Mind Like Child

Chris' SAT Questions: Permutations, Combinations, Oh My! (Part 1)

Guest Post by Chris Seymour

Some math problems on the SAT ask you to count the number of possible ways to do something. Now, you might be thinking, “Come on, Chris, everyone already knows how to count!” But counting problems on the SAT are not as simple as counting on your fingers and toes. Take the following example question: 

In order to solve this particular kind of problem, we'll have to tackle two concepts, permutations and combinations. Sound intimidating? You're not alone. Permutation and combination problems are a unique sort of problem on the SAT. To make matters worse, most math classes will tell you to solve these problems using exotic-looking formulas, such as:

Our Favorite Posts from 2012

Posted by Karen and Damon O'Hanlon

Well, it certainly was a wonderful year. We're all very sad to see 2012 go... But hey, at least the posts won't go anywhere! So before we continue down the road of time, let's take a look at some of our favorite posts from the year past.

 

First, we have Karen's favorite posts.

Organizing for Art So That Your Classroom Won't Explode

 

7 Tips for Giving Awesome Praise to Kids

 

Karen's Classroom: Snowman Art

TV Review: Superman, The Animated Series Season 1

 

Karen's Classroom: Teddy Bear Graphing

 

 

And now, Damon's favorite posts.

Video Game Review: Pikmin

Damon's Blacktop: Bull and the Teacup

 

Why Fail Children?

3 Guidelines for Teaching Kids Mindfulness

Damon's Blacktop: Animal World

Meet A Theorist: Lev Vygotsky

Posted by Karen O'Hanlon

Born: 1896

Died: 1934

Area of Expertise: Socio-Cultural Psychology (also termed cultural-historical psychology)

Influenced by: Georg Wilhelm Friedrich Hegel, Karl Marx, Ivan Pavlov

Concepts and Terms: Social constructivism, zone of proximal development (ZPD) [learning leads development], cross cultural variation, scaffolding, behaviorism, interior speech, symbolic play

Born in the same year as Piaget, Lev Vygotsky had a stellar, if tragically short, career. He died at age 37, leaving behind a substantial amount of unpublished work. Recognized early as a genius, by the time he was a teenager Vygotsky’s family and teachers marveled at his extraordinary intellect and vast memory. Naturally, those around him began to expect great things from Vygotsky.

However, due to political circumstances in Czarist Russia, the place of his birth, Vygotsky found himself in a frustrating situation. Only three percent of all university openings were allowed to be filled by Jews. With so much competition for a very small number of slots, you can imagine only those with exceedingly high entrance exam scores were admitted. But no sooner had Vygotsky achieved these scores, than the game was changed. Suddenly the government decreed that the three percent would instead be chosen by lottery. Vygotsky was grief-stricken, but in the end luck won him a lottery placement at Moscow University. Even after his admittance, as a Jew he was discouraged from studying his favored academic subjects. Philosophy or history might’ve jump-started an academic career, but the laws of the time would have barred him from holding a paid government position such as teaching. So instead of pursuing his primary interests, Vygotsky began studying law.

While enrolled at Moscow University, Vygotsky also went to classes at Shaniavsky People’s University, a free university established by professors who had been dismissed from teaching because of their opposition to the Czarist government. At Shaniavsky, Vygotsky encountered the writings of Hegel and Marx. In 1917, Vygotsky graduated from both universities.

Share Time: Where Have All the Playgrounds Gone?

Posted by Karen O'Hanlon

So what is going on with playgrounds these days? That’s basically all I want to know. Even a lot of the spiffy modern ones don’t seem to offer the kind of possibilities for fun and development that I remember as a child.

When I look back at the community playgrounds that I knew in the '50s and '60s, they weren’t necessarily all that remarkable looking. The swings and climbing bars were made of ugly steel pipe. Seesaws were made using straight wooden boards with indentations cut out near the ends for legs. And the merry-go-rounds were just flat metal disks, mounted to a center post, with a few arched bars on top. Most kids would sit in the middle while a few ran around the outside, pushing to get it spinning fast, before hopping up onto the edges for the best ride of all. For me, the greatest swings were the really tall ones, with their long metal chains and a half-circle strip of rubber that snugged up against my bottom and made me feel securely tucked-in. I could swing so high that I had a clear view of the tops of the trees. It made me feel like I was flying. The tall slides were great, too, although their metal sometimes got too hot in the burning mid-day sun. Even that was okay, though. I learned to test the metal before I got on, or use a piece of cloth to sit on before zooming down.

