Improving Number Sense with the Divisibility Rules

I decided to make divisibility my first lesson of the year for 7th grade next year for a couple of different reasons: it is a skill used in many different concepts throughout the year and it really helps promote overall number sense in students.

The divisibility rules (hopefully) help students be less dependent on their calculators, which is an area I am hoping to improve on this year.  So even though it isn’t the most exciting lesson of the year, it is an important one, and I think it’s a good way to begin the year.

I go over the rules for 1 – 10 with my students.  (I used to skip over 7 but every year students ask me if there is a rule for 7….so now I give them the rule along with an example for which the rule could be useful (i.e. 231) but then explain that 9 times out of 10 it is easier to just do the division than it is to use the rule).

In addition to the rules I give students tips, like if a number isn’t divisible by 2 then it isn’t divisible by any other even number.  Tips like that help with divisibility by 8, since that isn’t the most useful rule.  I tell students to only check for divisibility by 8 if (1) the number is divisible by 2 and then (2) if it is divisible by 4.

I made half-sheets the students can stick in their notebook with the divisibility rules.  You can download it free by clicking the picture below.

divisibility rules half sheets

To make the lesson more exciting I enlisted the help of my programmer husband.  He was able to build an interactive divisibility rules game that my students can play on their chromebooks, which I am super excited about!  In the “Divisibility Challenge” game, you can choose which rules you want to practice and then either play for mastery, speed, or just for practice.  I think I am going to originally have students play for mastery, where they need to play until they get 10 questions correct.  I plan to begin class the next day with a speed competition to see who can get the most questions correct in 3 minutes.  Competition always seems to get middle school students involved and engaged!  Click below to try a round free.  (It should open right in your browser).

divisibility demo

If you are interested in getting the full game for your class it can be purchased in my tpt store for $6.

divisibility pic1

 

Thanks for reading,

Christina

Share Button

Related Post

Getting Organized with Google Drive

In my last post I shared with you my process for writing long-term plans and I shared my scope & sequence for Algebra I.  (Click here to read that post if you missed it!)  Today I thought I would share my long term plans for my other classes (7th grade math, 7th grade Pre-Algebra, and 8th grade math),  as well as my next step in the planning process – setting up Google Drive.

Click below to download my scopes & sequences.  (They are editable word docs).

7th Grade Scope & Sequence

Pre-Algebra Scope & Sequence

8th Grade Scope & Sequence

As I said in my last post, my next step after writing my plans was to set up Pinterest boards for each unit in each class to store any ideas I find online that I don’t want to forget.  (So far I have boards set up for each of my Algebra and Pre-Algebra units, which you can check out here).

The next thing I am doing is setting up folders in my Google Drive.

google drive organization

I am terrible with physical file cabinets – mine is literally just filled with junk and stacks of extra copies, etc. and is not organized in the least.  That’s why using Google Drive works so well for me!   Google Drive has been great in helping me be more organized but in the past I haven’t used it to its full potential.  I truly believe that folders and sub-folders are the key to making the most of this awesome organizational tool!

For the coming school year I started by making a main folder for each class I teach:

google drive

Within each folder I put general class information and am in the process of creating sub folders for each unit.  (So far I only have the first two units done).

class gdrive

I am putting any and all resources I might need for the unit in each unit sub-folder, so to keep them manageable,  I plan to put sub folders within each unit folder for each lesson.  So in my Unit 1 Algebra Basics folder I will have a folder for Adding & Subtracting Rational Numbers, Multiplying & Dividing Rational Numbers, Writing & Evaluating Expressions, etc…  One folder for each lesson within the unit.  Within each of those folders I will put the classwork, task cards, worksheets, homework assignments, etc. that I am using for that lesson.  Then everything is in one place and easy to find.

Seems like such a simple idea (and it is!) but for some reason I never thought to actually organize my Drive the same way I organize my lesson plans.

While my plan is to try to fill my unit folders as much as possible ahead of time, I will also add to them throughout the year as I determine what resources I am using for each lesson.  The nice thing about that is that once it’s done I will have it ready to go for future years, as well!

The other nice thing about using Google Drive as opposed to a physical filing cabinet is that I can use the search bar at the top to type in what I am looking for in case I can’t remember the name of it or what folder I put it in.  It makes it soo easy to find what I am looking for, which is exactly what someone who is not naturally organized (like me!) needs!!

