“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.
Lesson 7 with a quick peek at 8.
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.
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)
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.
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.
I thumbed through Unit 6 when the books were first delivered and saw it included scatter plots and two way frequency tables. No problem. I’ve taught those things before. And scatter plots means line of best fit, so we’ll be reviewing equations of line and slope shortly before testing. Perfect.
And I put the book away.
In our first year we ended up moving unit 6 to last, because unit 8 seemed more important to get in before testing. So it turned out we barely touched any of unit 6 the first year, and definitely did not take time to get the big picture of the unit.
So here I am in year 2, finally digging into Unit 6. And I can say there was a lot I didn’t get first time around. Let’s begin with the title: Associations in Data. This title fits the two ideas I formally thought of as separate into one overarching concept, and that understanding frames everything you are doing in the unit. (I wish this was a little more explicitly discussed in the unit overview – maybe next edition??).
In grade 6 students work with displaying and analyzing numerical data around a single attribute. They recognize that data can be described in terms of it’s central tendency and it’s spread, and use line plots, box plots, and histograms to display the data and make visible these important features. Early in the unit, students are asked to recall these terms as ways to display single variable data sets.
What is different in 8th grade is the addition of a second set of data that may or may not be associated in a predicable way with the first set. When the data is numerical, we can use a scatter plot to see if there is a predictable pattern or association between set 1 and set 2. If the data is categorical, we can use the two way frequency table and relative frequency to determine whether there is an association we can use to make predictions.
Numerical Data and Associations
Using numerical data on a scatter plot we can ask does the daily high temperature have any connection to the number of snow cones sold on that day? We might say:
As temperature increases, the number of snow cones sold increases.
We might describe the association using more precise mathematical vocabulary:
There is a positive linear association between temperature and number of snow cones sold.
Or we might describe the association using the equation of a line of best fit:
The relationship between temperature, T, and cones sold, c, can be modeled using the equation T = 2c -70.
Or potentially just in terms of the slope of the line of best fit:
Two additional cones are sold for each 1 degree increase in temperature.
In all cases we are implying that there is an association between that two sets that allows us, knowing something about 1 set, to make a prediction about the other set.
(So far, nothing dramatically different from what I expected, except perhaps the opportunity to describe the relations ship in terms of just the slope of the line of best fit.)
Below are some additional practice and review problems we made to focus on this part of the unit. Questions circled in red emphasized using the slope (and units) to describe the association. Our first time working through the unit test we were worried that students might struggle with that. After working through the unit I see this idea comes up a fair amount, but I do like the chance to re-emphasize the meaning of slope of a line in terms of units.
Graphs That Don’t Begin At (0,0)
Sometimes it is just not convenient for a graph to begin at (0,0). A graph where the x axis is labeled with the year is a perfect example.
Watch for situations in the unit where this comes up. The first is in the lesson 2 summary, then twice in lesson 3 and 5 times in lesson 4. This same graph and context from lesson 2’s summary comes up in the lesson 3 and lesson 4 summaries. The fact the graph is familiar when you summarize 3 and 4 lets students focus on the mathematics from that day. Don’t skip the Lesson 2 graph – be sure to deal with the “not (0,0)” issue when it first arises.
Below part of our review problem 6 is shown, where we took some time to focus on the fact that not all graphs begin at (0,0). (see green circle). There is a spot on the test where the line of best fit exits the left side of the graph above the x axis, but the y intercept is actually negative. We added this problem to create a discussion about y-intercepts that don’t show on the piece of graph that is given.
Categorical Data and Associations
Lessons 9 and 10 focus on categorical data. My teaching in the past has focused primarily on displaying this data using bar graphs and two way frequency tables, but in this curriculum, the question is consistent throughout the unit: Is there an association between the two sets of data? In categorical data, this can be interpreted: If we know a subjects answer to question A, can we predict their answer to question B? How reliable is that prediction?
Lesson 9 begins with a chance for students to notice such and association. Students who do not play sports are more likely to watch a fair amount of TV.
