Tag Archives: high school math

Scrap Paper Pocket INB

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I am one of “those” who prefer to keep students on same page of INB.  If you’re not, that’s great, too.

Anyway, the question arose how do you keep everyone on the same page?  Some students take up more space to answer questions or write larger than others.

I teach them how to do extension pages (thanks Megan!) -you can read more here.

You could also begin each unit with a 2 page pocket-directions and post here.

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But last spring, we created quick pockets out of scrap paper.  

1.  Start with scrap paper, slightly smaller than width of page and twice as tall as you want your pocket.

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2.  Fold it in half, tape down top/back.

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3. Tape each side to INB.

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4.  Fold work in half and insert into pocket.

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5.  Thanks to @druinok, I used her outline & layout with Chapter Essential Questions, vocabulary, Learning Targets and suggested practice all on a half sheet.  I then taped it to top of page so I can quickly flip up to access work stored in pocket.

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As always, this may or a not be of help to you, but just another tool available.

Trig Ratios – #made4math

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Through the years, I’ve seen students struggling trying to remember which Trig Ratio is which.  I have a colleague who draws a big bucket with a toe dipped into the water.  She says she tells the students “Soak-a-Toe” to help them recall SOH-CAH-TOA.  Another has described the “Native American”  SOH-CAH-TOA tribe as the one who constructs their teepees using Right Triangles.  The most entertaining though is the rap from WCHS Math Department “Gettin’ Triggy Wit It” on youtube.

I wanted to use an inquiry activity to help them develop the definitions of the Trig Ratios.  Basically, they constructed 4 similar triangles, found the side measures, then recorded ratios of specific side lengths.  Next, I had them measure the acute angles, then we used the calculator to evaluate the sin, cos and tan for each angle measure.  Students were asked to compare each value to the ratios they had recorded in the table and determine which ratio was closest to their value.  Here’s the file https://www.dropbox.com/s/gfvhnictujfj2ik/similar%20triangles%20intro%20trig.docx?dl=0 Similar Triangles Trig Ratios.  Anyway, its not a perfect lesson, but a starting point.  If you use it, please comment to let me know how you modified it to make it a better learning experience for students.

In the past, students sometimes struggle trying to decide which ratio they need to use when solving a problem. I put together an activity adapted from a strategy called  Mix-Pair-Freeze I’ve used from my KaganCooperative Learning and Geometry book.  This book offers numerous, quality activities for engaging your students.

You can make copies of this file, Trig Ratio Cards File, then cut cards apart to use.

Trig Ratio Cards

Each student gets a card.  They figure out which Trig Ratio is illustrated on their card (& why).  They mix around the room (with some fun music would make it better), then pair up with someone.  Each person tells which Trig Ratio and why (can be peer assessment, if one is mistaken).  They swap cards, mix and pair with another classmate.  This continues for several minutes, allowing students to pair with several different people.

When I call “Freeze!” Students are to go to a corner of the room which is designated Sin, Cos or Tan.  Within the group in each corner, students double check one-another’s card to determine if they are at the right location.  Again, peer assessment, if someone is wrong, they coach to explain why, then help them determine where they belong.

Students swap cards, mix-pair-freeze again.

I like this activity for several reasons:

  • 1. Students are out of their seats and active.
  • 2.  Students are talking about math.
  • 3.  It allows them to both self-peer assess in a low-stress situation.
  • 4.  I can listen to their descriptions and address any misconceptions as a whole-class as a follow-up.

 

To clarify, the intent of this activity is for students to determine what information they are given in relation to a given angle, then decide which ratio it illustrates. It is meant to help students who struggle deciphering what information is given.

Representing Polynomials FAL & Open Card Sorts

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After an assessment last week, it seemed to me what I was doing wasn’t sticking for my students with polynomials.  So let’s just scrap plan A.  Plan B – I pulled out my Discovering Algebra book, came up with a box-building data collection that lead into the FAL I have linked  below.

Formative Assessment Lesson – Representing Polynomials

Thursday, students were given a 16 x 20 piece of grid paper and asked to cut out square corners and create a box with the largest volume possible.  We combined our data as a class.  Recording the corner size removed, length, width and height.  Students were asked to observe the data and respond I notice…  & I wonder… and that’s where our class began on Tuesday.

