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Math Problem Solving Strategies to Steal from the Language Arts Classroom

Problem solving strategies to steal from language arts

There is a common saying among teachers, that you must learn how to “beg, borrow, and steal” to be able to find the best resources and practices for your classroom.  Today I am going to tell you to steal…some strategies from the reading and language arts teacher down the hall to help your students in math!

I love working with students in small group because the conversations you have with them really open your mind to their perception of understanding when it comes to problem solving.  I work with fifth and sixth graders at a high at risk campus.  Many of the students are second language learners or just lack the prior knowledge that helps so much with understanding complex word problems.  A small group focus is often modeling and teaching students to move through problem solving in an efficient and meaningful way.  If you teach, you already know…this is hard work!

Here is an example of a problem and the conversations my students and I have had while problem solving.

Me:  “What is happening in this situation?”
Student: “Subtraction.”
Me: “No, I mean in the story, what’s happening?”
Student: “Addition???”
Me: “I don’t mean what operation you might do to solve the problem, I mean what is going on in the problem?”
Student: “Multiplication!”
Then I realize that my head is starting to hurt and I want to hide under the small group table.

That was once upon a time, but now I have a different method to use while modeling problem solving with “story problems”… and I stole it… from the language arts department!  Here it is...

Close Reading for Story Problems

This strategy is based on a few ideas I have seen my fellow teachers do with students in their reading groups with their English language arts classrooms.
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Click to download a free problem solving template!

Step One: Help students determine which type of problem this strategy works best for.

Ask students to decide whether their math problem is a "math problem or a "story problem."  Explain that a math problem is one where the problem tells you exactly what to do, while a story problem was a situation that is like a short story.  Story problems will often involve people that are doing something and, although it is very short, it has a beginning, middle, and an end, just like a story they might analyze in reading class.  A multi-step problem is really just a super short, super boring story!

Step Two: Read the problem like a story.

When we have determined that we have a story problem, our next step is to read it carefully.  We read it once, fluently, from start to finish.  Once we have read the whole problem, ask the students what happened first, what happened after that, and so on.  This helps the students to slow down and stop jumping to conclusions about what to do when solving the problem.  Also make sure to visualize the problem.  This can be done by sketching quick pictures and diagrams or mentally.  This time will also help you to determine if there are students who are unable to solve the problem because they have no concept for what is happening in the problem.  This can happen with students are second language learners of English or do not have a large prior knowledge set.  

Step Three: Differentiate between details and inferences.

Many students do not transfer their learning well between one discipline and another.  Although students learn about details and inferences in language arts class, and about observations and inferences in the science classroom, they cannot always automatically apply that in the math classroom.  They need to be able to, though, to be effective problem solvers.  In math we call the details or observations the "givens".  Ask your students to summarize the givens in the problem without making any inferences at first.  This will serve as a checklist while problem solving to make sure they stay on track and complete the entire problem.

teach a student to problem solve
After listing the givens, start making inferences.  You should make make sure the students understand that these inferences did not come from the problem, but rather from their interpretation and analysis of the problem.  Why is this so important?  Many students enter the upper grades with a "keyword" understanding of problem solving.  They think that there are certain words that will always signal certain operations.  For example, every time they see the word "total" they think they must add or every time they see the word "difference" or "less than" they should subtract.  In reality there are no "keywords" that will always tell us what operation to do.  Depending on how the words are used in the problem could completely change the way they need to be interpreted.  You don't want your students to rely on keywords, you want them to rely on their understanding of the specific problem.

Step Four: Use the inferences to solve the problem.

The students' ability to make good inferences is the real key to creating good problem solving.  Inferences that are based on a strong list of given information from the problem, will be a good base to the students work.  Some great ways for students to express their inferences in a math setting are writing expressions and equations, using strip diagrams and bar models to evaluate problems, or creating an organized list of steps for problems solving.

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Differentiating True and False Questioning for the Math Classroom

True and false questioning is probably not what pops into your head when you think about a differentiated and rigorous math activity.  The most plain of all question types really can become a thought provoking and multi-level activity that you can use with almost any topic in your math curriculum.

