I put together these hyper docs in Spring of 2020, when over one weekend we went from face to face instructors to managers of distance learning opportunities. The subjects highlighted reflect the priorities for the course and the topics that are usually taught during spring semester.
I have been asked to share these resources so many times that I decided it was easier to put them all in one place for easy access.
Are you tired? These last six weeks have been a brutal baptism by fire into digital teaching and learning.
I know some of you are doing more than one of these at a time. If that’s you, those slopes are additive, so you are probably at the top of that list. Be comforted that it will never get steeper than this. ☺
One place you may consider upping your game, either now or for the future, is in the way you are recording and delivering content. I spent the week exploring different tech options and found a few that are much easier than what I previously have done, and some that help students to be more interactive and engaged while watching. I took one 8th grade lesson introducing the idea of volume and tried it out with each of the following platforms. (My purpose here is to show off the tech, so forgive less than optimal teaching. Also, I committed to posting my first product for each platform, so you could see the way it might actually come out without spending a week of redoing and perfecting. All that to say, watch with grace.)
Record in Powerpoint: The record feature in Powerpoint makes editing or adding to your video a breeze. Your speech and animation clicks are captured and saved slide by slide. Done and realize something you forgot? Add a slide and record whatever talk goes with it. Mess something up? Don’t start over. Just re-record that one slide. In the end, it can be converted to a single video.
Nearpod: This is another great option for those of you who teach using slides, but also is very adaptable to loading small videos you have saved from your computer or found online. The great thing about Nearpod is the wide variety of ways you have to choose from for students to answer questions or interact with the content you create. As you explore this self-paced lesson, be sure to see the variety of student interaction types on slide 3, 5,13, 24, and 26.
EdPuzzle: This platform is great for uploading any old or new video you have made or found online and inserting questions to check for understanding during the video. You can upload your classes from google classroom if you wish to, which allows you to keep a gradebook of viewing time and % correct for each video you assign. The video editing tools are simple and wonderful for clipping out little misspeaks or random interruptions. I’ll never have to stop and re-record again! Also, EdPuzzle has the option to set it so students cannot skip over video or questions – your choice.
Screencastify: Every How To video above was made with Screencastify. Like EdPuzzle it allows me to crop out random bits, but it also allows me to add in video pieces, merging them to one relatively seamless video.
Use this if
More info here
+You teach from slides
+You want to be able to and edit audio to your slides easily
+You’d like an option to show your face in a small corner screen
In mid March 2020, with virtually no time to plan and prepare, schools around the nation shut their doors and moved instruction online. Amazing, wonderful teachers scrambled to learn how to set up google classrooms, record lessons, magically get lessons on usb drives uploaded and assigned in those shiny new online classrooms. They learned or relearned how to assign work digitally through Khan Academy, Delta Math, iReady, or 100 other digital math practice platforms.
The first weeks were trial and error. Teachers put all 5 classes into one classroom, and told students to upload pictures of the work for each assignment. At a minimum of 2 photos per student per assignment, 4 assignments a week to 160 students made a huge number of images to be opened 1 at a time and sorted onto the right grade sheets. And some students had trouble uploading into google classroom because, hey, it was new for them too. So teachers found themselves recipients of photos via email, text, and tweet. Some were dropped in the school office on potentially germ covered paper, which meant driving over to pick up and grading with gloves.
One teacher team that works exceptionally well together but had never used google classroom divided labor. One person made worksheets, one made videos, one photocopied packets for pick up and one made the google classroom. One google classroom. For all 650 students. As a place for info to go out it was fine. But as things started getting turned in, well, it was pretty overwhelming.
In the midst of all this craziness, and a learning curve that rivaled any pandemic’s exponential growth, the easiest thing to do was to create a learning experience that, as much as possible, mirrored what was happening already in their classrooms. It was comfortable for teachers. It was comfortable for students. At a time when everything about life was different than what it had been one week before, stability and consistency in classroom structure was what we all felt like we needed.
And then it began popping up . . .
What is missing?
The kids. Being together. The laughter. The high fives.
The humanity. The heart. The community.
The sounds of discovery and failure and success!
The reason we went into teaching. The reason we stay.
We miss our kids.
And no matter how independent we try to raise our learners to be, they miss us. Kids who never thought they would miss us, miss us. The feeling is mutual.
And honestly, what were we thinking??
Some of us have been recording lessons and providing them online for absent students for years. How much has that worked for that 6th period baseball player who misses 2 days a week all spring? Uni-directional imparting of information, no matter how slick your videoing or screen casting skills, can never replace the richness of our classroom communities. And at this time, when things are scary, and resources are scarce for some of us, when life itself seems uncertain, maybe it is this community that is the most important thing to keep constantfor our students.
How do we make room for humanity in our digital classrooms?
