Thinking Ahead – Planning for Next Year’s Classroom Culture

I was in Austin all last week training for UT Dana Center (@UTdanacenter) International Fellows
(#UTDCIFF) and Department of Education Activities (@DoDEA) College and Career Ready Initiative teacher workshops happening this summer. A major focus for the week was on classroom culture and how important this is to mathematical learning and student discourse. Everyone at this training was either a current math teacher, a supervisor, mentor, coach, professional development provider, etc., so naturally, as part of the conversation, the following questions/concerns arose:

  1. What is classroom culture and why does it matter?
  2. How do you get students to talk to each other and engage in productive learning?
  3. How do you respond to teachers who say things like, “well, this would never work with my students” or “I can’t get my students to talk about math when we are in groups”…

You get the picture, and I am sure you have either thought these things or heard these from teachers you work with.

The short answer – it takes planning, training, and consistency. If a teacher thinks that they can just put students into groups, give them a problem, and they are going to immediately start talking and working together, they are very quickly in for a big surprise. Especially that first time, and especially if you have never done these types of collaborative learning with your students. Which brings us back to classroom culture.  What is it and why does it matter?

There are many definitions out there of classroom culture. I will give you my perspective. Classroom culture is a classroom environment where students feel safe making mistakes, they are comfortable sharing their thinking process with other students and with the teacher, and all ideas are entertained and acknowledged. Everyone’s voice is heard, everyone gets a chance to participate, and there is respectful conversations and debate about the work being done.  This matters because then students are given permission to persevere in problem solving situations where they may not know the answer, or may have a different approach then someone else or may have a question about something another student or the teacher has shared. It ties into those mathematical practices (#1 & #3, just to name a couple):

  • Make sense of problems and persevere in solving them
  • Construct viable arguments and critique the reasoning of others

But, this type of engagement, discourse and collaboration with and among students doesn’t just happen. Here are what I consider the three basic elements:

1. Planning

Planning entails thinking about the structures you want to use with students (so pairs, small groups, whole class) and the types of discussions and work you want to students to engage in. There is more to it than this, but some things to think about are

  • What task are students working on and what is the goal (a worksheet of 40 problems is NOT going to promote student discussion). Provide a rich task that fosters critical thinking, questioning, problem-solving.
  • How do you want students to engage? Are they talking in pairs first and then sharing with the small group? Does each pair/group need to show some product (i.e. their work, their thinking, the end result).
  • How will you bring the whole class together at the end? Will each group share out? Will you hang work and have a ‘gallery walk’ and come together to share?
  • How will you know that students have learned or reached the goal? What should students be able to do?

You need to think of these things ahead of time, most importantly because without an engaging, rich, though provoking problem, the conversations students have won’t be productive (and can lead to all the issues mentioned previously).

2. Training

How do you get students to talk about math (or any subject?) How do you get students to work in pairs or small groups and stay focused on a task? How do you get students to listen to each other and to provide critiques without insult (i.e. no ‘that’s stupid’ or “you’re an idiot”). It takes training.  I mean that literally. You have to show and model what it is you expect of them and practice, practice, practice.  Again, there is more to this than what I am listing, but here are some ideas:

  • Start those first few days/weeks of school with non-content related activities that are non-threatening, fun, and where everyone feels comfortable sharing (so talk about ‘the best horror movie’ or argue for/against a ‘beach is the best place to vacation’)
  • Set up group norms – i.e. if someone is talking, everyone else is listening; everyone makes mistakes, and that’s okay, you can support them and provide alternatives, but never insult them; everyone must contribute one idea; everyone’s idea should be heard; you can disagree but must provide a reason why; etc.
  • Show them how to get into small groups (so physically moving desks back and forth – it’s fun to do this a timed game); show them and practice how to talk with elbow partners, or face-partners, or the people next to them.  Practice sharing talk-time (a time works here).
  • Show them and practice group ‘roles’ – i.e. timer, recorder, controller, group spokesperson, etc. Switch roles up.
  • Practice different ways of calling on students (so they know they are all responsible at any time) – so person in the group/pair with the shortest hair, or the darkest colored shirt, or blue eyes….really anything works.

There are obviously lots more ways to set up these collaborative processes, but the idea behind training is that there are some expectations for talking, sharing, and working together, and if we practice these and adhere to these, then our time learning is going to be more positive and productive. Practice, practice, practice.  Which leads to consistency.

