Quadrilaterals are a big topic in geometry. There are so many things to know that it tends to get confusing for students. Students have misconceptions from their middle school math classes that are hard to overcome such as that a square is a square and only a square. A square is no way, now how a RECTANGLE! OH My!

I created a quadrilateral unit where I begin with a card sort activity. The cards have different shapes on them and the students are asked to separate them into parallelograms, trapezoids and other major shapes. This year, I decided that they should have a “for sure” pile and also make a “not sure” pile. I love listening to the conversations. Below is a pic of some slides I show:

The next thing that I like to do is discuss the Venn Diagram for Quadrilaterals. For some students, this is a breeze but for others, they are totally confused on why I’m using ovals in a diagram to represent groups of quadrilaterals. It’s best to make sure your students remember what a Venn Diagram is. I like to give an example of a region with math students overlapping a region of biology students to show that the overlap means all students taking both math and biology. Look at the Venn Diagram below. Can you figure out what quadrilaterals go in each region? Can some go in more than one spot?

This unit is the best place to use always, sometimes and never questions and if the students understand the Venn Diagram, then the always, sometimes and never questions are pretty obvious. It’s also a good time to talk about what does opposite and consecutive mean? Many of the definitions and properties use this terminology, so I spend time helping them understand where opposite sides and angles are versus consecutive sides and angles.

I like to get the kite and trapezoid out of the way first, so I can spend most of my time on the parallelograms. Students are not familiar with the kite, so this is usually a brand new topic for them. They think they know what a kite is but usually they are getting a rhombus confused with a kite. Each time I present a new quadrilateral, I give the definition and then we try to find other things that are always true about the shape. This is cool, because you get to talk about the diagonals and how they create congruent triangles. I also try to put proofs into the lesson as much as possible.

During the trapezoid part of the lesson, there is a discussion on isosceles trapezoids, midsegments of triangles and medians of trapezoids. A good reminder at this time is how trapezoids are related to parallel lines cut by a transversal, so that they can understand that there are some same side interiors that will be supplementary. Again, there is so much information, that its hard to know when to stop. Trapezoids could be a two week lesson if you let it, but I keep it to two pages. After the trapezoid lesson and the kite lesson, I give the students some practice on finding various parts of the shapes.

The rest of this unit is spent on parallelograms. Each time that I get to a new shape, I call it a “Parallelogram Study” or “Rectangle Study” etc. I let the students work through the definitions, properties and proofs. The other aspect of this lesson is discussing the coverses of the definitions and properties. This helps the students realize that if you see a shape and you are not sure what it is, then what is the least information you need to decide it is a rectangle for example.

The lesson concludes with practice on the parallelograms. There is a page of work where some major algebra topics are practiced. For instance, there is a rectangle problem where the students have to set up and solve a system. There is a rhombus problem where the students have to solve a quadratic. There is a square where the students find the length of the diagonal using the variable “s” for a side. This problem is a lead up to 45-45-90 triangles. I usually have to help the students with this whole page, but I don’t mind. Since I’m an Algebra II teacher as well, I like my geometry students to see as much algebra as time allows.

There is a set of task cards that act as a review for the Quadrilateral Test at the end of the unit. The test is only two pages long, but it is pretty involved. There is a major problem where the students have to find quite a few things. The picture of the problem is seen below:

Finally, there is another quick assessment that I use as a retest. All answer keys are included. It usually takes me about two weeks to get through all the work plus a couple of extra days to review and take the test. I love this unit. The information is extensive and I love how it hits on previous geometry and algebra topics. If you are interested, please check it out in my store. Click the pic below to go see the Quadrilateral Unit. If you would like to read more about my geometry curriculum, I have a blog post that you can read here: Geometry Curriculum for the Year.

## Algebra – 2nd and 3rd Grading Periods – Moving Toward the STAAR!

This is a continuation of some of my earlier posts. I was so proud of my students last year for passing the Algebra STAAR, so I wrote about it here: How I Got a 100% Passing Rate on the Algebra EOC Part 1 and Part 2.