9.2 introduces students to 3 ways that data might be displayed: two way frequency tables, bar graphs, and segmented bar graphs. There is a short card sort included in 9.2 to let student practice matching data sets that are displayed differently. In activity 9.3, students learn to find relative frequencies from data in a two way frequency table. they are asked to make conclusion about whether an association exists based on these relative frequencies.
The next day, they will actually create segmented bar graphs with these relative frequencies and most students will find the visual helpful in determining the association. If the percents are very close to the same in each segmented bar, there is no association to help make a prediction about one thing given the other.
Here are a few practice and review problems we made for this section of the unit:
The questions circled in blue really ask the same thing in two different ways, and are meant to create a discussion around what it means for their to be an association between the data. Several of the other questions offer chances to continue this conversation.
I first saw the “One Incorrect” activity model on Greta B’s blogCount It All Joy. I believe her original inspiration is from the work of Don Stewart here.
Since Unit 7 includes so much “Which of the following are the same as . . . ?”, this practice structure seemed like a perfect fit. Use whichever pages fit your needs. Note the open middle style appear on this unit assessment. Chose the level of openness you are comfortable with.
Some details on construction: Each sheet has a picture over the top of the actual editable equations/text. So if you would like to edit problems, or make a different page for each group and have students rotate and find the “one incorrect” on their classmate’s creations, you can easily duplicate a slide and edit it.
Tips for those feeling pressed for time as they approach unit 7:
Lesson 1-3 can be condensed. Use the pre-assessment to decide how much. 2 days? 1 day?
Lesson 4 and 5 are worth slowing down on a bit. The tables are amazing in lesson 5. Take time to fully understand all that first table does for you and reference back to it through the lesson.
Lesson 6 and 7 extend exponent rules to other bases. If you have an anchor poster for your work from lessons 1-5, update it here. See the note below from the Activity Synthesis for lesson 6, activity 2:
I would do this by using post-its to replace the 10’s with 2’s, then 5’s, and eventually rewriting with x’s. These are the lessons where lots of exponent rule practice is built in.
Lesson 8 and Lesson 9: Great lessons, but if time is an issue and you are able to revise a test question, this is a spot where the time desperate can cut. Note that test question number 6 part c is covered in lesson 8.
If you are minimizing lesson 9, be sure to include a moment for big and small number vocabulary, maybe including an anchor chart for reference.
Lesson 10: Spend the time here. There are some amazing resources to explore with this lesson.
For Activity 10.2, check out the applet in the on-line materials for placing numbers on the number line. This is something to consider having students interact with in pairs.
Also in Lesson 10, Activity 3 has a must-use applet to visualize the expanding number line.
In lessons 8 – 10 the focus was on modeling functional relationships with proportional functions, linear functions, and piecewise linear functions. In addition to providing students practice with the important skill of writing equations for these types of functions, students were repeatedly asked to make connections between graphs and the real life situations they model.
Throughout the remainder of the unit, students will begin working on extending the learning about volume from Grade 7, where they learned the formula for the volume of a right rectangular prism. Although this is not a priority standard for the grade, it is important in the progression of student learning and will be tested in state testing here in California, which is an SBAC state. In addition, students will continue to be given opportunity to explore linear and nonlinear function relationships in tables and graphs.
At this point in the year, many teachers are feeling behind and wondering about what they can condense to help them finish all the content they are responsible to teach before the end of the year. Non-priority standards seem like a great place to look for cuts. I will point out a few condensing points in this walk through.
Lesson 11 Filling Containers
11.1 Which one doesn’t belong.
Take this opportunity to refresh/introduce vocabulary students will use during this half of the unit by creating a list of vocabulary for geometric solids as students explain their thinking.
11.2 Height and volume
Because the textbook suggests using a lab setting with graduated cylinders that not everyone has access to, I know some teachers consider skipping this portion of the lesson. If you typically use print resources and slides from Open Up Resources community resources you may not notice there is also an awesome digital application available for students to explore the relationship between volume and height. Interacting with lab materials or this app will ensure that all of your students have a solid conceptual understanding of volume before we begin working on formulas.
Spend the time to do a thorough activity synthesis here, even if it means you skip 11.3. You can get the essential math learning here.
