About Leeanne Branham

Secondary Math Teacher on Special Assignment Curriculum and Instruction, CUSD Clovis, CA *Opinions expressed entirely my own.

All In with OUR/IM – Grade 8 Unit 3 Lesson 5-7

Not all Relationships are Proportional – Making sense of y intercepts not equal to zero

Warning – There are so many chances to over-teach in this set of lessons.  Stay focused on what you are trying to accomplish.
Your main goal here is to introduce students to situations where the relationship is not proportional.  Live in the context of these situations and help students think about what that y- intercept represents in that situation.

Maybe this wasn’t a problem for you, but for many of the teachers using this curriculum for the first time, the Ant and Ladybug problem from Lesson 3.1 felt awkward. Lots of us found it uncomfortable that time was on the y axis.  In our experience, time is usually the independent variable, recorded along the x axis. We had years of experience with time measured along the x axis. Some of us had told previous classes of students that time was ALWAYS along the x axis. So when we ran into this problem that challenged what we had always seen, it was hard to adjust our thinking. We have mostly worked through those issues now, but my point here is that it felt awkward, because it was not what we had seen before, and we had to spend a little extra time thinking about and making sense of the situation.  

Your students are about to have that same experience. They had no trouble with the Ant and the Ladybug, because they didn’t have the weight of previous experience to get in their way.  But non proportional relationships, lines that don’t go through (0,0)? That just feels wrong. Just like us, students need to spend a little time in a context to help them make sense of this unfamiliar situation. Lessons 5-7 are meant to let them do that


Lesson 5 – Stacking Cups

My advice is to run activity 5.2 with the book closed.  I love this set of slides made by Morgan Stipe.1 We’ll look at a few to help you imagine the conversation you are leading. (You’ll have to imagine the animations).

This first slide starts as “low floor” as you can get.  Everyone should be able to engage with counting the cups and agreeing on the measure indicated by the ruler.  Then everyone is invited to think to themselves: 

“If 6 cups is 15 cm, how tall will 12 cups be?” 

The baiting to use proportional reasoning is strong, and students will almost all say 30cm. The

The moment of “Wait, what??” when it doesn’t work out as they expect is the setting of the hook.  Now they are invested in figuring out what is going on. After a little individual think time student should work in partners or groups to try to figure out what is going on, and then create an estimate for number of cups in a stack that is 50 cm high. Groups may benefit from having 3 cups to manipulate as they think about the situation.

This slide shares the final answer and serves to focus us on the purpose of this activity in the lesson:

The next activity brings back the term rate of change that they learned in lesson 3 and connects it graphically to the slope of the line. This one I had students work in their workbooks. To close the lesson they learn the term linear relationship can be used to describe any relationship between two quantities with a constant rate of change. 

No equation writing yet. Just “hmm, it’s possible to have a relationship that is not proportional. Some don’t go through (0,0), but they still have a constant rate of change, which is the slope of the line.”


Lesson 6- More Linear Relationships: Slopes, Vertical Intercepts, and Graphs

The warm up gives you a chance to continue to talk about rates of change. Use the opportunity to connect the idea that some patterns they notice are growing at a constant rate of change and others are not. 

Red (+1, +1, +1) constant rate of change = 1

Blue( +3, +3, +3) constant rate of change = 3

Yellow ( +1, +3, +5) not a constant rate of change.  If we graph this it will not be linear. 

The bulk of lesson 6 is a set of situations that students are supposed to match to graphs.  For every teacher who says there is not enough practice, here it is – a day of practice. 

Because students have a fairly strong understanding of slope from a graph and a tenuous understanding of the meaning of the vertical intercept, matching is more easily done focusing on slope.  After they are matched, have a discussion of what the vertical intercept means in this situation. Morgan Stipe’s slides are again a wonderful resource.

At the end of the slide deck, Morgan has some great slides to consolidate and bring the learning together. (Keep imagining those animations. Answers appear after questions).



Lesson 7 – Representations of Linear Relationships

Activity 7.2 gives one more opportunity for students to think about a situation that is not a proportional relationship and make sense of both the slope and the vertical intercept (aka y-intercept). You have the choice of running the activity as a lab or as a whole class demo that you lead.  7th grade teachers will tell you, labs are memorable experiences you can refer back to for the whole year. ( Drink mix anyone?2) But running it as a lab will take about twice the time and require gathering equipment. 

The digital version linked in the curriculum is great (pictured below).

Activity 7.3  practices finding slope from 2 given points, and generalizes the slope formula in question 2 and 3. It also refers back to writing equations of lines as we did at the end of unit 2, using this slope formula.  

If you are condensing or combining lessons for block days, these two activities could easily be done on separate days.  

