Adding Conceptual Understanding to Your Math Class


Conceptual understanding: what is it, and how do we teach it?

An ideal picture of conceptual understanding in math class:

Students exploring the patterns they see.  Students making sense and reasoning about what they know.  Students predicting and solving based on ideas that they can explain.  Students proving answers with models.  Students teaching each other.  It is a beautiful thing.


How do we teach conceptual understanding? 

One thing I have found is that I need to stop teaching (in the traditional sense).  Teaching often translates to telling, which will not result in the picture above.

Here are four ways to build students’ conceptual understanding in your math class:


Models allow students to see why formulas work and how to solve problems.

Students struggling with basic operations?  Use physical models to build reasoning and number sense.

Students struggling with fractions?  Use fraction tiles to model comparisons and operations.

Students struggling with area of triangles, circles, parallelograms or composite figures?  Cut paper models to prove the formulas.

Students struggling with volume?  Use cubes to build prisms and fill containers.

Students struggling with polynomials?  Use algebra tiles to model multiplication and factoring.



Instead of giving students a rule or formula to use, show examples and let students figure out the rule.  Not only will this build conceptual understanding, but it will also boost students’ confidence as mathematicians.

Students struggling with adding and subtracting integers?  Talk about examples using temperature or depth before talking about any rules to follow.

Students struggling to remember laws of exponents?  Work out examples, and ask students to develop their own rules based on solutions.

Students struggling to remember angle relationship rules?  Have students measure examples and tell you what they notice.

Students struggling with transformations on the coordinate plane?  Have students transform examples and find out the rules for coordinates.

Students struggling with graphing quadratics in vertex form?  Show students examples and ask them to find shortcuts on their own.

Students struggling with parent function transformations?  Graph changes to parent functions and look for patterns.



This simple question has a powerful impact.  If students are forced to explain why, they are pushed into conceptual understanding.

Students may know to move the decimal when dividing, but do they know why?

Students may know to cross multiply to solve a proportion, but do they know why?

Students may know a zero exponent results in 1, but do they know why?

Students may know the formula for circumference is C=πd, but do they know why?



Experiments are not just for science class.  Experiments allow students to explore, think, discover, predict, and develop conceptual understanding in math.


Want to design your own experiment?  The scientific method is a great guide to get started:

  • Ask students a question.
  • Students make a hypothesis.
  • Students test their hypothesis and collect data.
  • Students analyze results and draw conclusions.

Experiments are engaging and lead to deep, lasting understanding.


Conceptual understanding is true understanding.  For many of us, it is not how math class was presented when we were in school.  Therefore, we have to think outside the box and even push our own understanding to promote it in our own classrooms.


Thanks for reading!



Halloween Resource Roundup for Middle and High School Math


halloween cover

Halloween is just around the corner.  I love getting students engaged with Halloween-related activities in math class.  I’ve gathered some of my favorite resources for middle and high schoolers.   These are so fun, students might forget they’re learning.  Read along to find a Halloween activity for the topic you are teaching in your math class.  (Click the images below to go directly to each resource.)




Students go trick-or-treating around the room as they answer scientific notation questions.  I love this activity from 8th Grade Math Teacher.  (And, she has other trick-or-treating topics in her Teachers Pay Teacher store.)





Students translate, reflect, and rotate shapes to create images of decorated pumpkins.  The rigor is high, but the fun is too.  You can find this one in my store.





Students love mystery graphs, and Hayley Cain is the queen of mystery graphs.  This skeleton one is the perfect way to practice graphing this month.







Literal equations just got a little less painful with this resource from Math by the Mountain. Students “carve” their pumpkins as they solve each problem.  Check the reviews; teachers and students love this activity!





Students square numbers, and you end up with the perfect door decoration for October.  If iteachalgebra made it, you can bet it is going to be perfectly cute.






Practice integer operations and graphing on the coordinate plane.  Students solve integer problems to reveal ordered pairs, and then graph the points to reveal Halloween pictures.  Find this in my TpT store.





Engage students in simplifying and identifying equivalent algebraic expressions with this spooky mystery picture resource.  Find it in my TpT store.





Happy October, and Happy Halloween!  I hope you have been inspired to add a little Halloween fun to your math class.

