Write Your Own Story Problems

The purpose of Math is to teach people to think.  It teaches patterns that people can use in real-life situations to resolve problems.  Unfortunately, I think a lot of that purpose is lost in today's world of worksheets and lists of isolated "math problems."  Children (and adults) do not connect the value of higher level math to everyday life.  It is important that you teach your child (or children) how to connect what they are doing to the world around them.

If you are using the Concrete-Pictorial-Abstract method of teaching, you are well on your way of teaching connections from "school work" to the "real world."  When I teach parents how to teach, I separate the 3 angles into separate lessons.  When I am using that method to teach children, however, they do not even notice that I've changed gears.  They just know I am asking them to do the next step.  Here's an example of a lesson I used this week with both my daughter and a 2nd grader whom I tutor.

I gave them the (abstract) problem of 17+26.  I write it vertically on a piece of paper (small paper, actually - a note pad).  I then asked them to solve the problem.  Both girls struggled a little, even though we've done similar problems in the past. 

Not waiting long for them get frustrated, I pulled out some dimes and pennies and had them show me 17 cents (one dime, seven pennies). Under those coins, I had them show me 26 cents (two dimes, six pennies).  I then asked them to put the pennies together and see if they had enough to trade for a dime. They, of course, could and I physically got a dime to trade for ten of the pennies. We stopped and marked on the paper that we now had 3 pennies (in the answer spot) and added one dime (over the tens place in the problem).  I then had them count the dimes and they added that answer (4) under the answer bar so that the answer was 43. 

I then took it a step farther.  In the past, we stopped at that point, but now we needed more to develop the concept.  I had both girls write their own story problem for the math equation I had given them. I did the writing (no reason to make it more frustrating - they can do the writing after we've done this a few times).  I asked them each to name something that there could be 17 of.  My daughter said stuffed animals, my tutoring student said diamond rings.  I then asked them to tell me who has the stuffed animals or rings. My daughter said they were hers, my student said her grandmother had them.  I then asked if the addition symbol meant they needed more or less of the toys or rings, and they both said more. After writing 2 sentences, I told them the answer bar meant we needed a question.  They both needed help coming up with a question related to the problem, but I happily helped them at that point by giving them 2 different questions which would be acceptable and let them choose which one to use. I know that in the near future, I will not always need to help them so much, but it's okay to give a lot of help when learning a new concept.  Here are the Math problems we ended up with:

(1)  I had 17 stuffed animals.  I had 26 more in my tent.  How many stuffed animals did I have in all?

(2)  My grandmother had 17 diamond rings.  She had a girl party and got 26 more.  How many diamond rings did she have for the party?

(hmmm...If it's going to get me 26 diamond rings, I might want to figure out what this girl party is and have one myself -haha!)

We then read over the problem a couple of times, allowing the girls to each read aloud the story problem she had written and making sure they saw that it matched the numbers in the math problem we had solved first.  After that, we did one more problem in the same way, though for my tutoring student, we chose a subtraction problem (with regrouping).

NOTE:  these lessons were separate for each girl, but it would have been educational, also, if they had been at the same time so they could compare their story problems.

That was the entire Math lesson.  It took about 20 - 30 minutes, but I believe they got much more out of it than if I would have given them a worksheet with 25 addition problems on it.  I want them to understand how math connects to the real world, because when they become adults, I want them to be able to think for themselves, and math is a monumental piece to that incredibly complex problem of developing reason.

Regrouping with Addition Problems

Teaching some math concepts can be difficult.  If you have read this blog much, you know that for Math skills, I like to use the concrete-pictorial-abstract pattern for every skill.  Some basic math, though, needs a creative method to be able to make an idea "stick."  Regrouping, in both addition and subtraction, is one of those skills.

 Regrouping is a better word to describe what we used to call "borrowing" or "carrying over."  It is used in both adding and subtracting where you have to exchange values.