Coming next in Lesson 8: Connecting all of this (along with unit 1 learning about translating lines) to express equations of linear relationships as y = mx+b, where m is the slope and b is the vertical intercept. 

1 Slides by Morgan Stipe are available for each lesson. The are linked on the teacher page under Community Created Resources.

2 Mystery Mixture lab from Grade 7 Unit 2:

Motivating simplest radical form

From deep in my archives – this is a lesson I used long ago to motive simplest radical form. It was one of my early ventures with Math Practice 7 – Look for and make use of structure.

So you are about to teach special right triangles, and to do that you want students to understand simplest radical form. The problem with simplest radical form from your students’ perspective is that it is a needless complication that does not add any meaning to the situation.  7.071 is much easier to understand as a quantity than 5√2. But we know that the patterns are easier to see if they use simplest radical form.

So . . .

What if you started with trying to get them to notice the pattern in the sides of a 45-45-90 triangle using decimals?  Here’s how I envision a possible intro lesson going.

  1. Give them two angles in each triangle and have them notice they are similar. Then give them one side on each triangle and have them find other two sides (isosceles, pythag).

Here are some possible triangles you could use, and the answers they would get in decimal form:

image 1

It will be handy if the Pythagorean theorem work for these is on board to come back to, so you could have students put up this work.

 

  1. Teacher says: “Using the patterns you see, can you predict the length of the hypotenuse for a 45-45-90 triangle with leg =7? No pencils allowed – predict in head.” Take guesses . . . .too high, too low, just right

Now use math to check who was closest.  Answer: 9.899

Students may use similar triangles to set up proportions and solve between triangles if they have worked with those recently. That is fine.

 

  1. Tell students: “In this next unit we will be learning how to use the consistent ratio between two sides of the one triangle (for instance hypotenuse to leg in the ones we have been looking at) to find all the angles and all the sides of any triangles similar to it. So what is the ratio in this case?” Have students find.

image 2  

 

  1. Teacher talk: So decimal is consistent but not catchy. Could we have used it?

7(1.414) =9.898

Close. Off due to rounding.

Perhaps play with including more decimals to increase accuracy.

Would be nice if there was a way to be more exact without having to write out hundreds of decimals.

 

  1. Tell students: “There is another way to express radicals to make pattern easier to see and use.” Go back to pythag work on board and break down the square roots. I would suggest for this moment that you use the greatest square method so they can understand what you are doing.

 √50= √25∙2=√25∙√2=5√2

The point now is not to teach them how, but just to let them see helps see the pattern, so that they might see some value in learning how.

triangle 1:  5, 5, 5√2         triangle 2:  3, 3, 3√2         triangle 3:  4, 4, 4√2

 

  1. Teacher: “Anyone want to guess the hypotenuse for a 45-45-90 triangle with leg 7 now? No pencil!    Answer:  7√2 !  

How accurate is that?”

(Show them how to use calculator to check.) Pretty darn accurate!

“Can anyone guess the decimal  for   √2  ?”

 

  1. Transition . . . “Before we look for more patterns, let me teach you how to rewrite square roots to help you see the patterns.” Or if you’re on a roll and feel experimental, cut a equilateral triangle in half to make a 30-60-90 and see if they can find the 30-60-90 ratio themselves in radical form. They may end up “inventing” how to break down simple square roots all by themselves.

All in with OUR/ IM Math 6-8 : New for Year 2 – Extra Practice pages for each lesson

The teachers of our district worked together to create resources to address a variety of problems they had during year 1.

Made to meet a variety of teacher needs:

  • I keep running out of time for the cool down
  • I want to allow students a chance to reflect and revise cool downs
  • I don’t know how to collect homework/ hold students accountable
  • I wish my students had a little more basic practice available for home
  • I want problems aligned with the curriculum that I can
    •  use for whiteboard practice during my extended block 
    •  use for practice during my support class
    • Send home for students who work with tutors
  • I want to spiral in more review of previous grade level content before students need it in the curriculum

Here is our set for the first half of grade 8 unit 1.  Here is a copy in word for easy editing.  We are spiraling in material students will need for unit 4 throughout the first 3 units.*  You can see that in the first half of unit 1 we focused on one step equations, using fractions and decimals with comfort, and the distributive property. Our goal is to have students comfortably at grade level as they begin unit 4’s equation work. We never want these reviews to distract from the lesson at hand, and so this review is short and builds slowly.

How teachers might choose to use this resource:

We hope it is flexible enough to fit each teacher and classroom. Here are a few ideas:

  • Keep current cooldown routine but limit time – this is their rough draft only. Do a quick Critique, Correct and Clarify (MLR 3) with one of the rough drafts, perhaps structured as a “My Favorite No”.  Students do a Stronger and Clearer (MLR 1) version of the cool down as part of homework.**
  • If you need to collect something from students daily – give one page per day.  Blank back side can be used to work the 3 to 5 practice problems provided by the curriculum
  • If you just want students to have access to extra resources for mastery – hand out as a packet at the beginning of the unit. 
  • If you want extra problems to draw on for classroom activities (stations, white boards, intervention), use problems as a teacher resource to create these activities. 