My Favorite Math Game

gamecover1Games can be an awesome way to engage students.

I am always on the hunt for games that utilize my class time well by challenging students and making class fun.  My favorite game is Quiz Quiz Trade!

Why I like Quiz Quiz Trade:

  • It only takes about ten minutes to play, so I do not need to devote an entire class period.
  • It gets students up and moving around.
  • It makes students talk about math.
  • Students help each other out with difficult math problems.
  • Everyone in engaged the entire time we play!


If you have never played Quiz Quiz Trade, here’s how:

  • Prep by printing or writing problems on cards.  (I like to write the answers on the back, but that is optional.)
  • Give each student a card.
  • All students stand up and raise a hand to begin the game.
  • Each student should find another student with a hand up.
  • Partner 1 shows the problem on his/her card, and partner 2 tries to solve. (Quiz)
  • Partner 2 then shows the problem on his/her card, and partner 1 tries to solve. (Quiz)
  • Then partners switch cards. (Trade)
  • They raise a hand to find a new partner, beginning the process again.


Ideas for Implementation:

  • This game is a great way to re-energize a class. I recommend using the game for a short break during class. Movement is great for the brain!
  • If students are a little shy to interact, you might set a requirement for how many different students everyone must trade cards with. For example, tell students they must interact with 10 different people before going back to their seats.
  • Mark a couple of the cards with a star. The students who have these cards when time is called win a prize.


I love this game in math class, and it can work for any subject!



Check my Teachers Pay Teachers store for several pre-made sets for Quiz Quiz Trade. (And get this Algebra Vocab set for free!)



quiz cover nbt qqt quiz7th

algqqt original-2416683-1 original-3034503-1



Create Your Own Pacing Guide

Slide1Many schools and districts provide pacing guides to save teachers the trouble of making their own and to keep everyone on the same schedule.  I strongly discourage this practice.  While I do see some benefit of keeping everyone on the same pace, I have rarely seen the same pace work for everyone.  I have greatly benefited from creating my own pacing guide for my own class (or with the other math teachers at my school).


When a pacing guide is created with a specific group of students in mind, decisions can be made to help those students succeed and make gains on end-of-year tests.


Slide2The first step in creating your own pacing guide is to know your standards.  Knowing your standards may sound obvious.  However, we sometimes think we know the standards well enough, so we do not spend time looking deeper at them.  Other times, we may be teaching a new grade or subject and decide we will study the standards as we get to them during the year.  Whatever the reason, not knowing the standards extremely well at the start of the year is a huge mistake.


What does it mean to know your standards?  If you were to see a random test question, would you be able to tell if it fits a standard you teach and know if it should be on the final test? If your textbook has a chapter that has always been taught at your school, do you have the confidence to skip it if it does not align to a standard?  “Yes!” should be the answer to both.


Knowing your standards well will give you the assurance you are doing the right things throughout the year.  To be most effective, you need to know what you are teaching now, what you are teaching next, how long you can spend on each topic, how deep the students need to understand each topic, and how important that topic is on the final test.


I will guide you through the process I use to gain a deeper understanding of the standards and how I use that understanding to build a custom pacing guide. 


I use two colors of paper when digging into my standards.  One color is for major standards and one for supporting standards.  Color-coding helps keep track of what is most important.  (You might even choose three colors if your standards are divided into three levels of importance.)

I rewrite each standard in very plain words on the correct color paper so that I end up with a list of standards in my own words.  I label each one with the standard code so I can reference it later.  Here is an example from 8th grade math standards.

The standard: CCSS.Math.Content.8.NS.A.1
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
What I write: NS.A.1 Rational vs irrational numbers, decimal expansions and conversions

The point of this list is to be able to glance at a standard and immediately know what it is about.  There are no rules to how long your descriptions should be.  Some will be short, and others may need to be lengthy.

Standards can be written in interesting ways.  Lookout for footnotes.  Look for sample questions if your state provides them.  Not only do you want to be sure you are teaching everything a standard addresses, but you also do not want to spend time teaching more than needed.  Standards can be quite confusing at times, so do not be afraid to ask for clarification from experienced teachers or even state test-makers. 