3. Consistency

I know teachers here this all the time – if you set boundaries for your classroom, you need to be consistent or students will not follow them (heck, this is true for parents as well!). Again – those first few days and weeks of school are where you set these boundaries up and start practicing with students and modeling both behaviors and actions. More importantly, follow through on any consequences. For classroom culture, this means if you have an expectation that students listen when others are talking, whether that be student or teacher, then be consistent.  If you are talking and they are not listening, stop – call it out – and then talk again. Same thing for students talking. Acknowledge when something is not adhering to expectations and call it out and then refer back to your expectations. Students very quickly learn what is expected, and if they realize that you are going to consistently hold them to these expectations, such as listening, allowing for mistakes, everyone’s ideas matter, etc., then they are going to feel comfortable speaking up and sharing their questions and their solutions/ideas. It becomes a classroom where learning is up front and center and ‘we are in this together’ becomes the norm.

CHALLENGE

I plan to do some more specific posts about classroom culture and provide some resources connected to planning and training. For now, I brought this idea of classroom culture up at the end of a school year because as teachers, you are about to embark on a summer of rest and relaxation. For most teachers I know, it is also a time where we do some personal learning and planning for next year. I would like to challenge all of you to really think about how you want your classroom culture to be next year. You need to start on day one of school creating this classroom culture, so spend some time this summer planning for that. What structures do you feel you could incorporate (i.e. pair work, small groups, etc.) and learn about those structures. What are rich tasks and go find some that would work for the content you teach. What do you want students doing when they are learning together? Go find some tips and ideas for how to create those collaborative discussions and problem-solving environments.

Only YOU can change the classroom culture in your own classroom – so think about what you want that to look like and sound like, and spend some of your summer learning and finding ways to foster this culture in your classroom when school starts in September (or August).

Fractions with a Calculator – Looking for Patterns

calculatorI have been working with teachers and using manipulatives, both physical and virtual, to help students think about fractions and develop conceptual understanding about fractional operations, versus just memorizing rules or tricks, as we so often do with students. There are fraction circles or fraction strips that work well as physical manipulatives, and there are several virtual manipulatives as well (i.e. DynamicNumber.org for any Sketchpad users out there, and the National Library of Virtual Manipulatives to give just a couple resources).

Manipulatives are a valuable resource in math class as they allow students to visually represent numbers, manipulate them, get hands-on with the math, and make some connections before moving into just the numerical representation alone. When working with fraction manipulatives, from my own experiences and those I have had with students, the manipulatives can constrain the number of possible examples we can provide students (either because a teacher might not physically have enough for all students or the manipulatives themselves only go up to certain values). As an example, most physical fraction circle manipulatives allow you to work with a limited range of fractional values – halves, thirds, fourths, fifths, sixths, eighths, tenths and twelfths. Virtual manipulatives offer more options, which is nice because students should see more than just common fractional pieces or ‘nice’ fractions – sevenths, or elevenths or twenty-fifths as an example. Obviously, the idea of manipulatives is to provide that hands-on experience, visually see what’s happening, and then create conjectures.

Another tool that is often overlooked, particularly at the elementary level, is the calculator. Obviously, when dealing with fractions, you want a calculator that uses natural display, showing fractions in their numerator over denominator form so students recognize the fractional number. I realize many of you might be thinking that the calculator is a bad choice because it provides the answers….but that in fact is an advantage here when trying to help students recognize patterns and develop their own understanding of fractional operations.  We want students to recognize what seems to be happening – test it out on many examples before they come to a conclusion.  A calculator (like the fx-55Plus shown above) is a great way to do this.  If you don’t have manipulatives, you can actually use a calculator like the fx-55Plus to help students understand fractional operations.

Let’s take fraction addition. Obviously, we are going to start with adding fractions with like denominators.  You can put several different problems into the calculator and students can observe both the added fractions and the answers. Students can talk and share what they notice about the multitude of fractions they are adding (all with like denominators). They can make up their own addition problems and see if the pattern or things they notice hold true. Fraction and answers showing up quickly help them discern patterns because they can quickly see many examples, and use ‘funky’ fractions, not just the typical ones we tend to always rely on (i.e. halves, thirds, etc.). It’s even okay that the numerator might occasionally end up larger than the denominator – the pattern still holds true (i.e. the denominator remains the same, the numerators are added together).

With a calculator, you can use messy fractions with not your typical denominators and even numerators larger than the denominator. For addition, our focus is on what patterns do the students see with the numerator and denominator and do those patterns hold true no matter what fractions we are adding? We can get into simplifying the answers at some point, but at first, the focus is on the addition.

