last year in 8th grade, mostly focusing on solving systems using graphs. I started this unit by focusing on creating models for situations with two conditions and two unknowns (see this blog post). We then learned how to solve the algebraic models without graphing and I incorporated applications problems as I went. Mixture problems are their own little category of applications though, so I introduce those separately.

This year I decided to introduce mixtures visually with something all students love - cereal! Last year my student teacher and I talked about this idea and she implemented it in her classes (thank you Tess!) so I took her good work and tweaked it a bit for my class this year. 

I set up a little "Research & Development Lab" in the back of my classroom:

Those ziploc bags each hold an ounce of the labeled cereal (about a heaping cup of each).

And provided some facts that our research team is using:

The question was how many ounces of each type (Kix & Cocoa Puffs) would have to be in this blend to achieve the size and cost we are aiming for? Students saw that this was a situation with two unknowns and two conditions. So it was time to organize this information into a system of equations (all this comes from student responses):






The tricky bit of course is in setting up the equation that represents the cost. Students have to remember that the sum represents the cost of each individual type of cereal (cost per ounce times number of ounces) so that must be equal to the cost of the new cocoa kix cereal box (cost per ounce times the total number of ounces in our prototype box).

Once we got this all set up students solve the system. Now the fun part - we actually measured out a "box" of this special blend together.  And then students got to sample the cereal (yum). I gave everyone a small cup full of the cereal to munch on.


So how did this work? Well it was certainly fun and I have a bright bunch of kids this year so they were able to create the system pretty easily. I handed out the cereal samples for students to munch on while we did some problems together  but there were only 10 minutes left to class and I think everyone was too excited snacking to really think about any more problems. I started to work on an ISN foldable looking at different types of mixture problems but gave that up pretty quickly. Instead I gave them two problems to try for homework and figured the next day we would pull things together.

DAY #2 on Mixtures

Before I went over homework and got any feedback from students I had them work in small groups at the board on mixture problems similar in nature to what we did in class with the cereal. After the group solved theirs they had to switch and analyze the solution and work of the problem next to theirs.

This worked really well and students said they were understanding mixture problems better.

Then we focused on mixture problems that contain percents (including interest problems).

After that we used our ISN to summarize our learning with the left page containing a foldable showing 4 examples that the students set up systems for...



(inside bits here:)





and the right page solving those systems.

All files used in this lesson (worksheets, board problems & ISN foldables) can be found HERE.

Start a Lesson With a Question

Whenever possible I try to start any lesson with a question to lead into what we are learning that day. This is a real shift from when I started teaching in the 80s - teaching training then focused on "tell tell tell" while students listened. Being the kind of learner that needs to understand I was never a big fan of that and figured out how to make questioning a bigger focus of my teaching. Marilyn Burns was a great mentor for this and I used her book

Collection of Math Lessons, A: Grades 6-8  a great deal when I taught middle school.

Anyhow back to this blog post...Here I am focusing on the 3rd week of the MTSoB blogging initiative

So to repeat myself - Whenever possible I try to start any lesson with a question.

Algebra One last week we were starting systems of equations. I knew that students had dabbled in this in 8th grade - they had solved systems of equations using graphing and perhaps had been exposed to some algebraic methods. But did they really know what a system was? I love really knowing what certain math concepts ARE and what they are good for (not always real life applications, just how they interconnect with other things I know about math), It's really the only way I can learn math. 

So I started class with this question..

What are some mathematical models that we can use to solve problems?

Students came up with great responses (almost like there were cue cards or something) - Graphs, Tables, Equations.  Yay! But this wasn't really the question to lead the lesson, this was the warm up. Next I asked them

 You have a bunch of quarters and dimes. 

The value of your coins is $3.00. 

How many of each coin do you have?

My students sit in small groups so I let them work on that. They gathered up some graph paper and made beautiful linear graphs of this situation with a lovely equation to accompany it. We talked about whether this graph was continuous or discrete and whether we would consider values in all four quadrants. Also with some prodding they came up with a table showing the possible solutions. Great work!

Then I told them that this question presented them with a situation that had two unknowns (number of quarters & number of dimes) and one condition that established a relationship between the unknowns. It resulted in a number of possible solutions. What if I gave them another condition to partner with this information?

 What if the number of dimes is two more than the number of quarters?

Then students worked on narrowing things down with this condition. They added on to the models they already created (another line on the graph, another equation for that condition). They found that this second condition resulted in a unique solution. Cool.

These initial questions allowed us to explore situations with two unknowns and two conditions that would have a unique solution. From here we could define systems of equations (after we did a few more examples) and keep in mind that is what drives a system of equations in two variables. Two unknowns and Two conditions. 