Unit 1 Lessons 1-10 pdf word

Unit 1 Lessons 11-17 pdf word

Unit 2 Lesson 1-12 pdf word

Unit 3 Lessons 1-14 pdf word

Unit 4 Lessons 1-15 pdf word

Please share other ways you find to use these!

* I’ll continue to add future units here as they are ready.

**For more on math language routines see the course guide – tons of helpful information is waiting for you there!

All in with OUR/ IM Math 6-8 – Looking back at Year 1

I work closely with the 8th grade teachers of our district and can say without hesitation: We are excited as we move into year 2 with this curriculum.  

But true confessions: Sometimes year 1 felt bumpy. 

It was hard for many of our team to adjust to this different way of teaching. It was hard for our students who wanted so much for us to revert to old methods and tell them what to do, and were stubbornly determined to wait us out. Sometimes the connections were not what we expected or were used to, and it felt a little uncomfortable. And sometime unit test scores were not what we wanted.  

Time was tight and we had no real time for reviewing before state testing. In fact we had to cut short things that were priority standards. After testing we covered things that featured prominently in the state performance task. 

We pushed onward and did our very best for our students. But I’d be lying if I didn’t say we were worried going into state testing, and even more worried when we watched them taking it. 

And in spite of what was sometimes uncomfortable, state test scores showed amazing program growth.  

  • Our average growth per teacher was 7%
  • 47% of our teachers experienced double digit growth
  • More than half our teachers had over 50% of their students meet or exceed standards. (for comparison the district averages for the last 4 years in 8th grade math were: 45%, 43%, 38% and 41%)

So while sometimes we had days where our lessons felt like a hot mess, somehow we got better.  How is that possible??

  1. Our curriculum was better aligned to the grade level standards. Teachers spent time teaching the things they were supposed to teach at the correct level of rigor. 
  2. Teachers learned more about the math content. Teachers who planned using the teacher materials were pushed to think more deeply about the math they were teaching. They begin to see connections they did not see before. It was amazing PD to dig in and learn this course. 
  3. Teachers began to understand the vertical connections and their place in the progression of student learning. The curriculum emphasized building on previous year’s learning all year long. 
  4. Spiraling allowed mastery over time.  Topics came up again and again, which allowed students to revisit and deepen their understanding. Research shows that this has a profound impact on retention.
  5. Pedagogy and access. The curriculum is built to support teaching differently than we have before, in a way that promotes access and understanding for all students.   Some teachers experimented with this more than others, but for all of us we are just at the beginning of this learning. Note – lots of teacher and student growth could happen without this piece, but the greatest growth came in classrooms that experimented with these new methods. 

We can’t wait to see what happens next year now that we know the curriculum and are addressing some of the problems with pacing and practice that we had during the first year.


So the moral of our story is don’t be afraid to jump in. Even with our less than perfect execution, the curriculum was good for our students.

And that is what we are all about.

Fighting Courage

A personal post having absolutely nothing to do with math, but everything to do with teaching.

 

So as a kid I was a wimp. Actually I don’t know how much that has changed in the 45 years since my 10th birthday.
Fact about me: My pain threshold is record-breakingly low. As a kid this translated into crying and whining at even the tiniest injury, and being paralyzingly afraid of one thousand everyday things.
My mom would tell you I did have redeeming and lovable qualities, because, you know, she’s my mom. Even so, parenting what some might describe as a whiny crybaby and others might describe as a hysterical brat, couldn’t have been all sunshine and roses. The day the dentist said he wouldn’t see us anymore because 8 year old me was hysterical about the shot he was trying to give and impossible to manage comes to mind.
From my current vantage point, I am amazed I lived to adulthood. Mom is a better woman by far than I. Think about it for a moment. . . How would you parent this hot mess of a kid?
I can’t say her solution would work for every kid, because each child is unique. But for me her solution was pure genius. She couldn’t change the fact that things would hurt me. So she chose to teach me about courage. Not to “toughen up” or to change my perception of pain (“That does NOT hurt!” was something I heard from others). She taught me that being brave is not about not being afraid, but about facing that fear and doing what needs to be done.
She told me stories of children who saved their siblings from tidal waves and wild animals. She bought me books like Children of the Resistance, The Diary of Anne Frank, and The Hiding Place. She told me stories of Harriet Tutman and the Underground Railroad. . . stories of the Little Rock Nine and Rosa Parks . . . stories about Susan B Anthony and Elizabeth Cady Stanton.
She called them stories of Fighting Courage. That became the theme of my childhood.

