Function Notation Same/Different

I have seen other people use the prompt “What is the same?  What is different?” around Twitter (Check out #samediffmath), but I’d never formally used it in my classroom.  I have asked those questions before on the fly, but I’d never created something to put in front of my students where those questions were the main focus.

As I was driving to school Friday and was thinking about what I was teaching that day, I had this thought to create one for function notation.  Function notation is something that some students struggle with, and it sort of surprises me every year because it is so similar to things they’ve already done.  Here’s what I came up with.

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I printed these out on half sheets of paper and had my students do a Stand and Talk with them.

Here are some of the responses I got for “What is the same?”

  • Both have an answer of 57
  • The last 5 rows are the same.
  • Both replace the x with 4.
  • Both have 3x^2 + 2x + 1

For “What is different?” we talked about how on the right it has f(x) and asks to find f(4), and on the left instead it says “evaluate…for x = 4”

We talked about how so much of the problems are the same, but if I just gave my students the top row, they would know how to do the left one, but would feel completely lost with the right one.

In my second class, I explained how these questions are asking something very, very similar but the notation is different.  I thought of the example of in elementary school if they were given 5 × 3, they would know exactly what to do.  However, if they were given 5 • 3, they wouldn’t, even though it is asking the same thing.  As I was explaining that, I could see some students making the connection to the two problems we were looking at.

I LOVED using these prompts intentionally in my classroom, and I’m looking forward to finding more ways to incorporate this into my classes.

(Here are some of the other things I’ve done with function notation.)

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Equations and Inequalities with No Solution or Infinite Solutions

I was looking for something a little bit different than what I had done in the past to introduce equations that have no solution or infinite solutions.  I came across this post from Sarah who blogs at Everybody is a Genius, and it was exactly what I was looking for.  I also liked this because when I had these students as 6th graders, I used scales to introduce them to solving equations, so this wasn’t a new idea for them.

I gave this sheet to students and told them to fill in the boxes to keep the scales balanced, and that for each scale, the number in the box must be the same.  Students have done a few different Open Middle problems this year, so some students struggled with the idea that they could no reuse numbers since they are used to not being able to reuse them for those problems, but they eventually understood what to do.

Equation Scales

As I was walking around, exactly what I hoped would happen, happened.  Students got two number 3 and I heard, “What?  This doesn’t make sense.”  “This is impossible.”

As we went over what students came up with, we discussed how in #1 and #4, we could pick any number we wanted, in #2 and #5 only one number works, and in #3, and #6 no numbers work.  Then we took some notes on this.  In the notes sheet I handed out to students, I included a picture of the scale and we wrote out the equation and showed what was happening to the scale as we did the algebra.

I liked that introducing this topic this way to students gave students a visual to help them understand these types of equations.


The next day we did some practice at the whiteboards.  I always include some problems that have one solution (especially ones where x = 0) because some students want to start saying every single equation either has no solution or infinite solutions, even though I stress that this only happens when the variables are eliminated.

Sarah Carter has created a nice Open Middle style problem to go with this topic.  Here students can use the numbers -4 to 4.

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Last week we worked on solving inequalities with infinite solutions and no solution.  I really liked what I did last year for this, so I did something similar this year.  I started the day by having students solve an equation that had no solution.  Then, I asked students which inequalities would make that true and which would make it false.

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We briefly discussed which would make it true and which would make it false, and that was pretty much the only instruction I gave students that day.  They had little to no trouble transferring the idea of equations with no solutions or infinite solutions to inequalities.

I shared at the end of this post a Desmos card sort I use as well as another Open Middle style problem on this topic.

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Overall, I’m really happy with how students are doing with these types of problems.  I think that introducing this idea using the scales really helped my students to see what was going on.

8th Grade Unit 6: Exponents (Part 2 Scientific Notation)

I shared part 1 of our unit on exponents here.


I got most of my notes from Sarah’s blog.  She also has a ton of activities on her blog here.