I put all our questions on a powerpoint. I also have a homework sheet to accompany this. We created a cool foldable for our notebook of an example (this was used the next class block as our warm up).  All supporting materials can be found at this link.

Here's the foldable I used with my students to illustrate these models. It's a cool "4 square fold" that my student teacher found last year (thanks Tess!). So it allows you to get a lot of information down in once place.

Here is the front all folded up into 4s (see how it folds up to use only half the page?):

Open it once and you get two places to have information:

Completely open it up and there are 4 places to have information:

Systems of Equations Project - Business Letters

This is not really a performance task (which is what we are trying to do one per unit in Algebra One), it's more of a project. But it's interdisciplinary, gets students to apply math in a real life situation, easy to differentiate and really great! The other Algebra One teachers in my school created a performance task for 2013-2014 and although it was okay they went back and used this again in 2014-2015.

Overall the "problem" is not so hard for students. The tricky part is communication and attention to detail. Two not so strong skills for high school students! But it's also fun for students because it has role playing and some creativity. And if you have students who are far better in English than in math, then this is their time to shine!

The Premise

Students are given a letter from a business that contains a problem that can be solved using a system of equations. They take on the role of the business person who received the letter. They solve the problem and write a letter back to the business clearly communicating the solution including an attachment with all the algebra details. Here is a sample letter that a student might receive:

Teacher Prep

I have a whole bunch of letters at varying levels of difficulty. Prior to handing out the student letters I determine who should get what letter (here is the differentiation) with my advanced class all receiving a more complicated problem (mixture). I personalize the letters, typing each student's name where it says "your name". And then I print them all out on the only color copy machine in the school. Since I want a really nice polished letter back from the students I think it's important that what I give them is full color. 

The Process

Students receive their letters in class. They are given general instructions on what their task is. With an average level class I then have them solve the problem at home and come in the next day with their solution. As students are doing class/group work on some other task I go around the room and check their solution. You could also collect them and hand them back the next day after checking.

I then go over the whole business letter concept. Stressing that they are playing a role and need to be professional. They are not writing this letter to a friend (so they should not have pleasantries such as "how are you doing?" in the letter). Spelling and grammar should be perfect, they want to make a good impression. 

A tricky bit for students is understanding the whole address placement and general format of greeting and closure. For example the letter above has the address header for"Gourmet Boutique" and they need to understand that that address now goes under the date in the place of "Coffee Distributors". Basically the two addresses switch places. I point out that they should have the same format with the address blocks, the date, the salutation ("dear....the person who wrote them the letter they got - but don't include their title here) and the closure ("sincerely....their name and title - they make up a title).

The creativity comes in now with a cool header with a fun typeface and a graphic at the end of the letter.

Another formatting comment - the letter is single spaced and in English classes they are used to doing essays in double space.

After we go over the general formatting I talk about content. They should see that a letter has a general format in its paragraphs.

There should be three paragraphs in their letter. 

The first paragraph has a professional greeting of some sort and restates the problem that the student received. They can lay on the schmooze pretty thick because they are wooing a client. (I am honored that you have chosen Coffee Distributors for your coffee needs. I am sorry that I was not clear on the correct mixture quantities for your premium South American blend. I am happy to provide the details regarding the exact quantities of Brazilian coffee and Colombian for your personalized blend.) I would not write out a correct first paragraph but would definitely "say" it for students so they can hear how it might sound. Students are told that this paragraph is similar to an introduction sentence in an English class essay. 

The second paragraph is the "meat" of their letter. Here they state the question (again!) and communicate the answer. They comment on how they solved it in a general way with a reference to the attachment. (This result was calculated by using an algebraic system of equations, see the attachment for full details and mathematical explanation). 

The third paragraph just wraps things up - there are professional pleasantries, possibly a direct reference to the problem and reassurances that if they have any other questions they just need to ask. 

(Thank you again for your interest in working with Coffee Distributors. I assure you that we have only the highest quality product and our blends are extremely well rated. If you have any additional questions please do not hesitate to contact me. Thank you again, and we look forward to working with you in supplying all your coffee needs.)

Sadly, in general, I find that students are not especially good writers. And this professional business letter format is so foreign to them. So it's important to stress that they are not writing an email or a text to a friend. They are trying to grow their business and develop trust and respect. So they must be very professional and have impeccable spelling and grammar. One error could lose a client for them. 

