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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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 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.

8th Grade Unit 2: Inequalities (Part 2 -Compound Inequalities and Absolute Value Inequalities)

You can read about the first part of our unit on inequalities here.  In the next part of the unit we do some word problems, compound inequalities, and absolute value inequalities.


Word Problems

I know the word problems I give students aren’t very “real world” and that this is an area I need to work on -finding/creating better word problems for students and doing a better job of teaching them as well as incorporating them into class.  I don’t have anything fancy I do for these other than a couple examples together as a class and then partner practice.

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Compound Inequalities

I use Notice/Wonder to start our conversation on compound inequalities.  Then we do this Desmos activity.  I also like this Polygraph activity for compound inequalities.

This year I also realized I could make a connection between “compound inequalities” and “compound words” and “compound sentences” that students are familiar with already from their English classes.  Why did it take me so long to do this?

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The following day I use this Which One Doesn’t Belong? for a warm-up and then we go back to the image from the notice/wonder the day before.  I put numbers on the graphs and students write inequalities for each.

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I used one of Sarah Carter’s awesome questions stacks for practice on solving these types of inequalities.  You can download the file she shared here.

One of my classes got to Point Collector.  This was my first time using this activity with students.  It’s SO fun!

One of my students came up with this for the last challenge.  He didn’t quite follow the directions exactly, but I love that he wanted to get the maximum number of points.  I overheard him telling another student about it later on during the class period when they were working on something else.  His friend goes, “Oh, so you cheated the system?” 😉

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Absolute Value Inequalities

I tried a couple different ways of introducing absolute value inequalities this year.  In a couple classes I started with Notice/Wonder.  Then in another class I started with an absolute value equation such as 3|x – 1| + 4 = 19 and had students solve that.  Then I changed it to an inequality and asked students what they thought would be the same/different about solving the problem.  Both ways of introducing the topic were good for different reasons.  I think for next year I may try to find a good combination of both.

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We did some vertical non-permanent surface practice with solving absolute value inequalities at the whiteboards around my room.

I also used this Open Middle type problem I made.  You can download the file here.

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Here is the link to download the test review.

8th Grade Unit 2: Inequalities (Part 1)

After solving many different types of equations in 8th grade, inequalities are up next.  We start by reviewing graphing inequalities before getting into solving them.  Then we also work on inequalities that have all real numbers and no solution as answers.


Review of Graphing

Although students have seen inequalities and graphed them in the past, I’ve found that it is worth my time to spend a day or so giving students a quick refresher on this.  There are several great Desmos activities for this.  Here are a few that I’ve used and like.

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Solving

In the past I had an activity I used to get students to discover when the inequality symbol needs to be switched when solving inequalities.  It was sort of lengthy and cumbersome, but I didn’t know how to improve it more than I already had.  Then I saw Sarah Tweet the picture below.  It was EXACTLY what I was looking for!  Thanks Sarah!  Here is the link to download Sarah’s file.

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Then for practice students do a Tarsia puzzle.  I created the puzzle a while ago and don’t know where the file is that I can share.  If you’re unfamiliar with Tarsia puzzles, you can learn more about them here.

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I also have a question stack that I use for these types of problems.  You can read about Sarah Carter’s question stacks here.

 

 

 


All Real Numbers/No Solution

To introduce inequalities that have No Solution or All Real Numbers as the solution, I went back to what students already knew about equations like these.  I had students solve a problem similar to the one below and then asked them what inequality symbol we could replace the equal sign with that would make the inequality have no solution and the same for all real numbers.

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Then for practice, I had students work on this Desmos activity.

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I also tried creating an Open Middle problem for these types of problems after seeing a similar one Sarah created for equations.  I had one of my co-workers take a look at a different Open Middle problem I made, and he had a great idea from when he has used Open Middle problems in the past.  He suggested to start by letting students use whatever numbers they want, and then after they come up with a solution to restrict them to only using certain numbers.  I thought this was a great idea, so that’s what I did.  I started by telling students they could use any integers they wanted as long as they didn’t repeat any of the 12 numbers.  When a student came up with a solution, I said they could only use the integers -6 to 6.

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You can download the files for the Open Middle puzzle here.

6th Grade Unit 2: Intro to Algebra (Part 2 -Evaluating Expressions)

I blogged about the first part of our Intro to Algebra unit in 6th grade here.  In this part of the unit, we finally get into the algebra stuff.


Word Phrases

Before we start evaluating expressions, we talk about what variables are and what the purpose of them is.  We review word phrases.  I started this year by putting these words up on the board and having students classify them based on their operation.

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After we have done a bit of large group practice with that, I have students do a card sort/matching activity.  Once I have checked their answers, they can play memory.  Some students have also played Go Fish with the cards as well.

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I also use this Desmos card sort on another day to review this concept.

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Evaluate Expressions

Then we get into evaluating expressions.  This year I made a small tweak to how I introduce this to students.  In the past,  I would stand up in front of the class and tell students what you do when you’re asked to evaluate an expression.  I’m really trying to get away from being the teller of information in my classroom and instead be the asker of questions to get students to explain the mathematical concepts to each other.  Here’s the small change I made this year.

I put “x + 2″ on the board and asked if we could come up with a number answer for this.  I saw several heads shaking no, and when I asked why, they told me, “We don’t know what x is.”  Then I added to the board “x = 7″ and asked if we were able to come up with a number answer for this now.  They told me we could and that the answer was 9 and then explained how they got that for an answer.  I could tell that not all of my students had caught on yet or weren’t fully paying attention, so instead of me rephrasing what the student had just said, I asked, “Can someone else explain to the class how ‘Sue’ got 9 for an answer?”   Then I put something like 4x + 3 up and repeated the same process.

It was a small change, but it felt SO much better than standing up in front of the class telling them a process to follow.


The first activity I do is something I created several years ago, and I realized last year when I did this that it’s pretty similar to Sara’s Add-Em-Up activity.  I created 4 different sets of 5 cards.  Each set of cards is printed off on a different color paper so that I can tell which set students are working on.  Below is a picture of the first page of the document for this activity which has 4 copies (each column) of the first set.  I have 8-10 copies of each set.  Because all 8-10 sets are the same color, before I laminated the cards, I wrote a number on the back of each card in a set.  This way, when I find a random card on the floor I can ask, “Who has the red 3s?” and easily figure out where the card goes.

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Students are put in groups of 2-3 and in their group they work together to evaluate each of the cards.  When they are done, they add up their 5 answers and come tell me what they got.  If they are correct, I will give them the next set.  If they are incorrect, I don’t tell them which one they got wrong, and they go back to their group and work to figure out what they did wrong.  I’ve tried to level the cards so that each set gets increasingly difficult.  Set 1 has one to two operations on each card.  Set 2 has three steps to each problem.  The next set incorporates decimal operations and the final set has the variable in the problem more than once.

One of the reason I like having a different color for each of the sets is that it is easy for me to see where students are when we are doing this activity.  If I look around my room and see one group on the red set (the first set for me) and every other group is on orange or green (the 2nd and 3rd set for me), I know I need to check in with the group working on the red cards.

Below is an example of each of the four sets.

 

 

 


As I was looking through my stuff to find the file for that activity, I came across another activity I created and had forgotten about.  For this activity, I put students in groups and have them start at a problem.  They can choose to solve either problem on the card.  I encourage all students to try at least one of the more challenging problems.  Once students solve the problem, they look for the answer on another card that I have hanging around the room and then solve either problem on that card.  Eventually, they will loop through to all of the problems and end back where they started.

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I also used this Desmos card sort towards the end of the unit to review with students.

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Find the Flub warm-ups are great for evaluating expressions.

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Tables

In the past, I haven’t done much with tables in any of my classes, and I know that this is something I should do more of.  I added this Desmos activity to this unit, and overall I was pleased with how it went.  It was challenging for some of my students, which was my goal when creating it.

 

 

 

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Here is the link to download the activities from this post.