About Leeanne Branham

Secondary Math Teacher on Special Assignment Curriculum and Instruction, CUSD Clovis, CA *Opinions expressed entirely my own.

All in with IM: A day in the life- Grade 8 Unit 7, Lessons 9-11


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

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

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

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

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


Previous day:

Lesson 7 with a quick peek at 8.

During announcements:

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

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

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

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

Opening Discussion:

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

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

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

Part 2:


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

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

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

Desmos Application:


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

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

All In with IM – Grade 8 Unit 6 Associations in Data

True confessions:

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.

Image result for average home price vs year

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.

Links to questions pictured in this post: review as pdf

review word version

extra practice

All in with IM – Grade 8 Unit 7 Exponent and Scientific Notation Practice

Here’s a quick resource for exponent and scientific notation practice that aligns nicely with Illustrative Math / Open Up Resources Grade 8 Unit 7.

I first saw the “One Incorrect” activity model on Greta B’s blog Count 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.

Lesson 11 goes through the same thinking, only with very small numbers. Desmos has a activity, The Solar System, Test Tubes, and Scientific Notation, that combines lesson 10 and 11, as well as some fun Scientific Notation practice.

In the Activity attached at the top, there are also 4 pages practicing the second half of the unit.

All In with IM: Grade 8 Unit 5 Lessons 11-16

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.

All In with OUR/IM – Grade 8 Unit 3 Lesson 5-7

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. 

The digital version linked in the curriculum is great (pictured below).

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.

2 Mystery Mixture lab from Grade 7 Unit 2:

Motivating simplest radical form

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.

  1. 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:

image 1

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.

 

  1. 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.

 

  1. 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.

image 2  

 

  1. Teacher talk: So decimal is consistent but not catchy. Could we have used it?

7(1.414) =9.898

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.

 

  1. 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.

 √50= √25∙2=√25∙√2=5√2

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.

triangle 1:  5, 5, 5√2         triangle 2:  3, 3, 3√2         triangle 3:  4, 4, 4√2

 

  1. 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  ?”

 

  1. 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.

All in with OUR/ IM Math 6-8 : New for Year 2 – Extra Practice pages for each lesson

The teachers of our district worked together to create resources to address a variety of problems they had during year 1.

Made to meet a variety of teacher needs:

  • I keep running out of time for the cool down
  • I want to allow students a chance to reflect and revise cool downs
  • I don’t know how to collect homework/ hold students accountable
  • I wish my students had a little more basic practice available for home
  • I want problems aligned with the curriculum that I can
    •  use for whiteboard practice during my extended block 
    •  use for practice during my support class
    • Send home for students who work with tutors
  • I want to spiral in more review of previous grade level content before students need it in the curriculum

Here is our set for the first half of grade 8 unit 1.  Here is a copy in word for easy editing.  We are spiraling in material students will need for unit 4 throughout the first 3 units.*  You can see that in the first half of unit 1 we focused on one step equations, using fractions and decimals with comfort, and the distributive property. Our goal is to have students comfortably at grade level as they begin unit 4’s equation work. We never want these reviews to distract from the lesson at hand, and so this review is short and builds slowly.

How teachers might choose to use this resource:

We hope it is flexible enough to fit each teacher and classroom. Here are a few ideas:

  • Keep current cooldown routine but limit time – this is their rough draft only. Do a quick Critique, Correct and Clarify (MLR 3) with one of the rough drafts, perhaps structured as a “My Favorite No”.  Students do a Stronger and Clearer (MLR 1) version of the cool down as part of homework.**
  • If you need to collect something from students daily – give one page per day.  Blank back side can be used to work the 3 to 5 practice problems provided by the curriculum
  • If you just want students to have access to extra resources for mastery – hand out as a packet at the beginning of the unit. 
  • If you want extra problems to draw on for classroom activities (stations, white boards, intervention), use problems as a teacher resource to create these activities. 