She knew she couldn’t prevent pain from coming into my life, but she could help me choose my response.

She also chose to live fighting courage in front of me. Once we witnessed a road rage incident that turned into a brutal attack. A driver was pulled from his car through the open passenger window and pounded into the pavement by the other angry driver. 3 lanes of traffic were stopped at the light, and two full gas stations full of people looked on. My 5’3″ mother was the only one to get out of her car and yell at them to stop. “Everyone of these people can identify you and will testify against you! All those cars behind you are writing down your license plate right now!” I was terrified as she stepped from the car, but also incredibly proud. Her actions saved that man’s life.

In the wild, she taught me not to run from wild animals. When a big bear entered our camp, she stood tall and puffed up, and made noise to scare them off. I truly do not know what the correct move is in that situation, but what I learned was that my mom had fighting courage in the face of both man and beast. Not that she had no fear, but that she was willing to do what needed to be done to save herself and protect those around her.

At 56, mom continued her lessons about fighting courage as she battled stage 3 colon cancer. She confides now that there were times she thought she couldn’t take the chemo any more, and she would have to die. But I watched her advocate for herself in her care, chose nurses and people around her that were positive and encouraging, and decide to live.

From then until now, I have watched her attack the natural challenges of aging with the same fighting spirit. Exercise, healthy eating, mental exercise, and choosing to reach out and become an active part of her new community when she moved are all choices she modeled for me.

You don’t have power over evil in the world or natural disasters or disease or the aches and pains of time. But you do have the power to choose your response when those things come your way. The ability to face these things head on instead of hiding or wallowing in self pity is Fighting Courage. It is the greatest gift she has ever given me.
I am sure there are parenting moments she regrets, because every parent who ever lived has those. But this piece of parenting is what I needed more than any other. I strive to let it define my life as it has hers.

Today, at 77, she continues her lessons, this time with stage 3 breast cancer. Chemo starts this month. Here she is: my hero.

All In with OUR/IM – Equations of Lines based on Slope

Grade 8 Unit 2 Lessons 10-12

 

When I say “equations of lines” what do you think of? Go ahead, write down the first 5 or so things that come to your mind .

Here is a collection of words and ideas I got from a few of my coworkers:

20181017_165230.jpg

If I asked your students, what would their list look like?

I dug into the 7th grade curriculum to get an idea. A search of the course guide for the phrase equation of a line got zero hits. When I searched for just line, I got “vertical line”  – not exactly what I was looking for.  When I search for equations I got 29 hits.  Many of them were about solving equations, but at least an equal number were about writing equations to describe a relationship. The focus for most of these equations was making sense of patterns specific to a context, but they also wrote equations to express a proportional relationship between numbers in a ratio table. Students also learned that graphing a proportional relationship created a line going through (0,0), but they never use the term equation of a line. That is a new idea for us to develop this year.

So my advice: Set aside all your preconceived notions about equations of lines.  Because we will not be starting where our thinking about them begins. We are starting at where your students are currently at.  And we will get to all those goodies eventually, but not during Unit 2. Be patient.

So in Unit 2 lesson 9, the IM curriculum introduces students to the idea that “the quotient of a pair of side lengths in a triangle is equal to the quotient of the corresponding side lengths in a similar triangle.”   So in the figure below we could say:

In lesson 10, they focus our attention on two specific sides of a right triangle drawn along a given line, the vertical side and the horizontal side.  See what a tiny step in thinking it is to make those the two sides we are comparing?  And for every right triangle we draw along the same line, we find the ratio of the vertical side/ horizontal side are equivalent.

20181017_172605.jpg

And finally in lesson 11 we talk about writing an equation to show the relationship of all the points that are on that line.  Notice we are connecting to their 7th grade work that revolved around using equations to express relationships. So any two points you chose on that line can be used to draw a slope triangle. All of those triangles will have the exact same ratio of vertical side to horizontal side.  So in unit 2, THIS is what we mean when we talk about equations of lines:

Ratio of the sides of one triangle = ratio of the corresponding sides of a similar triangle

More specific to slope:

Ratio of the vertical side to the horizontal side on a slope triangle drawn along a line = ratio of the vertical side to the horizontal side on another slope triangle drawn along the same line

And more general to make an equation that shows the relationship for any points on the line

Using the general point (x,y) as one of the points creating the slope triangle, the ratio of the vertical side to the horizontal side on that slope triangle drawn along the line = ratio of the vertical side to the horizontal side on another slope triangle drawn along the same line.

So shake your head and try hard not to see a messy version of y=mx+b.  The equation we are writing is two equivalent ratios.