11.3 What is the Shape
If time allows, this is a nice next step in volume explorations which connects the modeling from lessons 9 and 10 to our volume explorations. The Desmos activity Waterline goes great with this activity and could be used any time after this lesson if you include activity 11.3 in your student’s learning experiences.
11.4 Which Cylinder?
A great cool down and follows nicely from the activity synthesis of 11.2.
Lesson 12 How Much Will Fit?
12.1 and 12.2 tap into the fun and challenge of Estimation 180 type activities to:
help students think about volume,
expose them to different shaped containers we will be finding volumes for this unit
Practice using correct academic language to describe these solids
Think about units of measure appropriate to volume
Create a curiosity about how we might calculate volume of solids that are not rectangular prisms.
Keep the activities fun and light.
12.3 Do you know these figures?
This connects back to the Which one doesn’t belong? from 11.1. Students learn/practice more with academic vocabulary and learn to draw these figures on their papers.
The Lesson Summary is a good time to have students make notes about what they have learned so far about geometric solids. Have students do an individual write for these, then use a stand and talk to have them share and add on to these notes. Close with a class discussion.
Extra time? Work ahead to do 13.1 ( a review of 7th grade circle work) This will ease the time crunch for lesson 13, which is a key lesson in this portion of the unit.
Tight on time? 11.1, 11.2, 12.3, and 11.4 make a nice single combined lesson if you are feeling behind at this point.
(11.3 could be pulled back right before state testing to review several topics)
Lesson 13 The Volume of a Cylinder (A full day with nothing to skip)
13.1 A Circle’s Dimensions
This warmup is meant to bring up students 7th grade learning about circles including words like radius, diameter, the number pi, and how to calculate a circle’s area. Don’t skip the launch, which reviews many of these things. If no student can come up with the formula for the area of a circle, or if students can produce formulas for both area and circumference, this is a great time to model using digital resources, including asking Siri, Google, or Alexa.
13.2 Circular Volumes
This activity connects previous learning about volume of prisms to the new volumes they will be learning. Spend the time to be sure they get this. A physical models (using unifix cubes, stacking boxes or stacking cylinders) are extremely helpful to students visualizing this learning.
13.3 A Cylinder’s Dimensions
This doesn’t take long, and is more important than you realize. Circulate and catch errors as students try to sketch the radius and height for each of these. Watch on D and E for students labeling the diameter as the height. In activity synthesis, discuss this confusion.
13.4 A Cylinder’s Volume/ 13.5 Liquid Volume
If you end up getting sucked into doing 13.4 together as a class, both the cooldown and practice problem number 1 give immediate chances for students to practice and apply this learning. Be sure to make time for at least one of those before students leave your room.
Lesson 14: Finding Cylinder’s Dimensions
14.1 A Cylinder of Unknown Height
This is a great set for the rest of the lesson. I lie adding the questions “If I told you the height was 3, could you figure out the volume?” If I told you the volume was 32pi , could you figure out the height?”
14.2 What’s the Dimension?
Use this to discuss methods students use to find the missing dimensions. The curriculum suggests using the Math Language Routine “Stronger and Clearer” to focus the attention on refining their explanations. For partners or groups that finish quickly, the Are you ready for more is a nice extension. As you circulate, keep a paper with possible answers handy for those groups, so you can focus your time on those struggling with their explanations of #1 and 2. Since there are an infinite number of solutions to the Are you ready for more, all students should be able to continue working until you are ready to lead the synthesis.
14.3 Cylinders with Unknown Dimensions
For those who are always feeling the need for more student practice, here’s a great opportunity for students to develop fluency with the formula work. Don’t get trapped into feeling that all students must finish every line of this table during class. After all students have finished the first 4 rows, they have done enough thinking to be able to follow along with the discussion. You may need to project a completed table to allow all students to participate in a discussion of patterns that emerged in the table. (see activity synthesis questions in the teacher materials).
14.4 Find the Height
Take time to have students complete individually and turn in this cooldown. Use it to evaluate how students are doing on their work with cylinders and address common mistakes or misconceptions the next day.
Lesson 15 The Volume of a Cone
15.1 Which has a larger volume?
The estimations in #1 and 2 will be answers in activity 15.2. Now is the time to get students to invest in their guesses, as well as help them be able to successfully sketch a cone.