We shared out our responses, some adding ideas as we continued the discussion.  Work with our data on TI84s – we saw a connection between our constraints 0, 8, 10 and the graph of the regression equation.  This was not new, during the discussion, a question was brought up about what values would result in a volume of zero.  Students were able answer that with confidence and a reasonable explanation.

The FAL pre-assessment confirmed my students weren’t quite ready for the full blown lesson.  With discussion of rigor and relevance the past few days, I wanted to offer students something engaging but not so over their head, it was a flop.

I backed up and did a bit of prep work yesterday – with the following discussions in class:

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Following with whiteboards / sharing for this slide from FAL:  FALreppoly3

and a simple practice set to ensure they were on track.   FALreppoly4

 

We began class today with a quick check of the 6 practice – with a focus on similarities / differences.  Noting the double root of #5.

Prior to the actual FAL, I decided to use the same equations and graphs they were to match during the FAL, except I would have them do a card sort.  Originally, I had planned to ask them to sort cards into 2 groups.  While pondering how I could make it better, I recalled a colleague sharing ideas about open card sorts from a John Antonetti training she had attended.  So, this is what I did.

I told students I wanted them to sort the 11 equations – any way they wanted – they just needed to be able to share out their reasoning behind their choices.  After a few moments, I called on different groups and we looked at their sorts.  I should have snapped pics / documented their responses.  I was amazed – not that they did it – but how well they did it.   The things they were looking at – were much better than my original idea to sort in to 2 groups.  Students were asking students – why they put one in one group instead of another. Pausing after we had the cards sorted on the board – giving other opportunity to look others’ groups…some were obvious, others were not.   I even had groups who had the exact same sorts, but with completely different reasoning.  Wow.

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At some point we began talking about “What does that tell us about the graph?”  Almost everyone was engaged and comments added to the discussion.  Next we went on to the graphs to sort.  Again, any way they wanted…just be ready to share reasons.

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Most of the sorts were better than ANYTHING I would have suggested.  My eyes were opened – I could see their thinking.  And others did as well – it was obvious in the eye brows raised and head nods.  In both classes, there was one equation that never seemed to “fit in” the other sorts – but students were confident suggesting it belonged to a particular graph (& they were correct).

When I realized the sharing took more time than I had planned – I ran copies of the equations and graphs to send home with students and asked them to match on their own.  My plan is to put them back in their pairs for the actual pairing of the FAL.  They also had blank graphs for any without a match.

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I learned so much listening to my students today…  I am looking forward to the assessment of this standard.

I didn’t feel like I taught anything today…

…but I did feel like my students left with a better understanding…because I chose to step aside and give them the opportunity to share their thinking…

It was a great day.

 

 

Linear Equations Card Match #made4math

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Let me first say – I did NOT create this set of cards.  I received them in a session at KCM about 3 years ago.  Kudos to whomever they belong. 

 I was looking for resources to use during my RTI and ran across a box I had used in the past. 

LinearEquationsMatch – the file of the cards.

You can do several different sorts with them.  POINTS-SLOPE, POINTS-EQUATION, GRAPHS-EQUATIONS, etc.

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I have each complete set on different colors of cardstock, so I can have several sets out at once, but none of them get shuffled.

Midpoint – on a different day than Distance

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In years past, I’ve usually taught Midpoint and Distance on the same day or at least on consecutive days.  After a reminder of some brain research last fall – how our brains store information by similarities but retrieves information by differences – I decided to try things in split them up this semester – hoping to lessen the confusion students often face (do I add or subtract with midpoint/distance formulas?).  Again, this confusion stems from teaching a procedure without paying close attention to in-depth student understanding.

I chose to introduce “Midpoint with Coordinates” the same day we were working with segments, bisectors, midpoints of segments.  No bells/whistles here – just the basics

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I gave students a grid index card and the points A(2, 1) B(8, 11) and C(8, 1) to attach to their INB RPH.  Simply starting with locating the midpoint of the AC and BC.  But also asking them to compare/contrast the coordinates of ACE and BCF each time.