 

One thing I learned early on when I was a teacher is that children have to value something before they want to learn about it. I have also found that children (as well as adults) value money.  I use money often as a manipulative to teach difficult skills.  As much as possible, I use real coins.  This was very difficult to do when I had 27 students, as it was hard to gather enough coins needed and not worry about theft, but as a homeschooler or as a tutor, I just grab some coins out of our "change cup" which we keep on the refrigerator.  (My husband will never miss anything except the quarters!  And besides, they usually go right back into the cup when the lesson is over)  I discovered that before the child sees the skill itself as important, s/he will value the coins and/or bills that you are using to teach the concept. Either way, it gets their attention and, therefore, makes the skill easier to teach.   Also, when I had to reteach this skill to 4th or 5th graders, they did not feel embarrassed that they did not understand something which they should have picked up in 1st or 2nd grades.  In fact, the students who did not need to relearn this concept were often jealous that they were not included in this lesson! 

 

Before using these coins in your Regrouping lesson, make sure you have taught your child that a penny is worth one cent and a dime is worth ten cents.  Also help them understand that you can exchange ten pennies for one dime.  They don't have to fully understand that, as this lesson will have the added benefit of helping them understand that also, but it helps if you are not teaching two different new skills at once.  They also need to understand how to add two digit numbers without regrouping.

 

THE LESSON - Addition with Regrouping

Concrete:

Write the problem on a piece of paper:

  34

+28

 

Then pull out dimes and pennies.  I usually make sure I have 20 pennies and 10 dimes (per child) as a general rule.

 

Have the child show you 34 cents using dimes and pennies.  Have some divider to put the dimes on the left side and the pennies on the right side.  (I use the line down the center of my expandable table, but if you don't have that you can use a ruler, a licorice stick, a long toy- anything that is straight). 

 

About an inch below those coins, using the same dividing line as the 34, have your child show you 28 cents. 

 

Explain that they just showed you the problem. To add them, push the pennies together and the dimes together.

 

Now ask if they have enough pennies to trade you for one dime (which you take from the "extras" pile).  They should carefully count 10 pennies and push them to the penny "extra" pile.  You give them a dime, but put it on the dime side.

 

Pictorial:

Tell your child that you have changed the number of pennies and dimes you have, so you need to change the problem to show what you now have.   You added a dime, so you have to write a 1 to the dime side of the problem to show what you just did.

 

Now count how many pennies are left. Have your child write the 2 under the 8.  (for younger children, or even some older children, they have had hard time lining things up. Draw a line on the problem to separate the ones from the tens, extending it into the answer space.  This will resemble the separating line on the table, which helps).  Tell them you now have 2 "ones" because pennies are worth one cent.  (When I say an important lesson, I usually make them repeat it verbally a few times to make sure they heard me and know it's important). 

 

Now remind the child that they added a dime and point again to the 1 he added on the paper.  Then have him count the dimes.  Have him write the "6" in the tens column.  Tell him he has six "tens" because each dime is worth ten cents.  

 

Have him tell you how many "tens" he has.  Then tell you how many "ones" he has.  Then have him read the answer.  Many children, especially at first, will say something like "six and two."  Help them see that if you take away the line, it is "sixty-two."  You write the word "sixty-two" under the answer.

 

Abstract:

Now go back and solve the problem you just did together.  I use "touch math," so at this point, I or my child writes the dots on the digits of the problem.  (you can skip that step if you do not want to use touch math.  If you don't know about it, look it up.  It is a fabulous method of teaching counting, and it ends up being much more accurate than fingers.  They have some free materials so that you can see where to put the dots, which for me is all that I used from the program). 

 

Then have them add the ones.  I usually say this several times, trading out the words "pennies" and "ones" so they can connect them in their brain.  Do NOT use the actual pennies at this stage unless your child is completely lost, in which case you probably need to spend more time counting single digits instead of going on to double digits.  Remind them you traded ten pennies for one dime, so they should trace over the 2 in the answer place and the one that you added above the dimes.  Get them to explain to you why that "1" was added.  (Even if you just said the answer, the more your child can explain, the more you know he understood the concept).  We put the dot in the center of that "1" for touch math.

 

Next, add the tens.  Talk about what they are doing, using the words "dimes" and "tens" often and interchangeably. 

 

Have them read the answer (which you already wrote in words).  Then have them copy the words below where you wrote them.

 

Do 3 - 4 more problems, then stop for the day.  You want to give your child time to "stew over" the method of solving the problem.  This should be done  3 - 5 times per week for at least 2 weeks (about 5 problems per day) before you use problem with hundreds in them.  After a few days, you can start having problems with a hundred in the answer, but you will need to have a dollar to trade for ten dimes.  Use the pennies and dimes every day, but as your child understands the concept better, you will find that they are using them less and less.  It's okay for them to like playing with the manipulatives (the dimes and pennies), as long as they are using them correctly to solve the problems.  He will stop playing with them as he gets excited about solving the problem correctly.  It shouldn't be rushed because this will actually hurt the learning process. 