Lesson 11 takes on the work of saying how long the vertical and horizontal sides are if one of the points is going to be (x,y).  

Teaching tip:

Let one point be at (3,1). “Slide” your (x,y) around and keep asking for the vertical distance.

What if y was at 4?  

What if y was up at 7?

What if it was down at 2?

Instead of just writing

3

6

1

on those vertical sides, write

4 -1 = 3

7 – 1 = 6

2 – 1 = 1.

Then slide (x,y) way up past the edge of the grid, so they cannot tell the y value. Label it (x, y) and encourage students to use a variable expression to name that vertical length

y – 1

Do a rerun of the same logic with x.

The big idea for these equations of lines is that EVERY point that is on that line will make this proportional relationship true.  Because no matter what point on the line you pick to draw your slope triangle, the slope ratio will be the same. 

Here is an extra practice problem for the ideas from this section. Notice in the final part we use the equation to check and see if a point is one the line.

Screenshot (71)

 

Practice Strategies and Games – Alternatives to 1-31 odd

Welcome to my virtual filing cabinet!

I needed a place to store all the wonderful practice strategies and games I find posted online. Some of these include huge repositories of pre-made activities for every math topic under the sun. Be sure to add them to your personal storage.

Enjoy!

Practice Strategies

Question stacks

Explained here by Sarah Carter at Math = Love.

Repository of ready made activities on a variety of topics here.

Add ‘em Up

Explained here by Greta B at Count it all Joy.

Repository of ready made activities on a variety of topics here.

More high school examples from Sara VanDerwerf here.

Open middle style version here.

Menu Math

Explained here by Nat Banting at Musing Mathematically.

Ready made activities on a wide variety of topics included at bottom of post.

One Incorrect Activity

Explained here by Greta B at Count it all Joy. Samples at end of this post.

More be Don Stewart here.

Row Games

(really a practice structure)

Explained here by Kate Nowak at Function of Time.

Repository of SO MANY ready made activities sorted by topic here .

Games

Math survivor Game

Explained here by Julie Reulbach at I Speak Math.

Trashketball

Explained here by Julie Reulbach at I Speak Math.

Speed Dating

Explained here by Kate Nowak at Function of Time.

Fly swatter review game

Explained here by Julie Reulbach at I Speak Math.

War

Explained here by Denise Gaskins at Let’s Play Math. Many samples available in post.

Matho

Explained here by Julie Reulbach at I Speak Math.

Connect four

Explained here by Audrey at Math by the Mountain.

Math relay

Explained here by Sue Benhardus at A Blank Sheet of Paper.

1 to 100 grid

Explained here by Julie Morgan at Fraction Fanatic.

A more wild and crazy version explained here by Jon Orr at Mr Orr is a Geek.

The Unfair game

Explained in a tweet here by Jennifer Abel. Someone really needs to blog this!

Draw it

Explained here by Julie Reulbach at I Speak Math.

Pass and Fold

Explained here by Michelle at Unscrambling Math.

Ghosts in the Graveyard (adapts to any holiday or season)

Explained here by Sarah Carter at Math = Love.

Math “Paper Telephone”

Suggested here in a tweet by Berkeley Everett. Original game blogged here by Paige at The Game Gal. Hoping for a blog post with more mathy examples soon!

Tech games:

Kahoot

Explained here by Julie Reulbach at I Speak Math.

Quizizz

Explained here by Marissa G at La Vie Mathematique.

All in with OUR/IM – Grade 8 tips for mid unit 2

My thoughts compiled to create a tip sheet for the teachers from my district. These tips do not replace a thorough reading of the amazing teacher notes provided by the authors at Illustrative Mathematics. Begin your lesson planning there. You can find them linked HERE.

Lesson 6 Similarity

6.1 Warm up

  • My first notice: The teacher notes before “Launch” say specifically that students are not expected to use the formal terms ( commutative, distributive, inverse). They are mentioned for your preparation in case students happen to bring them up. Don’t get tangled up in a vocabulary lesson. Focus on that there was only 1 answer, but more than one way to get there.
  • My second notice: “Much of the discussion takes place between partners.” Don’t let this eat up class time. Pick one or 2 they don’t agree about and then move on.

6.2 – 6.4 Similarity transformations
My big notice – This lesson has the teaching first! The launch of 6.2 is basically the notes, then 6.2 – 6.4 activities practice the definition of similar. Each one is easier and more scaffolded than the one before.
6.2 Good for class discussion. Students really seemed to see that there were multiple points that could be translated onto each other, and that the point they chose changed the center of dilation.
6.3 Good to walk around and help kids while they work in groups. Accept rough sketches in this activity.
Questions I would ask to wrap up more quickly if you are in a time crunch:

  • Which picture is oriented the same as Figure A?
  • Which picture is larger than the original?
  • Which is smaller?
  • Does anyone want to check their thinking? (project and discuss – check with tracing paper or cut out.