Why were these simple playground structures so enjoyable? It's not just because they were the only thing available. There was something more to them than that.

At least in part, it was for the same reason that I loved cardboard boxes. The materials were simple, but open with possibility. What the playground didn’t offer visually, my friends and I filled in with imagination. The tall swings became an airplane where my friends and I were the crew. (We pulled those planes out of countless near crashes!) We became actual monkeys on those monkey bars. We hung from them like the branches of high trees, contentedly swinging, perching, and twirling from bar to bar. Sometimes the merry-go-round transformed into a boat out on the ocean, caught in a whirling current. Or maybe it was a spaceship hurtling through the stars, the launchpad for whatever new adventure we could imagine around the playground.

Every piece of playground equipment, just like every cardboard box, offered endless opportunities for imaginative play. Nothing about them needed to be fancy. Yet in today’s world, the kinds of equipment I played on have been labeled far too dangerous for kids. This is a shame, and generally, I think playground safety concerns have been blown out of proportion. Now, I’d be the first to admit that as a kid I got my fair share of bumps, bruises, skinned knees, and splinters along the way. Yet I don’t remember a single broken arm or leg. (I do remember someone falling on the blacktop during a chase game and ending up with a severe sprain. Ironically, no play equipment was necessary for that injury!)

As an adult who became a teacher, I remember pieces of equipment vanishing one by one from the school playgrounds where I worked. The merry-go-round was the first to go. Then the slides disappeared. By then the seesaw was already long gone. The monkey bars lasted the longest, but the modern playground became devastatingly bare. For much of my career, all the kids had to play with during recess were a few balls, a handful of jump ropes, and each other. Without props to bolster their imaginations, kids fell into degraded games filled with competition over limited equipment. Their frustration was compounded by too many chasing games, often leading to pushing and name calling. 

There’s always a certain amount of aggression that arises when you have a group of kids immersed in unstructured play—regardless of the environment—but I think it was less pronounced back when I was a kid. We were more interested in preserving the pretend games we had created, leveraging the equipment that was available, than in bickering with one another. The magic of our imaginations gave all of us something to work towards together.

And when I think about the loss of the old play equipment, it’s more than just the loss of imaginative stimulation that I lament. By taking away play equipment (and often not even replacing it with anything at all), kids are being denied some great developmental and problem solving opportunities.

For instance, why does a seesaw work beautifully sometimes and not at all other times? If two similarly sized kids wanted to ride the seesaw together, you were in business. But what if two kids were not the same size? Then you really had to think. Sometimes the littler kid could scoot more toward the end of their side of the seesaw, or the bigger kid might be able to slide in more to the center. Other times we would invite an extra small child to join the first small child. Then two little ones equalled the weight of one big one. We figured out the balancing problems in order to have more fun. (And again, this was not a source of constant, serious injury.)

Then there’s the merry-go-round, which taught a whole different set of skills. When little kids would negotiate with the older and bigger ones about who should push and who should sit in the middle, play on the merry-go-round became a socially interactive exercise. True, the older kids who pushed got the satisfaction of being in control but they also became responsible for making everyone’s ride more fun. It was an early leadership role, and they learned to feel pretty good about taking care of the smaller children. For the smaller kids, there was modeling involved. They got to see what it meant to be a bigger kid, experiencing both the joy of independence and important roles of responsibility. 

On our playground, my friends and I were just having a ton of fun, not really conscious of the fact that we were also experimenting—learning math, science and social skills in the process. It’s hard to name any piece of modern playground equipment that fosters these same skills. As a teacher, I felt tasked with a whole lot more than just physical safety. We’re failing our kids when we don’t recognize that there’s more to playground time than mere physical movement. I can’t help thinking that we've sacrificed too much in the name of safety, without really recognizing everything we lost.

Related links:

• Can you have seesaw math without riding a real seesaw?
• Can a playground be too safe?

Chris' SAT Questions: Shortcutting Systems of Equations

Guest Post by Chris Seymour

In most algebra problems, you have only one equation with only one variable. However, some algebra problems can give you multiple equations with multiple variables. This is called a system of equations. Solving a single equation is easy, but when you have more than one equation, it can be difficult to know where to begin. Here's an example:

Most high school students learn two methods for solving the problem above. One of these methods is substitution, where you solve for one variable, and plug the result for that variable into the other equation. The other method is elimination, where you stack the equations, one on top of the other, and find a way to combine the equations so that one variable is eliminated. Are you familiar with either of these methods?