The Set Up

Begin by thinking of statements that are always true, sometimes true, or never true for the topic you want to cover.  Grab a free set of cards to go with an integer lesson by clicking the image to the right!  A simple way to create this is by creating a word document with a table with three columns and as many rows as can fit on the page.  Type statements in each of the cells.  By using a three columned table, you can make sure you have a good mix of each type of statement: type the "always true" statements in the first column, the "sometimes true" in the second column, and the "never true" in the last column.  It is okay if there are not equal amounts of each type, but having a variety keeps the activity more interesting.  When writing the statements, write about vocabulary, algorithms, mathematical rules, and even equations or expressions to simplify.  Before having the students do the activity, cut apart the statements, mix them up, and place in an envelop or bag to organize the cards for the students.

Let's Differentiate!

The card sets can be used in many ways, some will be easier and others more challenging.  Look at the ideas described below to see some of the many level you can take your class to using just one set of cards. 

Starting out easy...

  • Sort the cards before passing them out.  Give each student or group just one type of cards.  Have the students read through the cards and decide whether the their set is the always true set, the sometimes true, or the never true set.
  • Give students only "Never" cards.  Ask them to determine what in the statement makes it false.
  • With the whole set of shuffled cards, have the students sort them into categories with the titles "Always", "Sometimes", or "Never."

Take it farther...

  • Use the never true only cards again.  This time have the students rewrite the statement to make it into a true statement.
  • Have the students use the sometimes true statements.  For each statement have them determine a time when the statement is true and an example of when the statement is false.  They can write or draw an illustration explaining their choices.

Really challenge them!

  • With the statements on the always true cards, have the students prove why the statement is always true using mathematical rules, examples, and illustrations.  This is often more challenging than it seems since a really deep understanding of the concept is needed more to prove and answer true than false.  To prove something false you need only one example!
  • Have the students add to the set with their own statements.  Have the students write some addition examples of always true, sometimes true, and never true statements.  Have the student justify why it is in the category and debate with other students.

After reading this, do you have some more ideas of how to use "Always, Sometimes, Never" cards?  I would love to hear about it!

Also look at the links below to keep reading from some of my colleagues for more differentiation tips for the middle school math classroom.

My Number One Classroom Management Tip for Back to School

I am not someone who would claim to be a classroom management expert.  Student discipline is actually one of my least favorite aspects of teaching, so having simple things that I could do to improve my students' behavior is important to me.

After ten years of teaching, what is my number one tip for improving student behavior?  Here it is...  Ask students frequently, "What are you doing?"

I don't mean when they are doing something that they are not supposed to be doing.  I mean all the time.  Any time.  Random times.  Why?

First think about what happens when you typically ask a student that question.  Here are some of the most common responses I have heard from my students:

"What?  Me?"

"I'm not doing anything!"

"He's doing it too!"

The answer is almost always hostile, defiant, confused, or defensive.  Unfortunately by the time a lot of kids have gotten to fifth or sixth grade, some have been so conditioned to being in trouble, that defensiveness is pretty much their default setting when interacting with adults.

So why would asking this more often help?  I started asking students this all the time, especially when I knew that they were doing what they were supposed to be doing.  Here is an example:  I see that a couple of students are doing a great job doing their daily warm up.  I walk over and ask, "What are your doing?"  At the beginning of the school year, I would get confused looks and sometimes even "I'm not doing anything!"  Then I would say, "But you are doing something.  You are all doing a great job working on your warm up.  Thank you!  Great job!"  Eventually the students will be used to the idea that when you ask the question, you really mean it.  You really want to find out about what they are doing.  This gives you the opportunity to praise your students for positive behavior.

Procedures and Rules Task Cards
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Another benefit to this, is you will find out things that you didn't even think to ask.  Let's use the same example of the students working on their warm up.  You ask the same question and the student who is comfortable with the question might tell you something that you need to know.  You ask, "What are you doing?"  This time they answer something like, "Well, I'm trying to do my warm up, but I can't remember how to find a common denominator."  Now you know that this kid needs some help, and since you know what they need, you can fix it!

The most important result of getting your students comfortable with this question comes into play when they are not doing what they are supposed to be doing.  Eventually, even the kid who at the beginning of the year would yell or get upset when asked the question, will come to the point that they know they can answer you honestly.  So now when you walk up and ask, "What are you doing?"  They might actually give you an answer!  Back to the students doing their warm ups...

