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:

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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.

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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.

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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:

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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:

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

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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.

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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All in with OUR/IM – Making It All Fit

It’s a huge learning curve for lots of us. We see the potential, but there are so many things to figure out – how am I going to collect homework – or am I? What do I do with all these cool downs? How do I carve time out of an already packed lesson to make the time spent on homework and the things I learn from cool downs valuable?

 

Time could be a question to figure out all by itself. The minutes per lesson are tight. The students are slow to engage in conversation and productive discussions so everything takes longer than it says.  People say to trust the curriculum and I do . . . I know each part is carefully inserted to bring students to conceptual understanding AND procedural fluency . . . . but all that just makes it harder to leave anything out.  I feel unprepared to make the important teacher decisions that often have to happen on the fly in front of kids. What can I skip? What do I stop and address? I know when something seems important, but you keep saying it is all important.  And the quote in my signature tagline mocks me every time I send an email:

 

“The greatest influences in the quality of the education that a student receives are the decisions that a teacher makes on a daily basis.”

 

I do not have a classroom of my own, but I am working with some amazing, caring, and talented teachers who are rolling out OUR/IM 6-8 math this year, and I hear and see all of this. We listen to the teachers and districts who are on their second year and hear:

“It was like that for us too at the start.”

“Don’t worry it gets better as you all learn the routines.”

And that is great and encouraging, but many of teachers feel like they are drowning now. Waiting a month for the rescue boat that is coming seems a long, painful wait away. They see individual great stuff, but, bottom line, they are awesome experienced teachers and they just aren’t used to not knowing what they are doing.

And on facebook and twitter I hear other teachers saying the same things. So let’s address this head on and get you a life preserver TODAY.

 

Idea 1: Use a time machine

It would be awesome if this was actually an option.  If we had a year of teaching this curriculum under our belt, we would know what was coming, when something was a big deal that we had to address right then, and when we could wait for it to come up again later.

 

Idea 2: Accept not knowing and keep treading water

This just isn’t okay. If we keep teaching without knowing where we are going and trying to do everything and go over everything, we are going to get desperately behind and not finish the curriculum. And we may collapse from exhaustion.

 

Idea 3: Prep a week at a time

I get it. Prep is a ton of work at this point.  Every weekend I sit down and prep for the week just as if I was going in at 7:45 Monday morning to deliver the lesson. I have done it a couple of different ways, but I have to say nothing beats:

  1. Working through the student lessons and homework and then reading the teacher guides to see what I was supposed to get out of them.
  2. Doing 4 or 5 lessons in one sitting to see how things will flow throughout the week.

 

With the vast experience of being on my 4th week, I suggest:

Sit and do 4 or 5 student lessons including homework, peaking a teacher’s guide only when necessary to clarity. You can really feel the overlap and spiraling when you do that.  Then go back and read as many of those teacher lesson notes as you can manage time for. Read the last couple as you progress through the week.

 

Some adaptations I was able to make this past week when I planned that way:

For 7th grade, I realized the second activity from lesson 11, which contrasted scales with units and scales without units, was a nice lead in to lesson 12, which had as a learning target:

  • I can write scales with units as scales without units.
  • I can tell whether two scales are equivalent.

If part of lesson 11 needed to run over the the next day, that was going to be fine. If the students have done well on pre-diagnostic test problems 1 and 2, skip the first activity of lesson 12 to get that extra time. If not, consider skipping optional activity 12.3.

For 8th grade, which I have been doing all along, it seemed there were 100 things that bled together like that.  I could condense lesson 7 a ton, because the ideas were repeated in 8, 9 and 10. However, doing them all together I noticed the rigid transformation in activity 7.3 was the same set of moves need in 8.3. If I skipped 7.3 originally because of time, I could have students look back at that page if they were struggling with 8.3.

 

If you want to get as close as possible to the time machine idea, this batch lesson prep is the way to go.

 

Words of wisdom from OUR/IM 6–8 Math Guru Sara Vaughn during her first year with the curriculum:

“I finally got into a groove and became much more efficient in preparing for my Open Up classes. Rather than preparing daily as I had done September through January, in February I started batching my lesson preps.”

(Here is the rest of what Sara had to say at the beginning of her OUR/IM journey.)

 

And this was Sara at the end of her year 1:

I could write for days about how jazzed I was each day as we learned math in an entirely different way this year. I could tell you how I learned something new each and everyday, not only about student learning, but also about math. You need to experience that for yourself though. Please be smart enough to do that the week or at least day before your students do. It will make you so much more efficient and effective than I was. I eventually got ahead of them, but not far. I am excited for next year for sure!

OUR made me love, adore, and treasure teaching Math 8 for the first time ever. It was fun. It was meaningful. It was amazing. I cannot thank OUR enough for bringing joy into my classes through quality curriculum. I would have never thought that possible, but I lived it.”

All in with OUR/IM: Grade 8 Tips of the Week #3

COMPILING THE WISDOM OF OUR/IM 8TH GRADE TEACHERS FROM AROUND THE COUNTRY TO MAKE THESE TIP SHEETS FOR TEACHER IN 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 THOSE LINKED HERE.