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As I was writing this, I remembered this image that Heather shared from one of Sara’s presentations.  I think this would be a GREAT way to introduce scientific notation next year.  I’ve got to remember to do that!

The last couple years, I’ve used tables similar to those below to help students notice patterns.

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After we talk about converting between standard form and scientific notation, I’ve used this Desmos activity.  I also like this Desmos activity.

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Then we get into multiplying and dividing numbers in scientific notation.

I made this Desmos activity for practice.

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The biggest thing my students struggle with at this point in the unit is when they multiply or divide and get a number that isn’t in scientific notation.  Something like 64 x 10^6.  They know the exponent will change by one, but many students get mixed up on whether it gets bigger or smaller.  I always, “Don’t try to memorize a “shortcut”.  Think about what 64 x 10^6 is.  Write it out in standard form, and then convert it to scientific notation.  Then you don’t have to try to memorize anything.”  The students that listen and follow my advice, usually have no issues with this, but it’s the students who want to take a “shortcut” that end up not getting these problems correct.  Please tell me I’m not the only one who has this issue!


I’ve got a couple Which One Doesn’t Belong? warm-ups for scientific notation.  I know I pulled the second one from Twitter.  I can’t remember who shared it.  If it’s yours, please let me know so I can give you credit for it.

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I’ve used this scavenger hunt as well.  I like that it gets students up and moving around.

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I created this worksheet for students to practice.  (I think I created it.  I may have modified it from somewhere.  Again, if you recognize it, please let me know so I can give credit to who originally created it.) . You can download it here.  I’ve created a few other worksheets of this format and like that it’s self checking for students.

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8th Grade Unit 5: Systems of Equations (Part 2 – Elimination & Choosing Method)

I shared here part 1 of our unit on systems of equations -solving by graphing and by substitution.


Elimination

I started this unit with the following warm-up.

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Then I followed it up with this notice/wonder.

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One of my students this past year noticed that, “These are like the problems we did for warm-up.”  I love when my students notice connections between the warm-up and the lesson for the day and that I’m not just having them do random stuff.

I was really pleased with how this led nicely into the lesson for the day.  Students noticed that in each problem there was a zero.  They were able to tell me why that happened.  When I told them that this is another method for solving systems of equations called elimination, at least someone in each class was able to explain why they thought that elimination was a good name for this method.

It had been a while since we had done Vertical Non-Permanent Surfaces, and these problems work great for that.

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Choosing Method

The last part of this unit was having students choose the best/most efficient method for solving a system of equations.

I started with this Desmos Activity.  I didn’t have students solve the systems the first day.  I wanted them to just think through what method they would want to use to solve each.  The next day they actually solved some of the systems.

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I love this Desmos Activity from Paul Jorgens.

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I’ve uploaded some of the worksheets and notes I used from this unit here.

8th Grade Unit 5: Systems of Equations (Part 1 – Graphing & Substitution)

Graphing

Even though this is Unit 5 in 8th grade, this is the unit I’ve done last the past couple years I’ve taught it.  It just worked out that way my first year teaching 8th grade, and I liked it that way so I did it again this past year.  I like having it at the end of the year because it’s a good way to review graphing lines.

I start the unit by spending a half day or so on just a review of graphing lines.  I almost didn’t do that this year, but was SO glad that I did.

Then we do Desmos Polygraph and this Desmos activity to intro systems of equations.

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Then we spend some more time practicing.  I’ve used this Desmos activity in the past.

After that we get into systems of equations that have no solution or infinite solutions.  I usually start by putting up one of those types of problems without giving students any explanation.

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Students have had a lot of experience solving problems with no solution or infinite solutions by this point in the year, so I usually have some students who come up with the fact that because both equations are the same line, there are infinite solutions.


Substitution

I blogged here about how I introduce substitution to students.  I’ve used this method 3 years in a row and am still amazed every. time. how smoothly it goes.


Here is the link to download some of the worksheets I use for this portion of our unit.

8th Grade Unit 4: Applications of Lines

We start our unit on applications of lines by discussing independent and dependent variables.  I have a note to myself to remember to use the following language next year because it worked well this year.  Nothing earth shattering, I know.