You see that one of your students is not doing their warm up, so you ask the question, "What are you doing?"  His reply, "I'm playing with this eraser that my friend gave me last class."  So what do you say now?  Ask the student "What are you supposed to be doing?"  He answers, "My warm ups?"

At this point the problem is likely solved.  He knows he is supposed to do his warm ups and that the expectation is that he will get to work.  Not only that, the emphasis is on his awareness of his behavior, not on your awareness of his behavior.  When students are aware of their own behavior, and your expectation of their behavior, they are more likely to fix it on their own.  The issue was solved without escalating the situation, especially with a student who is used to getting in trouble.  You got what you wanted and the student was able to save face, very important for the preteen, teenage student.

So, what are you doing this school year?  What is your go to classroom management technique?
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Pi Day in the Upper Elementary Math Classroom

Pi Day is coming soon!  Did I make a delicious typo?  No we're not talking about pie--we're talking about pi--the irrational number the represents the ratio between a circle's circumference and its diameter.  Pi Day is celebrated on March 14th each year because the numeric way of writing the date is 3.14, which are the first three digits of pi!

Are you feeling more blue than a blueberry pie because you don't get to celebrate Pi Day with your students since they are too young to learn about circumference and area of circles?  There are some fun ways to celebrate your elementary students as well.

Your upper elementary students may not study circumference, but they do study the properties of two-dimensional figures.  Use this fun day to review the attributes of polygons versus non-polygons.  Circles have special attributes that make them different from polygons.  Instead of a perimeter measurement like polygons, circles have a circumference measurement.  While polygons have base and height measurements, a circle has a radius and a diameter.  All of these special properties of a circle relate to--you guessed it--pi!

What is the number pi?  As an elementary school teacher, it may have been a while since you have taught about this aspect of circles, and how do you explain this concept in a way that will have any meaning to an elementary school students.  Here is a brief and age appropriate way to describe pi to your students:

  • Pi is a ratio, or a special fraction, made when the circumference of a circle is compared to its diameter.  The circumference is like the circle's perimeter and the diameter is like its width, except it goes from one edge to the other through the center.
  • Pi is an irrational number.  This means that although it is close to the whole number 3, it has decimal digits that go on forever!  Kids love this fact because they enjoy seeing real life examples of infinity.

Yes, I know that a fraction is a special ratio and not the other way around, but if you are working with third, fourth, or fifth graders, they will not have a clue about ratios.

What about some activities for your class?  Of course I've got those for you, too! Click the image to download a free game and and activity for your upper elementary Pi Day celebration!

If you want even more Pi Day fun for your elementary classroom, check out this comic book, comprehension questions, and game from my store on Teachers Pay Teachers.

Tips to LOVE Teaching, Even During Test Prep Season

When state testing is around the corner, two things seem to happen: stress levels rise, but student and teacher happiness falls.  Some of this is inevitable.  High stakes testing is by definition stressful because it is a quantitative measure of things that are near and dear to our hearts.

As a teacher you love your students, feel satisfaction in their learning, despair when they do not seem to understand, and rejoice when they do.  How can these be quantitatively be measured by one test?

When test prep starts, your love of teaching doesn't have to end.  Even though you are working toward a specific and demanding goal, there are some ways to like, even love, teaching during the review weeks leading up to your state test.

Tip 1: Don't sweat the small stuff

Not all content is created equal.  Are there some things that your state tests more often than other skills?  Is there one topic you can review that will help build many other concepts?  For example, in the Texas sixth grade math state test, the computations and algebraic reasoning is more highly tested than geometry.  If I am pressed for time, what should I probably review, equations or area?  Based on what I know about my state test, sixth grade has a stronger focus on algebra than geometry.  If I review algebraic concepts I am more likely review a greater percentage of materials on the test.   Does your grade and state have a focus?  This should be your focus during review, too.  Another thing to think about is to find a topic with far reaching value.  My fifth graders will be well served to review multiplication of decimals because this topic is an important base to other fifth grade topics like modeling multiplication, area, volume, and even division and order of operations!