Do you have any tips for organizing school materials or using Google Drive?  Feel free to share by leaving a comment!  (I believe I finally got the comments working again after they had been broken for months!  yay!)

Thanks for reading,

Christina

Share Button

Related Post

Long-Term Planning (Algebra I)

I personally believe that it’s a good idea to start thinking ahead to the next school year around this time of year.  I admit that I don’t normally start planning until August but this year I am getting a head start because I believe it will allow me to be much less stressed come September.

If you are looking to start planning out your year and have no idea where to start (which was me a couple of weeks ago before I just jumped in), I’ll share my process for long-term planning.  The first class I worked on was Algebra I, so I’ll share that one today.

long term planning blog pic

Step 1:  Look over everything you need to teach and break it up into units.  I try to use as few units as possible while keeping each one a manageable size.  For Algebra I, I came up with 11 different units.

Step 2:  Determine the order in which you want to teach those units.  This can be tricky because you need to make sure that students have all the prerequisite skills for each unit and you want the year to have a good flow.

Step 3:  Determine all of the different lessons that will be included in each unit and the order in which you want to teach them.  This is the most time-consuming part, in my opinion, but it will save you a lot of time throughout the year if you get the whole year figured out before school starts.

Step 4:  Determine an approximate length of time it will take you to teach each unit.  Since my Algebra I class is an advanced class for 8th graders, I am able to move fairly quickly.  Therefore, I normally plan to teach a lesson a day.  I plan on 2 days for topics that I know students will find challenging.  I add 3 days to the end of a unit since I typically spend 2 days on review and 1 day for the unit test.

Here is my long-term plan (scope & sequence/curriculum map) for Algebra I.  Feel free to download it and edit/adjust it to meet your needs.

algebra scope

The next thing I did is something I should have done long ago…I organized my Pinterest boards.  I love getting ideas on Pinterest but I previously put all the great teaching ideas I found on my “Math Teaching Ideas” board…and then forgot all about them.  So I setup a separate Pinterest board for each Algebra I unit and moved pins from my (useless) “math teaching ideas” board onto the appropriate boards.  Now when I see great Algebra ideas on pinterest or read about them on blogs, I pin them onto the board that corresponds to that unit.  I forsee this being very valuable when I am actually writing my lesson plans throughout the years – I can glance through the unit Pinterest board to remind myself of all the great ideas I want to implement.

Click the picture below if you want to check out my Algebra I Units Pinterest Boards:

pinterest

Step 5: (I won’t get around to this step until closer to the start of the year when I have my schedule…)  Translate your estimated time frames into calendar dates.  I have a blank school year calendar that is weekdays only.  I write in school holidays/half days, etc. and then write in my approximate start dates for each unit based off the estimated time frames I came up with.  I then adjust as necessary around long breaks like Christmas and Easter.

You can download my calendar FREE from my TpT store by clicking the image below:

calendar pic1

There you have it…my process for writing  long-term plans.  I have had years when I haven’t done detailed year-long plans and years when I have done them and I can say from experience that it really does pay off to do them because it helps you have a smoother, less stressful school year when you have a map to follow.

Do you have any tips or tricks for planning out a school year?  Please feel free to leave a comment sharing your ideas!

Thanks for reading,

Christina

Share Button

Related Post

Planning Ahead for Next Year (and a Free Poster!)

I’m trying something new this year…I’m planning ahead! 🙂  I typically don’t even start thinking about the next school year until August (school starts for me in September), but my goal this year is to have my year pretty much figured out BEFORE August so that I can relax and enjoy my last month of Summer.  I know I will still have things to do in August like setting up my classroom, but I am hoping to have as much done by then as possible.

So far I have brainstormed some classroom policy ideas that I want to change next year, which I will write about in future blog posts, and I have started writing out my curriculum maps/scope and sequences for each of my classes.  To keep me motivated I am breaking up the more “serious” planning with some fun projects for my classroom that I enjoy doing, like making posters & decorations.

Here is the first poster I made for my classroom that you can download FREE from my TpT store!  I made it because I am tired of having students show me “completed” homework that is NOT up to my standards and having them look at me blankly when I tell them it needs to be redone….so I thought it would be helpful to post the components of an acceptable homework assignment.

hw poster

In my next blog post I’ll start sharing my curriculum maps and other plans for the year!