I promised that I would keep anyone interested up-to-date this year on my progress. I love doing this because I’m going to be able to look back and see where I want to improve after I get my results this year. Here’s my post on the first six weeks: First Six Weeks in Algebra I

I lumped the 2nd and 3rd 6 weeks together in one post because I have so many interruptions during this time. Every time I turn around there’s a field trip, a district benchmark, PSAT, TSI or other disruptions. My strategy has been to get through as much content as possible. I know my students very well at this point and I know who to keep an eye on.

The content that we’ve covered heavily is seen below:

I’ve pressed on and given lots of quizzes, tests and homework.

Some of my favorite activities have been some boom cards that I’ve made. The set of cards in the the resource below has 20 questions. Click here to go do the first four cards in the student view.

So far I haven’t pulled many questions of old STAAR test. Their minds were blown when I was explaining the recursive formula in arithmetic sequences. I did look back into old STAAR test to see how often sequences have been tested. The only question I could find from the the past three years was this problem from the 2017 released test:

In general, arithmetic (and geometric) sequences are not a big part of the test. The 2019 test did not have any questions on the topic. This question below is from the 2018 test:

I suspect that they rotate questions from the TEKS and that next year there will be at least one question like #22 or #9 above.

I wanted to start some recycling of the first six weeks through practice sheets like I did last year but life happens and I did not start this. (Side note: I’ve been teaching for 33 years. I always make plans to do this or that, but I’ve learned that I cannot always get to everything. Please don’t beat yourself up if you do this too. A lot of things in education and the school environment cannot be controlled. Don’t worry if you have visions of grandeur but it doesn’t always work out.) One thing I do feel good about is that I do not let the students use a calculator every day. They have to do math in their head. They did a lot of solving for y and manipulating formulas so they did get a taste of some of the things from the first six weeks which was mostly solving equations. Another thing I feel ok about is I know that I’m about to do systems which will also be good for practicing solving equations. We will also hit inequalities again through systems, so recycling information is going to happen naturally!

While on the topic of systems, if you are behind in your curriculum, this is a good time to try to catch up. Systems are important, but you can save solving for systems for after the STAAR test. Teach them how to set them up and solve them on the calculator for now. I hate this, but at the same time you have to make sure you cover all the material. Save solving systems algebraically for later if you need to.

I promised myself that I would make sure and have students explain the math they were using more. I wanted to know if they really understood how to solve for y and graph equations, so I made a flipgrid question when we got to solving and graphing inequalities. The students really enjoyed it and it was an eye opener for me. Students have a hard time with the vocabulary and I could tell who was bluffing their way through explaining the process.

The second semester has started and now it’s crunch time. I have to be deliberate in everything we do. We are starting with graphing and writing linear systems and then on to exponent rules. Check back to see what happens next!

## Geometry Curriculum for the Year

It took me a year and a half to complete my geometry curriculum, but I finally did it. The thing is though, I’m constantly updating and adding to it. I guess you could call it a living and growing curriculum. I’m using the curriculum myself, so I find things I want to update or make better all of the time. I’ve started a facebook group for those that purchase this curriculum. When you gain access to the group, it will be an opportunity to work with other teachers using the same curriculum. The group is new and small now, but I’m hoping that more people will want to get involved soon.

(By the way, don’t forget to look for a special right triangle freebie at the bottom of this post!)

On the main page of my Geometry Curriculum on TpT, you will see every resources in the curriculum. I also sell each one of these resources separately in my store. A third way that I have the curriculum broken down is by units which I also sell in my store.

Why did I feel the need to write this curriculum? Many of my students come to me after taking Geometry over the summer. In my district, students have the opportunity to get ahead in math. If the student does not pass Geometry during the summer, they end up using the exact same curriculum again. They get to keep their book from the summer with all of the work already done. It doesn’t make sense. Also, if you don’t know about slader.com, you should check it out. All of the answers to the math book that we use are here! So frustrating! Kids know about this site. You should see if your geometry book is here.

You can find me on Instagram @timefliesedu . I like to show student work in action on my Instagram posts. Check it out if you have time. It will help you get a feel for some of the lessons. To see the curriculum in my TpT store, click the picture below. The picture shows one of my favorite conditional statement activities. As you can see, we do lots of cutting and pasting.