15.2 From Cylinders to Cones
If you have a geometric solids set like these, testing out students’ estimations from activity 1 can be great fun. Usually students think it will take 2 cones to fill one cylinder with the same base, and the fact that it takes three ( and only a few students got that correct) makes the formula especially memorable.
If you do not have a set of solids, a nice video is included in the materials. You can watch it here.
The questions in this activity mirror ones students will see on the unit assessment so take time to fully synthesize this, recording the formula for the volume of a cone on classroom anchor charts and in student notes.
15.3 Calculate that Cone/ Are you Ready for More? /15.4 Cool-down
Here is an opportunity for students to put this learning into practice. Be sure to include time for the compound solid in the “Are you Ready for More” activity, which uses both cylinder and cone volume formulas.
The cooldown in Activity 15.4 also asks students to calculate volumes with both formulas.
Lesson 16 – Finding Cone Dimensions
What would you anticipate students having difficulty with as they work from a given volume and height to find the radius of a cone? Whatever you thought of ( undoing, the fraction, the pi, the squared term), it is addressed in advance in the warm up- don’t skip it!
16.1 Number Talk: Thirds
Do this activity with books closed, revealing and discussing equations one at a time.
If number talks are still new to you, here is a Number Talk Cheat Sheet with handy prompts for both you and students as you adjust to the routine.
16.2, 16.3, and 16.4
These activities mirror the work in lesson 14. All tips listed there apply in this lesson as well.
Community created resources Lesson 16:
Shout out to Rachel H who created a great review worksheet for volume formulas, similar to the charts in 14.3 and 16.3. Since this includes spheres, you may choose to use this as part of your unit review.
Next up in the unit –
Lessons 17 and 18 connect volume patterns to functions and give students an opportunity to identify linear and non-linear functions. Time crunched teachers could consider combining these lessons.
Lesson 19 – a beautiful lesson that builds an intuition for the volume formula of a sphere. Don’t cut this.
Lesson 20 finishes and formalizes the volume of a sphere work. Could be shortened or combined with 21.3 and 21.4 to review the unit.
Lesson 22 – wraps the unit nicely, connecting functions and their graphs with the volume learning. Could be skipped if time is an issue.
Not all Relationships are Proportional – Making sense of y intercepts not equal to zero
Warning – There are so many chances to over-teach in this set of lessons. Stay focused on what you are trying to accomplish. Your main goal here is to introduce students to situations where the relationship is not proportional. Live in the context of these situations and help students think about what that y- intercept represents in that situation.
Maybe this wasn’t a problem for you, but for many of the teachers using this curriculum for the first time, the Ant and Ladybug problem from Lesson 3.1 felt awkward. Lots of us found it uncomfortable that time was on the y axis. In our experience, time is usually the independent variable, recorded along the x axis. We had years of experience with time measured along the x axis. Some of us had told previous classes of students that time was ALWAYS along the x axis. So when we ran into this problem that challenged what we had always seen, it was hard to adjust our thinking. We have mostly worked through those issues now, but my point here is that it felt awkward, because it was not what we had seen before, and we had to spend a little extra time thinking about and making sense of the situation.
Your students are about to have that same experience. They had no trouble with the Ant and the Ladybug, because they didn’t have the weight of previous experience to get in their way. But non proportional relationships, lines that don’t go through (0,0)? That just feels wrong. Just like us, students need to spend a little time in a context to help them make sense of this unfamiliar situation. Lessons 5-7 are meant to let them do that
Lesson 5 – Stacking Cups
My advice is to run activity 5.2 with the book closed. I love this set of slides made by Morgan Stipe.1 We’ll look at a few to help you imagine the conversation you are leading. (You’ll have to imagine the animations).
This first slide starts as “low floor” as you can get. Everyone should be able to engage with counting the cups and agreeing on the measure indicated by the ruler. Then everyone is invited to think to themselves:
“If 6 cups is 15 cm, how tall will 12 cups be?”