 

This short week we will finish the first half of unit 1, and since it is our first time through, in many classrooms there is a little clean up and review necessary.

Lesson 8 was focused on rotation patterns, giving students more practice with this most difficult-to-master transformation, and give them opportunity to gain experience and noticing  begin things about rotation line segments. Any other things that might be discovered are NOT the main learning goal of the lesson, which is simply:

  • Introduce figures which are built by applying several transformations to one starting figure.
  • Practice rotating line segments around various points.

Moral of the story – Don’t over-teach!

Hints for 8.3 –

  • have students who are struggling  look back at how they did 7.3 last week.
  • Some students see the warm up has a center point they all rotate around and may look for a center point for this figure as well.  Watch for that method as you circulate so you can call on that student to share their thinking ( mini 5 Practices opportunity for those of you working on using this teacher move)  

 

Lesson 9 builds on these experiences, focusing on how translations effect parallel lines.

Both these lessons are full of opportunity to build a strong class culture full of curiosity and cooperation. Students should play and discover ways to move a figure from on place to another, and should appreciate the variety of ways found by their classmates. They should practice explaining their thinking and trying to understand the thinking of others.(MP 3) Having classmates try their methods out should naturally uncover their need for increased precision (MP 6) in their language and any misunderstandings about how the different transformations work.

Something to focus on:

One of the three learning goals of lesson 9 is

Understand that parallel lines are taken to parallel lines under rigid transformations.

This is something our previous curriculum did not emphasize, but the explanation for problem 7 from the end of unit assessment is built on this fact, so don’t skip it.

Lesson 10 finishes the idea that transformations preserve angle measure and segment lengths.  Note that they are to be finding segment lengths and angle measures WITHOUT MEASURING. In the mid unit assessment they will see a problem very similar to the cooldown from lesson 10, so make sure your lesson synthesis hammers home they fact that they can figure out these measures without using rulers or protractors.

Teachers around the country are creating review materials for this mid unit assessment.

Sarah Kallis made Kahoot review for mid unit assessment.

Erica Ympa made quiz for this first half of the unit.

Matt Parker used Desmos Transformation Golf with his class to give them extra practice with their transformations.

Anne Agostinelli shared this great review strategy for revisiting cool downs. Maybe using it as you review for a test would be a good way to try it out.

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Here are the slides she used with her students for this “Icon Feedback” Activity.

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All in with OUR/IM: Grade 8 tips of the week #2

Compiling the wisdom of OUR/IM 8th grade teachers from around the country to make these tip sheets for teacher in 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 those linked here.

The number one tip that applies to every lesson, every week all year is be clear what math you are trying to teach in that lesson.

From my post entitled Digging into Planning:

Having a clear grasp of what exactly you are trying to accomplish in the day will make sense of everything you are doing. It will also help your activity and lesson synthesis be clearer and to the point.”

Now to the details . . .

Lesson 1.5

Several teachers reported that students were happy to be doing instead of describing transformations in this lesson, and teachers were happy to be back to the more comfortable rectangular grid paper.

Tip one – One of the biggest huddles seems to be getting students to read and decide for themselves what they are supposed to do.  Take some time as a class to read and decode the directions for 5.2 before turning students loose.

Tip two – Don’t over teach. Look back at the learning goals:

  • Apply transformations on a coordinate grid.
  • Apply transformations on a line segment on a coordinate grid.

Kids do have a chance to notice patterns as they reflect over the x and y axis, which is a great opportunity for higher DOK thinking, but it is not the goal of the lesson.  If you do the translations so they can notice the pattern, you instruction is not aligned to your goal.  

Tip three – If  number 3 of your pre-assessment showed many students struggling with naming and using the coordinate grid, this lesson is an opportunity to address that before you get heavily into using the coordinate axes to explore slope and graph lines. Start that with the warm up. In 5.2, dwell on part c of question 1. In question 2, think together about where (13, 10) would be, and where (13, -20) and (13, 570) would be in relation to (13,10)

Tip four – Some students are confused about why activity 5.3 doesn’t use A’, B’ notation. Here is why: Since AB is going to be transformed several times, we would have lots of points with the same name.

This comes up again on the cool down, where students have difficulty with naming corresponding points using letters other than A’, B’, etc.  Before handing out the cool down, you could used the graph from the warm up to practice this by labeling the original triangle CAT and telling them the translation took triangle CAT to triangle DOG.

An extra after visiting classrooms: Did you notice that the warm up offers an alternative way to describe direction and distance in a translation? Read again. The test implies that either way is a complete  and precise method for defining a translation.