6.4 This is optional. It is the most scaffolded of the 3 practice activities. Students just have to pick from a list of possible transformations instead of thinking them up themselves. For an RSP class or support class, this may be a great starting place. Or it might be one to save for during review.

Practice problems (HW) for lesson 6 – second part of number 1 could be a challenge problem. If you decide to assign it to everyone you may need to discuss the degree spacing of the rays. Since there are 6 in a quarter circle, they are each 15 degrees apart (90/6 = 15)

Lesson 7 – Similar polygons
Note – Polygons give extra things you can notice about similar figures . . . things that you can not see in “curvy” figures. Today we will find that for polygons, you do not have to write a series of transformations if you can show

  1. all the corresponding angles are equal, and
  2. all the corresponding sides are multiplied by the same scale factor.

7.1 Important vocabulary consolidating
Here is an extra example:
If you are in my period 2 class, then you are in 8th grade.
If you are in 8th grade, then you are in my period 2 class.

7.2 Use this to launch the class discussion. Don’t spend forever here. 10 minutes is probably more than you need. Get these two key facts out of the activity synthesis and write them on board.

  1. If corresponding angles are not equal, then shapes are not similar.
  2. If corresponding sides do not all have same scale factor then shapes are not similar.

Keep referring back throughout the period.

7.3 Needs Black line master cut up. This is a fun movement opportunity. Use the activity synthesis as is and then add notes to for their lesson synthesis.

Practice problems (HW) for lesson 7 – If there is time, use 2 and 3 to discuss activity 7.3 and what almost fooled them. If you do not have time, you might have students skip those problems.

Lesson 8 – Similar Triangles
Big picture for how this lesson fits in:

Screenshot (61)

8.1 Warmup
Allows review of positives, negatives, order of operations, grouping. If time is tight, you could save this for your test or pretest day.
Connection to what we are doing today: more than one right way, but only one right answer.

8.2 Pasta Triangles -main thing for the day
I spoke with a teacher whose students really loved doing this with pasta. She did say it required front-loading that “We don’t eat our tools. We don’t throw our tools. We don’t leave our tools all over the floor. Just like you don’t eat your calculator or your ruler, this pasta does NOT go in your mouth.”
For me personally I felt like messing with the pasta and trying to get it just perfect ended up distracting me from the math takeaways of the activity.
Here is what my work looked like when I built triangles using pasta:

20181009_173838.jpg
And here is what it looked like when I just drew in the sides using a straight edge:

20181009_173941.jpg
In a moment I will explain how specifically I did it sans pasta.
But first . . .one other choice you have to make is do you want everyone at the group to have the same set of triangles or different sets? If you have different sets there needs to be a point in the class where they get to compare with other people who had their same set. This could be accomplished one of these ways:

  • Have students with different sets use different color scratch paper to tape onto. Then stand and make a group of 3 or 4 that have the same color to compare.
  • Ask students to tape the sets on a back cupboard or the board. “Set A tape your triangles here, Set B here, Set C here, Set D here.” Then look at the results together as a class. (Remember this is really the only activity you have to get through on this day.)
  • Have students separate into the 4 corners of the room to compare, then return to their seats.

Even though you have the find your match issue, I think the “oh wow” factor is much greater when they are not sitting at the table together.

Here is how I ran it without pasta:

  • I handed one black line master to each group and had them cut them into 4 strips of 3 angles each. Each person in the group got 1 set.
  • I had them measure their three angles, rounding to nearest 5 degrees to move it along. Asked what they noticed, or what they added up to for those who noticed nothing. (Surprise! 180 degrees!)
  • I asked what they could make with those three angles. Hear a straight angle and a triangle.
  • I told them today we would be working with triangles. Our purpose today is to find a shortcut for checking if two triangles are similar. . . one that didn’t make us write an whole sequence of transformations and didn’t make us measure every side and angle.

Question 1 Is knowing one pair of angles are equal enough to guarantee two triangles are similar? I had them predict silently – write yes or no in the bottom corner of their scratch paper. Then we proceeded to check:

  • Everyone take your angle A.
  • Tape it lightly to the scratch paper.
  • Use your straight edge to extend the sides. You are the boss of the sides. Make them as long as you want. They do not have to be the same length.
  • Use your straight edge to connect your two sides to form a triangle. Do you think your triangle is going to be similar to the other people that had your same angle? Let’s check.
  • The first two sets had 30 degrees for angle A and the second two sets had 50 degrees for angle A. I had students hold theirs up toward me and (looking for variety) I picked 4 to 6 kids to stand up and hold their triangle so the class could see. We rotated the paper so that all the given angles were oriented the same way. We all agreed they were not similar.