Thanks for reading,

Christina

Share Button

Related Post

Ideas for Keeping the Math Fresh in Students’ Minds

The end of the year is often a time for reviewing the math learned throughout the year, whether to prepare for state tests, final exams or cumulative projects, or just to fill the days after testing is done.  However, I have come to see how important it is to review all year long and not just wait until the end.

My first year teaching, my students seemed very receptive to my lessons, seemed to grasp the concepts, and did well on the tests I gave them after each chapter.  However, I was surprised and disappointed in the standardized test scores of some of my best students.  I realized that the problem was that once we finished a chapter and moved on to the next chapter, they never saw the material from that first chapter again and so by the time standardized tests came around they had forgotten some of the things that they used to know really well.  I learned from that experience and have since incorporated various ways to keep the math fresh throughout the year:

keeping math fresh

  • I know different teachers have different opinions on calculator usage, but my personal feeling is that once students prove that they can do something by hand, I allow them to use calculators so that they aren’t spending too much time on the computational aspect of a complex problem. However, I don’t want them to forget how to do problems by hand, so once a week I give them a “no calculator review” in place of my traditional “do now” questions.  They usually cover topics like fraction, decimal, integer, & rational number operations.  (You can read more about my No Calculator Reviews in this post).
  • After each chapter test, I give my students a cumulative review that I count as a quiz grade. It covers things from all previous units that we covered.  Students know that the cumulative quiz is coming so they know that they can’t just forget the material after they learn it.  Sometimes I allow students to use their notes for the cumulative quiz or give it as a take-home quiz, but most of the time I give it as a traditional quiz.  Not only does this encourage students to (hopefully) retain the math they are learning, but it shows me if there are particular concepts that a lot of students seem to have forgotten that I may need to revisit.
  • This is, in my opinion, the most important strategy. I give my students problems and ask them questions that require them to use skills learned previously in the year.  It is so important for students to see connections between the different units they learn, so any time I can incorporate an “old” skill/concept into a “new” one is a win in my book!  Here’s an example of a task card I made for a lesson on finding the area & perimeter of irregular figures that requires students to use previously learned skills:

irregular figure

To find the perimeter students need to use the Pythagorean Theorem and to find the area & perimeter students need to perform operations with mixed numbers, so this one problem reinforces a couple of different skills learned throughout the year.  Giving students lots of problems like this makes it virtually impossible for them to forget the math they learned earlier in the year since they are constantly using it.

 

I hope this gave you some ideas for helping students keep the math fresh!  If you are looking for ways to keep it fresh over summer break or just need a good end of year review packet, I have math review packets for students going from 5th to 6th grade, 6th to 7th grade, 7th to 8th grade, and pre-algebra to algebra I in my tpt store for $4 each.    Each packet contains detailed explanations of how to do the various types of problems, worked-out examples (showing each step), and 100 practice problems.  Click the pictures below for more information on each packet.

Slide1 math review packet new cover math review packet for 7th-8th grade pic1 Slide1

 

Thanks for reading,

Christina

Share Button

Helping Students See Their Strengths in Math

Well, it’s here.  The official turning point in the year when students and teachers (whether they admit it or not) turn their thoughts to summer.  That’s not to say that teachers are done teaching or students are done learning…but middle schoolers – especially 8th graders – come back from spring break ready for summer.

So i figured I’d share an end of year idea I had for my classes in case anyone reading this is thinking about those last days of school, too. 🙂

We’ve all had kids in our class that have shown tremendous growth over the course of the year.  We’ve all had kids who come in for extra help because they really want to understand.  Then there are the kids who are always helping their classmates who are confused, the ones who always pay attention in class, complete their homework, etc.

The problem I’ve always had is that for my school award ceremony I am required to pick 2 students from my math class to receive an award: one is the student with the highest average and the other is for a student who displayed great effort.  The issue is that there are often multiple students who I feel are deserving of the effort award but I’m only able to pick 1.

math awards pic3

This is such a common sense idea that I honestly don’t know how I haven’t thought to do this before…but I decided to do my own certificates in addition to the school awards.  This way I can recognize multiple deserving students.  In fact, I challenged myself to come up with a strength to recognize in each of my students so I can give every student an award.  I think that it is important for every student, especially the ones who believe that they are “bad at math”, to realize that they have potential, a skill, or talent, that can help them be successful in math.