 

Lesson 1.6

This is your first day using MLR4: Info gap  (Math Language Routine 4)

Ideas for how to introduce from around the web:

  • Morgan Stipe shared a video to help train your students on how the info Gap Routine works.
  • Norma Gordon and Sheila Jaung shared an activity builder on Desmos that uses a 7th grade info gap activity to train 8th graders before their first 8th grade info gap in lesson 1.6.
  • Several teachers talked about doing the first card together as a class, having a student or pair of students come up and be the teacher’s partner. Once the class agrees there is enough information, they all solve it. For the second card, they worked with their partner.
  • Beth Pope Hill shared ”For 1.6, I laminated the info gap cards and gave the groups 2 different color dry erase markers so they could do the transformations on the card. I think that helped them move a little faster.”

 

Matt Parker reminds us that the Desmos Activity Transformation Golf is a great companion activity to provide extra practice at this point.

 

Lesson 1.7

The lesson synthesis brings out the question, ‘How can you look at two shapes and tell one is not the image of the other?” This is a question on the mid unit assessment, so take the time to bring this out.

In one classroom a teacher added this to their lesson summary (you could also have them take down as notes in the back of the book):

  • If one shape IS the image of another, then all corresponding sides and angles will be equal.
  • If two shapes DO NOT have all corresponding sides and angles equal, then one is not the image of the other.

Practice with a few shapes: “Could A be the image of B? Convince me.” Problem 1 and 3 in the practice exercises do this as well.

 

Lesson 1.8

Note the “Building towards” standard- understanding this is important for the final question of the end of unit assessment.  

Not much talk about this lesson on line yet, except this encouraging note from Kent Haines on twitter:

Screenshot (22)

And one more piece of encouragement from Chrysty Hunt Clarkson in the 8th grade OUR/IM facebook group :

Screenshot (23)

All in with OUR/IM: Grade 8 tips of the week #1

Compiling the wisdom of OUR/IM 8th grade teachers from around the country to make these tip sheets for teacher in 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 those linked here.

Lesson 1.1

Prompts for the which one doesn’t belong day 1 from the amazing @MrsNewell:

Screenshot (21)
Some people on this lesson played a you tube video of the Cha Cha Slide and either had their students dance or notice all the motion direction words. Here is one you might choose to use: https://youtu.be/wZv62ShoStY

People suggested modifying the black line master for Triangle Square Dance to all be on one page, so it is harder to see which picture your partner is looking at. Here is a one page version you can use. Here is another one formatted in a single line.

Lesson 1.2

On this warm up discussion, have students highlight one set of corresponding sides to see the angle of rotation. You will want to have them turn back to this during lesson 1.3
Today is the day you give formal vocabulary. How will you ask student to participate in this? Ideas I saw shared:

  • Take notes in back of workbooks on note pages
  • Highlight words and phrases in the student lesson summary
  • Add informal language from 1.1 to lesson summary
  • Write definitions for the glossary words underneath the summary
  • Do a stand and talk, switching partners, to define the glossary words to each other
  • Add an entry to an interactive notebook
  • Find a real life example of at least 3 words in the list with your group and share with the class.

Lesson 1.3

We were hearing from lots of teachers that they had trouble with lesson 3. Monday, our district OUR/IM implementation team sat down and did the student lesson together. You definitely need tracing paper to make this lesson work. Even with the tracing paper, it was interesting to see how we approached it differently. We each initially thought our way was the easiest and most natural way to do it, which means kids in your class will probably be using alot of different ways. We strongly advise doing the second activity together in your plc time to help you know what to expect as students work the task. There were also ones for each of us where we said, ” Wait, how are you doing that?”, especially as we got to the isometric graph paper.

Having students stop, check answers, and share how they did it with a partner or group should help students increase efficiency. As students get to #5-8, you may need to put someone at the doc cam to show how they did something if you have a big group stuck. It is REALLY important that you brought out that the angles are all 60 degrees during the warm up. If no one notices that, you may have to notice it yourself. . . ” did anyone notice anything about the angles in these triangles? (silence) I noticed that they were each 60 degrees.” (point out 3 to make a straight angle or 6 to make 360 degrees).

Hints/scaffolds to offer when needed:

#1 and 2: Students can use the corner of the patty paper to help locate the shape squarely on the grid.

#3: drawing a cross at the point of rotation can help students visualize angle of rotation

#4: Method 1- line tracing paper along line of reflection. Trace and flip over the line.
Method 2- trace, fold paper along line of reflection to flip. This method slightly more affected by inaccuracies in tracing.

#5: Start by focusing on rotating segment AB 60 degrees. Refer back to angle you highlighted in warm up on lesson 2. (So many different ways to see this. Might be good to stop and share strategies.)

#6: Same hint as #5. . .focus on only one side. “ what if you just rotated side CD clockwise 60? Where would it’s image be?”

#7: same hint as #4. For me personally, my inaccuracy in tracing and folding made method 2 a fail here. After switching to method 1, we went back and looked at my valid method and why it didn’t work. I became acutely aware of my need to attend to precision. Might be worth a class pause and discuss on that point.

#8: everyone felt so happy to get to this one after struggling with adjusting to the isometric paper for awhile. This is where we felt “ ok, this grid isn’t that bad.” If you are running low on time and your class is frustrated, this might be a good spot to skip to as you close the activity.