As you make your list, think about pre-requisite skills.  On a separate sheet on paper, note any pre-requisite skills that your students typically lack but will need.  For example, on the standard above, I would write fractions.  My students always need brushing up on fractions before we learn rational and irrational numbers.  The list of pre-requisite skills should be custom to your students based on your experience in past years.  If you are new to teaching, new to a school, or new to a grade level, seek an experienced teacher to help you with this process.

Slide3When you have finished this process, you should have a list of simply-worded major standards, a list of simply-worded minor standards, and a list of pre-requisite skills that you know your students will need to review.




Making Your Custom Pacing Guide

Making your own pacing guide will allow you to take ownership of how your class runs throughout the year.  If you are an administrator, please allow your teachers time to complete this process before the school year begins.  County pacing guides or online pacing guides are fine for ideas, but they were not created with your students in mind.  If you are forced to use one of those, talk with your administrators and see what options you have.

You will need the lists you created in the previous section for this exercise.  Cut apart each standard so you have strips of paper (color-coded) with one standard per piece.  Then begin to group and arrange them into units.  A unit does not have to be a specific length or number of standards.  It could be one standard or several.  Group them as it makes sense to you!

Do not worry about what you have done previously or the order the textbook goes.  You know your students best and what will work for your classroom.

Some questions to consider as you group and arrange:

  • Do you want to cover a whole topic at once? Or would introducing it early in the year and going deeper later work for your students?
  • Do some standards involve multiple topics? (If so, make an additional strip of paper for that standard and put one in each group it applies.)


Once you have a rough draft of your units (basically piles of paper strips at this point), look over your list of pre-requisite skills.  Now is the time to decide if you need to spend a bug chunk of time early in the year teaching these skills or if you can review them as needed throughout the year.

When teaching students who are several years behind based on achievement tests, I have heard many teachers say, “With so many standards to cover, I do not have time to review old material.” However, I like to consider the following situation.  What if you were tutoring a struggling student for a whole year?  Would you begin with grade level material?  I would not.  A student cannot retain difficult math lessons without a solid foundation of the basics.

In my own experience, I have had years where I began with review, and I have had years where I reviewed as needed.  Both can work.  When teaching 8th grade math to students who predominately had scored less than proficient on state math tests, my team decided to devote about five weeks to pre-requisite skills.  We knew it was the right thing to do for those students.  The topics we covered included multiplication, fractions, integer operations, order of operations, and graphing basics.  Issues we had the previous year helped us determine what to cover.  Those students made huge gains that year and were better prepared to understand 8th grade math when we got there.

When I began teaching high school geometry, I knew students would have some skill deficits.  However, because I was not sure where those issues would lie, I chose to review throughout the year instead of in the beginning.  This method allowed me to take note as I saw misconceptions and plan to work on them between units or as bell ringers.

Look over your pre-requisite skills and think about your students.  What will benefit your class the most: a review unit or review scaffolded throughout the year? I have had success with both.  The only mistake you can make here is not planning to review at all. 


Now arrange your groups into the order that makes sense to you.  Some questions to consider:

  • What topic is approachable, engaging, and would work great at the beginning of the year?
  • If a unit is heavily tested, should it be at the end of the year? Or should it be at the beginning, so you have time to reteach as needed?

If your school has given you a pacing guide, see if you can arrange your units into roughly the same order as the guide.  If you disagree with the school’s pacing guide, discuss it with your administrators.

Now decide a rough timeline for each unit.  Count how many weeks you have from the start of school until your state test takes place.  Also subtract a couple weeks (or more) if you know you will lose time due to things like weather, field trips, testing, etc.  Then assign each unit a number of days or weeks.  Pay attention to your color-coded standards.  Be sure you are allotting more time to major standards.  Tweak your numbers until they fit in your school year.

What if they do not fit? I always feel like I could use double the time I am given.  You may consider eliminating a few minor standards from your pacing guide.  When teaching Geometry to students who averaged around the 30th-40th percentile, I chose to skip a couple very difficult minor standards.  In this case, I knew we would need a lot of time to learn those standards that would only appear in one or two questions on the test.  Do not cut out anything that the next grade level depends on you teaching.