Once students have the idea that with a like denominator, you add the numerators, you can then switch it up. Let’s add fractions with unlike denominators.  You can encourage smaller numbers in the denominator and numerator to start, and then once students think they have the pattern, they can ‘test it out’ with some larger digits in the numerator and denominator. The thing here is the denominators are different and so how does the end result differ (if does) from when the denominators are the same? What might be happening? Test it out.

The beauty of the calculator (again, one like the fx-55plus that quickly and easily shows fractions in their natural display), is that students can create many examples to look for patterns and then quickly test their conjectures on different problems to see if it works. You are encouraging critical thinking, problem solving, and communication using a simple tool that provides much more diverse fraction examples than you can provide with manipulatives alone.

My point – when helping students develop number sense, especially with fractions, don’t rule the calculator out as a tool. You should use multiple tools with students to provide them with different ways to develop their own conceptual understanding. Calculators can be a tool, even at the elementary level.

 

 

 

Math Magic or Calculators?

I was perusing my news feed trying to find something of interest to write about, and came across an article entitled “The Common High School Tool That is Banned in College” i.e. the calculator. It’s an interesting article, worth a read,  basically comparing the high school perspective on the use of calculators to the college perspective or non-use of calculators. There is no right or wrong answer – I think it depends on the math content, what you want students to do (i.e. basic algorithms to solve problems or using mathematics to solve deeper problems).  Depending on your goals, the use of calculators and technology differs. As with any technology, calculators are a resource that needs to be used appropriately, and we need to be teaching that.  Common Core Mathematical Practice #5 – Using Appropriate Tools Strategically is all about this. Calculators have their place and are important to help explore and expand mathematical understanding, but we have to help students understand when their use is necessary and not a ‘crutch’, as stated in the article.

This was on my mind obviously, when I then ran across a tweet post by Go!Math Videos @gomathvideos that shared a TedX talk by Arthur Benjamin entitled “Faster than a Calculator”, which naturally sparked my interest and seemed related to the question of should we be using calculators. In the video, Arthur Benjamin has members of the audience use calculators while he does calculations in his head. He then goes on to wow everyone with his math ‘tricks’ (what he calls mathemagics). He ends by doing a 5-digit square calculation by thinking out loud as he ‘solves’ a problem. It’s fascinating – he changes numbers to words to help him solve – he is definitely using his own ‘algorithm’. The video does not answer the question should we be using calculators – but it definitely shows that calculators are just one way to get a solution and it may not always be the fastest. Anyway – just some fun for this last post of 2016. Enjoy!

Wishing everyone a Happy and Safe New Years!

Using Connections to Build Understanding

I am teaching a Geometry & Spatial Reasoning course for Drexel this semester for their math masters program for teachers. Absolutely love it because I am learning so much from my students/peers, but because it really is bringing home the importance of prior knowledge to help build connections and real-world connections in helping students learn versus memorize, and construct and reconstruct based on their ability to make connections.

My students, who are a mix of very new math teachers, experienced teachers, and even some career-switchers still in the early stages of teaching, are having this great discussions on the importance of using prior knowledge to help student make their own connections. Some have been doing this all along, but others, as they themselves struggle with some of the geometric concepts we are ‘learning’ (relearning in some cases), are coming to understand the value in helping students use what they know to build on and connect to new information. Makes it easier to recall, and builds a confidence in students that when faced with an unknown situation/problem, they have the skills and confidence to look at it, break it down or add in things to make the unknown familiar and then look for and make use of structure (see what I did there….Common Core Math Practice #7!) to help reach a solution or develop a new conjecture/conclusion.

As an example, we’ve been doing a lot of work with inscribed angles in circles and how do you help students use prior knowledge to build the idea that an inscribed angle is half the measure of it’s intercepted arc if you don’t want students just memorizing formulas? Basically, the conversations revolve around constantly using prior knowledge to make connections, which might mean you need to add in an auxiliary line to a given shape to ‘see’ something familiar (i.e. a linear pair or a triangle, as examples). A strategy that really helps students look for and make use of the structures they are familiar with to help them make sense of a problem.  Here’s an example of just one way to explore inscribed angles, using previously knowledge about triangles:

inscribed-angle

  • In Fig 1, we have an inscribed angle and its intercepted arc a. How could you show that angle 1 (the inscribed angle) is half the measure of it’s intercepted arc? Here’s where students need to make sense of this structure – what prior knowledge can they use to help them?
  • In Fig 2, they add in a radius (auxiliary line), because they know all radii in this circle (any circle are equal – doesn’t change the original inscribed angle….but now – we have a triangle and a central angle (angle 2).  What do they already know? Well, they know the central angle 2 is the same as the measure of the intercepted arc, which is the same intercepted arc as angle 1 (inscribed angle).
  • In Fig 3, students are looking at the triangle created and using prior knowledge – we can mark the two radii equal, making this triangle an isosceles triangle, which they already know from prior knowledge has two base angles that measure the same (angle 1 & 3). Angle 2 is an exterior angle to the triangle, and angles 1 & 3 are remote interior, which they know from prior knowledge sum to the measure of angle 2. Since angle 1 & 2 are equal (isosceles triangle), that makes them each half of angle 2 (Sum divided by 2). Angle 2 is equal in measure to the intercepted arc, so angles 1 & 2 are each half of that, so the inscribed angle 1 is half the intercepted arc.
  • Fig 4 shows that the relationship holds true even if you change the size of the inscribed angle.

This is of course just one example for an inscribed angle, but they can then use this to show that inscribed angles that are not going through the center of the circle have the same relationships – ie add in auxiliary lines, use linear pairs, or triangles or other known things to help make sense and show new things. Prior knowledge, connections – they really matter.

As teachers, it is our duty to make sure we are modeling and helping students use what they know to build these connections and see the relationships. It takes deliberateness on our part, it requires modeling, it requires setting expectations for students till it becomes a habit (habits of mind) to look for and make sense by pulling in previous knowledge.

Another thing we need to do is make connections to real-world. My students are sometimes struggling with this idea of relating prior knowledge and new ideas to real-world applications, but if you get in the habit, its not so hard to do. Since I am focused on circles and the lines that intersect them now with my class, I pulled up a ready-to-use lesson from Casio’s lesson library that is a great example of a real-world connection to circle concepts that would force the use of previous knowledge.  The lesson is briefly described below:

The Perfect Glass Dome: (here’s a link to the complete, downloadable activity)

  • – Use coordinate geometry to represent and examine the properties of geometric shapes.
  • – Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

This activity uses the Prizm Graphing calculator and picture capability to help build understanding.

dispcap1 The kinds of questions and connections to prior knowledge that can be asked of students just by looking at the image are pretty endless. What relationships do you see (i.e. lots of diameters, or straight angles, lots of central angles, all the angles are 360, are their auxiliary lines we could add to find the areas or relationships or angle measures, etc.).

If you look around, you can probably find a real-world example of most math concepts your are working on with your students. Show them pictures, show them real objects they can get their hands on. Start asking questions. Ask them what they recognize or think they already know. Ask them if they could add something or take away something to see a familiar object/concept. How does that help them? What relationships and connections help them get to something new or interesting?

My Drexel course and student are reemphasizing for me (and them) the importance of prior knowledge to help build connections on a continuous basis, all the time, every day. It helps students think mathematically and consistently use vocabulary and math concepts to deepen and create new understanding and relationships. It also promotes logical reasoning and problem solving – win-win!

 

 

Math and The Electoral College

With the election looming, and yet another Presidential Debate this evening (anyone else dreading it?) and more polls than you can shake a stick at, it seems appropriate to think about the math behind the Electoral College. I admit to really not understanding this whole system – and I know I am not alone. With the rampant conspiracy theories about the November 8 election, and a ‘rigged election’ and cries to eliminate the Electoral College and go to a popular vote only, it had me diving into “what does the Electoral College mean, why do we vote this way, and is it fair?” I think this is a GREAT conversation and critical thinking activity to have with students, especially in classes like statistics where you can actually study and do ‘mock votes’ and see what the outcomes are with or without the Electoral College.

A quick summary of what the Electoral College is – and please note, I still am a little iffy about whether I truly get it. In 1787 the delegates of the Constitutional Congress made the decision to do this indirect way of voting for the President of the USA. It was a compromise between those who wanted a) individual citizens to vote for President (1 person, 1 vote, majority wins); b) letting State legislators choose the President; or c) letting congress choose the next President. The idea was to create a method where the best candidate was chosen. Individuals in a state vote for President – the winner in that state gets all the states electoral votes (though some split the electoral votes now), and the electors (who are elected by voters),  put in the final vote for President. The person who gets the majority of Electoral Votes (270 or more) wins. Still confused? How is this fair?  Bear with me….I am hoping I can figure that out myself!