Also great day to remind your students that struggling is part of learning. Here is a great read here that would make an awesome writing prompt for hw the either day 2 or day 3. http://bit.ly/gloryinstruggle
I personally chose it for the night of day 2 so they come in with good thoughts about struggle as they start this lesson.

FYI If you are using the technology apps built into the curriculum, lots of teachers had struggles with students not reading directions and every hand being up once they start. Adjust your intro to tech appropriately.

Added later after watching classrooms doing this lesson:

Students DO NOT intuitively know how to use the tracing paper.  This is a lesson that should help them develop these skills. Telling them how is not enough. They need to see it.  Don’t miss this tip in the teacher lesson plan:

“For students using print materials: Optionally, before students start working, demonstrate the mechanics of performing each type of transformation using tracing paper.”

The most successful classrooms had students telling teachers the transformations they used  while teachers followed those directions using tracing paper on the board or under a document camera. Just orally sharing the steps without visually modeling with precision left a large majority of the class confused.

Lesson 1.4

You need to make your 5 practices anticipation sheet before this lesson(Download a copy here), and try to understand each of the methods listed in the teacher notes. I never would have thought of a couple, and would have had a hard time thinking on my feet if students shared one of the more unusual ones, so I was grateful the teachers notes had them all written out. The lesson notes for the teacher also do a great job emphasizing how to get the most out of students sharing their methods.. My favorite tip:

“Each time a reflection is mentioned, ask students where the line of reflection is located and when a rotation is mentioned, ask for the center of the rotation and the number of degrees.”
If you do this and then the lesson synthesis, they should be 100% ready for the cool down.

Added after observing classrooms working this lesson:

Students are struggling with vocabulary.  Having the terms prominently on the board seemed to really help students internalize and incorporate them into their own working vocabulary. (Most word walls I observed were small or less prominent and so did not have the same effect).

20180828_110057

In general, lessons 3 and 4 provide many great opportunities to notice that people see things in different ways and there is more than one right path to get to the solution
( or in this case, the image). 
If you want to draw out this idea a bit, try a dot talk or two. See this Jo Boaler video of a dot talk in action. You can also use these resources by Steve Wyborney, here  and here to help you prepare to lead one yourself.

This week, most sites did not get through all 4 of these because of other back to school community building activities, so hints will end there for this week.

If you have other tips for these 4 lessons or pictures of your classroom in action, please share in the comments.

Extra tips as more teachers get to these lessons:

No patty paper? Here are some things teachers have made work in a pinch:

  • cut up old transparencies from overhead projector days. Students will need an overhead pen.
  • clear plastic dessert plate (someone was getting creative there!)
  • wax paper
  • wax free food service paper from the school kitchen
  • Dry wax paper
  • parchment baking paper
  • tracing paper
  • tissue paper from their christmas wrap box
  • plain white printer paper ( not sure I would be able to see through this, but it worked in at least one classroom.)

Here and here are completed versions of the 5 practices sheet for 1.4. Chose whichever is easier for you. Use the time these save you to actually try each method so you are ready to lead the conversation.

All in with OUR/IM – Digging in to Planning – Part 1

Spinning my wheels

Is anyone else is finding themselves filling their time with things besides actually planning lessons? I have definitely been falling into that. In the last couple weeks I have made carefully color coded pacing calendars for 8th grade. Here was my first version:

Notice how awesomely the unit colors match the book colors. Made me so happy.
This one looks great on my computer screen, but when I printed it, my printer did not translate colors well. So I spent about 3 more hours and what seemed like a full ream of paper until I got it to print in colors I was happy with. Here is that version.

Then I made a binder with color coded tabs by unit for my teaching notes. Then I added lesson tabs to my copy of the Unit 1 teacher’s edition. Then I decided I wanted to do the same for my student edition. Now I have decided that tabs on the side were maybe a bad idea, because they get bent on the shelf. So I begin contemplating moving all my tabs to the top.

If I was a classroom teacher I am sure I would also be spending time on labeling my storage bins, creating file folders, deciding on how to arrange desks, handle geometry toolkits, etc, etc. And all those things would be awesome and make it easier to stay organized as the year progressed. But none of those things alone help me know what I am going to say when the students are sitting in front of me.

So wherever you are in your prep, at some point you need to evaluate the time you have left and say, “Enough of the side dishes. Time to dig in to the lessons themselves.”

How to begin

The experience of planning feels a little different to me now that I have a print teacher’s edition. This January I planned a bit of Grade 8 Unit 1 using materials I was looking at online. This summer I am planning with a hard copy of my teacher’s edition in hand. To be fair, 100% of the teacher edition material is on line. But the way the teachers editions are put together helped me personally not skip important steps, and allowed me to make notes and highlight questions and ideas I wanted to make sure to include when I ran the lesson. Here is a picture of the table of contents for my Grade 8 Unit 1 teacher’s edition:

table of contents

Step 1: Unit Overview

So I started at the beginning and read the Unit Overview. I underlined, circled and highlighted the things that I wanted to remember and come back to just before we start the unit. I made a list of terms that students would be learning in the unit, and I added post-its to things I did not understand or wanted to read more about.