Finally, asking them to locate G, the midpoint of AB.  Walking around the room, it was quite fun watching the various strategies.  The great thing was asking students to share their different strategies.  One used rise/run, several “counted diagonals” from A and B until they got to the middle, one used the midpoints of AC and BC and traced up from E / over from F until he found where G was located.  After discussing methods using the graph, a student stated “I just added my x’s then divide by 2 and added my y’s then divide by 2.”  When discussing how the coordinates were alike/different, a student asked “Isn’t that, what C____ did? Just averaging the x’s and averaging the y’s?”

So, I never actually gave them the “Midpoint Formula.”  Awesome.  Of course, we went on to practice the skill a few times.  I also chose 8 questions from Key Curriculum’s Discovering Geometry (did I mention, I *LOVE* this book?!?!? And have since the mid-90’s!)  – that required a  bit more thinking beyond skill/drill.  Two questions that led to some great discussion today was:

Find two points on segment AB that divide the segment into three congruent parts.  A(0,0) and B(9,6).  Explain your method.

Describe a way to find points that divide a segment into fourths.

But in class, I offered another – what about if I need to divide it into fifths?  Students worked individually, pair-share – then class discussion.  Quite different approaches.  I loved it.

What was even better, a student asked, “But  the examples we’ve used all have an end point at the origin.  Will it still work if the endpoint is not at the origin?”  Aaahhhhhhhhhhhh! That’s music to my ears!  Wow. Wow.  I love it.  I love it.  I love it.

This is a nice little open question to share with your students.  It definitely allowed me to see student understanding of the task by their work / responses / discussion.

#Made4Math Monday – Parallelogram Foldable

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Its been a while since I’ve sumbitted #made4math Monday post.  I really like the idea of foldables – a kinesthetic graphic organizer…I believe they have a positive impact on student learning when used purposefully.

This one (found here parallelogram foldable) for parallelograms, rectangle, square and rhombus.  I wanted a foldable that somehow showed all were all in the parallelogram family, but still kept them separate – I chose a trifold.

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When I saw an example of the tri-cut Venn Diagram, I knew I wanted to incorporate it somehow to show squares as the overlap of rectangle and rhombus.  This picture does not show the cuts between rectangle/square and square/rhombus, but I think its visible in the last picture.

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The file is simply the skeleton, please feel free to make it your own (ha, just don’t go selling it as your own!)

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I am still debating what should go in the center – thinking of examples / non-examples.   Possibly even giving students a couple of example problems using properties of quadrilaterals.  Istuck area formulas in at the last second – but think it may be more effective to let students discover area of a rhombus on the own.  Suggestions are always welcome!

Station Activities – Geometry first attempt

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Geometry Station Activities by Walch

Geometry Station Activities for Common Core Standards (Station Activities for Common Core High School Math)

With little time to plan, I jumped right in to a set of station activities for my semester – block Geometry classes!

My first run was with the parallel lines / transversals stations (I know its a bit out of order, but it will be okay!).

I instructed students to take out one sheet of graph paper and we folded them in half, labeling Station 1 & 2 sections on one side and Station 3 & 4 sections on the back side.   Students were in groups of 3 or 4.  I know the big idea is to move around to the various stations – but my new room is too small :(.  Rather than running a ton of copies, I made 3 complete sets of the statin instructions and placed them into page protectors.  Students completed their work on the graph paper.  When complete, they would exchange their station instructions for another station set located in front of the room.  This way students do not have to wait on other groups to complete before moving on.

Using a different color for each station, I highlighted the station # and any Words Worth Knowing (thanks everybody is a genius blog!).  Two of the lessons called for spaghetti, I used toothpicks.  Each student will also need protractors.  The stations are not dependent on one another, so order of completion did not matter.

The discussions were great because students’ angle measures were not equal to their group members’ but the same “patterns” occurred.  I probably like the discussion questions component of the activities best.  Each student responds to a given set of questions in writing.  Then they must pair up with someone who was not in their original group to discuss their responses.  Simple misconceptions are quickly cleared up during this time.