Slide4Now you have your units made up of standards arranged in order and given a specific amount of time.  Title a piece of paper with each unit title, write the length of time, and attach your standards for that unit.  (You can type these up and make them nice and neat, or do like I did and just tape those strips of paper down.)  These individual papers will be used for notes as you continue planning your year.  As you reach each unit, reread the full-length standards to make sure you plan lessons to align.


This is the process I have used to help me plan for successful years.  I wish every school provided teachers with the time to create a pacing guide before school started.  Teams of teachers collaborating to create pacing guides for their own students is a great way to start the year.  Every teacher should be involved.  The process does take time and effort, but you will create a guide for your year that works best for your students.  I truly believe this is one of the best ways to take ownership of your classroom and start the year right.


Thank you for visiting my blog.


What to Do (& Not Do) the First Day

FIRSTDAYHave you heard you set the tone for the whole year on the first day of school?  Recently, I discussed the first day of school with a group of math teachers at a workshop.  We shared ideas and what it means to set the tone the first day.

Think about what you want to see your students doing throughout the year in your class.  In your ideal classroom, what are students doing and how?

Make a list.

If the list describes your ideal classroom, these are the things you need to cultivate the first day!

In the workshop, many teachers said they want to do more group work, but it is scary.  Therefore, we sometimes hold off on groupwork until later in the year.  We talked about how putting students in groups day one would set the tone for collaborative work for the whole year.

The first day students are nervous and attentive.  This may be the time they behave better than at any other time of year.  They do not know me yet, and they have not decided how they will behave in my class.  They will decide within the first week, and it will be difficult to change that decision.  Students will see what activities and behaviors happen in my class, and they will expect those same activities and behaviors throughout the year.  If my class is extremely boring the first week, it will be difficult to get them excited later.  If students do not work in groups in the beginning, they may complain when I try it later.



Let me help you plan your first day to make it successful.  (These items may be stretched out over a few days, especially if your first day is very short.)

Student Arrival

Students need to meet you right when they arrive.  Introduce yourself so they know they are in the right place.  I have seen recommendations to shake their hands.  I think that is great!  Do what feels natural to you.  Be yourself, which is hopefully at least pleasant and happy to be there.

seatsI highly recommend assigned seats.  This helps you learn names and takes away the anxiety some students feel about picking a seat the first day.  I have my seats numbered.  When a student arrives, I introduce myself, ask the student’s name, say it back, and give a seat number.  This allows me to take roll, helps me start learning names, and it prevents students who do not belong from entering my room.  I also have a seating chart displayed on the board in case they forget their numbers (which happens a lot).

One of the items in my ideal classroom is that students start working on something immediately when they enter each day.  Therefore, I have something for them to do immediately the first day.  This could be a student survey or a notecard with a little information.  I do not recommend they just sit and wait for you to do something.  This will give them the idea that your class will have a lot of sitting around and waiting.  Also, they are more likely to move around if they do not have something to do.

On the board, I have my name and clear instructions, such as, “Welcome to Math!  Have a seat in your assigned desk.  Begin working quietly filling out your student survey.”  Students are nervous the first day.  I believe they want to please you.  Make it easy for them!



Students want to know about you.  Be genuine and tell them a little about you and the class.  If you like to crack jokes, do it.  If you are not funny, do not try to be.  If you are quiet, do not feel like you need to ramble on.  I think students just want to know you are pleasant, which hopefully you are.

Get them excited about the class.  If you are doing a cool project soon, give them a hint.  Tell them the amazing things they are going to learn.  One year in my Algebra I class, I asked for volunteers to come up and try to tell me about a random term or topic that we were going to learn that year.  For example, I put the quadratic formula on the board, and a student tried to explain to us what it was.  Of course, the class had never seen the quadratic formula before, so it was pretty funny.  One student tried to define “parent function,” and the whole class got excited about getting their moms and dads to do math.  Introduce the class to students in a way that makes them excited to come back.

I also like to include student introductions, but be careful.  My first year, I asked my middle school students to tell their names and something about themselves.   I still remember a student I will call Joseph who stood up and said, “I’m Joseph, and I’m the class clown.”  Then, he proceeded to prove that point the rest of the day and the rest of the year.  Now I like to do short, structured introductions.  I ask my high schoolers to say their names, and then the rest of the class waves or snaps or something silly.