 

Don't use the virtual money (meaning the programs on the internet) until AT LEAST 2 weeks of successfully using the real or play pennies and dimes.  Anything online requires a higher level of understanding and your child will lose a lot of the value of this type of learning if you ignore the physical money and go straight to the virtual world.

 

Have fun!

Best Math Method

My favorite method of doing math is to use the concrete-pictorial-abstract method with every problem. Concrete means they use something they can move around to show the concept. Pictorial means they draw a picture showing the concept. Abstract means they use numbers and symbols to show the concept.

 

When they are first learning a concept (at the beginning of the year), you do all or most of the work in the concrete stage. As they progress, you add the pictorial and abstract stages, doing less with the concrete stage but not eliminating it all together, even after they show they fully understand. This type of learning is not a quick "how many can you finish in 2 minutes" type of learning. I tried to never do more than 10 problems per day, sometimes only doing one to three problems. The children will spend a lot of time on whatever stage they are most comfortable with. In the concrete stage, all they want to do is play with the manipulatives (whatever you are using to demonstrate the concept). When they start to understand the concept better, they will get very detailed in their drawings for the pictorial stage. They want to put whiskers on the cats or make the "x" representing the dogs have a face to match that animal in the concrete stage. That's okay. It means they are progressing in understanding the concept. When they start to feel comfortable in the abstract stage, you still want them to demonstrate the other stages, but you will want to start using story problems rather than just the raw numbers and symbols. If a child understands the basic concept, go on to harder numbers but don't change the overall concept you are working on for the year. Also, don't forget to include learning about money at each stage. That will have to be after they've worked with the concrete items and are starting to feel comfortable with the pictorial or abstract stage. Measurements can also be added, but be careful not to go too fast. If you're teaching the counting concept that year, think about using other objects besides rulers, etc., when you measure. "That couch is 10 of Daddy's shoes long!" Don't introduce fractions of a measurement before you have talked about fractions, for example. It will be very frustrating and your child will probably regress.

As a teacher, I couldn't believe how many 4th and 5th graders did not fully understand counting! Teaching it this way ensures that they know what they are doing it and can use these basic math concepts when they get to harder problems. Let me give you a couple of examples at different ages.

(1) When my daughter first stayed home, she could count but she could not match the numbers with objects (one-to-one correspondence). For the "concrete" step, I gave her a small box and had her fill it with stuffed animals. She lined them up in the living room (this took an excruciatingly long amount of time because she had to explain to each one why he was chosen before the others, but that was okay because she needed to do that). We then took some store-bought number flash cards and put one number next to an animal, in order. Then we went down the line, counting them as we went. I would do things like trade the places for the animals and we would count again to see if the total changed. That was math for the day (it took about 30 - 45 min. to do it). We did this several days using My Little Ponies, Barbies, Noah's Ark animals, etc. so she could get the concept. She would never have gotten tired of playing with her toys, so I had to decide when to change the activity - which was about 2 weeks. (I was bored, but I wanted her to really understand so I had to find a way to be excited about it). At the time, she had a severe fear of writing (thanks to public school), so we had to skip the pictorial and abstract, but if we didn't skip it, the pictorial would have been to draw a circle or an "x" for each animal and write the total number next to it. The abstract step would have been to write tally marks (teaching her to bundle them together) and put "=18", or whatever the total was.

 

(2) Addition was done with the same toys. Concrete: we got 3 plush dogs and 2 plush cats and put them side by side. We counted how many dogs we had, then how many cats and then talked about how many animals there were total. (I picked out the toys we used this time). Since my daughter was still afraid of writing, I did the drawing of the pictures, but I was just very basic. I used circles for dogs and x's for cats. Then I wrote the number 3 under the dogs, the number 2 under the cats. Then I circled them and wrote 5 outside the circle. It would have been better if she had drawn it, but with her fears, I had to take what she was willing to give, so she leaned against my arm and we talked through everything I drew. On the same page, I wrote 3+2=5. We talked about the problem, I had her point to the right animals, the drawings, and the abstract numbers so that I knew she understood the concept. I think 5 was the maximum we did. As she learned how to add, I let her pick the toys we would use. For subtraction, we did the opposite.