Well, guess what—sadly, for the purposes of the SAT, it doesn't matter whether you know how to use substitution or elimination, because both methods are virtually useless.

Now, don't get me wrong. Substitution and elimination are great methods when you need to solve for each individual variable in a system of equations. But on the SAT math sections, time is scarce. Solving for every variable, one at a time, is far too time-consuming. That's why the SAT will rarely ask you to solve for every variable separately. Instead, SAT questions will ask you to solve for a single, particular variable or expression, and your job is to find a way to solve for it directly. So the question is, how does one do that?

The good news is, in this type of problem, there's almost always some kind of shortcut you can use to solve a seemingly difficult system of equations in just a couple of easy steps. That means you won't have to do any fancy manipulation or rearrangement of the equations. Usually, all you have to do is line up the equations and apply one of the four basic arithmetic operations: addition, subtraction, multiplication, or division.

Karen's Classroom: Coffee Can Ice Cream

Posted by Karen O'Hanlon

Age Group: 4 to 6 years old

Skills taught: Observation, predicting outcomes, chemical changes in materials, vocabulary (gas, liquid, solid), following directions.

Pre-requisite skills: Ability to take turns.

Activity length: About 1 hour. It takes about 10 minutes to introduce the activity, 15 minutes to prep ice cream mix, and somewhere between 15 to 30 minutes to freeze the ice cream. Making it can take longer depending on how consistently and vigorously children roll the ice cream cans. It can be tough work and your hands get good and cold.

 

Materials:

• Small metal coffee can (11 oz.) with plastic lid

• Large metal coffee can (33 oz.) with plastic lid

• Duct tape or wide masking tape to keep lids on

• Ice cream ingredients (1 cup whole milk, 1 cup heavy cream, 1/2 cup sugar, 1 teaspoon vanilla)

• Small bag of crushed ice (the equivalent of 3 gallons)

• Ice cream salt (I had to ask the grocery clerk to get it from the stock room. My local Safeway had it, but didn't keep it on the shelf.)

• Measuring cups

• Measuring spoons

• Mixing spoon

• Pitcher or mixing container with spout for pouring

• Large, flat surface on which to roll cans (such as a floor, a very big table, an outdoor sidewalk, or playground blacktop)

• Bowls, spoons, and napkins to serve the finished ice cream

• Ice cream scoop or large spoon

• (Optional) Plastic table cloth as a workspace if children will be sitting on the ground

• (Optional) Activities for kids who are waiting for their turn, or finish earlier than others

• (Optional) Gloves to keep kids hands' warm while they roll ice cream can

 

Procedure: Introduce the activity by telling the kids they will be learning how to make ice cream.
 A nice way to get the activity started is to spend 10 minutes brainstorming what kids already know about ice cream. Ask questions like: How is ice cream made? What does ice cream have in it? How does it taste? Where does ice cream in the store come from? For this part of the activity, encourage every kind of contribution. There are no right or wrong answers, this isn’t a teaching moment. It’s just an exercise to get them thinking. 

Steps:

1. Show the kids the ingredients, the equipment, and give them a preview of the process.
2. Divide kids into groups. A bigger group means less rolling for each individual child when you begin the rolling process. A smaller group will mean more rolling for each child. Take this into consideration when establishing group size. Younger kids get tired faster, so in their case, you may want more kids per group. Mix ice cream ingredients in the spouted bowl or pitcher.
3. After mixing, pour into the small coffee can. When the mixture is poured into the small can, the can should be no more that 1/2 to 2/3 full.
4. Secure the lid to the small coffee can using tape.
5. Place the small coffee can inside the large coffee can. Then, in the space around the small can, pack alternating layers of ice and salt. Continue the layers of ice and salt all the way to the brim of the large can, covering the small can completely.
6. Secure the lid to the large coffee can using tape.
7. Time to freeze the ice cream. Have the kids take turns rolling the can back and forth to one another. The teacher monitors to make sure every child gets a turn. Usually, the ice cream will freeze in 15-20 minutes, but sometimes it can take longer. It is ok to check the consistency of the ice cream after about 15 minutes. You can repack the edges with another round of ice and salt if there is too much liquid left in the ice cream concoction. Then continue rolling as necessary.
8. Once ice cream is frozen, serve into individual small bowls and distribute to kids along with spoons and napkins.