Be intentional about connecting personally. Take time to ask, “How are you doing?” communicate back and forth with students when they take you up on those opportunities and share with you. One of my favorite ways right now is using this great set of distance learning starter screens from Desmos. Turns out editing a Desmos activity or creating one of your own is not very hard.(Who knew!) Choose a slide or two each week to give your students a chance to share if they need to. If not Desmos, give them a writing prompt or two to chose from each week. They can’t just hang after class to talk. Let them know your door is still open.
Allow students some flexibility and choice. We are in a time where we have so little personal control. Give students the opportunity to choose which way they are going to show they have learned something. Hyperdoc lessons are a great way to build in self-pacing flexibility and/or activity choice.
Build in opportunities for creativity and play. Learning activities don’t have to look exactly like a math book. There are lots of ways to practice the mathematics they should be learning in creative and playful ways. And because creativity and play are two well proven stress relievers, adding opportunities for those in your classwork is taking care of your students in a uniquely human way.
Make room for mistakes and second chances. I know they are moving ever closer to adulthood, and teaching them responsibilities and natural consequences is important. But right now is a time to hold a little more loosely to those parts of our teaching. Instead take the opportunity to teach them about kindness and grace. We as teachers may never know all the extra stressors that are attacking each child’s home life.
Do they have a parent who is separated from the family because they are a health care provider?
Is a relative or loved one who is far, or even in the home, critically ill or especially endangered by this pandemic?
Have one or both adults in the home lost their jobs?
Do they have new responsibilities as an instructor of younger siblings?
Are they simply a child that takes the worries of the scary world we live in deeply to heart?
Any of these things could affect their ability to spend time on learning, or the likelihood of the learning they spend time on sticking with them. It is time for grace.
And honestly, it is time to spiral back to things again and again. Let’s do less, but do it with love, playfulness, humanity and grace.
Face the fact that students might not learn all of the math that you would have taught them if this was a different year. Embrace the fact that you have more to give them than facts and formulas. Remember why we stay in this stressful, crazy, exhausting profession and give yourself permission to make caring for your students the first priority during this once in a lifetime moment in your teaching career.
“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.
From deep in my archives – this is a lesson I used long ago to motive simplest radical form. It was one of my early ventures with Math Practice 7 – Look for and make use of structure.
So you are about to teach special right triangles, and to do that you want students to understand simplest radical form. The problem with simplest radical form from your students’ perspective is that it is a needless complication that does not add any meaning to the situation. 7.071 is much easier to understand as a quantity than 5√2. But we know that the patterns are easier to see if they use simplest radical form.
So . . .
What if you started with trying to get them to notice the pattern in the sides of a 45-45-90 triangle using decimals? Here’s how I envision a possible intro lesson going.
Give them two angles in each triangle and have them notice they are similar. Then give them one side on each triangle and have them find other two sides (isosceles, pythag).
Here are some possible triangles you could use, and the answers they would get in decimal form:
It will be handy if the Pythagorean theorem work for these is on board to come back to, so you could have students put up this work.
Teacher says: “Using the patterns you see, can you predict the length of the hypotenuse for a 45-45-90 triangle with leg =7? No pencils allowed – predict in head.” Take guesses . . . .too high, too low, just right
Now use math to check who was closest. Answer: 9.899
Students may use similar triangles to set up proportions and solve between triangles if they have worked with those recently. That is fine.
Tell students: “In this next unit we will be learning how to use the consistent ratio between two sides of the one triangle (for instance hypotenuse to leg in the ones we have been looking at) to find all the angles and all the sides of any triangles similar to it. So what is the ratio in this case?” Have students find.
Teacher talk: So decimal is consistent but not catchy. Could we have used it?
Close. Off due to rounding.
Perhaps play with including more decimals to increase accuracy.
Would be nice if there was a way to be more exact without having to write out hundreds of decimals.
Tell students: “There is another way to express radicals to make pattern easier to see and use.” Go back to pythag work on board and break down the square roots. I would suggest for this moment that you use the greatest square method so they can understand what you are doing.
The point now is not to teach them how, but just to let them see helps see the pattern, so that they might see some value in learning how.
Teacher: “Anyone want to guess the hypotenuse for a 45-45-90 triangle with leg 7 now? No pencil! Answer: 7√2 !
How accurate is that?”
(Show them how to use calculator to check.) Pretty darn accurate!
“Can anyone guess the decimal for √2 ?”
Transition . . . “Before we look for more patterns, let me teach you how to rewrite square roots to help you see the patterns.” Or if you’re on a roll and feel experimental, cut a equilateral triangle in half to make a 30-60-90 and see if they can find the 30-60-90 ratio themselves in radical form. They may end up “inventing” how to break down simple square roots all by themselves.
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.