For multiplication and division, I will use what I did when I taught school (my daughter isn't there yet). We would take objects and group them.

(3)   Multiplication is simply repeated addition (3+3+3=9 or 3x3=9). I taught them that the "x" symbol meant "groups of," so 3x3 should be read "three groups of three." Then we used these objects (we used much smaller items in the classroom than stuffed animals because we're dealing with large numbers) to create the problems. 7x8 meant 7 groups of cheerios with 8 in each group. Then we practiced counting by multiples to find out how many of the items we actually had. Pictorial stage was shown by putting an "x" inside a circle for each group, separating the groups and writing the numbers outside each circle, then drawing a large circle around the entire group and putting the total number near that circle. Abstract would be to write a multiplication problem. After they understand this concept very, very well, then you can teach the entirely abstract concept of larger multiplication. Look into Lattice multiplication when you get to this stage. I taught both the traditional method and Lattice Multiplication. Every problem had to be done twice. We only did 10 problems per day, but they knew what they were doing!

(4) Division is taught just the opposite of multiplication. Division is simply repeated subtraction (9-3-3-3=0). You start with the total amount and divide the manipulatives into smaller, equal groups with remainders. Draw the pictures by drawing the total amount and then circling groups out of the whole. In the abstract stage, though, you will want them to write every division problem 3 different ways.
          
54÷6=9

         9
 54√‾₆‾ (sorry, this one didn't come out on Word well)

54/6=9 (should look like a fraction)

Go on to "long" division only after they fully understand the basic concept. I did not teach "short" division to every child because it should only be taught if they fully know what they are doing with long division.

(5) This method can be continued with the higher math skills, as well. In Singapore, they require all three stages (concrete-pictorial-abstract) to be used through the top grade in High School. One vital concept in algebra is to make sure both sides of the equation are equal. Use a balance scale to get this concept across. The pictorial stage is very important, because it's hard to find different weights to show these concepts. You can use different lego blocks sitting on a scale drawn on paper to show the concept

Money Exchange Game

            Money is – without a doubt – an essential life skill which all children need to understand.  Unfortunately, telling a “concrete” child that a dime is equal to ten pennies, or five nickels are equal to one quarter just doesn’t make sense.  They can see that there are more pennies than dimes, so how is it possible for them to be equal?  While the child really won’t be able to understand it until he reaches the “formal” stage of thinking, around age 12, you can do some things so that your child can use money and explain it, even though they may not fully understand it.  One of these methods is using the Money Exchange Game.

            For the Money Exchange Game, you need at least 20 pennies per person, at least 10 dimes per person, 2 dice per person, and one dollar bill.  The coins should be put in two piles in the center of the playing area, and the dollar bill should be placed where everyone can see it in the center. 

            Players take turns rolling the two dice.  For each number rolled, they take one penny.  When a player has 10 pennies, they exchange them for one dime.  It makes rolling a 10 or above on the dice very exciting because they can choose a dime instead of a handful of pennies.  Of course, a roll of an 11 or 12 allows them to choose a dime and one or two pennies to match the number – which is even better.

            When a player gets 10 dimes, s/he can exchange them for the dollar bill.  This person is the winner of the game – congratulations!  It is up to you, as the parent or teacher, to decide if this child gets to keep the dollar or if s/he has to give it back. 

            This game can be very fast paced, but it’s a lot of counting.  If your child is very young or has great difficulty counting, you might want to lower it to 50 cents instead of a dollar.  It is a fabulous game which addresses many different skills and will keep the interest of children of many different ages and abilities. 

            Playing games can be fun, but they’re much better if they can learn while they are playing them.  This is not a game you want to play more than once per day, but maybe once or twice a week would provide great skill practice for counting

Teaching Patterns

            Patterns are a part of every Math program, particularly in the elementary ages.  Many teachers – whether public school, private school, or homeschool – quickly teach a pattern and then go on to another topic.  Others will use the lessons of patterns as an art activity and not connect the lesson to any other form of math.   Is patterning really worth teaching? 

            In a word, my answer is ABSOLUTELY!  Patterning is the basis of algebra, which, in turn, teaches logical reasoning, which is vital in adulthood.  Skipping patterning skills, or only regulating them to an occasional art project, is missing a cornerstone in mathematics.