 

Tips:

• Incorporate as much "hands on" as you can. Allowing kids to feel the slippery coldness of ice and the rocky grittiness of the salt are instructive for them. They will learn vocabulary, connecting the words to experience.

• When you are mixing the ingredients, kids can count the number of stirs or they can each have a turn (maybe two stirs each). Everyone can help count. This eases the waiting whether you decide to have everyone stir or the teacher does the stirring.

• Depending on age, abilities, and interest level, there are lots of possibilities for how to structure groups and rolling. For instance, two kids (supervised by an adult helper) can roll at a time, while other children are engaged in a different activity. Keep a list and check off names as each group of two takes their turn. Alternatively, three to six kids can sit at a table rolling. Using this strategy, I've had four tables going at once with an adult at each table. (One benefit of this method is more ice cream at the end to share.) Or you can do one big group rolling, with kids seated around a table cloth. If you do one big group, there can be a lot of waiting so it is wise to engage the kids with counting, songs, or whatever you can think of to keep the kids involved. Even so, 15 to 20 minutes is probably as much focus as you can expect with younger kids.

• If the ice cream is taking too long to freeze, it's ok to stir it by hand and put it in the freezer. Before you stir, be sure to take the time to show the kids the frozen ice cream starting to accumulate around the edges of the can.

• While kids and adults are enjoying the ice cream, it's a great time to discuss and review vocabulary and the kids' experience of making the ice cream.

 

What I love about this activity: Of course, the ice cream! And the fact that the kids love ice cream and are excited to make their own so the activity is very engaging. Kids who have never thought about how ice cream is made are often surprised. Said another way, the ice cream is rich but so is the experience. You can get creative and vary the ingredients by adding fruits or other flavors (like cinnamon, anise, mint, lavendar). Each group could then make a different flavor. You can organize a tasting party and make a graph of favorite flavors, then guide a host of counting and comparing activities. This is a great end-of-the-year, hot weather activity. But it can be done indoors during colder months as well. Any season is the right season for ice cream!

Related links:

• The science behind ice cream making
• How to make ice cream using a plasic jar and plastic bags
• Why do you need salt to make ice cream?

Chris' SAT Questions: Variables Gone Wild

Guest Post by Chris Seymour

How long have you known what a variable is? If you're like most students, you've known about variables since middle school algebra, or maybe even since late elementary school pre-algebra. That's a pretty long time.

But now let me ask you: How long have you known what a number is?

This time, if you're like most students, you can't even remember a time when you didn't know what a number was. It seems ridiculous to even bother answering such a question.

No matter how comfortable we are working with variables, we will never be more comfortable with variables than we are with numbers. We've been working with numbers our entire lives, and even today, we use concrete numbers far more often than we use abstract variables to solve everyday problems.

Suppose you came across the following statement in a math problem: "John weighs x pounds." When we read that statement, it's impossible to get a mental picture of what John might look like. It's too abstract.

However, suppose you were to say, "Okay, let's just make x equal 300 for now." Suddenly, you have a clear image of John as a hulking linebacker or sumo wrestler. This image allows you to think about the problem in concrete, real-life terms. It also makes it easier to figure out other pieces of information that you might need to solve the problem.

Perhaps you'd like to figure out how much John weighs compared to other people. Perhaps you want to know how much weight he'd have to lose to get to a certain goal. Solving for these pieces of information no longer requires any complicated algebra if you imagine the variable as a number; all it takes is simple arithmetic.

This technique has many different names, but we'll just call it 'Replacing Variables with Numbers'. You might be surprised how often you can take advantage of this trick.

Here's an example:

Yuck! This doesn't seem like a very easy question. Most people would have to reread it several times just to wrap their heads around the given information. But take heart. The problem is much simpler than it seems, and the only things that make it seem more complicated are those pesky variables. Let's try replacing those variables with some concrete numbers:

Now let's plug those numbers into the original question: 

Unfortunately, we have a problem with the numbers we chose. The person will be 2 years old in 3 years—that means the person hasn't been born yet! But even though our numbers don't work, we've already learned something valuable about the problem: x must be greater than y. Now let's revise the numbers we've chosen:

 

Again, let's plug those numbers into the original question:

Look at that! Suddenly, the problem is much simpler. The person will be 10 years old in 3 years, so the person must be 7 years old right now. Then, in 4 years, the person will be 11 years old. So our answer is 11. 