The layout of this lesson allows students to talk about and look for patterns during the station groups.  They process their new information as they write responses and allowed to share verbally again with a partner.  Finally, as a whole class we debrief the entire lesson(s).   This format really supports the literacy strategies discussed this summer in our twitter book chat #lit4math.

I like that no prior knowledge was required for students to successfully learn about transversals and the special angle relationships formed when parallel lines are present.

I have compared the listed CCSS for Geometry Station Activities to the suggested Geometry standards of Appendix A and this book addressed over 75% of those standards.  Only the measurement and any probability suggested for Geometry are not included in this book.  There are 16 station sets and I have my students for 18 weeks…my thought is to use at least one per week, as appropriate…  I’ll share more as we get in to the semester.  But for this first run, I say 2 thumbs up.

*Station 4 deals with corresponding angles – and I reworded Question #1, because it was misleading.  Anytime you use investigations, you should definitely go through the entire lesson / activity before presenting it to your students.  (duh?) I see this happen too often, teachers just pull out an activity and pass out to students with little/no knowledge of what students will expect / questions they will ask.  The book also gives a list of possible student misconceptions to watch for.

If your students are not used to this layout of lesson – it may take a little more time to get them through it.  Once students got a feel for it, the last stations went more smoothly and quickly.

I hope to hear more from others who are using station style lessons. @tbanks06 also shared some experiences with stations for #myfavfriday and said its the best $40 you’ll spend this year!  Shop around – I found all 3 of my station books for under $85 total.

Got to give a little shout out to HoppeNinjaMath – welcome her to math teacher blogging!

Station Activities for Algebra I

I began working on creating cards for the activities needed in this book:.  I typed the “index cards” needed for several of the lessons.  Feel free to borrow/tweak and use in your classroom – and share – please, just don’t sell “my cards.”  You can find the card sets I completed here.

I am now teaching Algebra 2 and Geometry, so the Algebra I project is not going to get completed anytime soon.  Sorry.

Geo-board Investigations

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I was clearing out some files this weekend and ran across this packet from a presentation at KCTM in 2002.  I had just completed my initial National Board Certification earlier that spring (still didn’t know if I had certified yet) and thought these lessons were worth sharing.

I’m not sure if you’ll be able to read the first two pages – orginal files are long gone and just by happenstance I rance across this packet.  Reading through it – its almost like I was “blogging” 10 years ago – but it reminds how important reflection on your lesson will always be – how much you can learn about teaching by pausing to think about student thinking/responses.  Whether you use actual geo-boards, paper/pencil or modify to www.geogebra.org – maybe they will give you some ideas for your classroom.

Geo-board Investigations

  • Parallel & Perpendicular Investigation – use rectangle properties to find relationship with slopes
  • Amusement Park – distance between 2 points (I hate using distance formula and often allow students to find slope triangle, then apply Pythagorean Theorem)
  • Midpoint Investigation
  • Midsegment Investigation
 
 

*I used the reinforcement tabs for students to write coordinates/label points on geo-boards.  BUT don’t let them peel and stick…just leave on paper and drop over the geo-board tab.

Made4Math #5 Polynomial Station Activities

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Its been one of those busy weeks, so I’ve not actually created anything “new” but decided to share something I used last spring.  The idea developed after @lmhenry9 tweeted a need for ideas to use with polynomial stations.  A month or so later – I decided to use a similar idea.

I purchased a bag of 8 wooden blocks from Hobby Lobby ~ $3.  Used my sharpie to add expressions to the blocks.  Created instruction cards for each station.  Based on a pre-assessment, I grouped kids by similar struggles – those who were a step ahead could “play” more game-like activites – while I could spend time with groups who needed some extra support.  We spent a couple of days in class rotating activities.  I think most pictures are self explanatory.

1.  Collecting Like Terms

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2.  Adding / Subtracting Polynomials* – let students know which “color” block is the first polynomial.  For a little discussion, ask if it really matters?  If so, when/why?

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3.  Multiply Monomial x Polynomial

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4.  Binomial x Binomial

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5.  Factor Match – I didn’t have orginal copies with me to scan – but will get them posted here asap.

