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.
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.
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.
A personal post having absolutely nothing to do with math, but everything to do with teaching.
So as a kid I was a wimp. Actually I don’t know how much that has changed in the 45 years since my 10th birthday.
Fact about me: My pain threshold is record-breakingly low. As a kid this translated into crying and whining at even the tiniest injury, and being paralyzingly afraid of one thousand everyday things.
My mom would tell you I did have redeeming and lovable qualities, because, you know, she’s my mom. Even so, parenting what some might describe as a whiny crybaby and others might describe as a hysterical brat, couldn’t have been all sunshine and roses. The day the dentist said he wouldn’t see us anymore because 8 year old me was hysterical about the shot he was trying to give and impossible to manage comes to mind.
From my current vantage point, I am amazed I lived to adulthood. Mom is a better woman by far than I. Think about it for a moment. . . How would you parent this hot mess of a kid?
I can’t say her solution would work for every kid, because each child is unique. But for me her solution was pure genius. She couldn’t change the fact that things would hurt me. So she chose to teach me about courage. Not to “toughen up” or to change my perception of pain (“That does NOT hurt!” was something I heard from others). She taught me that being brave is not about not being afraid, but about facing that fear and doing what needs to be done.
She told me stories of children who saved their siblings from tidal waves and wild animals. She bought me books like Children of the Resistance, The Diary of Anne Frank, and The Hiding Place. She told me stories of Harriet Tutman and the Underground Railroad. . . stories of the Little Rock Nine and Rosa Parks . . . stories about Susan B Anthony and Elizabeth Cady Stanton.
She called them stories of Fighting Courage. That became the theme of my childhood.
She knew she couldn’t prevent pain from coming into my life, but she could help me choose my response.
She also chose to live fighting courage in front of me. Once we witnessed a road rage incident that turned into a brutal attack. A driver was pulled from his car through the open passenger window and pounded into the pavement by the other angry driver. 3 lanes of traffic were stopped at the light, and two full gas stations full of people looked on. My 5’3″ mother was the only one to get out of her car and yell at them to stop. “Everyone of these people can identify you and will testify against you! All those cars behind you are writing down your license plate right now!” I was terrified as she stepped from the car, but also incredibly proud. Her actions saved that man’s life.
In the wild, she taught me not to run from wild animals. When a big bear entered our camp, she stood tall and puffed up, and made noise to scare them off. I truly do not know what the correct move is in that situation, but what I learned was that my mom had fighting courage in the face of both man and beast. Not that she had no fear, but that she was willing to do what needed to be done to save herself and protect those around her.
At 56, mom continued her lessons about fighting courage as she battled stage 3 colon cancer. She confides now that there were times she thought she couldn’t take the chemo any more, and she would have to die. But I watched her advocate for herself in her care, chose nurses and people around her that were positive and encouraging, and decide to live.
From then until now, I have watched her attack the natural challenges of aging with the same fighting spirit. Exercise, healthy eating, mental exercise, and choosing to reach out and become an active part of her new community when she moved are all choices she modeled for me.
You don’t have power over evil in the world or natural disasters or disease or the aches and pains of time. But you do have the power to choose your response when those things come your way. The ability to face these things head on instead of hiding or wallowing in self pity is Fighting Courage. It is the greatest gift she has ever given me.
I am sure there are parenting moments she regrets, because every parent who ever lived has those. But this piece of parenting is what I needed more than any other. I strive to let it define my life as it has hers.
Today, at 77, she continues her lessons, this time with stage 3 breast cancer. Chemo starts this month. Here she is: my hero.
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.