But oops, we don't have 11 in the answer choices! Don't worry, all we have to do is replace the variables in the answer choices with the same numbers we used in the problem:

  

Choice B is the only one that comes out to 11, so B is the correct answer.

 

Let's try a tougher one now:

Take a look at the answer choices. When the answer choices contain variables, replacing the variables with numbers is almost always a good option. Since the total area is 5,000, the math will work out more easily if we choose something that goes evenly into 5,000. Let's say x = 100. Area of a rectangle is base multiplied by height. So that would mean that y = 50 (since 100 x 50 = 5,000). Let's label the diagram with our values for x and y:

Now, just add up the lengths of the fences. We have two fences of length 100, and three fences of length 50, so the total length of the fencing will be 350 feet. Once again, all we have to do now is replace the variables in the answer choices with the number we've chosen and find out which one gives us 350.

 

D'oh! We have 350 for both C and D. Although this is not an ideal situation, it's nothing to worry about. It just means that we will have to replace our variable with a different number to figure out which of these two answers is correct. Let's say x = 200 this time. Then you know y = 25 to give us an area of 5,000 (200 x 25 = 5,000). This time, if we add up the lengths of the fences, we get 475. Now we can go back to the answer choices. Remember, as a time saver, you only have to check choices C and D—the other choices are already out of the running.

Therefore, C is the correct answer.

So just remember these steps:

1. Choose numbers to replace variables.
2. Use these numbers to solve for the answer.
3. Replace variables in the answer-choices you're given with the numbers you chose to find which ones match your answer from Step 2.
4. If get more than one answer-choice that matches your Step 2 answer, repeat Steps 1-3 using new numbers.

I have to say that replacing variables with numbers is probably the single most valuable trick you can learn for the math section of the SAT. Much like magic, you're making all of the variables in the problem simply disappear! So, the next time you do a math section, look for opportunities to replace variables with numbers. You just might find that the problems are much easier than you thought.

Related links:

• Chris' SAT Questions: Misplaced Modifiers
• Chris' SAT Questions: Dealing with Funky Shapes

Chris' SAT Questions: Dealing with Funky Shapes

Guest Post by Chris Seymour

Today we start a recurring guest post by my good friend, Chris Seymour. Chris was a Valedictorian at my high school, and got a score of 2330 out of 2400 on his SAT! Next he graduated from UC Berkeley, and these days he works as an SAT prep tutor. So grab a pencil, relax, and enjoy unlocking the SAT's secrets. - Damon

If you like the idea of not doing math on the math section of the SAT, listen closely! In some cases, understanding a math problem is just a matter of approaching the problem in a different way. Today I'll teach you the best approach for dealing with the weird shapes the SAT sometimes throws at you. This approach can take a difficult geometry problem and turn it into a simple problem that requires virtually no math to solve.

Here's an example of a geometry problem that uses weird shapes to try and confuse you:

The problem asks us to find the perimeter of the diagram, but the diagram is a weird shape. We don't have any formulas for finding out the side lengths of a weird shape like this, so what can we do?

The answer is simple: if we don't know how to work with this weird shape, then we'll just have to break the shape down into other shapes that we're more familiar with. There are actually only a few basic shapes that you need to be familiar with for the SAT: triangles, squares, rectangles, and circles. Virtually all unfamiliar shapes on the SAT can be broken down into these four shapes.

Let's take another look at the diagram. Imagine drawing a line somewhere on this diagram that will break this large shape into more familiar shapes. Where would be the best place to draw such a line? (Remember, on the test you can actually draw in your test booklet. The test graders don't care how you mark up your test.)

It turns out that if you draw a line connecting the bottom left and the bottom right corners of the diagram, the problem becomes much simpler. The two angles at the top are already labeled as right angles (90 degrees), so you can see that we just have a triangle taken out of a square.

The bottom two angles of the square must also be right angles. Using our knowledge of complementary angles, we can figure out that the bottom two angles of the triangle are each 60 degrees (90° - 30° = 60°). The angles of a triangle add up to 180 degrees, so the third angle must also be 60 degrees.

The three angles of the triangle are equal, so we know it is an equilateral triangle (which tells us that all of its sides are also equal in length). The bottom side of the triangle is the same length as the bottom side of the square, so each side of the triangle is 7.

