“Is there anywhere close we could see this in action?”

“Is there maybe a video that could help us imagine how this looks with kids?”

“How do these lessons work on a block schedule?”

“The lessons seem so scripted. I need to be able to adjust and make a lesson my own, and I am not sure that works with this curriculum.”

I just observed a wonderful lesson on a 113 minute block that answered so many of these questions beautifully. The teacher, Ms. D, agreed to let me share it with you.

Previous day:

Lesson 7 with a quick peek at 8.

During announcements:

Students read and annotate lesson 9 summary.
The teacher leads discussion: “What did you chose to circle or make note of?” etc.
She focuses students on middle line:

“The advantage of using powers of 10 to write large numbers is that they help us see right away how large a number is by looking at the exponent.”

and the final paragraph, circling: “easier“ and “avoid errors“.

This, Ms D said, is the why of what we are learning today.

Opening Discussion:

To open the day they discussed together Lesson 9 Practice Problems 1 and 2. Problem 1 was a chance to check in on vocabulary for large and small numbers. During problem 2 she focused on all the different ways students could come up with to express each number as a multiple of a power of ten, and had lots of discussion until they felt comfortable moving back and forth.

Pro move– when students misspoke, she wrote exactly what she heard, and invited classmates to agree or disagree. This brought out simple things like the difference between “8 to the 7th” and “8 x 10 to the 7th”

As they finished each part a – e, the teacher circled the answer that was written in scientific notation without defining what that meant and said they would talk about definition later. On part f, she asked students to decide which was scientific notation. They were able to see the pattern and the need, since otherwise there are so many right ways. (MP8 in action)

Part 2:

Then she moved to 10.1. After a first round circulating and checking students work, she brought up a common misunderstanding she was seeing, and discussed the importance of having the tick marks evenly spaced. She showed that labeling with powers of 10 created tick marks where the first two were 10 apart (from 0 to 10), and the next two were 90 apart (from 10 to 100), etc. She asked, “How many tick marks are there?” “What if the last number was 20? (cover 10^7 with a post-it that said 20). A student was able to suggest dividing by 10 since there were 10 tick marks, and they successfully labeled the number line counting by 2s. Then they went back to the number line ending with 10^7, and applied the same process of dividing by 10. Ms D wrote out the division problem, and asked students to apply exponent rules to simplify. They found that they needed to count by 10^6. At this point she translated, “That’s 1 million, right? Can you count by one million up to 10 million?” Students all began counting out loud. She wrote 1,000,000 and 2,000,000 over the first two tick marks. “How could we write those as multiples of a power of 10?” They did those two together and then she asked them to complete numbering that line using multiples of a power of 10.

From here they moved to 10.3 (10.2 appears in the Desmos activity they did later). Ms. D asked student to work silently to express each number as a multiple of a power of 10. After a few minutes working with her circulating, she asked them to convert each of their answers to be written with same power of 10, to make them easier to compare. “Share your answers with your partner. Did you both chose same power of 10?” and finally, “Let’s all change to 10^8, to match number line we will use next.”
She showed the zoom in digital app and asked how they could label the zoom-in line. A student quickly suggested using decimals, and Ms. D encouraged them to complete that labeling with their partner.

Zooming out the rest of the way using the applet gives and chance for them to check their work ( see below). From here students added the points to the line (#3) and the class discussed #4-5.

Desmos Application:

From here they moved to this Desmos activity (If you haven’t played this yourself yet, follow the link and check it out!). Students worked individually all the way through, sometimes going back and revising their thinking as they saw what classmates had entered ( in slide 3 for instance). Occasionally a quiet collaborative conversation broke out.
Students who had looked less engaged during class discussion were super engaged and talking math 100% during this activity. For students who finished early, she had them listen with ear buds to Mr. Aaron’s lesson 11 video. On the last slide of the Desmos activity, which was her cool-down for the day, all but 2 students correctly answered the final question by themselves.

If your counting, that was 3 lessons with 94% mastery in one block.

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

Unit 5 Lessons 1-10 pdf word

Unit 5 Lessons 11-21 pdf word

Unit 6 Lessons 1-10  pdf word

Unit 7 Lessons 1-15   pdf  word

Unit 8 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!

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 8^th 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.

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:

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.

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.

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:

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:

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

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.

Conclusion – one 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.