# Systems of Equations (part 2)

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None of what I’m sharing is new…but its me reflecting on the week…so I can reference back and make adjustments in building a better unit  of learning experiences for next time around.

To address some student questions, here are examples used in class to follow-up.

On white boards:

• y=x+1
• Pick a value for x.  Find y.  (Ex. (3, 4)
• Now, let’s double our equation.  What?!?  Yep, double it. 2(y=x+1)
• Okay. 2y=2x+2
• Use your same value for x from above and find y. (3, 4)
• What do you notice?
• Let’s multiply our first equation by 5. 5(y=x+1)
• 5y=5x+5.
• Use your same x value from above, find y.  (3,4).
• Did that happen for everyone?  Turn and talk…
• What if we took half of our equation?  .5y=.5x+.5
• for the same value of x, it works again (3,4)
• Then we go to Desmos to see the graph of our equation along with ALL of our versions of the equation.

Its a big idea that I don’t tell them.  They have observed why we can use this “magical” math thing is actually just a different version of the same equation…as one student put it “its the same equation, in disguise!”

But I also feel there is value in diverting from my original plan here to address the student’s struggle to figure out WHY? we do this in elimination, otherwise, it is literally, a “magical math thing” that just happens.

I need to do a better job of this – equivalent expressions / equations – earlier in the year, when we are looking at equations of lines…but also, how can I connect it with scale factors and similarity?  It all comes back to proportionality, but what strategies and tasks can I use to help my students make the connection and really develop a deep understanding?

Next on the list, we graphed our systems we’ve solved in Desmos.  Noticed and wonder…comparing our graphs to the work we’ve done algebraically.  Ohhhhh.  We found the intersection point!  Again, not me telling them, but they see it on their own.  I love that Desmos allows us to graph an equation in standard form.

Finally, I asked students to solve these equations and discuss their results in their groups:

•   4x -6y   = 12                 and             7x – 4y = -11
• -2x + 3y = -6                                     14x – 8y = 16

When does 0 = 0? ALWAYS!                   When does 0 = -6?  NEVER!

Again, we looked at the graphs in Desmos…

Several quickly stated the first set was only a multiple of the first equation, so it would be the SAME line!  (yes. secret happy dance!)

And the parallel lines never intersect…the equations were multiples on one side, but NOT on the other, a student noticed.  Its a translation, just moving one line up or down – another student stated.  So, how can I use their intuitive thoughts to build a better lesson?

I found Racing Dots on teacher.desmos.com  –  based on an activity, Great Collide by Jon Orr – to bridge between special situations, graphing solutions, substitution and algebraic solutions – will share more on this task later!

# Card Tossing & Spiraling Curriculum #tmc14

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Awesome session Mary and Alex!  Thank you. Thank you. Thank you.

The session focused on their experiences with Grade 10 Applied students ( Canada).  The entire course is activity based which allows students to not miss out on big ideas as they would in a traditional unit by unit aligned course.
Students have repeated opportunities to experience big ideas. The tasks are rich  with multiple entry points and different approaches to solving.  It’s a collaborative environment with accountable talk.  There are fewer disciplinary issues with increased engagement.

Each 6 weeks a mini – exam over entire course up to that point takes place.  Questions are in context and tied to activities they have completed.

We began with beads and pennies on our desks and this task… Cole has 2 smarties and 3 juju bed for \$.18 while Noah has 4 smarties and 2 juju be for \$.20.  They shared that systems are presented this way – no algebraic forms- for the first several weeks of class.  I, personally, can see how effective this strategy could be.

The next activity shared was Sum of Squares (he doesn’t refer to it as Pythagoras Theorem, yet – or did he say ever?)

Students are asked to cut all squares from side length 1 to side length 26.  Each square is labeled with side length, perimeter, area.  Then they build with them.

Basically students explore and eventually they focus on triangles formed with question, are there 3 you cannot make a triangle with?   Which combinations form different types of triangles. Begin looking at 3-4-5 triangle families, similar triangles (Kate suggested dilations here), discuss opposite side and adjacent sides, then give them a TRIG table and allow them to figure it out.

Compare side lengths with perimeter, or side length with areas.  The possibilities of math concepts are endless.
We ended the day with Card Tossing by collecting data, then using rates to make some predictions.