8.1 overview

In the picture above, the post-it quotes “Experimentally verify the properties of transformations, reflections and rotations.” I wondered which properties in specific. To resolve this I looked at the 8th grade common core standards for Geometry. Because I am obsessively thorough, I also talked to some knowledgeable friends who suggested I read the 6-8 Geometry progression documents from Achieve the Core. Any of these are good options for your questions as you plan. The facebook community, Open Up Resources 6–8, by Illustrative Mathematics is a great place to look for knowledgable friends.

Here is what I found:

Screenshot (7)

“Which properties?” was answered right there, under the main standard. Yeah!

Being an old high school teacher, I wanted to also include that reflections carried a point perpendicularly and an equal distance across the line of reflection. When experimenting, students might notice that, but it is not an 8th grade standard and I do not need to add that detailed definition at this point. I share this because no matter which grade or which unit you are teaching, it is easy to overteach. So as you begin to go through the unit, it is important to decide what you are actually responsible for. This is the number one thing you can do to make sure you are able to make it through the whole year of curriculum. Good news – The reason there is only one post-it on that page is because all my other questions were answered as I read the materials included with the curriculum and worked through the rest of my planning.

Step 2: Assessments

Next, with the overview fresh in my mind, I worked through the assessments – pre, mid- unit, and end-of-unit. Most units do not have a Mid Unit Assessment, but if they do, realize that most of that material will not again be tested on the End of Unit assessment. You and your PLC need to come to some decision about whether those will be counted as two separate unit exams, two half exams, or a quiz and a full assessment. In 8th grade, because the first unit is so long and we need to send report cards at the end of 6 weeks, I am leaning toward weighting them equally just to get more grades in the grade book.

When you are looking at the assessments in the teacher’s edition or teacher materials online, you are looking at the full answer key. In the Pre-assessment that includes why each question topic was chosen and when they are going to need that information in the unit. In the later assessments, solutions include discussion about what misunderstandings would cause students to pick each multiple choice answer, and rubrics that helped me fine tune my understanding of what was expected from students.

As I worked through these I removed many of my question post-its and added details to my notes on the overview page.

Step 3: Sections

Next I opened the teacher materials on the computer and marked out section grouping on the table of contents ( see pencil brackets in picture above). The lessons in the same sections are developing the same set of ideas. I found it helped to think of them as one giant lesson, because one day does not complete one idea like it did in my previous textbooks.

In one sitting, I:

  • Read through each lessons learning goals and learning targets,
  • Looked at the Alignments in the print version (called CCSS Standards online), and
  • Read through the cool down and possible answers.

Then I moved to the next day. In the Alignment section, I noted whether the standards were listed as Building on, Addressing, or Building Toward. This really helped me get a better feeling what I needed to talk about in each day, and what could wait for a day or two.

2018_07_20 3_40 PM Office Lens

2018_07_20 3_41 PM Office Lens

2018_07_20 3_47 PM Office Lens

I especially liked this one from 8.1.3, which should me they were addressing standard 8.G.A.1 in that lesson, but there would be more work in subsequent days on that same standard.

2018_07_20 4_47 PM Office Lens

Why so many days on a single standard?

According to course guide, each sequence of lessons includes the following:

  • Activate prior knowledge – invites all students in
  • Introduce representations and contexts – essential for developing an understanding of the mathematics
  • Concepts – where I used to start if I did a great lesson
  • Language and notation – where I started if I was pressed for time
  • Connections and consolidation – the place where students can move to the most efficient method of solving problems
  • Procedural fluency – the goal

As you look through the section, think about which of parts of this progression are happening during each lesson. Having a clear grasp of what exactly you are trying to accomplish in the day will make sense of everything you are doing. It will also help your activity and lesson synthesis be clearer and to the point.

Under Pressure

In a perfect world, you would do this for all of the sections of the unit before you begin you lesson planning and teaching for the unit. Having the assessments fresh in your mind will be a huge help in figuring out where you are headed. If it’s summer and school is still two weeks away, invest that time. If you need to know what you are teaching tomorrow, that is probably not realistic. Finish your one section and then begin digging into the lessons. Not optimal but it will probably happen sometime this year.

So Much Time

Yes, planning for new curriculum is going to take time. But until now, you have probably been in the curriculum writing business. Writing warm ups, searching out and adapting activities to support your lessons, finding or writing performance tasks or other problem solving opportunities – all that took time that you no longer have to spend. And the thing that is making you nervous now is the nagging feeling that you have no idea what you’re getting into. So let’s fix that. Sit down and put in the time. The confidence you gain in knowing where you are headed and what you are talking about from day one of the unit will definitely be worth it.

Up next- Planning a lesson

All in with IM/OUR – Planning your year

In the last few posts in the series I encouraged you to:
✓ download your blackline masters and cool-downs and start your materials prep
✓ Dig in to IM resources to get the big picture of how this curriculum is built to be taught
If you haven’t read The Big Picture post, I encourage you to stop and do that first. Without a grasp of how differently this curriculum is put together, the specific planning you do will be less effective.