Conclusionone pair of matching angles does not guarantee the triangles are similar. I let them celebrate if they had predicted correctly, then I made a big deal of how bummed I was. That would have been such a great short cut! No measuring any sides, only measuring one pair of angles. I think this eventually made them value the shortcut when we got it.

Question 2 – what if we use all 3 angles? Does that guarantee they are similar? At least we wouldn’t have to measure sides. Again we predicted.

  • Carefully peel your angle A off, turn your paper over, and tape it down on the back side. If it rips, use your protractor to measure and draw the correct angle.
  • Use a straight edge to extend the sides for as long as you want. You are the boss of your sides. Try to make something other people won’t. Warning, really short sides are pretty difficult to use.
  • On your angle B and C, use your straight edge to continue the sides to the edge of the small piece of paper.
  • Take angle B and tape it down on the end of one of angle A’s sides. Be very precise in lining up the angle sides.
  • Extend the other side of angle B. Keep extending until you can fit angle C onto both the side from A and the side from B. Students should notice that they need to draw all the way until the two sides meet to be able to place angle C. You will want to use this notice to draw out that only 2 angles are really needed. (I wonder about having half the class put on the third angle and the other half just have them draw to where the sides on angle A and B meet. Maybe that would be a good way to show that you don’t really need the third angle to get similar triangles.)
  • Again have 4 to 6 students stand and show triangles. Since there are now 4 different triangle sets, this is where taping on a back cupboard with all the similar triangles on one door might be a nice way to debrief.

Question 3. I skipped question 3. Students got that only 2 were needed already.

8.3 – A chance to use the shortcut
If time allows, students can try out the shortcut, finding and proving three different triangles that are similar to triangle DJI by eyeballing and then measuring angles.
If time is short, doing a sample like problem 1 and problem 3 from the homework could also help consolidate the learning and prepare students for their practice problems.

Getting the most out of Formative Assessment – A Best Practice Story

I work with an amazing group of dedicated, talented teachers. One of the great joys of my position is seeing how they put the things we learn about together into practice. This guest post was from one such classroom.

 

In Math 3, we are learning about parent functions and translations.  The functions students struggle with the most are exponential functions.  We had been working with all of the function families for a few days. I really wanted to see how the students were doing with the exponentials, so I gave an exit ticket, which is pictured below.

cool down problem

I collected their work as the students left the class and looked through the cards.  About half the class got the problem wrong. It was a little discouraging, but I felt everyone could learn from the mistakes.  The next day when they came into class, I started out by saying  that there were a lot of misconceptions with exponential functions. I told them that we would be using everyone’s cards as our warm-up and learn from some of my “favorite mistakes”.  I had pre-sorted the cards into piles of correct and incorrect answers. I started by projecting a card with a wrong answer. In groups, I had them talk about the answer and figure out what was right and wrong. As I circulated I reiterated it was important to find out what was right and wrong in the problem and that everyone had done something right.  After giving them a minute or so in their groups, I had them popcorn out what the student did correct. We then talked about what was wrong. I put up another wrong card, and repeated the same instructions. We did this a few times, also including an answer that was correct. Suddenly, there was a shift in the class, and they could quickly see what was right and wrong in the graphs. It was amazing! We begin moving rapidly through every student’s card in the class.  All of a sudden, students could quickly see the translation and asymptote (or lack thereof) and were able to state why the graph was correct, and/or why the graph was incorrect. It was such an awesome experience for all of us, and so much better than going over a homework problem together as a whole class! It took about 10-15 minutes, but the time spent was well worth it. I gave a quiz that day and no one missed the exponential function graph!!  Here are a few pictures of the students cards (I am sharing mostly incorrect responses that gleaned more discussion than the correct answers).

example 1example 3example 2

 

All in with OUR/IM: Week 5 – Wrapping up unit 1

Not sure how everyone else is doing, but as we approach the end of unit 1 and begin looking at the assessment, we are feeling the need for a little clarity. For confidentiality, I will not talk directly about test items, but I do want to focus you on a few ideas that need to be coming out as you wrap up this unit. You may get the most out of this post if you sit with a copy of the assessment next to you and have a little scavenger hunt through the document to find how the various math topics I discuss come up in the assessment. Then check out some of the resources I have at the end, and see what you can use to revisit what you missed on the first time around.

Ready? Here we go. . .

Are two shapes Congruent?

Screenshot (41)In lessons 1-13, you dug deeply into congruence. This was not a brand new, unconnected topic. The understanding of congruence is built on the rigid transformations we learned about in the first half of the unit. Justifying congruence is the WHY of those lessons. The anchor chart above makes some of those connections.