Video of Alex & Nathan picture below is only a screenshot.

@AlexOverwijk downed by @nathankraft 75 to 72

Each person in the room completed several trials of tossing our cards for 20 seconds.  We found our average rate of success, then determined who we thought might beat King Card Tosser.

Alex asked us to predict how long they needed to toss if he gave Nathan a 35 (?) card advantage so it would be super close and exciting.  Our prediction 38 seconds about 75 cards. Many ways of making the predictions were possible. Not to shabby, huh?

This task was fun, exciting, engaging.  Definitely on the to-do list.

This approach is definitely something I would like to consider, if administration will allow it!

# Gallery Walk #ppschat Challenge

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A common theme in many chapters of Powerful Problem Solving is Gallery Walks. Several techniques are offered throughout the book, but the common goal is to allow students to view their classmates’ approaches to problems.

One of my faults with online book chats is lack of follow-through. I can sometimes use an extra nudge of accountability. There are often so many great ideas and strategies in the books we are chatting that I get overwhelmed and not sure where to begin. Advice: pick 1 thing. Try it. Reflect. Revise. Try it again.

So here is my attempt at a gallery walk. I simply cut apart a pre-assessment for a Formative Assessment Lesson and each pair of students taped it to a large sticky note, discussed and responded. I was confident in many of the questions, but my goal was to identify the few some students were still struggling to understand completely, mostly questions involving transformations.

1. The large majority are fine with creating a possible equation, given the x-intercepts.

2. Initially these students tried -6, -4 and 2 as their intercepts. I asked them to graph their equation then reread the instructions. Oh. They had read write an equation, looked at the graph for possible intercepts and failed to read the y-intercept of (0, -6). One quickly stated the connection between y-intercept and factored terms and was able to adjust their response with ease. I believe it happens often to see a graph skim question and think we know what we’re supposed to do, only to realize skimming sometimes results in miseed information.

3. Within the lesson, many students quickly realized when a factor was squared it resulted in a “double root” and the graph would not actually pass through the x-axis at that point.

The 4 transformations seemed to causes the most disagreements. These were the ones we discussed folowing our gallery walk. However, it was during the gallery walk most students were able to adjust their thinking.

4.i. Listening to students as they were at the poster helped me realize there was not a solid understanding of the reflection across x-axis and maybe we needed to revisit. Possibly, they are confusing with across y-axis?

ii. A few students disagreed initially, but the convo I overheard was addressing that changing the x-intercepts was not sufficient, they looked at the graphs, then said, the functions needs to be decreasing at the begiining, that’s why you have negative coefficient.

iii. Horizontal translations always seem to trick students up. One disagreement actually stated ‘they subtracted and did not add.” Of course, we definitely followed up with this one.

iv. This pair of students argued over which one was right. The expaned version or factored form. Simple, graph the new equations and compare to see which one translates the original up 3 units.

A1 & A2 I believe they’ve got this one.

B1 & B2 some confusion here due to the extra vertical line in the graphic. This student was also interchanging graph & equation in their statement.

I thought the gallery walk was a good task to overview some common misconceotions. It was not intimidating, students were able to communicate their ideas, compare their own thinking to others. I truly tried to stand back and listen. They were on task, checking each other’s work. Each station allowed them to focus on one idea at a time. They were talking math. Most misconceptions were addressed through their discussions or written comments.

Having a moment to debrief the following day highlighted the big ideas students had addressed the previously and reinforced the corrections they had made. This was so much more valuable than me standing in front of the room telling them which mistakes to watch for. Their quick reflection writes revealed majority have a better ability to transform the functions, which was my initial goal for the gallery walk. A few still have minor misgivings that can be handled on an individual basis.

# Geometry, Baking & Decorating Cakes…

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A colleagues began a new venture last summer…Rich in Blessings, baking cakes.  She sent a link to this post
Perfect Buttercream Stripes to share the math needed to complete the task shown below.

Here are a few images in the post:

There could be some simple, yet nice questions arise in this setting, like:

If Mrs.D wanted 1.5″ stripes on a 10″ cake, approximately what cental angle measure would she need to use?

If she chose the slanted stripes with 1″ width, what angles would result in the guide strip for an 8″ cake?

If she used a 30º central angle for a 12″ cake, how wide would she need to cut the guiding stripes?