Pacing out the course

Finally it is time to answer everyone’s burning question – can I fit it all in? The best way to know is to sit down with a calendar of the year and start filling in squares. Everyone’s district calendar is a little different so this is something each site really needs to do for themselves. Before you begin, here are a few things to keep in mind:

Minutes Minutes Minutes

Not every lesson is written to take the same number of minutes. If you are in a self-contained classroom, this may not be a big deal. But for those who will teach 5 different sets of students per day, how things fit into a period is really important – even more so if your site runs a block schedule,where you see students in 110 minute chunks of time every other day. For you thinking about minutes per lesson is crucial.
In our district, 6th grade is self contained, but 7th and 8th grade students move classrooms period by period. To help get a grasp on the difference in lesson length, I made a spreadsheet for 7th and 8th grade. You can access this spreadsheet here.
Below find a picture of unit 1 from the 7th grade sheet.

7th minutes

A few important things to recognize –

  • Not all lessons are planned to take the same number of minutes. Notice as you look at the second column in the table above, the number of minutes per lesson varies. Especially if you are planning for a block schedule, some lessons are easier to put together on a block day than others.
  • The lesson synthesis is not included in the number of minutes. That lesson synthesis is where you lead discussion, drive home the important learning of the day, have students write down new vocabulary, orchestrate discussions, and tie ideas together. As you make decisions on whether you will include optional activities or discuss a previous cool down that students struggled with, you need to make sure you think about how long that Lesson Synthesis will take you. Most teachers say that for the first unit or so it took them longer than it did later in the year, when the they and their students were used to the structure and routines built into the curriculum.
  • Some lessons are optional. For myself, I begin by calendaring them all, but marked optional lessons in case I got behind or really wanted to add a day of error analysis and practice. Fyi 8th grade has less optional lessons than the other two.
  • Some activities are optional. We will worry more about choosing which activities you want to include as we plan individual lessons. For now, knowing you have some flexibility on those days will help you plan when you might have time to include any extras you are wanting to add (setting up classroom norms, error analysis, homework discussions, fluency games and review activities are all things that might be on my list.)
  • Units are divided into sections. Notice in Grade 8 Unit 1 pictured below, there are 4 separate sections plus a culminating lesson that is not pictured. Lesson from the same section will fit together more seamlessly on a block period.
  • Your previous experience If you have experience with the instructional routines embedded in the curriculum and with orchestrating class discussions that build on student thinking, the transitions to this curriculum will be easier for you that the rest of us. But for many, the transition will take some getting used to for both us and our students. Lesson synthesis time may just take longer at first. If that’s you, plan to move a little slower through the first unit.

Getting started

Chose a planning calendar you are comfortable with. Here are a few options you can download:
Year at a Glance – one page, back to back

year at a glance

I love that I can see the whole semester on one page with this calendar. There is room for 3 separate rows of text on each date, but two of them are pretty microscopic. Good for noting special schedules and lesson minutes.
2018-2019 traditional school calendar template

traditional school calendar

Handy for someone who likes to a little more space to write or who prefers to see only a month at a time.
Look up your own district calendar and put any holidays, staff development days or other special events on your document.

Some planning information and pacing guidelines to help you:

In our district, there is one teacher on special assignment supporting each grade level as they roll out the Illustrative Mathematics curriculum. These are some rough calendars we made to assist our school site PLCs in pacing out their year.
planning and pacing info 6th, 7th, 8th
The grade level I support is 8th grade, and I have reworked and revised this several times. I think it would be ideal if we could finish unit 4 before winter break, but that leaves little time to go slower at the beginning or establish class norms. You will see both versions reflected on this document.
Here is an example of how I paced 8th grade using our district calendar. As you can see, it really does all fit. Yeah!
8th grade calender draft This map is color coded to match the workbook covers. That makes me unreasonably happy, so if you are into details like that, you might enjoy this one.
Did you sit down and try it yet? No time like the present . . . everything is right here.

A Good Start

Take a moment to enjoy the sense of accomplishment as you look at your completed calendar. You know there will be little adjustments as things come up during the year, but with the number of minutes and the optional lessons noted on the document, it will be easy to make those adjustments when the need arises.

Sharing the work

Professional Learning Communities (PLCs) can be wonderfuls ways to share the workload and advance your practice. But it is easy to let them become bogged down in the details of calendars and the tasks that need to be done. This year, with this new curriculum, you are going to need to spend that precious meeting time on the math and the teaching. In fact, when discussing with school districts who used IM this last year what teachers needed most in the way of administrative and district support, they universally said time to talk and plan together. We encouraged out site leadership to be ready to provide substitutes so teachers could have this time to prepare. And as teachers, I encourage you to do whatever you can to take care of those routine tasks in advance or via email. Save meeting time for digging in and uncovering the mathematical purpose for each activity and warm up, so you can make sure your students get out of them what they should.
In support of that, there are some recurring jobs that will need to be done. Below I have listed a few tasks that your team might find helpful to delegate.You may able to be able to lighten your load even further by cooperating with the person that has a similar role at other sites. Check if there is someone in your district who can help facilitate this inter-site collaboration.

possible plc jobs

An aside about final exams

An aside about final exams. We are considering using this as the final retake opportunity. Since the curriculum is coherent and the practice problems are spiraled, student continue to have opportunities to improve their understanding of content from previous units. We want to honor their continued hard work at growing their understanding and computational fluency. If the final is loaded into our assessment program with questions grouped by unit, teachers can chose to replace a unit score with a higher score on that unit at final exam time.