But if you are like me, and have taught lots of years, this is probably not how you have spent your career thinking about congruence. At the high school level, congruence implied formal proofs based on matching corresponding parts. At the middle school level, it was an more informal set of thoughts around “exactly the same size and shape” and “fit perfectly on top of one another”. It’s this fits perfectly thing we are messing with here. If we are sitting together at the same table and you want to tell me two pattern blocks or puzzle pieces are congruent, you can pick one up and put it onto the other. If they are pictures on a piece of paper, you can trace one, move the tracing paper, and lay it right on the other to demonstrate that congruence.
But when you don’t have the person you are trying to convince right next to you, you need to write directions for how they need to move the figure so they stack up. Our precise mathematical language helps us write clear descriptions of that movement so they can follow those steps and be convinced.

 

Anyone done a puzzle with a toddler recently? If you haven’t, here’s an adorable 15 month old puzzle pro:

Notice his favorite transformation is translation. He slides the puzzle piece back and forth until they drop in. At around 1:25 seconds he has one that is not oriented correctly and his mother says “rotate counter-clockwise 180 degrees.” Well actually she says “ Flip it around.” He’s 15 months after all.

How would “flip it around” have worked for you? If you were blindfolded and trying to do that puzzle, would you have understood what she meant?

Now that our students are teenagers instead of toddlers, we are looking for better, more precise communication. But the idea is straight forward. Convince me they are congruent by helping me put the shapes on top of each other. Strong, well defined mathematical language is what makes that happen.

Landmarks

Sometimes the diagram they are working from lacks landmarks and that makes description difficult. Your students need to learn to be resourceful and add those landmarks as needed. That could mean drawing in and naming a line of reflection. That could mean adding point names to important vertices. Lesson 13 practice problem 4 is a great chance to practice adding what is needed to make their communication clear. Take a little time to make that a piece of conversation. What did you add? Why? In lesson 15 practice problem 4 it comes up again. Notice that both of these are on a grid, and that might not be true of all the problems your students are about to see (scavenger hunt time). Think about how you will prepare your students to address that situation.

But what if they are not congruent?

What if? How does that play out when we are sitting next to each other at the table with cut out shapes or pattern blocks? I imagine stacking them up and saying, “See? This angle is wider on the top shape,” or “See? The left side is longer on this one.”
That is how it works here as well. Show me what doesn’t match. That probably means you need to name, highlight, or circle the non congruent corresponding parts. That is practiced in Lesson 7 practice problems number 1 and 3.

Lesson 12 makes great practice on this topic. Consider introducing this anchor chart there.

Vocabulary anyone?

“So there is some geometry vocabulary from previous grade levels that my students just don’t know.” (scavenger hunt time)
If that’s not true in your classroom, yours may be the only one in the world like that. News flash- kids don’t remember everything from previous grades. Shocking, I know. ( Aside: I just read an awesome article, Addressing Unfinished Learning in the Context of Grade Level Work, if you are interested in how you could address that in your classroom.)

So if you find yourself wondering, “Do my students actually know what a rhombus is?”, optional activity 4 from lesson 12 is a great place to bring some of those words up in the context of congruence. Don’t make it too tricky for yourself. We used plastic place value sticks of 10 because that’s what we had. Toothpicks would have worked great. For those of you who have too much money and don’t know what to spend it on, Ang-legs would be awesome for this. We gave each student six, and then asked them to build things. Make a triangle. Can you build one not congruent to your partner’s? Someone thought of doubling the sides, and so I asked if that also doubled to angles. I used the phrase “Convince me” liberally as we worked through building square, rectangle, parallelogram, and rhombus. In both classes students build a regular hexagon while trying to make a parallelogram different from their partner’s. All of a sudden I knew we needed to talk more about what a parallelogram was. We brought up that ALL the corresponding parts had to be the same for the shapes to be congruent. Just equal sides ( square non-square rhombus) and just equal angles ( square and non-square rectangle) did not guarantee congruence.
It was fun, it was quick, and we got a chance to work a bit on some unfinished learning from previous grade levels that the kids really needed. It would fit great as part of a review day.

So define design . . .

Check out activity 13.4 again. This is a place to bring out that even though all the corresponding parts are equal, they are not arranged in the same way, and so the entire right face is not congruent to the entire left one. (scavenger hunt time) Also check out the extra diagram in your lesson synthesis for lesson 13. It reiterates this point in a way you could use as you reviewed if you didn’t use it before:

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If you did, here’s another sample that you can use to review this idea:

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There is not a single point of rotation that the entire first figure can be rotated around to give the second figure. Each piece of the first figure was rotated, but that did not create a rigid transformation of the entire figure.

More Review Resources:

So for most of us it’s our first year, and you might not have got it perfect the first time around. In case you are wanting to revisit some things, here are a few extra review activities to choose from:

 

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