If you look at the image above, which outlines the number of Electoral votes per state, you can see a huge difference – some states have an enormous number of Electoral Votes, and some very few. As you can see – size might matter (CA, TX, FL), but not always – VA is relatively small and yet has 13 Electoral Votes compared to say Montana, a larger state with only 3 Electoral Votes. So – how is the number of Electoral Votes determined? Hawaii gets 4 vs the very large state of Alaska only 3. So – it must have something to do with population numbers, which in fact is the case. The number of Electoral Votes is the number of state representatives in Congress (both Senate and House of Representatives), which are based on the population of the state. Every state will have at least 3 Electoral Votes (2 Senators, 1 Congressman). Obviously you now see why winning states like CA, TX, FL, PA, and NY are so crucial because of their large populations and large number of Electoral Votes.

I have been reading a lot and searching for good websites that might be helpful for teachers wanting to figure this out with their students. There are several sites that talk about the Electoral College and what it is – I didn’t find these too helpful from a teaching perspective, but they may be of interest to some of you from a historical, “why do we do this” perspective.

  1. Nice interactive map – http://www.270towin.com/
  2. Article about the ‘fair or unfair’ aspect of Electoral College and the funding – not sure it answers the questions but makes you think: https://blogs.scientificamerican.com/guest-blog/the-funky-math-of-the-electoral-college/
  3. This is a nice site with lots of historical perspective and answers to questions, like does my vote count? http://www.learnnc.org/lp/media/lessons/davidwalbert7232004-02/electoralcollege.html
  4. This article makes a case for the Electoral College system being fair: http://www.politico.com/story/2012/12/keep-electoral-college-for-fair-presidential-votes-084651

Here are some sites I found that would be helpful for doing some simulations and having interesting conversations with students. Many of these are interactive, with the ability to create election results (or simulated) to get a better understanding of how the Electoral College system works, and hopefully make a determination about whether it is fair or not.

  1. This was my favorite – be sure to check out the “Play Presidential Politics” link, as it has a simulation vote where you can create your own populations for states.  Would be great for students. http://www.sciencebuzz.org/topics/electoral-college-math
  2. Lots of information (some of which I used) in kid-friendly language: http://www.congressforkids.net/Elections_electoralcollege.htm
  3. Nice lesson here – some interaction/lesson plan info as well: http://www.scholastic.com/teachers/article/math-majority-rules
  4. Yummy Math – nice lesson here (using the picture above!) http://www.yummymath.com/2016/electoral-college-vs-the-popular-vote/
  5. From NCTM – a lesson on the “fairness” of the elections – Love Illuminations! https://illuminations.nctm.org/lesson.aspx?id=2825

So – is it fair or not fair? Does your vote count? I am not sure I can give you a definitive answer. It probably depends who you are, who you want to win, and where you live. But, in my readings looking at several charts that compared the ‘weights’ of individual votes toward the outcome (i.e. Does your vote count?), I think my personal opinion is yes, your vote does count, and yes, it is fair.

Notice Alaska, with a smaller population, has a much more weighted vote compared to CA. This may not seem fair – but, Alaska, a huge state with many diverse needs and interests but with a small population, deserves an equal representation in the government, which may not happen in a one-vote majority rule election, if we look at populations sizes of CA or NY, with their enormous populations. This is why the Electoral College was created in the first place – so every state gets a fair share of representation for their interests in the outcome, no matter their populations size, and those ‘states’ with larger populations don’t end up  deciding everything. NY’s interests are vastly different than Alaska’s, after all. A popular vote would be unfair because those larger states, who lean more one way or another, would control the outcome, leaving those states with fewer people, left out of the equation, and their interests not accounted for or lost in the process. My personal conclusion – I am actually for the Electoral College, after all my reading and my still foggy, but much better, understanding of the system.

One final note – look at the overall United States (in chart above) – the total population, the total Electoral votes, and the weight of each individual vote.  It’s 1.

So YES – your vote counts – get out and VOTE!  (And make it an educated vote, based on candidates proposed policies and plans – not based on emotion).

 

Conics – Casio Prizm vs. TI-84+CE

I am currently teaching a course at Drexel University and we are starting a unit on circles. I loved using Sketchpad when teaching because it allowed for dynamic manipulation of objects (shapes, functions) so that students could visually see the impact of variables to the shape, size, position of the object. Unfortunately, my students (math teachers in a Masters Math Teaching Program) do not have access to Sketchpad, though one does use Geogebra, and as this is a course focused on teaching, they need to use what they have access to in their own classrooms with their students. For many of them this does not involve any technology at all, which is sad, but for some, they do have access to graphing calculators.