Now remember, what was the question? That's right, perimeter! So add up all the sides.

So the perimeter of our funky shape is 7 + 7 + 7 + 7 + 7 = 35.

Here's another example of how you can break down a weird shape, this time to find its area:

The problem asks us to find the area of the shaded region, but just like in the previous problem, the shaded region is a funky shape. It resembles some kind of double-bladed axe head. Unfortunately, we don't have a formula for finding the area of a double-bladed axe head.

Look at the diagram again. Can you think of a way to divide this diagram into pieces so that each piece is a familiar shape?

This time the best solution is to draw two lines into the diagram itself. Draw one vertical line through the middle of the square, and one horizontal line, like so:

Now we've divided our diagram into four easy-to-work-with pieces. Each piece consists of a quarter of a circle taken out of a square (although the quarter-circle is the shaded part in two of the quadrants, and the unshaded part in the other two quadrants). At this point, you can use your formulas for area of a square, a², and area of a circle, πr², to solve the problem.

But wait! There's an even simpler way to solve this problem. Think of your four pieces as puzzle pieces. Can you find a way to fit them together?

Now you simply have two shaded squares and two unshaded squares, all of which are identical in size. Therefore, the area of the shaded region is exactly one half of the total area of the diagram. Applying our area of a square formula (A = a², where a = the length of one side), we get 6 x 6 = 36 for the area of the whole diagram. Then the area of the shaded region is 36÷2 = 18.

The 2008 PSAT featured a problem very similar to this one as the last problem in one of the math sections. Since math questions on the SAT are arranged in order of difficulty, this problem is supposed to be one of the most difficult ones you can encounter. But as you can see, this problem becomes very easy with the right approach.

So remember, when you're faced with weird shapes in a geometry problem, ask yourself two questions:

1. Can I break this weird shape down into pieces with more familiar shapes?

2. Can I fit these pieces together like puzzle pieces to simplify the problem?

If you use those two questions as a guide, those weird shapes suddenly won't seem so weird anymore.

Related Links:

• Get a Better SAT Score: 5 Tips to Defeat the Careless Error

Karen’s Classroom: Teddy Bear Graphing

Posted by Karen O'Hanlon

Age group: 5-7 year olds

Skills taught: Comparing and contrasting numbers, observing, estimating, recording and graphing.

Prerequisite skills: Experience playing with teddy bear figurines. Previous exposure to graphing materials. Counting to 20. Number sequencing. One-to-one correspondence. Color recognition for red, blue, yellow and green. Ability to hold pencil and crayons, and color (relatively) within lines.

Activity length: 30-40 minutes

Materials: Teddy bear figurines in four colors, large white paper for recording guesses, copies of recording sheet, crayons, pencil, small bags (plastic or paper), plastic containers for organizing bears, sorting mats.

Step One: Begin by demonstrating the activity for the entire group. I usually did this while children sat on the floor. Show the bear-filled bags and the recording sheets to the students (you can download the recording sheet here). The first step is to have the children color the bottom squares of the recording sheet to represent each bear color. Using bears that come in the four basic colors (red, yellow, green and blue), show the children how they will color four bears at the bottom of the recording sheet—one in each color. 

Step Two: Demonstrate removing the bears from the bag and spreading them out on a sorting mat. After laying them out, encourage the children to guess how many bears there are of each color. Write the guesses on a large sheet of white paper for everyone to see. You can use a matching colored pen for each guess so the children will get a visual of what color is being discussed. Be clear and explain that the red pen represents guesses for red bears, while blue stands for blue bears, etc. 

Step Three: Once the discussion of guesses is complete, place the bears in an organizing cup. Show the children how they will remove one bear at a time and color a corresponding color square on the graphing sheet to represent each bear. Model a few times so that they get the sense of how to do the task. This completes your demonstration, which should take a maximum 5 to 10 minutes.

Step Four: When you are finished demonstrating, break the class into small groups of 4 to 8 children. For kindergarten and first grade, you can even break the kids up into groups of two. By creating ‘partners’, you increase opportunities for peer discussion which will enhance math language development. If you are lucky enough to have adult helpers, each adult can oversee a group of four to eight students. The adult can also reinforce what has already been demonstrated, especially guiding the students into labeling the bottom square of each graphing column. Questioning the children about what they are doing is very important. Be sure to ask questions using words like “Why?”, “How many?”, and “How many more?”