            In the beginning, patterning should include simply recognizing a pattern.  These lessons include something like looking at a set of squares of different colors and the child must recognize that the pattern is “red, blue, red, blue,” and then tell that the next square should be “red.”  Developmentally, these lessons should really begin with concrete objects, like stuffed animals, different colored candies or cereal pieces, toy cars of varying sizes, colors, number of tires, etc.  The second step should be pictures on a paper, which is what most math programs include.  Colors, shapes, and pictures of objects are frequently used to demonstrate patterning skills.  Children should be able to tell what the pattern is and then be able to continue that pattern.

            This is a good beginning, but this is not where patterning lessons should end.  One step that is frequently skipped by teachers, particularly homeschool moms/teachers, is called patterning “transference.”  Once a child can recognize the pattern, the objects should be labeled with a letter or number.  For example, a “red, blue, red, blue” pattern can be transferred to an “AB” pattern.  A “square, circle, triangle, triangle” pattern can be called an “ABCC” pattern.  This is an important step because then the child can then transfer the pattern to a similar pattern, using different objects.  A “square, circle, triangle, triangle” pattern, once labeled “ABCC,” can then be connected to a “red, blue, yellow, yellow” pattern.  This is an essential step to developing logical reasoning.

            After a child is able to recognize a pattern, then transfer the same pattern to other objects, the next step is recognizing number patterns.  Numbers are more abstract, so that is why it is not in the beginning.  Patterning, like every other lesson objective, should be developmentally appropriate. Children should be able to explore different patterns, with increasingly more difficult steps.  Having children using number patterns of “add three” or “divide by 2” are examples, but you have to give them time to explore and create their own  number patterns before they are able to do more abstract patterns, such as find the rule of a table.

            When I taught fourth grade, we were required to teach something called a “function machine.”  For abstract thinkers, a function machine, or a table which lists numbers based on a set rule, makes it much easier to understand more complicated patterns.  An example of a pattern inserted in these tables would be, “2, 4, 6, 8.”  The recognizable pattern in this case is “add 2.”  The pattern is not stated – it must be recognized.  Then you can transfer this pattern, beginning with any number.  However, as stated in my article, “Not Just a Little Adult,” a fourth grader (generally ages 8 – 9) is firmly in the “Concrete” stage of development.  They are not ready for these non-stated patterns.  As a result, fourth and fifth grade teachers are very frustrated in teaching these concepts.

            To combat objectives the students were not developmentally ready for when I was a teacher and had no choice what to teach, I had to find a way to take this “abstract” idea and make it “concrete.”  What I did was teach the children that if the number pattern went up, the two choices were either addition or multiplication.  If the number pattern went down, the choices were subtraction or division.  Then they had to see what was needed to get from the first number to the second number.  I had them write the problem either above or below the number pattern, so they could physically see the pattern.  As you can tell, the above number pattern goes up, so my choices are either addition or multiplication.  Then I see what I can do to the first number to get to the second number.  2+2=4, so I can try to add 2.  However, 2x2=4 also, so then I have to check the next number.  If the pattern is add to, then 4+2=6.  Is 6 my next number?  Yes, so this is probably the answer, but I always need to check the other option.  4x2=8.  Is 8 my next answer?  No.  That means “add 2” is my pattern.  I took an abstract concept and made it concrete, so my students could understand it and do it.

            As a homeschooling mom, I recognize it’s a waste of my daughter’s time to teach her something she is completely not developmentally ready for.  At age 12, my daughter will probably be ready for this abstract concept, and it won’t take nearly as long for her to understand what she is doing.  If she is developmentally ready, she won’t have nearly the frustration level that she would have if it were taught too early.  In fourth grade, I will be teaching my daughter more complex, concrete problems which she will transfer to other objects, colors, or shapes.  I will not just check off a list that she understands it, but I will give her opportunities to explore these concepts so that she is ready when we start algebra and other more difficult concepts which rely on these base concepts.

            Patterning is a vital step in any mathematical program because it teaches logic.  Children should be able to transfer a pattern to other media, and they should be able to create their own patterns which they can then transfer.  Number patterns, tables, and function machines should be taught once a child is in the abstract level of comprehension.  These tables greatly simplify the concepts taught in algebra, so they should not be skipped, but they should be taught when the child is ready to learn.