If a batch of buttercream covers ____ 10″ smooth cakes, how many batches would I need to decorate ____ 12″ cakes.

Not sure at what level this is in the standards but I plan to sit down this afternoon and determine how I can use this context during the spring semester…I believe I can make this work for C.A.1, C.A.2 and C.B.5…

From corestandards.org

Understand and apply theorems about circles

CCSS.Math.Content.HSG-C.A.1 Prove that all circles are similar.

This could easily be done by constructing a “cake map” including 6″, 8″, 10″, 12″.  Allowing students to prove that all of their circles are similar, by showing they are dilations of one another.  Maybe I could even ask, if I wanted to enlarge my diameter 2″, what scale factor would be needed to accomplish this?

CCSS.Math.Content.HSG-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. (Thoughts here?)

CCSS.Math.Content.HSG-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

CCSS.Math.Content.HSG-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles

CCSS.Math.Content.HSG-C.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Maybe play off the idea: use different colors of fondant to create a pattern of the sectors…how many units^2 of each color are needed?

I believe with a bit of work, this could qualify as my Practical Living and Career Studies Program Review submission.  If I plan efficiently, collaborating with my Visual Arts department, I may be able to use it as the Arts/Humanities submission as well.

What ideas, suggestions can you offer that will push my thinking forward…make this a good, quality task?

# Color Coded Rotations

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Last fall I posted some experiences and conversations that took place during our work with transformations.  I learned something so simple from my students and this year, I have shared it with my geometry classes again.  When looking to rotate about the origin, simply rotation the graph and identify the “new” coordinates.

Sure, we do the typical, here’s a preimage, here’s the new image after rotating.  Now, what do you notice happened to the coordinates?  What’s the rule? (x, y) -> (-y, x) for 90º and so on…  but once again, they get so focused on remembering the rules, then they get the x’s and y’s and the  signs all jumbled and its a mess.  I wanted to give them something, a visual, they could go back to…

Shortly before fall break, while assessing some student work, I had an idea.  It seemed so obvious at the time, yet, I had never thought of it before.  Using color-coded axes to demonstrate the rotations about the origin.  I chose to make the positive side of each axis a different color than the negative side.  We would rotate, then discuss which color was now in place of (x, y) for the first quadrant.

For several students, you could hear their light bulb.

Again, nothing new, but something that gave my students a concrete reference rather than remembering the rules.

# Complete 180°… well, duh.

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Transformations …

My students do well with reflecting across x-axis or y-axis.  Though they struggle literally flipping a transparency or patty paper across the line y=x.  During whole class discussions today, someone stated “just read what it says in the equation.”  OK. “Y becomes the x. And read it the other direction, starting with x.  X becomes the y.”  True.  Simple.

Now, its rotation’s turn.

While using whiteboards to assess our FAL from Friday, it was obvious a handful were having trouble remembering “the rules” they had developed in their pair work.   They stated as long as they had the transparency, they could physically perform the transformation.   But wondered what they could do without those tools.  One student suggested, ‘I just rotate my paper/graph as it says.’ Another chimed in, ‘that’s what I do too.’  Well, duh. An example – the point (1, 4) rotated 90 degrees clockwise about the origin, would simply be one quarter turn to the right; resulting in (4, -1).  Rotate another 90 degrees for a 180 degree total results in (-1, -4).

So, as a class, we practice their strategy.  Genius.  Why can’t I think of the simple things?  So glad I finally figured out how to listen to my students – their way is so much better than mine.

Another strategy was to draw a line segment from the point through the origin to find the image rotated 180 degrees.  To find the 90 or 270, simply take your scrap paper and use a corner edge from the origin and draw a line perpendicular to the 180 line segment.  Students went on to suggest they used “slope” to place the image.

This is a sample of what they suggested.  Rotations around the origin.*

I like this last strategy, because it will also work when rotating around a given point.  Rotation around a given point.*  Another suggested that drawing a segment from the point through the given point of rotation…is kind of like the point of rotation is actually the midpoint of the image and pre-image.

I’m happy with the fact that they’re looking for a way that makes sense of the transformations.  They’re considering different strategies and our discussions often lead to whether it will work or not.

*created with screen recordings on Promethean ActivBoard – not sure if they will play otherwise.