I made awards for completing homework, participating in class, persisting in solving difficult problems, consistent effort, excellence in Algebraic thinking, outstanding critical thinking, excellence in graphing, good mental math skills, excellent overall achievement,…and many more (31 in all!).  I gave them all cute alliteration names, too, to make them more fun! 😉

math awards pic1

I want to show my students that I appreciate their hard work and effort. More importantly, though, I really hope this helps my students build confidence in their math skills and helps them see the strengths that I see in them.

If you like this idea but don’t want to make your own awards, you can purchase my math awards for $4 in my TpT store.  They are in editable PowerPoint form so you can type in names, and PDF form if you prefer to hand write the names.  I also included both color and black & white versions so there are options for everyone.   

Slide1

Thanks for reading,
Christina

Share Button

Discovery Lesson: Factoring Trinomials

Whenever possible I like to have students discover or figure out the lesson on their own (with some guidance from me, of course), rather than simply teaching them an algorithm.  One such topic is factoring trinomials.

I start my lesson on factoring trinomials with a = 1 by giving students 4 binomial multiplication problems and having them solve them, showing all of their work.

discover factoring pic1

I then tell them that factoring is the opposite of multiplying, so basically they are given the “answer” to a multiplication problem and they need to figure out the “problem”.  I have them look back at the previous 4 trinomial “answers” they got and to try to come up with a rule for factoring by figuring out where the b and c came from.

With a little time the students are always able to come up with the idea that the b is the sum of the 2nd terms in the binomials and c is their product.  So, when I give them a trinomial to factor they know that they will have (x +/- #)(x +/-#) and need to find 2 numbers whose product is c and sum is b to fill in the #s.

I like teaching factoring this way because the students understand that I didn’t just make up some rule.  They came up with the rule themselves by analyzing problems that they already knew how to solve.

I also teach factoring trinomials where a ≠ 1 this way.

discover factoring pic2

With these problems I have students come up with a rule to go from the trinomial answer to the “work” column.  They are usually pretty quick to notice that the first and last terms stay the same and that the middle term gets split into 2 terms.  When I ask how to know how to split up those middle terms they are able to come up with the idea that it needs to be split into 2 numbers whose sum is the original number but I usually have to push them to get them to see that the those 2 numbers also must have a product of ac.  I teach factoring by grouping earlier in the unit, so once students get to this point in the problem, they are able to just apply the factoring by grouping method to finish the problems.  (I know that there are other methods to factor trinomials where a ≠ 1, like guess & check and the “airplane” or “slip and slide” method but I personally find that factoring by grouping makes the most sense for students since they are using 2 things they already know: multiplication of binomials and the distributive property, in reverse.  I think this lends itself perfectly to helping students understand that factoring and multiplication are simply inverses).

Factoring is such an important part of algebra and is used in many different ways (solving quadratic equations, simplifying rational expressions, solving radical equations, etc.) that it is really important that students master it.  At least in my opinion, having students take some ownership of the process helps build their understanding and mastery.

[For students who understand the process of factoring but struggle to come up with the 2 numbers with the given sum and product I show them how to use a factor tree to help them find the numbers.  You can read about that here.]

Thanks for reading,

Christina

Share Button

Fun Algebra Easter Egg Hunt Activity

Just a quick post today to share a fun, quick Easter activity I’ve done in the past with Algebra/Pre-Algebra classes in case you want to try it…

Fill up an odd number of Easter eggs with pennies (put the same number of pennies in each egg).  Hide the eggs in the classroom before class begins.

When the students come in, ask for 2 volunteers.  Have those 2 students search for Easter eggs.  Since you have hidden an odd number of eggs the two students obviously will have found a different number.  Tell them that you want to be fair so you are going to even out the amount of money each student has by giving them some loose coins.

Easter Algebra Activity - math-in-the-middle.com

In the picture, you can see student A only found 1 egg and student B found 4.  So, I gave student A $0.25 and student B $0.04 to even out the total amount of money each student has.

Ask the class to figure out how much money is in each egg.  (You can give the money to the first student to get it correct as a prize if you want.)  Discuss how they figured it out and prove they are right by opening the eggs (in this case there is $0.07 in each egg).