I also talk about the attachment. This shows all the "algebra details". It should not just be work, it needs to have some words explaining the set up and implementation of their method. I am looking for the variables being defined. the system of equations clearly shown with some rationale for where it came from. The method used in solving should be stated (the linear combination method was used to solve this system). When the value of a variable is found state clearly what that represents. A short explanation of finding the other variable (that value is substituted into either original equation and then the other unknown is solved for). There should also be a full sentence stating the solution with respect to the problem originally posed. (you will need 80 pounds of the Brazilian coffee and 120 pound of the Colombian coffee). 

With all that in mind, I would have all the students in my average classes do a rough draft (of both the letter and the attachment). In the past I've had students work together on some peer editing. This sounds all wonderful and cooperative and collaborative. But I hate to say this, I find peer editing to be a waste of time. Their peers are going to say that their letter sounds great. They often do not see grammatical errors and don't recognize if something sounds unprofessional. So I would usually collect the rough drafts, make general comments on them and hand them back. If a rough draft letter needs a lot of work I make the general comments but then urge the student to come for one-on-one help. 

I've been working with advanced students for the last few years and the peer editing can be productive with them. They should work with a checklist when editting. And have a short amount of time (10-15 minutes). And I offer to those classes if they want direct teacher input on their rough draft to come during my extra help times and I will work with them. 

Then students have the final task of creating their letter and their attachment.  The final version of the letter must be typed and have an interesting header and a graphic/picture. The attachment does not need to be typed. I give them a few more days to get this completed and turned in.

Resources

Here are all my materials for this project:

General Information and Rubric

Peer Editting Checklist

Sample editted letter

How to format attachment

Regular (easy- medium) difficulty problems

Advanced (mixture) difficulty problems

Acknowledgment

Where does this project come from? I don't think I have ever had a truly original idea. Everything comes from something I've read or seen elsewhere. This project resulted from my experience in a session at an NCTM or ATOMIC convention in the 1980s. It was a very similar project but not so detailed and all the letters are mine.

Introducing Systems of Equations - Exploring Scenarios

http://www.thedogtrainingsecret.com/blog/teaching-dog-tolerate-love-children/ 

I always like to start a new unit with some understanding of why this might be useful and what the underlying idea is for the concept. What really is a system of equations? What does it represent?

Well a system of equations is usually defined as: A system of equations is a collection of two or more equations with a same set of unknowns. (SOS math)

What does that mean? How do we start exploring that? Dan Meyer has an interesting blog post (dy/dan) on this from February 2014 

[Fake World] Culture Beats Curriculum

I recommend reading it and considering how it fits into how you introduce systems. I like his approach and I also like to look at some scenarios that later we will describe algebraically with systems of equations. But right now we just consider them and try solving them without algebra (draw a picture, act it out, educated guess & check, make a table). 

Consider the classic "kids and dogs" problem: 

 In the park one day, I saw a bunch of kids and dogs.  Counting heads, I got twenty-two.  Counting legs, I got sixty-eight.  How many kids were in the park?  

(I bet you are trying to write a system to solve this right now aren't you? stop!!)

You can actually solve this by drawing a picture or more dramatically by acting it out. I saw this years ago in an issue of Mathematics Teacher (NCTM) but do not have the reference to it now. It's so so cool and can be done with younger kids. If you are going to act it out create a problem with less heads because that number of students will have to stand in front of the class.

Drawing a Picture

To draw a picture you draw twenty-two heads. And since you know that both kids and dogs have at least two legs per creature, draw in two legs for each. (I have students do this on the white board - one draws all the heads, then someone fills in the two pairs of legs each). 

How many legs have we drawn so far? 44 legs

How many legs did you originally count? 68

So what does that mean? (there are 24 more legs that we are not seeing yet in our drawing)

Someone will come up with that we have to draw those other legs on our creatures - doing two per creature until we use up all those 24 extra legs. (12 pairs)

Visually we see 12 creature with 4 legs (dogs) and 10 creatures with 2 legs (kids).

Students love this by the way - it is fun.

Acting It Out
You have the total number of heads acted out by that many students. So with this problem you have 22 students stand in front of the class (really a good idea to scale this down to a smaller number of heads for acting out). Now tell them that some of them are kids and some of them are dogs. But we'll start by saying that all of them will raise their arms (or front legs) in the air. Then we ask rest of the class - how many legs do you see on the ground? (44) But how many legs did we originally count? (68) So how many of those front limbs in the air are actually dog legs? (24) and how many pairs of front legs does a dog have? (2) so 24 legs is how many dogs? (12 dogs!) and from there you can calculate the number of kids (10).

This is a lot of fun too, just hard to do individually or at home. But it's a good way to introduce this and then show them the draw a picture method.