# Formative Assessment Lessons

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Its been 3 weeks since I’ve blogged.  Not because I didn’t want to – but life has just been head over heels busy.  The week following my last entry – I presented at KCTM – Literacy in Math Class.  I’ll blog about it soon.

In Kentucky, I was introduced to Formative Assessment Lessons about a year and a half ago.  I remember the first one I tried was not so successful.  But the more I learned, the more I realized, there was some good things embedded within these lessons.   At our last KLN meeting, we were asked to discuss our experiences with the FALs.  I hadn’t realized I had actually used as many as I have until we started running through the list.

My students can find some level of success as well as being challenged on the other end.  I observe student success with these lessons.  They are formatted in such a way, I am able to listen to student discussions, considering their ideas and able to pose questions that will foster more discussions.

Part of my session on literacy was to give students opportunities to talk, share and ask questions about their thinking.  Within the FALs, students are given either a problem solving task OR a conceptual development task.

In all lessons I’ve used, students respond to a given task as a pre-assessment, after completing the lesson, class discussions, they are given the opportunity to revisit the same or a similar task.

In the problem solving tasks, students are put into groups homogenously (based on similar approaches to solving a problem or even similar misconceptions/mistakes – not necessarily ability).  This allows students moving in the right direction to continue; while my time can be targeted to smaller groups of students, using questioning to guide their thinking, discussions.  Each group is given sample responses, and asked to think about the student’s reasoning – why they approached the problem as they did.  This gives the group an overview to see multiple ways to consider and opportuinties to critique the reasoning of others.

In the concept development tasks, students are usually given a task/questions and card sorts/matching activities.  Instructions will almost always require students to verbalize their reasoning, then their partners must either explain the reasoning in their own words OR why they disagree with their partner.  I feel verbalizing their thinking is a key component of literacy – helping them work through their own understanding but also listening to ideas of others, in a small group setting.  Many lessons offer extension suggestions if needed.

To complete the lesson, there is often a plenary discussion to wrap up, solidify the concepts.  Its very important to really listen to students – in some lessons, you are encouraged to scribe student comments/ideas with their names for ownership in the discussion.  White boards are a common component – seeking student responses – sharing different responses – asking questions – if others agree, disagree or have something to add to someone’s comments.

I am sharing about FALs because today, I left my geometry classes feeling good – that students were given an opportunity to think, discuss, share and learn – clear up some misconceptions.  I am looking forward to our whole-class discussion on Monday and the follow-up assessment!  Though there are still some mistakes – I think the sharing out will add/deepen to students’ understanding.

Representing and Combining Transformations was the lesson students worked on today.  I paired students based on similar responses on their pre-assessment.  I really enjoyed “sitting back” and listening to their discussions.  The particular task, they were given 6 different graphs with an L-shape and 8 different transformation cards.  They were asked to connect 2 shape graphs with a card describing the relationship between the two.

I’ll be honest in questioning the need for the transparency graphs – but after observing students, these were a key learning tool for most of them.  When they asked for help, I encouraged them to use their graphs to “see” what happens, then use what they noticed and apply it to their shapes.  I also found allowing students to place a push-pin at the center of a rotation was very beneficial to their understanding.  To observe how using different centers of rotation will affect the movement of the shape.

Recently, a colleague decided to try a FAL – Forming Quadratics with an Algebra II class.  In our last PLC, my colleague shared pros/cons observed during the lesson and that all but only a couple of students had improved / were very successful on the post-assessment.

FALs are idealy used about 2/3 the way through a corresponding unit of study.  This allows the teacher to view misconceptions and clear those up before finishing the unit.  Most lessons consist of a 10-15 minute pre-assessment, 1 hour for lesson/discussion (this can vary depending on students), 10-15 minute follow-up assessment.

Each lesson is aligned to 8 Mathematical Practices and outlines which CCS is addressed.

There is some prep-work involved, so don’t print a FAL and expect to use it immediately.  I use card-stock for the card sorts (each type of card gets its own color) – if you laminate them, maybe they will last longer.  Also, when it calls for a poster of student work, I don’t want them to glue pieces on a poster – then I’ll have to make an entirely new set next time.  I want to reuse them.

Today, I had students add a post-it note with their initials and I snapped a pic of their cards.  They can create an answer key on paper as well.

I would love to hear about others’ experiences with FALs – ways they’re using them in their classrooms!