To see each unit, click on the unit picture below:

I really like a hands-on curriculum. I use patty paper, compasses, protractors, tape and scissors on a regular basis. I also like to give google form tests as well as activities on google slides and Boom Cards.

Many of the activities are discovery lessons. I make sure there is plenty of algebra involved to keep these skills alive for the next math class. If you have any specific questions about this Geometry Curriculum, please leave a comment below, or ask a question in my TpT store.

Here is a freebie from one of my special right triangle lessons. CLICK HERE!

Enjoy and I hope that you visit my store to find more activities and lessons.

## Examples of Real Life Arithmetic Sequences

One of my goals as a math teacher is to present real life math every chance I get. It is not always easy, I have to admit. When I was in college and the earlier part of my teaching career, I was all about the math… not how I might could use it in real life. I’ve made it a goal of mine to find real life situations. I’ve also tried to catch the situation in action, but it’s not always possible especially since sometimes I think of an idea while driving or when I’m falling asleep at night.

My recent thoughts have been about arithmetic sequences. Seems easy, right? They are linear. There are a ton of linear situations. Yes, but I want visuals! I also did not want the situation to be a direct variation or always positive numbers and always increasing or positive slopes.

Below are some of the situations I’ve come up with along with a picture. I’m happy for you to use these situations with your classes. Enjoy!

Stacking cups, chairs, bowls etc. (Stacking anything works, but the situations is different when one thing fits inside the other.) The idea is comparing the number of objects to the height of the object.

Pyramid-like patterns, where objects are increasing or decreasing in a constant manner. Ideas for this are seats in a stadium or an auditorium. A situation might be that seats in each row are decreasing by 4 from the previous row. I use this in one of my arithmetic sequence worksheets.

Filling something is another good example. The container can be empty or already have stuff in it. An example could be a sink being filled or a pool being filled. (Draining should also be considered!) The rate at which the object is being filled versus time would be the variables.

Seating around tables. Think about a restaurant. A square table fits 4 people. When two square tables are put together, now 6 people are seated. Put 3 square tables together and now 8 people are seated. I really love this example. You can use a rectangular table as well and start off with 6 seats.

Fencing and perimeter examples are always nice. Discuss how adding a fence panel to each side of a rectangular fence would change the perimeter. Figure one could have one panel on each side (or change it so it isn’t square). Figure two could have two panels on each side. Each time find the new perimeter. The possibilities for fencing are endless. But how fun would it be to get actual toy fence pieces and do this in your classroom?!

Even thought this is not particularly a real life situations, it’s still good because the visual is real life. The students can touch the objects or even create the pattern themselves! Use toothpicks, paperclips or even cereal to make patterns. If you’d rather set them up somewhere in the room for math centers, then that would be good too! The following is an idea with cereal. If you count total Froot Loops, it’s not arithmetic, so it’s best to stick with rows, perimeter, or sides of the triangle to stay with a linear pattern. (Counting all of them is an area problem, so that would make it quadratic.)

Negative number patterns are not as easy to find. Our thoughts usually go to temperature or sea level. There are some fascinating places on earth that are below sea level. I think it would be cool to do a study on some of them. Once you’ve talked about some of these places, then various situations could be created like, during a rainfall the surface of the water started at 215 feet below sea level and rose at a rate of such and such per hour.

Situations involving diving in the ocean could be used as well. Did you know that a diver should descend at a rate no faster than 66 feet per minute or ascend at a rate of no more than 30 feet per minute? I’m sure many students don’t know why and this could certainly create some great accountable talk.

I hope I’ve given you plenty to think about. It’s really fun to create these problems. Students need to know that their math is real and useful! I hope this encourages you to use some of these examples or make up some of your own. I’d love to hear some of your examples. Leave a comment if you’d like. We can all learn from each other!

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Some of the examples I used above are in my Arithmetic Sequence Activity seen below. When I was creating this resource, it really stretched my thinking. I wanted to create something that students could learn from and see how these patterns are involved in real life situations. I’ve attached a couple more of my resources. I’m working on the geometric sequence activity now and hope to finish in a week or so. The second resource is an arithmetic and geometric sequence and series game. It is really suited for Algebra 2. The resource at the bottom is a formula chart for geometric and arithmetic sequences and series. It’s a freebie, so take advantage and download from my store!