This can also help you eliminate topics during review time.  For example, my fifth graders have a very difficult time rounding.  This is an important skill, but based on my state standards, this is not a fifth grade focus.  Its just a little piece of the curriculum.  Base on this, what I am not going to do is kill myself trying to teach my students to round numbers the weeks leading up to the state test.  It will make the kids frustrated and feel defeated, it will make me frustrated, and rounding is likely going to be a very small part of the test.

Summary:  Spend the most time on the most important topics, not the most difficult topics.

Tip 2: Keep building relationships with small group instruction

It is easy to enter "Teacher Panic Mode" when test prep season starts.  For many teachers this means moving the desk into rows and ending anything that resembles "fun" because this test is serious business, right?  Fight the urge, don't panic, and teach on!  Continue your small groups and if you don't normally use small group instruction in your math classroom, now is a good time to use it.  Small groups allow you to differentiate review for students with different needs.  You probably have some students who you are trying to get to an advanced or high level score while others you are just trying to ensure they pass or show growth.  These kinds of kids need different instruction, but you already knew that!  What you might not have thought of, is small groups are a great way to stay connected with your students.  They are getting nervous for the big test, too!  Spending time together in small groups will help reassure your students that they are ready and they can do well on the test.  Working with kids on a similar level will help let them know that they are not the only kid who still doesn't understand multiplying fractions and that they can learn how.  As a teacher, my favorite time in my classroom was working with small groups.  You just get to see the kids differently in that environment than in a whole group setting.

Summary: Small groups are not only academically beneficial, but they are good for you and your students' emotional well being, too.

Tip 3: Use engaging resources

An example of my leveled task cards.
Everyone loves a nice, long test prep packet, right?  Of course not!  There is little in this world that
the average middle school student finds more mind numbing than a packet.  Not only is it boring, it already looks like a test, so not only are students nervous about the test, but they get the testing experience for days before.  There are other types of academic and rigorous materials that are much more engaging.  One of my favorite is leveled task cards.  I have bought some and made my own depending on what I could find.  My favorite type for getting ready for a test have two important qualities: they have word problems, they have a variety of difficulty levels.  I also like collaborative problem solving activities that allow students to work in small groups to discover the answer to more complex problems.  When test strategies must be taught, limit the size and quantity.  Try just using one page at a time, rather than handing out a whole packet.

Summary:  Days and days of packets are boring, try task cards or other more engaging work.

Tip 4: Know where to start

When thinking of what to review first one thing that seems natural is to go over what the students had the most trouble with.  This is not necessarily a good plan.  Think about this: you have a short time to review what will help your students improve their achievement on the big test.  If many students had trouble with it the first time though and during regular class review after that there may be one of two issues.  One, the topic is really difficult, and your students may not be developmentally ready for it yet.  Two, (and this is one we don't want to admit as teachers) you may not be very good at teaching that topic.  Unless one of those two things have changed--maybe your students are more mentally mature by the second semester, or you attended some awesome professional development on teaching this topic, both of which are quite possible--the lesson will probably end the same way as it did the first time.  Instead of choosing the topic that your students had the most trouble with you are going to choose a "just right" topic.  Look for something that you middle of the road kind of kids did just "okay" on.  Your higher kids probably did pretty well on it and your lower kids probably struggled.  If you choose this type of topic, you are more likely to be able to make a big impact on your students' understanding and performance.  Why?  You are probably good at teaching this and your students are probably ready to learn it.  The average student is going to go from "sort of" understanding to a full understanding and your lower students are going to finally "get it."  You and your class will feel real success and that is the boost you need as you head into the test!  After you have covered some "just right" topics, if there is time, then you can move into some of the more difficult materials.

Summary: During review, time is short, choose topics that you can make a big impact in student understanding in a small time span.