Step Five: Following the hands-on activity, have the kids return to the location where they watched the initial demonstration. Review the lesson and compare the the kids’ results. This is a good time for individual children to share their graphs, have their questions answered, and discuss how their amounts were different from one another. Just a few examples should help solidify graphing concepts and give children the opportunity to re-hear and re-use the math language.

Tips: 

1. If the bears are new to the children, it will be difficult to get them to focus on completing a teacher-designed lesson. Allow them plenty of time to play with the bears before putting them into use as part of a goal-oriented lesson. 
2. Sorting mats are extremely useful in this kind of a lesson. You can simply use large pieces of construction paper for the children to work on and train them to keep their own materials on their own mat.
3. Use different bears to extend and modify the lesson. You can also get these teddy bears in six colors or three sizes. Multiple experiences with the same or similar materials will increase children’s learning of concepts. I like to follow the rule of three—three repetitions to help children truly absorb whatever language and concepts are being taught.
4. Modify the process of the lesson to suit younger children like 5-6 year olds. You can have these children put the bears directly on the graphing sheet, line them up in color rows matching the bottom colors and then have them remove one bear at a time to color. Instead of recording estimates with colored pens, make large circles with a colored bear in each circle and leave room to write the children’s estimates in the circle.
5. Telling a story about the different colored bears will help the children take an interest in the lesson. Adding elements of play and imagination to a lesson are a good hook for getting kids invested in the activity.

Things I like about this activity: I really enjoyed watching the children’s absorption in the activity when using these bears. I also like the way this lesson fits into a thematic unit that includes multiple subjects—for example, story telling for language comprehension and listening skills plus writing then reading child-created bear stories. A science unit exploring and learning about different kinds of bears also adds depth. I would also invite the kids to bring their teddy bears to school, and let them enjoy creating valentines for them (I had extra teddy bears just in case). We also made valentine cookies and had an indoor Teddy Bear Picnic, which is every bit as cute as it sounds. These activities were so interesting and relevant for the kids that it made teaching them fun.

Related Links:

• Don't have your own teddy bear worksheet? Download mine here
• Ideas for creating an integrated thematic unit
• More math and science activities

Get a Better SAT Score: 5 Tips to Defeat the Careless Error

Guest Post by Chris Seymour

Hello, all! Today we have a guest post by my good friend, Chris Seymour. Chris was a Valedictorian at our high school, and got a score of 2330 out of 2400 on his SAT. Next he went on to graduate from UC Berkeley, and these days he's working as an SAT prep tutor. Enjoy! - Damon

Succeeding in the math portion of the SAT is mostly a matter of knowing the content. You either know how to find the slope and y-intercept of a line, or you don't. If you know the right steps to take in order to solve a problem, you should get the correct answer. Simple, right?

But as a tutor specializing in SAT prep, I've come across many students who knew exactly how to solve every problem on the math section of the SAT, but were still getting wrong answers at times. These students were victims of the dreaded (and harshly named) Careless Error. This kind of error occurs not because of a lack of understanding of the material, but because of some breakdown in the procedure of solving the problem. Perhaps a simple arithmetic error, or a misreading of the question.

When it comes to careless errors, there seem to be two kinds of people. The first kind are not bothered by frequent careless errors because they know that they still understand the material. The second group, which tends to be higher-scoring individuals, are frustrated by careless errors because they feel powerless to avoid them. After all, you can't study for careless errors... or can you?

To the first group, I say, sorry, but there’s no partial credit in the real world. You may understand the material, but it's not showing up in your test scores. You’re going to run into many situations in life where the end result is the only thing that matters. This is one of those times, and you should be doing everything in your power to fix these careless errors.

To the second group, I say, fear not! It turns out that you certainly can study to avoid careless errors. It will mean changing lifelong problem-solving habits that may be difficult to break, but it is possible.

Here are five important ways to avoid careless errors on your SAT:

Tip #1: Identify every piece of information you are given in the problem.

When a student gets a question wrong, I'll often say, "Take another look at the question. Is there any piece of information here you haven't used?" That hint is usually enough for a student to pinpoint exactly what went wrong and solve the problem correctly.

These days, SAT math questions rarely give you information that is irrelevant to solving the problem. If a question tells you that the area of a triangle is 9, then the number 9 had better be involved in your calculations somewhere. Otherwise, it's likely that you've done something wrong.

