8th Grade Unit 3: Functions (Part 3 – Point Slope Form & Standard Form)

Here’s part 1 and part 2 of unit 3.


I used this warm-up the first day after our test on slope-intercept form to get students thinking about equations and graphs again.

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Then I do notice/wonder with this.

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I heard things like:

  • There are two x’s and two y’s.
  • There are little 1’s by one of the x’s and one of the y’s.
  • There are parenthesis.
  • There’s an m (slope).
  • There’s no y-intercept
  • Is it another form of a linear equation?

It leads nicely into discussing point-slope form and students realize that it isn’t as scary as it may look at first because they recognize the similarities between slope-intercept form.

When going over point-slope form, I make a point to emphasize to students why it’s named point-slope form -we can see the coordinates of a point and the slope from the equation.  I remind them that this is similar to slope-intercept form where we saw the slope and the y-intercept.

Then we go over a few examples of writing equations in point-slope form before doing an activity similar to what Sarah shared here.  I didn’t have big foam die like Sarah used, but I do have double dice, which students always think are fun.  Students rolled the dice to create two ordered pairs and wrote an equation in point-slope form of the line between those two points.  Then they checked their answers using Desmos.  Having students check with Desmos was key to helping them see what they were doing when writing the equation of the line.

I also modified this Desmos marbleslides activity to rearrange the equation so that they looked like what my students were used to seeing.  My modified version can be found here.


Then after some more practice using point-slope form, students are introduced to standard form.

I use notice/wonder again to get students thinking about this.

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Again, students came up with the following things:

  • There’s 2’s in all of them.
  • The two is always by the x.
  • One of the equations is in slope-intercept form.
  • One of the equations is in point-slope form.
  • In the purple one, the x and y are on the same side.

We also talk about how, unlike slope-intercept form and point-slope form, we don’t see the slope, the y-intercept, or a point.


Of the three parts to this unit, this one is takes up the fewest number of class periods.  Writing up this post made me realize that I could probably use a few more activities on these concepts.  If you have any ideas for me, please share!

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8th Grade Unit 3: Functions (Part 2 – Slope & Slope Intercept Form)

After talking about functions vs relations and linear vs nonlinear, we get into slope and slope intercept form in 8th grade.


Slope

To intro this I start by using a Desmos Polygraph.

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Then this year I tried this Desmos activity on steepness.  I liked we were able to move from talking about something students were familiar with, steepness, to a word students may not be familiar with in slope.

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Then I was able to use use this awesome activity Sarah shared here with some of my students, since they were a couple days behind the other class.  It was such a simple idea, but I absolutely LOVED how students figured out on their own how to find the slope of a line.  They also were able to explain when slope would be positive versus negative.  For next year, I might change some of the lines to include slopes such as 2/3 or 3/4.

And what would be a lesson on slope without Slope Dude?  I blogged about another activity I shamelessly stole from Sarah here.sd2

I used this Desmos activity I found online to have students practice finding the slope of lines.  After that, we talk about the slope of horizontal and vertical lines.  Students already know from playing Slope Dude that horizontal lines have a slope of zero and vertical lines have an undefined slope, but now we talk about why.

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Slope Intercept Form

A while ago on Twitter Mickie asked for ideas on introducing Slope Intercept Form to students.

I shared with her a worksheet I created a few years ago where students use Desmos to figure out slope-intercept form.  She had some great ideas of how to improve what I had created.  (Twitter and #MTBoS are so great!)  She shared with me what she and a colleague came up with.  I loved her addition of the table.  Here‘s the editable version of the worksheet she shared with me -I did make a few minor changes.

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When I’m teaching slope-intercept form, I try to make a big deal about how the name for this form of an equation makes sense based on the formula itself.  When we’re given an equation in this form, we can easily see the slope and the y-intercept, hence the name slope intercept form.  This is something that is obvious to me as the teacher, but I found that students don’t always make this connection.  Because of that, it’s important for me to help students make that connection.  This also helps later on when we talk about point-slope form.

One change I made this year to how I teach this is that I had students check their answers after graphing.  I decided to do this for a couple reasons.  One, I hoped that this would slow students down and help them catch mistakes they made when doing the slope of the line.  Two, I hoped that this would help students make the connection between the graph and the equation.  In the past, I don’t know that I have done a good job of helping students make this connection.

After students are comfortable with equations in slope intercept form, we go over writing an equation from a graph, writing equations for horizontal and vertical lines, as well as needing to get y by itself before graphing.

Here is the link to a Desmos activity I used to have students practice going from graph to equation.  I took the images from somewhere online, but I can’t remember where.  Sorry!  If anyone recognizes them, please share so that I can give whoever created them credit.

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You can download a worksheet and the test review here.

And last but certainly not least:  Desmos Marbleslides.  This is one of my absolute favorite activities from Desmos.

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Small Change

Do you ever make a small change to something you do every day that ends up making a HUGE difference and think to yourself, “Why didn’t I think of this years ago?!” or “Why don’t I remember to do this more often?”

This happened today, and it ended up to be pure magic.


Today in one of my classes I was introducing point-slope form.  I put the formula up on the board and asked students what they noticed and wondered about it.  Then we talked a bit more about it and did a few examples.

Then I gave students the coordinates of two points and said that we want an equation in point-slope form that goes through those to points.  Their task was to figure out how to do that.  They found a partner and got to work.  After a couple minutes, the chatter started dying down, and I could tell many groups were stuck.

This is where the magic happened.

I stopped the class and told them that for about 30 seconds they were going to find a new partner and talk about what they’ve been trying with the problem.  Then I was going to stop them again, and they would return to their original partners and keep working.

The room instantly came alive again with students talking about what they had tried.

Many groups were stuck at different places or had tried different things so my students had a lot to talk about with their new partner.  When they went back to their original partners, they had new ideas and were able to continue working.


I had done something similar to this last year, or maybe even longer ago than that, and why it took me this long to do it again, I don’t know.  I think my fear when I do things like this is that my students will be more off task; however, today the opposite happened.  The conversations within my groups were more on task for longer periods of time than they typically are in that class.

I want to incorporate this more into my classes, but at the same time be careful not to overuse it and be intentional about when I use it as I definitely think there are times something like this would be more effective than others.

 

8th Grade Unit 3: Functions (Part 1 -function vs relation, function notation, and linear vs nonlinear)

The first test in unit 3 for 8th grade covers the difference between a relation and function, function notation, and determining from a table whether something is linear or nonlinear.

I followed pretty closely to what I did last year.  You can read about that here.

I did use Sarah’s updated version of her representations of relations telephone activity.  I blogged about it here, and you can directly download her version here.

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Once we got into functions versus relations (again read more about what I did last year here), I used this Desmos Polygraph this year.  After I did it with one class, I ended up copying and editing the activity and changed the circle to another graph because my students kept thinking it was funny to pick the circle and have the other person guess on the first try…oh 8th graders!

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Then we started looking at tables and determined from a table whether or not the graph would be linear.  I feel like this portion of the unit is what I need to focus on improving the most for next year.  It went alright, but I didn’t love it.  I started with the following image and asked students what they noticed about the two and what made the green graph a straight line and not the red one.

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Usually someone will say that on the green one, the y’s go up by two’s.  Then I put this image up and ask why that theory doesn’t work in this case.

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We continued talking about what makes the graph linear, and the next day I used this Which One Doesn’t Belong? for a warm-up.

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When it came time to review for the test, I used a function vs relation Kahoot and this great open-middle type problem from Sarah to review functions versus relations.  I used it the same way Sarah did and had students use the numbers -4 to 4 and first had students place the numbers in the boxes so that the three relations were also functions.  Once students completed that, I had students place the numbers so that the three relations were not functions.

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I also have an Add ‘Em Up activity for function notation I created.  You can read about Add ‘Em Up here, and download the activity here.

6th Grade Unit 3: Fractions (Part 1 -Greatest Common Factor & Least Common Multiple

I start our unit on fractions by going over Greatest Common Factor and Least Common Multiple.  It seems that no matter how many times we go over what factors of a number are and what multiples of a number are and break apart what it means to find the “greatest common factor” and “least common multiple”, some students still get these mixed up.  I try to be super intentional about not saying “GCF” or “LCM” with students very often because I know there is likely at least one student who doesn’t know what those letters stand for -I try to use the vocabulary as much as possible with students instead of acronyms.  I want them to know what GCF and LCM mean because they will likely see that other places, but if I say “GCF” I always pair it with “Greatest Common Factor”.

We start with greatest common factor and first review what factors are, and then discuss what it means to find the greatest common factor of two numbers.  Once students get a quick review of this, they usually remember doing this in 5th grade.  I found this year that the methods my students used as 5th graders to find the greatest common factor were the “rainbow method” and T-charts.

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We do a few examples where students use one of those two methods before I put up an example with larger numbers like 192 and 320.  Students usually give me the reaction I’m looking for by groaning, and I ask why they’re groaning.  They tell me the numbers are big and there’s a lot of factors to list.  I then ask if they’d like a more efficient method of finding the greatest common factor and introduce students to what I call the “ladder method” to find the greatest common factor.  I think I first heard about it from Sarah here.  The high school teachers in my building use this method with polynomials, so I want to introduce it to students.