Naturally, this got me exploring what the graphing calculators could do, and surprise, surprise, I noticed quite a difference between the Casio Prizm and the TI-84+CE graphing calculators, which are the ones my class seems to have. I was investigating conics, and in particular circles, and what options the graphing calculators gave me, especially when thinking about dynamically modifying the variables to see how each impacts the graph of the circle. Here’s is a quick summary of what I found:

  1. Both TI-84 & Casio Prizm can graph conics (circles, ellipses, hyperbola, and parabola, though how to access these conic graphs is different on both.
    • It is more apparent/easy to find on the Casio (there is a Conic Graph menu).
    • TI requires knowing that there is a Conic app in the app menu, which is a button on the calculator. It is not seen from the main screen, and if you don’t know it exists, you won’t know it’s available.
  2. Both provide more than one equation form for each conic.
  3. Both show the graph of the conic, but how is very different.
    • Casio shows the graph on the coordinate grid, where you can see the whole grid, see values on each axis, and identify quadrant and key points on the graph
    • TI shows the graph in the entire window with a weird yellow frame around it. It is difficult to determine where on the coordinate grid the graph appears – there are axis marks, but no values, not origin, making it difficult for students to understand where on the coordinate grid the graph is. Very difficult to identify quadrant and key points on the graph.
  4. Both allow you to enter different values for each coefficient variable,
    • Casio has a modify feature that allows you to see the equation, graph, and coefficient variables on one screen. You can then modify one coefficient at a time and see it dynamically change on the graph, allowing students to visually see how each impacts the graph and see the conic change shape, location, and/or size.
    • TI84+CE only shows the graph or the equation/coefficients – never together.  You have to go back and forth between them when changing values. The TI does not clearly show where on the grid the graph is, does not show a size change (all conics look the same size, but the grid scale is changing). It’s actually very confusing and would be difficult to help student visually see the impact of changing coefficient variables on the size, location and shape.

Below is a video I made showing how to graph a circle and modify the coefficients on both calculators so you can see the differences I am talking about.

Hopefully you will come to the same conclusion I did – Casio Prizm is far superior when graphing conics than the TI84+CE.

Solving Equations with A Scientific Calculator

Solving  equations is a skill that students are expected to be able to do in pre-algebra and beyond. If we look at the Common Core State Standards, these skills actually come into play starting as early as 6th grade, with students expected to solve one-step equations and progressing to systems of equations by 8th grade. An important aspect of solving equations is connecting a real-world context to these and understanding what the ‘solution (s)’ mean in terms of that context.

The use of calculators or technology to help students solve equations is a controversial one at best, and as a math teacher, I do believe that students need to know the processes to solving equations without the use of technology first. But – when we get down to real-world application and problem-solving, the technology becomes a tool that allows students to go beyond just “getting the solution” and to making meaning out of those solutions, and using their solutions to make decisions – which is the ultimate purpose of finding those solutions, right? In these cases, I firmly believe that the use of technology, (more often than not a calculator), is a necessary tool so that students deepen their understanding and are not bogged down in the process of the calculation. Part of the practices – “use appropriate tools strategically”. 

As an example, let’s consider a simple real-world context that involves solving a system of equations, something required by the time students reach 8th grade (see Common Core Standards). Let’s say a scientist is mixing a saline solution and has one solutions that is 10% saline and the other 25%. He needs to make a 85 ml bottle that is 15% saline. How much of each of the two solutions should he mix to create the 85 ml bottle of 15% saline? This requires our two equations, with x = the amount of 10% solution and y= the amount of 25% solution.

  • x + y = 90 ml
  • .1x + .25y = 12.75 (15% of the 85 mL saline)

Perhaps students are actually in science class doing a lab and creating this new solution. While it would be reasonable to do this by hand using substitution, if this is part of an experiment, then using a calculator to get the answer quickly and therefore get on with the experiment might be a more logical step, especially when time is of the essence in classes. I am going to demonstrate on the fx-991Ex how to solve this problem.  I am using a scientific calculator because in middle school, students are more than likely going to have access to these versus a graphing calculator. This video shows how you can quickly solve the simultaneous equations, and also, with the QR code capabilities, also see a graphical representation of the solution.

If a scientific calculator is all your students have access to, remember that they can do a lot more than you might think.  I will explore more features of the ClassWiz in later posts as we continue to explore mathematics and using technology to support learning.