Now students get stuck on wanting to draw a picture but I want them to experience other methods too. You'll find in the next two methods students will really really want to start writing algebraic expressions and equations. I avoid doing much of that, however some little expressions for the conditions can be useful for some of the trickier conditions. But don't do any equations yet!

Educated Guess & Check
Guess and check seems so "random". But with educated guess and check students consider the result of a guess and then adjust in their next guess if the first guess is wrong. Making a table to organie their guesses is a good idea too. Have students consider this problem: 
The sum of Freida & Gene’s ages is 21. Freida’s age is 3 less than twice Gene’s age. How old are both Freida and Gene?
 After some discussion show students how to set up an organized guess & check table with four columns – Freida’s age, Gene’s age, check condition #1, check condition #2. Describe in words, do not get into writing equations (that’s a future lesson). Use the conditions to guess & check, continue until solution is found.

Make a Table

Now you might think - hey they made a table in the above problem. But that was just to organize their guesses. This table is to keep track of a pattern and look for a common result for two items. Consider this plant growth problem:

Plant A and Plant B are on different watering schedules. This affects their rate of growth. You will compare the growth of the two plants to determine when their heights will be the same.

Plant A starts at 5 inches tall and grows 2 inches per week.

Plant B starts at 2 inches tall and grows 3 inches per week.

When are the two plants the same height?

We can start by deciding what the independent and dependent variables are (IV # of weeks, DV total height of plant in inches). And then we can make a set of tables to keep track of the plant growth for each over a number of weeks.

After filling in the tables students can see a common set of values.

Be sure students can articulate what that ordered pair (3, 11) represents!

From these examples I then give them a practice worksheet of problems to work on in their small groups and to complete for homework.

Unit 6 Systems of Equations

The next few blog posts will not be as detailed because at this point in the school year I had a student teacher. So I am going to post her ISN work with some commentary. But after I teach these units (February - May 2016) I will update with my own materials and more detail. I don't think I have a student teacher this coming school year. (And my student teacher this past year was very good! What a trooper working with interactive student notebooks and creating her own materials! She did a great job and she has beautiful handwriting). 

Exploring Scenarios

Systems of Equations - we take a little different approach with starting this unit. We don't jump right into "here is a system and here is how we solve it". When planning I like to think about "what is a system good for? why does it exist as an algebraic process?". So we start off with "Exploring Scenarios" to get an idea of what it means to have two unknowns and two conditions. I'll do a separate blog post on that because I think it's a really valuable way to begin, for students to develop some overall understanding. We didn't have anything in our ISN on this but I will this coming school year.

ISN Foldables

One comment on coordinating your ISN with a student teacher. Let them put their inserts in your teacher notebook. I started off by giving my student teacher her own notebook to put her foldables in and I tried to keep up with her. But I wasn't always in the classroom when she taught so there were gaps. When she took over my lower level Algebra One she did use my notebook and it worked out so much better.

This was a cute idea for demonstrating substitution but I'm not sure if it's that useful in the ISN. I also came across the "blob" method for an insert on substitution method and used that with my lower level classes.

Cool summary foldable but the space got tight for showing work clearly. I like to have a lot of space.

Love the examples (go on right hand reflect pages)

My student teacher did a great introduction activity for mixture problems with Kix and CocoPuffs cereal. I'll have to post that when I use it. She had this foldable to show the table method for mixture problems. I've never been a fan of this because you have to remember how to set up the table and that can get confusing. I won't use this next year but someone might find it useful so I'm including it.

Really nice summary bit here - lots of time consuming cutting though, I'd suggest giving it to students the day before and they have to cut for homework. This would be left hand learning and then I'd include some algebraic examples for them to solve on the "right hand reflect" page.

Some good stuff here for systems of linear inequalities. I just was not in class for most of this and some things are blank and we've got an extra blank page. So this needs some reworking.

 Oh, in general I'm not a fan of all that text with "fill in the blanks". There is not a lot of space for filling in and I think when we give the students so many words they mechanically fill in the blanks without thinking about what it means.

And here is the folder with all my student teacher's foldables. 

In the Future....

Next year my tentative foldable plan for this unit is ...(L is for left side learning, R is for right side reflection).

L Exploring Scenarios  R example for warm up

L Systems of Equations defined, what a solution is, checking solutions R examples

L Solve by Graphing  R application example

L solve by substitution R application example

L solve by linear combination  R application example

L mixture problems  R example

L Types of Solutions  R algebraic examples

L systems of Linear inequalities  R examples

L systems of linear inequalities applications  R examples 

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