  • “(independent variable) causes change to (dependent variable)”
  • “(dependent variable) depends on (independent variable)”

I use a lot of Sarah’s resources found here for my notes, and I’m pretty sure that’s where I got the problems for this Desmos activity.

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The next day we do Sarah’s Ghosts in the Graveyard activity with independent and dependent variables.  Every time I use that activity I think to myself, “Why don’t I do this more often?  It’s great!”

After students have a pretty solid understanding of defining the dependent and independent variable, writing linear equations from word problems goes a lot better.


Then we get into parallel and perpendicular lines.  I blogged briefly about what I did last year here.

I start with parallel lines and use this Desmos activity.  One of the downfalls of starting with that activity is that when students are asked to solve problems where they need to write the equation of a line parallel to a given line through a specific point, they want to use Desmos to guess and check.  This is a good strategy, but I also want them to know another method.  I start the next day with a couple problems like these.

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After spending another day or so on parallel lines, we finally get into perpendicular lines.  I start with this Desmos activity.

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We spend another day or so practicing with perpendicular lines.  I’ve used this activity before and like how it brings back different forms of lines.

We also talk a little bit about parallel and perpendicular lines and quadrilaterals using this Desmos activity.

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Then we get into scatter plots.  I start with this Notice/Wonder

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We do Desmos Polygraph next.  Last year I had a student ask if there were two “loners”, and I will forever think of outliers as loners.

After students do that activity, I put the graphs up on the board and ask students to put them in groups.  They end up describing the different correlations to me.

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This Which One Doesn’t Belong? is great around this time in the unit.

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I took a couple tasks from this page and turned them into Desmos activities.  (I know she tweeted out links to the activities at one point, but I couldn’t find them.

Here’s one on correlation.

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And another on lines of best fit.

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Here is the link to download some of the worksheets I use in this unit.

VNPS Variation

(VNPS: Vertical Non-permanent surfaces)

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This is a different time we did VNPS, but my classroom looks like this nearly every time we do this.  Why don’t I do this more often??

Last year I created a “Two Truths and a Lie” worksheet on exponent worksheets.  Students simplify three problems.  Two of the answers are the same, and the third answer is similar but not quite the same.  I was really happy with how the worksheet turned out and how it went last year.

The day I had this worksheet in my lesson plan, I realized I wasn’t really looking forward to another worksheet with my 8th graders.  I try to mix things up, but lately they’ve had worksheets pretty consistently.  I was thinking through other options and was trying to come up with a way to change things up less than 2 hours before I had students in my room.

Screen Shot 2018-04-05 at 8.33.32 PMScreen Shot 2018-04-05 at 8.33.43 PMScreen Shot 2018-04-05 at 8.33.49 PMScreen Shot 2018-04-05 at 8.33.54 PMWhiteboards!  I realized it had been a while since I’d had students work at the whiteboards.  I took screenshots of the problems, projected them on the board, and had students work in groups through the four problem sets.

In the past when students worked at the whiteboards, I had them do one problem at a time and get it checked by me.  This time, I had students do all three problems in each set before I would check their answers.  They knew that two of the answers were going to be the same and one would be slightly different, so they already were doing some self-checking as they went along!  When students called me over to check their answers, I would tell them how many problems they had wrong but wouldn’t tell them which one.  I really liked the small change in how I had students do these problems.  This is something I do a lot when they’re working in Desmos or with other activities, but I hadn’t done it in this situation before and loved how it went.

I had more students engaged for more of the hour by doing the problems this way, and my students were talking through the problems more than they would have if they were working on the worksheet.  I actually had students cheer when I said we were going to the whiteboards.  🙂 It was a good reminder that I don’t have to always come up with some fancy activity to switch things up.  Something as simple as taking problems from a worksheet and having students complete them in a different way is enough sometimes.


Here’s the link to download the worksheet and others on exponents.  Here is where I shared other things I did in this unit last year.