But what about me?

If you are the only teacher on your site that is using IM, consider linking up with other users via the Facebook User Community Open Up Resources 6-8, by Illustrative Mathematics. It is a closed group, but if you ask to join you should receive acceptance pretty quickly.
Speaking of the user’s community, one of the hot topics there has been organization and storage. That would be another great thing to get set up before school begins. See a small piece of this conversation blelow:

facebook community 1
And here are some other recent posts:

facebook community 2
If you are on facebook, I encourage you to make this group a part of your year.

Next up – planning a unit, a lesson.

All in with OUR/IM – The Big Picture

The Big Picture

You can’t really get a picture of where you are going in a new city looking only at the zoomed in 30 feet of street that is shown on your phone’s navigation app. That’s handy and helpful, but almost everyone I know starts by zooming out and looking at the big picture. Where am I now and how does that relate to where I am going?

So time to zoom out a bit. Let understand a little about how the course is put together. I promise I’ll try to keep it brief, but I’ll give you places to look for more detail.

From Lecture to Problem Based Learning

For 20+ years a typical day in my classroom begin with going over homework questions, me “giving notes”, and then students working together in pairs or groups to practice the concepts and procedures I had just attempted to deposit in their heads. In my “giving notes” I made it a priority to make connections and help students really understand the mathematics. I tried hard to keep students engaged, asked them to discuss their thinking with their partners, and invited them to ask clarifying questions for the good of all.  “No such thing as a stupid question” and “If you are wondering, so is someone else” were repeated again and again.

This was how school had looked when I was a student, and I did my best to replicate and improve on the experiences I had as a learner.

Attempts to venture into changing the way class was organized were difficult.  Books were written to break things down into tiny separate procedures, and if I was going to make it through the curriculum, I needed to get through one of those procedures pretty much every day. My classroom structure was the most efficient way to get that done.

If you have been teaching very long, you know that not every student can gain complete understanding and fluency in a day.  And so, every year, some kids would get left behind. Not because I didn’t believe “All Means All”.  Not because I didn’t give every lunch and after school to support them. Just because the ideas were flying by so fast that it was hard to keep up.

Add to that those students that at 13 just can’t stay focused consistently on a math lecture for 20 to 30 minutes.  The in and out of the natural attention span of an adolescent created gaps in understanding, missing connections, and fuzziness regarding procedures even if my explanations were flawless.

It felt like a trap.  The only way we could “get through” the material was inadequate to actually have all students learn the material.

Until now.

This new curriculum from IM gives me hope that we can keep all of our students engaged. We can let our students’ understanding of the content deepen over time. We don’t have to leave any students behind. This curriculum is built differently, to allow it to be taught differently.  

From this point on, I will be referencing the curriculum course guide. I strongly recommend you take the time to read it. I have only pulled out a small portion of the many parts that I think will really help you get a better view of the big picture

How IM is Built

Unlike the many text book series that I taught from, which were designed for lecture based classrooms, Illustrative Mathematics 6-8 curriculum is designed to support Problem Based Learning (PBL). The overarching idea of PBL is that students learn math by doing math. This curriculum was written to set teachers up to make that happen.

“In a problem-based curriculum, students work on carefully crafted and sequenced mathematics problems during most of the instructional time. Teachers help students understand the problems and guide discussions to be sure that the mathematical takeaways are clear to all.

From the course guide

One of the pieces of the IM 6-8 curriculum that I am continually impressed with is how carefully crafted it actually is. Here are a few posts from the Illustrative Mathematics Blog that brought this idea home for me:

Warm Up Routines with a Purpose

Vocabulary Decisions

Respecting the Intellectual Work of the Grade

All together, the pieces of each unit tell a story, with each individual lesson bringing new dimensions of that story to the table. “The goal is to give students just enough background and tools to solve initial problems successfully, and then set them to increasingly sophisticated problems as their expertise increases.”**

From the course guide again

How IM is Taught

As a teacher new to problem based learning, the most common error is to focus on what students are doing (fold, cut, pass to the left) instead of what math students are learning.  Each activity and lesson has a purpose, and students are supposed to land somewhere. As a teacher your job as you prepare is to understand the mathematical idea behind each part of the lesson, and make sure at the close that you bring those thoughts to the forefront.

“Not all mathematical knowledge can be discovered, so direct instruction is sometimes appropriate. On the other hand, some concepts and procedures follow from definitions and prior knowledge and students can, with appropriately constructed problems, see this for themselves.”