Unit 1 Lessons 1-10 pdf word

Unit 1 Lessons 11-17 pdf word

Unit 2 Lesson 1-12 pdf word

Unit 3 Lessons 1-14 pdf word

Unit 4 Lessons 1-15 pdf word

Unit 5 Lessons 1-10 pdf word

Unit 5 Lessons 11-21 pdf word

Unit 6 Lessons 1-10  pdf word

Unit 7 Lessons 1-15   pdf  word

Unit 8 Lessons 1-15   pdf  word

Please share other ways you find to use these!

* I’ll continue to add future units here as they are ready.

**For more on math language routines see the course guide – tons of helpful information is waiting for you there!

All in with OUR/ IM Math 6-8 – Looking back at Year 1

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??

  1. Our curriculum was better aligned to the grade level standards. Teachers spent time teaching the things they were supposed to teach at the correct level of rigor. 
  2. Teachers learned more about the math content. Teachers who planned using the teacher materials were pushed to think more deeply about the math they were teaching. They begin to see connections they did not see before. It was amazing PD to dig in and learn this course. 
  3. Teachers began to understand the vertical connections and their place in the progression of student learning. The curriculum emphasized building on previous year’s learning all year long. 
  4. Spiraling allowed mastery over time.  Topics came up again and again, which allowed students to revisit and deepen their understanding. Research shows that this has a profound impact on retention.
  5. Pedagogy and access. The curriculum is built to support teaching differently than we have before, in a way that promotes access and understanding for all students.   Some teachers experimented with this more than others, but for all of us we are just at the beginning of this learning. Note – lots of teacher and student growth could happen without this piece, but the greatest growth came in classrooms that experimented with these new methods. 

We can’t wait to see what happens next year now that we know the curriculum and are addressing some of the problems with pacing and practice that we had during the first year.


So the moral of our story is don’t be afraid to jump in. Even with our less than perfect execution, the curriculum was good for our students.

And that is what we are all about.

Fighting Courage

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.

All In with OUR/IM – Equations of Lines based on Slope

Grade 8 Unit 2 Lessons 10-12

 

When I say “equations of lines” what do you think of? Go ahead, write down the first 5 or so things that come to your mind .

Here is a collection of words and ideas I got from a few of my coworkers:

20181017_165230.jpg

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.

20181017_172605.jpg

And finally in lesson 11 we talk about writing an equation to show the relationship of all the points that are on that line.  Notice we are connecting to their 7th grade work that revolved around using equations to express relationships. So any two points you chose on that line can be used to draw a slope triangle. All of those triangles will have the exact same ratio of vertical side to horizontal side.  So in unit 2, THIS is what we mean when we talk about equations of lines:

Ratio of the sides of one triangle = ratio of the corresponding sides of a similar triangle

More specific to slope:

Ratio of the vertical side to the horizontal side on a slope triangle drawn along a line = ratio of the vertical side to the horizontal side on another slope triangle drawn along the same line

And more general to make an equation that shows the relationship for any points on the line

Using the general point (x,y) as one of the points creating the slope triangle, the ratio of the vertical side to the horizontal side on that slope triangle drawn along the line = ratio of the vertical side to the horizontal side on another slope triangle drawn along the same line.

So shake your head and try hard not to see a messy version of y=mx+b.  The equation we are writing is two equivalent ratios.

Lesson 11 takes on the work of saying how long the vertical and horizontal sides are if one of the points is going to be (x,y).  

Teaching tip:

Let one point be at (3,1). “Slide” your (x,y) around and keep asking for the vertical distance.

What if y was at 4?  

What if y was up at 7?

What if it was down at 2?

Instead of just writing

3

6

1

on those vertical sides, write

4 -1 = 3

7 – 1 = 6

2 – 1 = 1.

Then slide (x,y) way up past the edge of the grid, so they cannot tell the y value. Label it (x, y) and encourage students to use a variable expression to name that vertical length

y – 1

Do a rerun of the same logic with x.

The big idea for these equations of lines is that EVERY point that is on that line will make this proportional relationship true.  Because no matter what point on the line you pick to draw your slope triangle, the slope ratio will be the same. 

Here is an extra practice problem for the ideas from this section. Notice in the final part we use the equation to check and see if a point is one the line.

Screenshot (71)