 

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I also had a station utilzing a Tarsia-style puzzle with variety of polynomial multiplication expressions.

Tic Tac Times – Students pick 2 factos listed at bottom of the page and multiply.  Place game piece on the product.  First player to get 3 or 4 (you pick the rules) in a row, wins!  For more challenge, each player must use one of the factors just used by their opponent.

* A sidebar – while creating my blocks – my daughter asked what I was doing.  I replied – making a game for my students to play.  She asked – can I play it to?  My first instinct was to tell her No – but I bit my tongue.  And then I remembered a problem she had left on my board one day afterschool and my students had asked me what it was… (After school, she and a couple of other “teachers’ kids” hang out in my room and play school.) I realized it was very similar to how she had been adding and subtracting 3 digit numbers in class this year.  So I explained how the x^2 was like her 100’s, x was like the 10’s and the # was just one’s.  She rolled the blocks and did a few problems…I’m thinking – if a 2nd grader can do it – so can 9th graders, right?

So I went in the next day – and shared “her lesson” with the class.    I gave an example like the one above – referring back to the problem they had seen on my board.  They understood the process of decomposing the numbers to add/subtract.  I connected the example to (3x^2+4x+2)+(2x^2+3x+5) to get (5x^2+7x+7) – good to go.  Then I asked, WHAT IF we let x = 10…  you know – not one student missed these problems again…

#myfavfriday Who Is Robert Wadlow & Super Size It!

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“My Favorite” was probably my favorite part of #TMC12, literally.  The snippets were quick ideas you could easily tweak for your own classroom.  So when @misscalcul8 suggested we continue – I was excited.  That is until I started thinking about what I would share.  How do I pick my favorite?   My favorite what?  I have a whole list of things I want to share – but today…a favorite unit I’ve used many, many times successfully with my students.

Sadly (for me), with CCSS, we have shifted the ratios/proportions completely to the middle school, so one of my favorite units Who Is Robert Wadlow? is no longer included in our Algebra I curriculum at the high school.  I would leave students with the question at the end of class the day before beginning the unit “Who is Robert Wadlow?”  Several would go home and look up – find information.  The following day, we would discuss, share his measurements (most in metric units) and as a class we would determine how to convert to standard units – so it would make the most since to our American Brains.  So my question – was he unnatural?  Or just a bigger version of us?  If you research, you’ll see how he was normal size baby when he was born.  We talk about how you go to the doctor for well-child visits and they measure you – plotting your height/weight on “that curve” and discuss why doctors do that.  How if we’re growing too fast/slow the doctors can run tests to see if something in our growth hormones need to be modified…

Anyway, to end the day we all measure our foot lengths and heights and create a scatterplot…surprised to see – its somewhat correlated (yes 9th graders are growing, so its not perfectly linear…) – then we add RW’s (ft, ht) to the plot…again, surprised to see, he fits the pattern…just a bigger version.  We calculate the height/foot length ratios for the class, then split the data out to boys and girls to see if there is indeed a common ratio…once again, surprised to see how close the ratios actually are.  We talk about people who are clumsy in while growing – what their ratios would look like – if they are too tall for their feet, etc.

I shoudl note I used this as opportunity to teach students how to enter data into lists on TI-84, L1=foot length, L2=height, L3 = (L2/L1) and how to create scatterplots on graphing calculator.

One year I even had students ask if this was related to Vertruvian Man and explore if they were similar to him.

As a final project in this unit, I would assign Super Size It as part of their unit assessment.

Y, B, H with their Super Size It projects.
Special K – scale factor of 5 …125 times more cereal!
Extra Gum – scale factor of 3 … 27 times more gum!
Chocolate Pudding – scale factor of 2 …8 times more pudding!
 
You could easily modify this activity to fit high school geometry – to determine how scale factors affect surface area / volume ratios.
 
Robert Wadlow Ratios & Proportions Unit Organizer
 
A few other files I have used within this unit –

So, for My Favorite Friday – one of my favorite units – I no longer get to use – hopefully one of you can use an idea or two and keep the spirit of Robert Wadlow & Super Size It alive!