Tip 5: Keep student anxiety low by keeping it positive

Many students are a nervous wreck when it comes to taking big tests.  Help your students feel at ease and confident as they prepare.  You may not believe it, since you an amazing teacher who reads cool educational blogs to improve yourself and your teaching, but there are actually teachers out there who repeated tell students that they are going to fail their big test!  As an instructional leader, I have actual dealt with this exact situation.  The teacher was in a panic that huge portions of her class were going to fail, and she told them that.  The kids were devastated or worse, indifferent.  Even if you are concerned that a student might not pass, focus on the positive.  Encourage the students that they can always show growth.  No matter at what point they are starting, a student can always show improvement and effort.  In Texas, the state testing even recognizes student progress as part of the testing results.  You can also encourage your students without making them anxious with friendly competition.  Let classes compete against another class for which can show the most improvement or students can even compete against their own past achievement and see what new accomplishments they can make.  Just remember to keep it friendly and fun.  One year we had a sports theme that the students loved.  Another time I was working with a group of students who had not passed the first round of testing and were having to take another test.  We looked at it like they were the underdogs and the whole stadium was rooting for them.  They loved the idea of making "the comeback" and showing the world they could do it!

Summary: Keep it positive!  No one wants to enter into the test already defeated.  Every student can show improvement.

Long division, a long battle

The dreaded words that send chills down the backs of students and teachers alike... Long Division! There are so many barriers to students effectively learning how to do long division, and each grade level that has to teach this skill has its own set of difficulties.

When students begin long division in fourth grade, it comes right on the heels of multiplication fluency.  For some students, so close behind that they have not had time to cement the multiplication fact fluency to the point to easily retrieve this information.  Long division becomes an agony of repeated skip counting.

By sixth grade, there are still some students who still struggle with the same issues that plagued them in fourth grade, but new problems crop up.  Students have often been told in early grades, "The big number goes in the house."  Well, its difficult to convince a preteen that their beloved fourth grade teacher was not exactly correct because now the size of the number has nothing to do with whether the number is the divisor or the dividend.  To complicate this, students are astonished to find out that there will no longer be writing remainders, but instead fractions and decimals.

It took me a long time to learn how to do all types of long division.  In fourth grade was the first time I was taught long division.  I will say taught and not learned because I definitely had not mastered it before I went to the next grade level.  It was so many steps, and I always seemed to mess them up.  The teacher tried to help by giving me mnemonics about famous restaurants and their cheeseburgers.  (Math teachers out there, I'm sure you're familiar with this one!)  She tried asking me how many times the number "went into" the other.  I have always been incredibly literal in my thinking, so I was picturing the dividend as a little house with a door in it and the divisor would walk up to the door, knock, and ask for a cheeseburger!  Hilarious, but that was not helping me learn to divide.

I finally did learn how to do most long divisions well enough to make it through until I was in high enough math classes to use a calculator.   I have a shocking confession to make though-- especially for a math educator-- I never learned how to divide when the divisor was two digits!  I finally learned how to divide by two digit numbers about a year ago-- after 10 years in the classroom, a successful college education in which I graduated at the top of my class, and three years of high school mathematics.

Because I was so bad at division myself, I feel a special connection with my students who also struggle with this skill.  I can often pick up on little things that are keeping students who are otherwise good math students from being able to divide. (...and also students who are not so great at math as well!)

Here are some of the things that might be tripping your students up and what you can do about it:

  • "The big number goes in the house."  The big number does NOT go in the house!  While this may be true of the word problems seen at lower grade levels, it is not a mathematically correct statement.  This causes all sorts of issues when students reach grades where they are dividing to find decimal or fraction values.  Rather it would be more appropriate to say that the number being divided goes in the house.
  • "How many times does this number go into that number."  Students who think very literally may not be able to see the meaning of this statement.  I have asked many students who have not be able to divide and are in higher grade levels if they can explain what that phrase means, and they cannot.  I ask if it would make more sense if I said something like "how many groups of this number can I make out of that number" and that usually makes more sense to them.  They can visualize the number being divided up into portions, but not the divisor repeating to make the dividend.
  • Dirty Monkeys and Cheeseburgers.  Mnemonics are great to help you remember something that you already have at least a basic understanding of, but they are not teaching tools.  Introducing mnemonics too soon can have a negative impact on students who need to cement concrete skills before they try to go to the abstract.  Although the mnemonics are fun and have their place for students who already understand the concept of division and are just needing reinforcement in the order of the steps, going to these devices too early won't help students who do not understand what each step actually means.  In case you are not familiar with the mnemonics above and you are curious: Dirty Monkeys Smell Bad (Divide, Multiply, Subtract, Bring down) or Does McDonald's Sell Cheese Burgers (Divide, Multiply, Subtract, Compare, Bring down).  There are of course many others and also variations of these two.
So what to do about this difficult skills?  Don't worry I won't leave you without some additional strategies to try!