Once more from the course guide (might be worth a peek)

You are the essential piece to the puzzle, the one that makes sure the intended math is clear to all. If you are running out of time, it can never be this lesson close that you skip. However in that close, you don’t have to do all the talking.* Since the kids are working for much of the lesson, you have time to be listening. Be listening for those key learnings that you might draw out from students.**  

If you are now hankering to get a peak at what a lesson might look like, guess what? The course guide is a good place to find that information. Time for you to break down and follow a link to the section of the course guide  titled How To Use These Materials.

Patience

Because ideas deepen over time, one lesson does not start and end the teaching on a subject.

The initial lesson in a unit is designed to activate prior knowledge and provide an easy entry to point to new concepts, so that students at different levels of both mathematical and English language proficiency can engage productively in the work. As the unit progresses, students are systematically introduced to representations, contexts, concepts, language and notation. As their learning progresses, they make connections between different representations and strategies, consolidating their conceptual understanding, and see and understand more efficient methods of solving problems, supporting the shift towards procedural fluency. The distributed practice problems give students ongoing practice, which also supports developing procedural proficiency.”

-From guess where? ( spoiler: the course guide!)

I am going to make a prediction here. You are going to worry at the end of a lesson that your students don’t seem as fluent in procedure as they did with your previous curriculum.  Be patient. Remember how carefully crafted this work is and trust a little bit. Notice in the quote above the learning progression:

Activate prior knowledge – invites all students in

Introduce representations and contexts – essential for developing an understanding of the mathematics

Concepts – where I used to start if I did a great lesson

Language and notation – where I started if I was pressed for time

Connections and consolidation – the place where students can move to the most efficient method of solving problems

Procedural fluency – what we tend to focus on as the goal.

Being patient enough to let your students have all those stops along the way is the best way to help them learn in a way that lasts past a unit test.

This video by Dr Jo Boaler from Stanford University explains that further:

Concerns

According to one IM trainer, the two biggest concerns that teachers seem to have with the IM curriculum are

  1. The pacing seems really tight

  2. There is not enough skill practice built in

I am not going to say to you that pacing will be a breeze or that you won’t want to add extra practice activities from time to time. But I am going to say that both these problems are exacerbated by a lack of patience.

The pacing will definitely be tight if we over-teach. If we think the first time something is mentioned you have to teach everything there is about that subject, you will consistently run out of time. One way to help yourself trust the curriculum is to stay ahead in your planning. First look at the big ideas of each week in a unit, so you can see when things will come up again.***

The pacing will definitely be tight if we try to keep everything else we always have done and add Illustrative Math. This is a complete curriculum. You do not need to be scrambling to make warm-ups and find performance tasks. It’s all in there. Each and every lesson starts with a thoughtfully chosen warm up ready to use. Each and every unit ends with a culminating task that applies the mathematics. And because topics continue on throughout the chapter, going over homework time isn’t the only chance you have to revisit the previous lesson’s ideas. As an example, look at the development of major work of 6th grade:

And what about that “not enough skill practice” worry? Students do need to practice and apply skills. But if you look at the lesson progression above, where do you think it is appropriate to have the bulk of the skill practice? It won’t be on that first day. By spiraling the homework we allow students to get practice over time, as they move through consolidation and toward procedural fluency.**** 

Trust

We all have a lot of learning to do, but based on the experiences of the teachers who used the curriculum this year, we have a lot to be excited about. We are part of an amazing shift in math education. Finally all really can mean ALL.

Extra Resources:

  • *One of my favorite ignite talks about learning to cede the role of sage on the stage:

The less I talk, the more they listen by Graham Fletcher

  • ** A fabulous book for helping us learn to lead discussions for math learning is The Five Practices for Orchestrating Mathematical Discussions. Much of the learning from that book is baked into the IM 6-8 curriculum. A couple Illustrative Math blog posts on the topic are linked here:

How the 5 practices changed my teaching

The 5 practices framework- explicit planning vs explicit teaching

  • *** Speaking of over-teaching, see this IM blog post on why teaching students to cross multiply is delayed in grades 6 and 7:

Why We Don’t Cross Multiply

  • ****A non-IM aside – if you do decide on a certain section that your students need more practice, there are a million wonderful resources for creative skill practice out there that can be just as effective as a 1-31 odd worksheet. Truth -sometimes they are just an old worksheet, just delivered in a more interactive way.  Here is a post that lists some popular practice strategies and games.
  • Much of what I am going to share in this post can be found in the Course Guide  provided by IM for each grade level. If you are looking online, you can find it here:

If you purchased the print Teacher’s Edition that same Course Guide is the small darker green book that came in the package.

Pro Tip:  if you did not purchase print versions of the material for your students, you may still enjoy getting the print Teacher’s Edition for yourself.  Everything it says is also online, but I find it easier to highlight suggested questions and add post-its to remind myself of something for next year if I have a hard copy.  It comes with a separate booklet for each unit, which makes it really nice for carrying back and forth to school.  Here is where to contact about any print option needs.