The best way to make sure you've accounted for every piece of information in a problem is to get out your pencil and underline or write out each piece of information separately. For example, if a problem says that a certain angle in a diagram is 30 degrees, label that angle in the diagram before you move on to the next sentence. As soon as you start reading a problem, you should start underlining and writing out information. Make a habit of it. If you have it built into your muscle memory to underline information, this step will not take any extra time at all.

Then, once you have your information clearly laid out, you can keep track of which pieces of information you've used in your calculations already, and which pieces of information you've yet to use. By the time you finish, every number given in the question should be found somewhere in your calculations.

Tip #2: Make sure you always know exactly what the question asks for.

SAT math questions are notorious for asking weird things. Instead of simply saying "Solve for x" like a normal math problem would, SAT questions will more often than not say things like "Solve for x²" or "Solve for x + y." A student may go through an entire problem doing every step correctly, with no errors in calculation, but still get the problem wrong because he or she forgot to square the answer at the end.

This kind of careless error is particularly painful because the student has done everything right until the very end of the problem. However, this error is easily avoidable. You just have to make a habit of stopping at the end of every question just before you are about to circle your answer and ask yourself, "What did the question ask for?"

You can even use Tip #1 to help you out here. The most important piece of information in the problem is the part that tells you what the question asks for, so before you even start on your calculations, draw a huge circle around that part of the problem. That way, you'll be more likely to remember it even after you've done all of your calculations.

Tip #3: Learn the potential hazards for specific kinds of problems.

If you know your math content well enough, then you should be able to start recognizing what kinds of careless errors are likely in any given problem. Is it a problem about circles? You can be sure that a huge majority of careless errors on that problem involve mixing up the  radius and the diameter. Is it a problem involving graphs, functions, or coordinate geometry? Better not mix up your x-values with your y-values.

If you can remember which kinds of errors are likely on which kinds of problems, you will be much more ready to deal with those errors.

Tip #4: Write out every step.

Most students learn to write out all of their steps in middle school. If a problem says x - 4 = 5, they write a +4 under each side of the equation to show that they are adding 4 to both sides. And out pops the answer:

By high school, however, most students have dropped that intermediate step. Instead of writing +4 under each side, their answer looks more like this:

And why not? The problem is too simple to mess up, right? But you'd be surprised at how many errors can be avoided by writing out the intermediate step in these situations. Just last week, I had a student trying to solve for x/y in a problem. His calculations got him all the way down to x = 3y. Here’s what he ended up with:

I guarantee that if he had explicitly written out the "divide by y" step, he would have arrived at the correct answer.

So let me make this lesson clear: Never, never, never perform an operation on an equation or expression without writing out exactly what that operation is.

This strategy also goes for labeling information. Having lots of variables in a problem can be confusing, so if you find out that x = 3, don't just write "3" in the middle of the problem and leave it hanging there amidst a mess of other calculations. You should explicitly write out "x = 3" in a place where it's clearly visible. It may seem silly, but the SAT grading machine won't have much sympathy for you when you accidentally insert 3 as your value for y instead of x.

Tip #5: As you go along, check each step to make sure it makes sense.

Going back and checking over your work is a good idea if you have extra time after a test, but it can be difficult to find careless errors this way. Students who check their work this way will often just do the same steps that they did the first time, which is an unproductive way to catch errors. Moreover, you shouldn't count on having extra time at the end of a test. In fact, if you do have a lot of extra time, it probably means you were rushing through too many problems.

A better method for checking work is to make sure each step makes sense as you go along. If you are dividing both sides of an equation by 2, try multiplying your result by 2 to see if it gets you back to the original equation.

You can also check your answer at the end to see if it makes sense. For example, if you have a diagram of an obtuse angle labeled "x," and your calculations tell you that x is 25 degrees, then you can tell just by looking at the diagram that you've done something wrong.



Many students believe that doing all of these extra steps will cause them to run out of time. It's true that at first remembering to do these things will slow you down, but that's why you want to build these habits as early as possible. If you can work these habits into your muscle memory, you won't even have to think about them anymore. They'll just come naturally. At that point, the small amount of time you've taken to apply these tips will be more than offset by the time you save by applying them. Just think, you'll never have to backtrack through an entire problem to figure out what you did wrong! So master these five tips, and enjoy waving goodbye to those nasty careless errors.