Easter Egg Equation Activity - math-in-the-middle.com

There you have it – a fun way to have students solve equations with variables on both sides (without them even realizing that’s what they are doing!)

Enjoy!

Thanks for reading,

Christina

Share Button

Related Post

Pi Day: Original (& Free) Ideas for Celebrating in Algebra I

Like most math teachers I love Pi Day!  I mean, who doesn’t love a day where you eat pie and celebrate math?!  I have always done a bunch of fun pi related activities with my students on Pi Day (you can read all about them and grab a Pi Day word problem freebie in this post from a couple of years ago), but the activities I usually do are geared towards 5th – 8th grade (pre-algebra) kids and aren’t really relevant to my Algebra I kids since area and circumference of circles are not in the Algebra curriculum.

So, I have set out to find ways to tie Pi Day into Algebra concepts and have come up with the following activities:

  1. Pi Day Literal Equations:  Literal Equations are a topic I teach towards the beginning of the year in Algebra I.  It is one of the harder concepts we do at the beginning of the year so I think Pi Day is the perfect time to revisit and review them.  I made a worksheet with a bunch of Geometry formulas that involve pi to have my students solve for pi.  I made it a bonus to see how many of the formulas students can identify.  The first student/group to finish and the student who correctly identifies the most formulas get prizes!  (Download the worksheet by clicking the image below).solve for pi pic
  2. Systems of Pies: Have students work with a partner to write a system of equations about pies.  Then have students walk around the room and solve each other’s systems of equations using the methods of their choosing.
  3. Pi Day Attack Review Game: Have each group draw a pie cut into 5 slices.  Ask students review questions on ANY Algebra I topic.  When a group gets an answer right, they can attack two different pies by “eating” a slice (coloring it in).  The last pie with any slices remaining is the winner!  (I wrote a detailed blog post about Attack here that you can read for a better explanation of the game and rules).

image

I hope this has given you some useful ideas for making Pi Day a success in your Algebra class!  What other activities have you done that bring Pi Day into the Algebra I classroom?

Thanks for reading,

Christina

Share Button

Linear Equations Walk Around Activity

I’m writing about another favorite activity of mine that I use for a few different topics throughout the year – walkarounds. They require minimal prep from the teacher and are a great, effective way to practice certain skills. This post is specifically about the linear equations walk-around activity I do with my Algebra kids after they have learned Standard Form.
image
Here’s how it works:
I have 6 different standard form linear equations that I copy enough times so that each student gets 1 equation. (You can download the equations at the bottom of the post). I give each student an equation and tell them to convert it to slope-intercept form and then graph it.
image
After a few minutes have passed and most students are done, I tell them to form groups based on the equations they were given (all the 1’s are together, 2’s are together, etc.). In their groups they need to compare answers and come to a consensus on the correct slope-intercept equation and graph. Once they are in agreement they need to get their answer approved by me and then transfer the correct graph to a mini whiteboard. (Large graph anchor chart paper would actually be ideal, but I don’t have any so I use the whiteboards). They should NOT write the equation on the mini whiteboard, just the graph and their problem number.
image

Once this is completed, I give students a recording sheet (download link is at the bottom of this post). I tell them to draw a big x through the number they graphed since they don’t have to do that one. Then the students walk around the room and have to look at the other groups’ graphs and determine the slope-intercept form of the equations that were graphed. They then need to convert those slope-intercept form equations into standard form. (The walkaround runs smoothest if you have a set order for students to walk around the room instead of letting them wander wherever. I tell them to go in order, so group 4 would start at the graph of 5, then go to 6, and then 1, 2, and end at 3). I also have found that it works best if students just write the slope-intercept form of the line while they are walking around, and then return to their seats to convert them to standard form.
image
I love this activity because it gives students an opportunity to work both independently and cooperatively and gives them practice converting standard form to slope-intercept form, graphing lines, writing equations from graphs, and converting slope-intercept form to standard form.

(If you are in need of additional activities to supplement your linear equations unit, you may be interested in the linear equations relay races I have available in my TpT store.)

You can download the 6 equation cards for the walk around activity (FREE) by clicking the picture below:

linear walkaround pic1

You can download the activity recording sheet (FREE) by clicking the picture below:

linear walkaround pic2

What activities have you done for linear equations? Please share in the comments!

Thanks for reading,
Christina

Share Button

Related Post