I want to mention learning names.  My first year, I called a student the wrong name the first day.  It was not a big deal, but I was mortified.  I was so afraid I was going to call someone the wrong name again, that I rarely said anyone’s name.  Then, after a couple months, a girl asked me why I never said her name.  She thought I did not know it.  I was even more mortified.  Of course, I knew her name, but she felt like I did not even know who she was.  Now I try to learn names as soon as possible and say names often.  Learning names is not easy.  Not many jobs require you to learn hundreds of names.

One of the ways I begin learning names is to print a seating chart of each class and have it with me constantly.  When students introduce themselves, I make notes.  When I need to call on someone, I look at my seating chart and call on someone by name.  When they are working on something, I study the seating chart while I look around until I have learned the names.  Students do not mind if you look at the chart to find their name the first week or two.  Also, I make notes, so I know who I have called on.  (I do this throughout the year.)


Procedures, Rules, Syllabus, Expectations, Etc.

I do think these are necessary things that should be discussed in the first few days.  The younger the students, the more time should be spent on these items.   Consider making these interactive instead of just reading through them.  Maybe students can fill in missing words or look for information in a scavenger hunt format.  Keep in mind that if students go to several classes their first day of school, they probably are going over rules, expectations, and syllabi multiple times.



I like to do an activity the first day of class.  In my ideal classroom, we are often doing activities and groupwork.  Get the year off on the right foot by doing the type of activities you will do throughout the year.  If you choose to do groupwork the first day, give clear detailed instructions.  Be sure it is an activity you are confident will go well and will not take too long.  Groupwork does not necessarily need to be math-related the first day.  The idea is to get students working together from day one in a collaborative, respectful way.

If you do not feel groupwork is best on the first day, that is fine.  You may choose to get an idea of your students before grouping them.  Instead of groupwork, I have had my high school geometry students measure angles with protractors and find angle relationships the first day.  Many have not used a protractor, so they learn a new skill day one.  They get to measure using a protractor, so it does not feel like a worksheet to them.  Geometry skills are great for the first day in any grade because students can draw, measure, and use tools, all of which are naturally engaging.


No Wasted Time

Set the expectation that no time will be wasted in your class.  If you have a few minutes at the end, do not make it free time.  Try to plan something for every minute of class.  Have an extra activity, video clip, or game ready to go if you happen to have extra time.  If students get to socialize at the end of class, they will expect to socialize at the end of every class.  They will not be upset if you never provide free time, but they will be very upset if you suddenly try to take free time they once had.


These are suggestions that have been successful in my classroom.  Every teacher and every class is different, so plan a first day that sets the tone for the kind of year you want to have in your class.  I hope you have a great start to the school year!



Below are resources you may find useful at the start of the school year:

original-2247647-1 original-3283399-1 original-3795372-1 original-3876462-1


Thank you to everyone who participated in the blog hop giveaway last week.  Camille Gelston won a math calendar and bulletin board kit!



An Interior Designer’s Guide to Classroom Design

Slide2Before I was a math teacher, I was an Interior Designer.  When I became a teacher, many people thought I would have a beautiful classroom, but it was not realistic for me then.   I did not have the time (or budget) to create a color-coordinated, themed, Pinterest-worthy classroom.

I admire teachers who create beautiful classrooms.  Dedicating time, effort, and often our own money to create a welcoming atmosphere is just one way we can show students we care about them.  However, I do not believe a classroom has to be expertly designed to be welcoming and functional.

If you are overwhelmed with the idea of decorating or unsure if you should hang something in your room, ask yourself: Is this a resource?  When you use your walls as a resource, you will get the most benefit from your decorations.

Ways to use your walls as a resource in a math classroom:

Slide6Hang a vertical number line.  One of the first things I made for my middle school math classroom was a number line that I painted on a long strip of banner paper.  I had it laminated, and I still have it.  I painted the numbers on a diagonal, thinking I might hang it vertically for part of the year and horizontally for part of the year.  I ended up leaving it vertical.

If you teach math, I highly recommend hanging a vertical number line.  I refer to mine at least a couple times every week.  It is useful for adding and subtracting integers, estimating value likes roots and irrational numbers, rounding, and the list goes on and on.  I love having a large number line that I can point to when explaining concepts.  And, vertical just makes sense.  Kids understand up and down much better than left and right when discussing values.