## Algebra Christmas Worksheet

Several years ago, I wanted a fun Algebra worksheet that my students could do during December. I created the worksheet and we did it that year and then I changed schools and forgot about the worksheet. A friend of mine at the old school I was at sent me an email: Hey, can you send me that worksheet we did a couple of years ago that had the Christmas tree on it? I had to think for a minute…Where did I save that? I found it and made it even better and sent it to her. I was so glad I found it. As I was looking through the problems, I was thinking about how great this worksheet is right before the semester final. It’s got so much good stuff on it.

The topics covered are:

• Knowing if a slope of a line is positive or negative.
• Finding slope and y-intercept from equations and graphs.
• Finding slope from two points.
• Using different forms of equations: slope-intercept, point-slope, standard
• Finding domain and range.
• Graphing a line from an equation.
• Writing an equation from a line on a graph.
• Taking a problem situation and writing an equation, then graphing it.

I’m very pleased with all the material covered and I’m looking forward to this worksheet being a part of my semester final review. Here are a couple of pics from the front and back of the worksheet:

This worksheet comes with an answer key. Click the picture of the product below to go check it out in my store!

## How I teach Factoring

Factoring is one of those skills that students must know how to do since they will use it in every high school math class. This skill is one of the most important skills and unfortunately some students never really get it. I hear calculus teachers complain about how their students can’t factor. Students should already know how to factor before entering calculus but why don’t they? Factoring will be in all the college entrance exams too because students need to know how factor for their college math classes. I know in Texas that students can know very little about factoring and still pass the Algebra EOC. Questions on the EOC can be figured out by working backwards from the answer choices. There usually is one question each year where students have to find one of the factors which does make it a tougher question. Students really do not learn how to really factor until Algebra 2.

When I was in high school the method that was taught was guess and check. We got pretty good at it but back then you knew your multiplication facts very well. When I first started teaching math, I honestly had no clue how to teach factoring. I’ve done every method or fad that came along but I have settled on a method after realizing that this is how it is taught in many college algebra classes. The method I use is GROUPING! I focus heavily on finding GCF’s and factoring by grouping and then when it’s time to factor the tough trinomial problems, we turn them into grouping problems.

I start the factoring unit by teaching the students how to factor out a GCF. To help them understand, I’ll sometimes call it “undistributing”. Once they understand how to factor out a GCF, then I give the students grouping problems. They are taught to group the first two terms and the last two terms and then factor out a GCF. I tell them that if they get the same answer in both parenthesis then they have worked the problem correctly. They factor out the common parenthesis and make another parenthesis with the leftovers. Once they get good at this then I talk about differences of squares and perfect square trinomials.

These are my notes from my Algebra 2 classes this year:

Next I teach them to spot easy trinomials and hard trinomials. I later explain that they are problems that either have an “a” equal to one or an “a” greater or less than one. I discuss how the signs work in trinomials:

• + + = ( + )( + )
• – + = ( – )( – )
• – – = ( – )( + ) the larger number gets the –
• + – = ( – )( + ) the larger number gets the +

When I teach the a = 1 problems, I tell the students to go to the last number and ask, “What multiplies to get the last number that will add or subtract to get the middle term.” Students can do this pretty well…especially if they know their multiplication facts.

When I teach the hard trinomials (a>1 or <1), I have the students draw a big X to the side of the problem. The students are directed to multiply the first term and the last term in the trinomial. They write that at the top of the X. The middle term goes at the bottom of the X. Next, they ask that same question about what multiplies to get the top number that adds or subtracts to get the bottom number. The students write it on the left and right side of the X. Now it’s time to turn the problem into a grouping problem. The students are told to write the first term of the original problem, then the two monomials they just found and the last term of the original problem goes on the end. Factor by grouping and they are done.

I know students should know this by the time they are in Algebra 2, but many of them don’t. I usually try to break these notes up into two days. I assign Games 1 – 6 of my Factoring Using Seek and Find. I love this activity because the students know if their answers are correct or not by finding the answers in the puzzle.