For practice, I have students write a number on a slip of paper that has many factors.  Then students pair up and find the greatest common factor of their numbers.  I have them check their answers with Desmos.  (Did you know that Desmos can find the greatest common factor of numbers?  It will also find the least common multiple.)  Then once they’ve checked their answers, they find a new partner.

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Then we get into finding the least common multiple of numbers.  Again, we review multiples, and how they are different from factors.

 

 

(We review the above nearly every day in this unit, so you would think I’ve got how I want to display it on the board for students down pat.  This is what I ended up liking the best, although I’m still not sold on how I showed factors.)

Then we talk about what it means to find the least common multiple of two numbers.  Students usually start to remember doing this as 5th graders.  We do a few examples of this before we talk about how we could use the ladder method to find the least common multiple.

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Next comes word problems.  I’ve modified this worksheet for practice for students.

Then I have a couple different loop activities that I use depending on when they best fit into the schedule.  The first has greatest common factor, least common multiple, and prime factorization, while the second loop activity has greatest common factor, least common multiple, and word problems.

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Each activity has 6 problems, which is nice because most groups can finish within a class period.  However, it’s not nice for keeping groups small if I want one group per problem. I ended up printing two copies of the activity on different colored paper, and this worked great.  Half of my students rotated through one color set while the other half rotated through the other color set.

You can download the files for both activities here.

8th Grade Unit 2: Inequalities (Part 3 -absolute value special cases and word problems)

The last part of our unit on inequalities covers absolute value special cases and word problems.  You can read about part 1 here and part 2 here.


Absolute Value Inequality Special Cases

To introduce solving absolute value inequalities that have no solution or all real numbers as the answer, I tried a couple different things this year.  In a couple classes I put this up on the board and we talked about each of them, which is what I had done in the past.

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In another class I put the following up on the board and asked how we could write an inequality that would never be true.

|x|______________

Then I asked how we could write an inequality that would never be true.  Once students gave me answers for that and understood, we talked about if it mattered what was inside  the absolute value bars.  Both ways worked fine, but I almost think I liked think I liked this new way better.  I liked that students were coming up with the inequalities.

The next day I started with the following warm-up problem.  I always like to include absolute value inequalities that don’t have all real numbers or no solution as the answer so that students don’t forget what they already knew about those types of problems.

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Here is the link to download a worksheet with these types of problems.

I also created an Open Middle type problem for these types of inequalities.  You can download it here.

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Absolute Value Inequality Word Problems

One of the things that my students need to know is how to solve word problems involving absolute value inequalities.  In the past, I just jumped into these types of problems, but this year I took a day to go over setting up the absolute value inequalities given a graph.  The day before this I had each student solve an absolute value inequality to use as this introduction.  You can download the problems I used here.

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I took several of these and had students find the midpoint of each graph and the distance from the midpoint.  Then students looked for a pattern.Screen Shot 2017-12-08 at 7.10.45 PM.png

Then I had students practice writing absolute value inequalities when given a graph and practice finding the midpoint of the graph and distance from the midpoint when given an absolute value inequality.

The next day when we got to solving word problems went so much better than it did last year having spent a day on this ahead of time.

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You can download the test review here.

6th Grade Unit 2: Intro to Algebra (Part 4 -Inequalities)

The last part of our Introduction to Algebra unit in 6th grade is inequalities.  I also wrote about part 1, part 2, and part 3.

I don’t spend a ton of time with this.  I start with a few Desmos activities.  Polygraph is the first thing we do, and then I come back to it a day or so later after students have done more with inequalities.  Then we review the inequality symbols before students work on this Desmos activity.  It is a basic introduction to graphing, and it is usually the first time that my students see the greater than or equal to symbols and less than or equal to symbols.

The next day to review we start with this Which One Doesn’t Belong.

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Then we work a bit on graphing inequalities when given a situation.  This Desmos activity is one of the first things we do to practice this.

The last way we review is Mathketball.  My students LOVE this game.  They put their desks in a circle around the room.  I put a garbage can in the middle of the room.  I put a problem up on the board, and if students get it correct they get to try to shoot their paper in the basket.  Students would play this for every concept if I let them, but I try to save this for problems that I anticipate all students completing in the same amount of time compared to problems that have multiple steps to get the answer.

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I want to find more pictures like these that students can write inequalities for.

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The day before the test a student asked how many problems were going to be on the quiz.  I said, “8 problems tops.”  Another student said, “You should make us write an inequality for that.”  I just love these kiddos so much.