Use word walls and math posters strategically.  Most math classrooms have at least a few math-related posters or vocabulary words on the wall, but are they useful?  When I first had a word wall, I had it because it seemed like a popular thing to have.  However, I did not use it.  Simply hanging words on the wall did not help my students learn the definitions.  (I should have known that!)

When deciding on word walls or reference posters, make a plan for how you will use them.  Make sure the students interact with these resources.  When I taught high school geometry, my students had a difficult time learning the vocabulary for all the types of angles associated with parallel lines cut by a transversal.  After they asked me for the hundredth time what a certain angle was called, I hung reference posters on the wall.  Afterwards, I did not need to tell them; they could study the posters and figure it out themselves.


Start each day with a math calendar.  I made my first math calendar by cutting cardstock and painting math problems for each day.  It got students’ attention from day one.  They were not sure how to work all the problems yet, but they were intrigued.

Now I have created several math calendar versions appropriate for a wide range of grade levels.  Teachers use them in different ways.  Some start the day by referring to the problem for the day.  Some have only the day’s problem posted, so when a student asks the date, they refer to the math problem.



Spread math into the hallways.  Many school hallways seem to have a few bulletin boards that go unchanged for long periods of time.  Students notice things in the hallway, especially if they have to wait in the hallway for any period of time.  If you have the opportunity, ask if you can decorate a bulletin board in the hall.  I have created a couple print-and-go designs that are appropriate for the whole school but also sneak in some math!





Hang student work.  I believe this is truly the best decoration.  Kids, no matter the age, love to see their work on the wall in your room.

Those are my best tips from an Interior-Design-turned-Math-Teacher.  I hope you are not disappointed that I did not discuss the latest trends and themes.  Choose a theme that makes you happy, or choose to keep it simple without a theme.  Choose decorations that support learning, and you can’t go wrong.



Thank you for reading my blog, and I hope you have a great start to the school year!

– Rachael


Below are additional resources referenced in this post:

mathcalendarexponents math cal integer angleposters
bulletinboardmath mistakes cover number line

Be sure to visit these other STEM blogs in our blog hop to ready your classroom for the new year!


Read on to ready your classroom:

Set High Expectations on Day One

How to Manage Your Calculators

Engage Them from Day One

How to Create Meaningful Anchor Charts to Decorate your Classroom with a Purpose

Back-to-School Checklist for Teachers

Managing Lab Supplies

Wake up! Thinking Outside the Box on Day One

5 Tips for Setting Up Your Math Classroom

Teaching Equations

Would this equation scare your students?
equation 1

When I first taught equations, I would never have asked my eighth graders to solve such a difficult problem.  However, now I like to start them with this type of problem.  Let me explain.

In eighth grade, students learn to work with multi-step equations.  They have been exposed to one-step and maybe two-step equations in previous grades.  When I first taught eighth graders equations, I told them things like, “What you do to one side, you must do to the other,” and “Use order of operations backwards.”  Students memorized lots of rules and how to solve familiar problems.  They did fine on the test at the end of the unit.  However, a few months later, I gave them the exact same test, and no one passed it.  They had forgotten how to solve equations.  Actually, they never knew why they were doing what they were doing.

The summer after my first year of teaching, my team met to plan for the next school year.  We needed students to understand equations.  The “Do / Undo Method” was our choice.  Students use their knowledge of order of operations and list everything that has been done to the variable in the correct order.  Then, they list the inverse of each step and perform them.  They understand an equation as having a variable on which operations have been performed.  If they can undo those operations correctly, they will find the variable’s value.  Below is an example.

equations pic

The students were able to make sense of the steps they were using.  They could apply the concept to equations with ten steps if needed.  I would not have dared to give my first year students problems like the one above.

I still think about a boy whom I will call “Sherman,” who struggled in most subjects and was in trouble most days in at least one class.  He lacked many math skills from previous grades, but he knew order of operations, so I knew he could solve equations.  Sherman learned the “Do / Undo Method” and was solving with the best of students!  I gave him a whole sheet of equations, and he solved them all correctly, much to the surprise of many of his peers.
equations blogpost