Top and Bottom Division Strategy

This is the one that I used that helped most for students who were not completely fact fluent or where having difficulty with the number of steps involved in the standard teaching method.

  • Step 1: Make a multiplication table for the divisor.  Label the multiplier column with a "T" for top and the product column with a "B" for bottom.  Begin the table with 0 rather than with 1 and go through 9.
  • Step 2: Ask what is the largest number I can subtract from the dividend.  Have the student identify this number from the "bottom" column.
  • Step 3: Write the top and bottom pair.  Write the "top" above the digit in the "house" and write the "bottom" below.
  • Step 4: Have the student subtract, bring down, and repeat until the entire division is solved.
Benefits to this method are that once a student as practiced a few times it is three steps: Top and Bottom, Subtract, Bring Down.  Also, the insistence on the fact table to begin the work is useful to students who are not fluent yet.  Also, in this process, the emphasized skills are writing the multiplication facts and subtraction.  The student does not have to be able to visualize the dividing of the number.  Because of this, though, I like this method for older students who have been otherwise unsuccessful with the traditional method rather than this being go to method for a whole class.

Partial Quotients

This is the wonderful method that finally let me be able to divide by two digit divisors!  This method, while not used much before, is now becoming more and more common because of the emphasis on alternative methods by new state standards as well as by the common core standards.

In this method, you do not have to use the largest possible quotient, but you get to pick the quotient.  It is similar to a repeated subtraction method of division.  The main stumbling block to this method is the user has to have some number sense about compatible numbers and multiplication to use it effectively.
  • Step 1: Choose a number to multiply by that is easy for you to compute in your head.  This will often be numbers that end in zeros, or are multiples or 1, 2, 5, or 10 because using number sense these can be done mentally.  Write this number above the "house".  Line up place values.
  • Step 2: Multiply the divisor by the number mentally.  Write this number under the number in the "house".  Line up place values
  • Step 3: Subtract making sure your place values are lined up.  You will not be "bringing down" in this method.  The difference is now your new dividend for the next repetition.
  • Step 4: Repeat with another compatible quotient.  Stack it above the other, lining up the place values.  Continue through with the steps until your division is complete
  • Step 5: Add up all of the partial quotients to get your final answer.
What do you do to help your students learn long division?

Expanded Form or Expanded Notation?

Click to preview a math center
 about expanded notation
Before Texas adopted its new TEKS and before the common core became common across the US,  place value instruction was generally presented differently that it is today.  Teachers all around the country have had to adjust to their new standards.

One place where the standards have changed is in the area of place value.  Rather than an emphasis on memorizing the names of each of the place value positions, the importance has shifted to knowing about the relationships between the place value positions.

One question that seems to have cropped up after these changes to standards has been "what is meant by the terms expanded form or expanded notation?"

In Texas, the standards differentiate between the terms expanded form and expanded notation.  The former being a less complex skill that is mastered by third grade and the later a more complex skill mastered by around fifth grade.  In the common core standards, only the term expanded form is used, but its use in conjunction with the emphasis on base ten system fluency define it as the expanded notation concept.

There are many ways to represent a
decimal number using place value properties!
So what is the difference between these two concepts?  Is there a difference, or is just a matter of semantics?  Here is what you need to know to make sure you are instructing your students to the depth and complexity expected by the standards.  Traditional expanded form would look something like a simple addition problem.  The number 734 = 700 + 30 + 4, would be an example of a whole number written in expanded form.  Expanded notation, on the other hand, is a bit more complicated.  It relies on the distributive property and knowledge of the relationships within the base ten system.  Using the same example, we would see 734 = (7 x 100) + (3 x 10) + 4, or something similar when writing an expanded notation.

Notice how, although similar, the expanded notation of the number distinctly demonstrates the relationship between the place value position of the digit and the value of that position in the number.

We also can see a variety of appearances in equivalent representations when using numbers with decimal digits.

Remember this school year to use a variety of representations when instructing your students during the place value unit.  Our standards have changed, and so should our instruction!  Have fun!