A quick tip on helping students that aren’t good with their multiplication facts. If they want to know for instance what multiplies to give you 300 that would subtract to get 44…Have students type 300/x into a graph of a graphing calculator, then go to the table. The table contains all the factors. I tell them to ignore all decimals. They will see a 50 and a 6 in the x and y columns. They can reason that if they subtract, they can get 44.

Factoring is a very important concept and students need this skill to survive in their upper level math courses. I finally feel confident that my students understand it since I now stick with a certain way of presenting it to them. I truly believe in the way I teach factoring and I hope that I have given you some ideas on how best to help your students successfully learn this concept!

For my Algebra I classes, I made some factoring matching cards you might be interested in: Factoring Matching Cards #1 and Factoring Matching Cards #2. Good luck!

## 9 Exponential Functions Activities That Are A Must!

I could do exponential functions all year. I really enjoy them and think they are super fun. When I got my master’s degree, I did a study on exponential functions. I learned so much and I found that I was really interested in them. I created this first lesson more than 10 years ago and have been using it ever since! Creating that first activity sparked me into creating more and experimenting with some other ideas. I’m excited to share with you my list of 9 Must Do Exponential Function Activities!

(1) Exponential Function Poster Activity:

This is my very first exponential function activity that I ever created. It’s not the first lesson I teach when I’m starting this content, but it’s my favorite. This activity is the ultimate collaborative and differentiable activity. There are so many interesting exponential function situations! It’s been tried, tested and tweaked. Basically this lesson is a collaborative activity where the students are given an exponential scenario. The groups must create a multi-rep poster where they collect data, draw a graph, write an equation and answer a question.

The lesson opener is a bacteria problem. I want every student to get a feel of how they should work through one of these scenarios. The bacteria problem talks about what bacteria are and how they can multiply very quickly. I help the students go through the multi-representations to make sure they know what is expected of them when they start their poster.

Next, I show them their choices which are:

• A Chain Letter Problem
• A Zombie Situation
• A Tournament Bracket
• A College Football Situation
• A Lovely Cockroach Scenario

Every situation usually gets chosen. You can entice students to create their own situation too. The college football situation was a student idea from years ago that I have improved upon to make it work better. Your students are amazing and creative, so don’t think that they wouldn’t be able to make up a situation of their own. The student of mine that created the football problem was not one of my top students, but because he was the one that thought of the scenario, he was interested and did a great job of completing the task.

After the bacteria problem, I turn the students loose and let them start their work. They are told to be creative and display the information in a way that is interesting and pleasing. I tell them to title the poster and make sure every person in the group writes on the poster. I supply the poster paper, the markers and the scenario sheets.

I’ve learned to watch out for misconceptions. Some students when creating graphs, will take the exact y-values and place those numbers on the y-axis. Here is an example below that I didn’t catch until it was too late. I cringe when I see this! (Not a very creative poster either…ugh!)

Once the posters have been created, it’s time for the Gallery Walk! I want the students to check out at least 4 posters. I’ve created a page that students fill in while looking at the posters. They have to write the title of the poster, determine the domain and range, decide if the situation is growth or decay and then write down one thing they may wonder about the situation. The conversations that I hear are amazing. They love getting to look at the other posters and they love to critique them as well.

I’ve had feedback from teachers that have taken my activity and changed it to fit their needs. One teacher used a speed dating strategy where the students worked through a problem on their own and became the expert. The possibilities are endless. Each teacher has their own unique way of teaching and their own unique classroom situation. If you have a group of rowdy kids that you don’t want up running around, then let them do their own problem on notebook paper or graph paper. You could even let them create the table and graph in excel and present the problem in a PowerPoint.

Check out this activity in my store: Exponential Functions Activity

(2) Exponential Function Activity in Google Slides Form

Out of necessity last year, I created a Google Sides version of the lesson above. I’m having a hard time deciding which one to use this year. Instead of making posters, the students create the table, graph and equation in Google Slides. This doesn’t sound very exciting except that my whole class was in the same Google Slides all working at the same time. I was 2000 miles away monitoring the activity. They asked me questions and I could see them working in real time. I loved it so much that I’m honestly going to have a hard time deciding what I should use. Maybe I’ll let one class do the posters and one class do the digital form and compare the two. If you are big into digital resources you will love this. I now have this version in my TpT store: Google Slides Exponential Functions Activity. Below is one of the slides that I graded. Looking at this now, I should have asked the students if this situation was discrete or continuous.

(3) Tower of Hanoi

Find a Tower of Hanoi game on the internet or have the students download an app on their phone. The object of the game is to move the discs from one stack to another stack in the least amount of moves. You can never put a larger disc on top of a smaller one. The number of discs and the least number of moves is an exponential function. It’s fun to let the students play a while and get them to create a table of the number of discs and the least number of moves and then see if they can figure out the exponential function.

(4) Twizzler Decay Activity

Tasty and fun. This is a freebie I’d like to share with you! I love using this as a quick lesson opener. Students measure a Twizzler and jot down the data in a chart. The student folds the Twizzler in 1/2, cuts it and measures it. Each time the student continues this step until there is not enough Twizzler left to work with. They plot the table and then lots of discussions can take place about decay or even the concept of half-life. Click Here for the Freebie: Exponential Function Twizzler Freebie

(5) Exponential Function Unit

This is the first thing I start with when I introduce Exponential Functions in Algebra 2. I refuse to stand up and lecture over this topic so I let the students work through this unit at their own pace. I copy the pages as a booklet. Students can use a calculator and even partner up if they want to work with someone. I let them work through the unit and figure out most of the information by graphing and using the information that they have already learned earlier in the year about transformations and domain and range. I do have to talk about asymptotes because we have not discussed this concept much up to this point. I teach on a block schedule and it takes most students a good 2 class periods to get this packet done. Topics covered are transformations, e, compound interest, 1/2 life, growth, decay, domain, range, y-intercepts, asymptotes, an inverse problem, writing equations from tables, growth and decay model scenarios, a paper folding activity, assessments and bell ringers and lesson closers. There’s a ton of information. I usually get the students to trade and grade after all is said and done. I feel like they learn a lot by working through this on their own. Students need to see that they can work on their own and figure things out. If you are interested, click the link: Exponential Functions Unit.

I have a set of 20 Exponential Functions Task Cards. For some reasons, students do very well with task cards. If you put these same 20 questions on a worksheet, some students will be bored or are overwhelmed with thinking about doing a 20 question worksheet and they will give up. Take the same 20 questions and put one on a card, now they will sit there and work through them. It’s amazing! This set of task cards would be a great review right before an assessment. The task cards cover recognizing growth and decay from an equation, transformations, key features of graphs, the growth and decay model and compound interest.

(7) Sierpenski’s Triangle

How do you get all of these activities done? Part of my strategy is to do them in stations. Really math labs or centers would be more accurate. It would be hard to time these stations and expect students to be completely finished with each task. The Sierpenski Triangle activity, the Tower of Hanoi and several more exponential phenomena are discovered and tinkered with during my Exponential Stations Resource.

I love the Sierpinski Triangle activity because not only do the students create beautiful art work, they have to collect data on the number of shaded or unshaded triangles. We then put all of the triangles together to make a giant Sierpinski Triangle!

(8) Compound Interest Study

Students are told that they have inherited some money but to receive it, they must follow some rules. Every student in the class will probably end up with a different situation. Each student gets 4 cards that tell them how much money they inherited, how long they have to invest it and 2 different compounding options to compare. They work through their problem and then share their information. This study sparks lots of good conversations and helps the students realize that compounding doesn’t make much difference but time invested does make a difference! Get the Compound Interest Study Here!

(9) Marble Slides Exponential Function Desmos Activity

If you aren’t using the Desmos Graphing Resources, you need to start. I love the Marble Slides Activities and so do the students. There are several Marble Slides Activities for various functions. The object is to change the equations so that when the marbles are dropped, they travel the correct route and hit all of the stars which means success. Students learn how to manipulate the equations so that the marbles do just what they want. Very fun and engaging!

So there you have it! If you can get most of these activities and lessons done, then your students will know tons of awesome math content. I have all of these activities bundled (except for the google slides activity) into one package for 20% off. If you are intersted, then click on the pic below. If there is something that you can’t find, please let me know. I’d love to add things that teachers are looking for. Thanks for visiting this article.

Happy Teaching…