The baiting to use proportional reasoning is strong, and students will almost all say 30cm. The
The moment of “Wait, what??” when it doesn’t work out as they expect is the setting of the hook. Now they are invested in figuring out what is going on. After a little individual think time student should work in partners or groups to try to figure out what is going on, and then create an estimate for number of cups in a stack that is 50 cm high. Groups may benefit from having 3 cups to manipulate as they think about the situation.
This slide shares the final answer and serves to focus us on the purpose of this activity in the lesson:
The next activity brings back the term rate of change that they learned in lesson 3 and connects it graphically to the slope of the line. This one I had students work in their workbooks. To close the lesson they learn the term linear relationship can be used to describe any relationship between two quantities with a constant rate of change.
No equation writing yet. Just “hmm, it’s possible to have a relationship that is not proportional. Some don’t go through (0,0), but they still have a constant rate of change, which is the slope of the line.”
Lesson 6- More Linear Relationships: Slopes, Vertical Intercepts, and Graphs
The warm up gives you a chance to continue to talk about rates of change. Use the opportunity to connect the idea that some patterns they notice are growing at a constant rate of change and others are not.
Red (+1, +1, +1) constant rate of change = 1
Blue( +3, +3, +3) constant rate of change = 3
Yellow ( +1, +3, +5) not a constant rate of change. If we graph this it will not be linear.
The bulk of lesson 6 is a set of situations that students are supposed to match to graphs. For every teacher who says there is not enough practice, here it is – a day of practice.
Because students have a fairly strong understanding of slope from a graph and a tenuous understanding of the meaning of the vertical intercept, matching is more easily done focusing on slope. After they are matched, have a discussion of what the vertical intercept means in this situation. Morgan Stipe’s slides are again a wonderful resource.
At the end of the slide deck, Morgan has some great slides to consolidate and bring the learning together. (Keep imagining those animations. Answers appear after questions).
Lesson 7 – Representations of Linear Relationships
Activity 7.2 gives one more opportunity for students to think about a situation that is not a proportional relationship and make sense of both the slope and the vertical intercept (aka y-intercept). You have the choice of running the activity as a lab or as a whole class demo that you lead. 7th grade teachers will tell you, labs are memorable experiences you can refer back to for the whole year. ( Drink mix anyone?2) But running it as a lab will take about twice the time and require gathering equipment.
Activity 7.3 practices finding slope from 2 given points, and generalizes the slope formula in question 2 and 3. It also refers back to writing equations of lines as we did at the end of unit 2, using this slope formula.
If you are condensing or combining lessons for block days, these two activities could easily be done on separate days.
Coming next in Lesson 8: Connecting all of this (along with unit 1 learning about translating lines) to express equations of linear relationships as y = mx+b, where m is the slope and b is the vertical intercept.
1 Slides by Morgan Stipe are available for each lesson. The are linked on the teacher page under Community Created Resources.
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.
I work closely with the 8th grade teachers of our district and can say without hesitation: We are excited as we move into year 2 with this curriculum.
But true confessions: Sometimes year 1 felt bumpy.
It was hard for many of our team to adjust to this different way of teaching. It was hard for our students who wanted so much for us to revert to old methods and tell them what to do, and were stubbornly determined to wait us out. Sometimes the connections were not what we expected or were used to, and it felt a little uncomfortable. And sometime unit test scores were not what we wanted.
Time was tight and we had no real time for reviewing before state testing. In fact we had to cut short things that were priority standards. After testing we covered things that featured prominently in the state performance task.
We pushed onward and did our very best for our students. But I’d be lying if I didn’t say we were worried going into state testing, and even more worried when we watched them taking it.
And in spite of what was sometimes uncomfortable, state test scores showed amazing program growth.
Our average growth per teacher was 7%
47% of our teachers experienced double digit growth
More than half our teachers had over 50% of their students meet or exceed standards. (for comparison the district averages for the last 4 years in 8th grade math were: 45%, 43%, 38% and 41%)
So while sometimes we had days where our lessons felt like a hot mess, somehow we got better. How is that possible??
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.
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.
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.
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.
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.
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).
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
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
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
all the corresponding angles are equal, and
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.
If corresponding angles are not equal, then shapes are not similar.
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.
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?
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.
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.
“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:
If you did, here’s another sample that you can use to review this idea:
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: