Patterned

A very significant Pre-math concept for kids to understand is patterns. Patterns are easy to make with anything you have on hand (fork, spoon, spoon, fork, spoon, spoon, etc), and can provide hours of entertainment for you and your child.

Work with your child on creating an extending patterns. Take turns starting a pattern and finishing it. For example, maybe you start a pattern first, then have your child extend it, then let your child start a pattern, and you extend it.

You should also help your child be able to start a pattern that you state. For example, you say “Can you make me a pattern that goes fork, knife, fork, knife?” And your child should be able to display that pattern for you.

I am working on creating some fun use-at-home math games and manipulatives that should be available soon.

Are there any concepts you would specifically like to have manipulatives for?

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Math Fun

This lady, Vi Hart, makes some really fun videos about math and some of the less useful, but seriously fascinating applications!

Here is a video for your enjoyment. Be sure to check out her YouTube page, and her website, for more fascinating math fun!

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Round and Round – trig functions

Last week we talked about some basic functions that you will deal with in algebra, calculus and beyond. Today I am going to introduce you to some basic trig functions.

One of the most fascinating applications of trig functions is that of daylight hours. Do you notice how the days get longer in the summer, then shorter in the winter? And then they get longer and shorter and longer and shorter, and the cycle just keeps going on.

Trig functions are just that – cycles.

This is a graph of the function f(x) = sin(x) (the one on the left is one period of the sine function, and the one on the right shows more what the graph does – it just keeps going over and over again – in a cycle.)

 

This is a graph of the function f(x) = cos(x). Cosine is a lot like sine – it just starts in a different place. Where sine starts at zero when x = 0, cosine starts at 1 when x = 0.

This is a graph of the function f(x) = tan(x). Tangent is a ratio of sine and cosine. The reason it is undefined at some places (see how the graph goes up and doesn’t come back down, and then it stars from far below?) is because sometimes cosine is zero, and you can’t divide by zero.

This is just a little introduction to show you what trig functions look like. There are also inverse trig functions, which we’ll talk about later. We’ll also talk about some really interesting applications of trig functions.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Prove It – how (and why) to write proofs

For some reason, when geometry students get to the proof writing segments of their classes, they freak out. “Why do we have to learn how to write proofs?” “This is useless.” Etc, etc.

Well, first off, let me tell you that writing proofs is not useless, and you do it every day (whether effectively or ineffectively) without even realizing it.

Remember how you wanted that new video game that just came out and your parents said “No” and you tried to convince them to let you buy the video game? Did you just say “I need the video game.”? Probably not. That’s not very convincing. What you probably did was talk about all the reasons you “needed” the video game (or car, movie ticket, date, you name it – whatever it is that you had to convince your parents you “needed”)

Well, little did you know – but you were (verbally) writing a proof! Who would’ve thought! You already knew how to write proofs!

The tricky part about writing proofs in geometry is that there are so many rules, and most of them don’t usually make sense (if you have a bad attitude about it, like so many kids do). The thing is, you know more math than you think, and most of the “reasons” for geometric things being the way they are come pretty naturally to us as human beings.

The way I like to think about all those rules we use in Geometry proofs are as tools that I keep in my Proof Toolbox. There are a lot of rules. Such as, when two parallel lines are cut by a transversal, the alternate exterior angles are congruent. (WHAT!?)

The fascinating thing about using words in Math is that we usually use words that make sense. Don’t think about all these crazy terms like “alternate exterior angle” and freak out – just calm down and look at each word in the term as an individual word, and check out what it means.

First let’s take the word “alternate” – what does the word “alternate” mean? Well, in a selection process, such as when someone gets chosen for a play or for a spot on a team, an “alternate” would be someone that would take that person’s place if for some reason they weren’t able to perform. But that’s not really what we’re looking for. What about when you skateboard through some cones in the road “alternating” sides of the cones? Now that’s more like it. So “alternate” just means on opposite sides of the transversal (the line that cuts the parallel lines).

Now let’s look at “exterior” – well, that one is pretty easy. It just means “on the outside”. If we think of our parallel lines like a tunnel, then the “exterior” angles are on the outside of the tunnel.

We’ll talk more about how to figure out what all these crazy math words mean in another post.

Now let’s talk about how to write a proof. Since we have been talking about parallel lines, let’s start there.

Usually a proof problem has two parts – the “Givens” (i.e., the things you know for sure) and the “Prove” (what you are supposed to show is true based on the things you know for sure)

Given: and are congruent

Prove: lines l and m are parallel.

Now there are a lot of theorems and properties (i.e. tools) that will help us write this proof, but a tool won’t help you unless you actually have it. In our case, that means that a theorem or a property isn’t going to help us unless we know it. So if you have been studying your tools, you will know right away that angles 1 and 2 are “alternate exterior angles.” We know that If two parallel lines are cut by a transversal, then their alternate exterior angles are congruent. The converse (which just means switching the parts of the if-then statement) is also true: If the alternate exterior angles of two lines cut by a transversal are congruent, then the two lines are parallel. Which means that this proof is easy – we basically just state the converse of the alternate exterior angles property.

I will be starting a series on proof writing soon that will have useful information for writing proofs all the way from beginning geometry into college analysis. If you have specific questions about writing proofs, let me know in a comment, via email or Facebook or twitter, and I will try to incorporate your answer into the posts for the series.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Double the Number, Double the Fun

For this post we are going to talk about the 2’s Times Tables. The twos are perhaps the easiest times tables to learn.

Multiplying by two is the same as doubling the number (that is, adding the number to itself)! Pretty easy, right? First, let’s practice counting by twos:

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24

That wasn’t so hard, right?

Now, let’s think about doubling numbers (adding them to themselves):

2 x 1 = 1 + 1 = 2

2 x 2 = 2 + 2 = 4

2 x 3 = 3 + 3 = 6

2 x 4 = 4 + 4 = 8

2 x 5 = 5 + 5 = 10

2 x 6 = 6 + 6 = 12

2 x 7 = 7 + 7 = 14

2 x 8 = 8 + 8 = 16

2 x 9 = 9 + 9 = 18

2 x 10 = 10 + 10 = 20

2 x 11 = 11 + 11 = 22

2 x 12 = 12 + 12 = 24

That’s pretty straightforward, right? So when you get a “twos” multiplication problem, it’s as easy as adding the number to itself! If you have problems with addition, not to worry! I’ll be posting some tips soon to help you with addition.

Find the rest of the Times Tables here: Times on the Table

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Loquacious Problems

We’ve talked a little bit this week about how math can sometimes seem like a foreign language. Well, this means that sometimes, we need to “translate” English into “math” – this phenomenon is also known as “solving word problems.”
A “word problem” is really just a math problem written in every day English speak (or German, or Italian, you know, depending on what language you speak). We solve word problems all the time. Like when you tell your mom you want five people at your birthday party, and that means you will need three pizzas, because each pizza has eight slices, and if everyone has three slices (you’re a teenage boy, and your friends are teenage boys), and three slices is 18, but two pieces would only be 16 slices, so you better go with three pizzas and you can save the rest for your little brother and sister’s lunch the next day. That is basically just one really long word problem. Or when you want to run a 5K (3.1 miles) and you know you run a 10 minute mile, so you tell your mom to meet you at the finish in about half an hour (10 minutes/mile * 3.1 miles = 31 minutes).
So really our lives are made up of word problems and they happen all the time, and we even write our own word problems without even being aware that we are doing it! Ha – how’s that for knowing more math than you thought you knew?
But for some reason, when we’re sitting in Mr. Hamlin’s math class staring at our pop quiz that is asking something about people washing cars and how fast can they wash the cars together, our brains freeze up and we feel like we have no clue what these people are talking about.
Well, the first thing you should do is stop and take a deep breath! You know how to do this – you’ve been doing it your whole life. Don’t let the fact that you are sitting in math class and not on the football field running laps freak you out. You can do this!
Let’s put together a sample problem so we can work through it together.

Joe, James, and Jessica are all pro car washers. Joe can wash a car in 40 minutes. James can wash a car in 47 minutes, and Jessica takes about an hour to wash a car. If they all work together washing a car, how long would it take them?

Now, it’s always good to read through the problem all the way before you start working on the problem, just to get a feel for what the situation is. Now, this time when you read it, read it as if you’re actually in the problem, and the situation is your situation. It’s much more motivating to solve a problem when it is your problem. Replace one of the characters with yourself – now it’s you, James, and Jessica, instead of Joe.
After you have read through the problem once (without hyperventilating), I want you to think about what you’re actually looking for. Are you trying to find out how many cars you can wash? How long it would take one person if the other two are helping? What is the question asking you to find?
In this particular “word problem”, the question says, “If they all work together washing a car, how long would it take them?” First off, let’s find out what kind of answer we’re looking for. Are we looking for a rate, a number, a distance? It looks like we are looking for an amount of time: “How long would it take them?” So chances are we’re going to come up with a number of minutes or hours. Since most of the rates given were in minutes (40 minutes, 47 minutes) let’s go with minutes. That means we need to change Jessica’s rate into minutes. This one should be pretty easy. How many minutes are in an hour? Sixty.
The next thing we need to do is write down some kind of relationship between what we know, and what we want to find out.
Well, since we’re talking about rates of washing cars, let’s look at the rates of car washing. Let’s call the rates R, and we’ll give subscripts to each person’s rate.

Now, to find their combined rate of washing the car, we need to add them all together.

So, together, you and your friends can wash 71 cars in 1128 minutes. Well, that’s great, but we don’t really want to know how many cars you can wash in a certain amount of time, we want to know how long it will take you to wash ONE car. So, the easiest way to do that would be to divide 1128 minutes by 71 cars (giving you the amount of time it takes to wash ONE car): 15.89 minutes.
So the basic steps for solving word problems are:
1.) Read the problem all the way through (without freaking out), and putting yourself in the problem, or pretending it is your situation, so you’re more motivated to solve the problem.
2.) Write down what you know (in this case, the rates of car washing).
3.) Figure out what you want to know (in this case, how long it would take to wash one car)
4.) Write an expression, using the information you have.
5.) Solve the expression.
I will have another post next week on word problems. Until then, check out this page over at PurpleMath.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Functional Graphs

There are some graphs of basic functions that every math student should know. If you can recognize what a function is going to look like when it is graphed on the xy plane, you’ll be a lot more likely to be able to figure out problems about that function, because you’ll be familiar with is.

Think of this – when someone is talking to you about a person, it’s hard to understand exactly what they are taking about, unless you have some kind of mental picture of the person they are talking about. Otherwise, you’ll be saying “Huh?”

So, to help you avoid saying “huh”, I am going to help you learn to recognize the graphs of basic functions.

First, , the most basic function. And probably the most boring. Blah.

And the next one, , this is a “parabola” – think “bowl.”

 

After the parabola, we must have a “cubic” (think, a “cube” has three dimensions – height, width, length)

Next, , also known as the “absolute value function”.

The slightly odd function, - a tricky function, because it is not defined at x=0!

Now for one of my very favorite functions,(an “exponential” with the famous number “e” – yes, it is a number.)

 

 

And we have have the exponential function without it’s inverse,(which is also not defined at x=0)

So there you have it – the seven basic functions (that are not trig functions – we’ll deal with those in another post). If you can memorize what these look like, you’ll be set through almost all of your math years. But I won’t let you get away with just memorizing them. Eventually we’ll study exactly why these functions look the way they do, and we’ll learn a lot of really amazing stuff about them.

Which is your favorite graph? Why?

(ps – I made these graphs using an amazing FREE graph drawing tool for your computer called WinPlot – if you don’t have a graphing calculator, this will be very useful for you. You can do a lot of the same things you can do on a graphing calculator using this program)

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Core Standards – what your child should know

What should a child know?

You can always use your state’s curriculum guidelines for a starting point of what your child should know at what grade level, but if you are homeschooling, or you just want a general sense of what math topics are out there, you can try some national cores that have been developed by different organizations.

First, there are the Standards and Focal Points developed by the National Council of Teachers of Math.

There are also Mathematics standards published by the Common Core State Standards Initiative.

Both look pretty useful and in depth, and I hope to use them myself in coming up with topics to write about on this blog, as well as in teaching my own children.

What standards do you use in your math teaching? Do you look at your state core? If you homeschool, which standards do you use?

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

you know more math than you think

Do you ever feel like math is a foreign language?

Think about how it feels to learn another language, or to be in a place where people are speaking a language you don’t understand.

Now think about how a child learns to speak their mother tongue (the language their parents speak). If a child is exposed to a language from the time they are born, and if someone interacts with them on a regular basis in that language, the child will eventually learn to speak that language. Are children born speaking a language? Maybe baby talk – but mostly they just cry.

Turns out, the ability to learn math is programmed in us similar to the way the ability to learn a language is programmed in us.

You can read about a study that was done on this subject here.

In the article, they mentioned that a lot of Euclidean Geometry (lines, space, shapes – all that basic stuff) concepts were “innate” in members of a remote tribal community in the Amazon who had no formal training in geometry. They got as many answers right about geometry as formally educated Americans.

Something that was interesting, however, was that younger children (5 and 6 years old) didn’t know the answers. The researchers weren’t sure why, but I have a hypothesis.

Since math is similar to a foreign language, and young children don’t often understand the concepts of grammar, it make sense that they also wouldn’t understand the concepts of math especially if math has not been “spoken” to them regularly. This goes back to the principle of teaching “pre-math” skills by exposing young children to concepts like classifying, sorting, number conservation, counting, etc.

If you want your child to understand math concepts better, speak math with them! Try to make it a point to “speak math” with your child every day. Whether it is by sorting and classifying, or by talking about shapes around you, or multiplying a recipe, or talking about how “steep” the stairs are, speak math with the children in your life!

How do you “speak math” with the children in your life?

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Pre-Math: One of These Things

Sometimes as parents, and as teachers, we discount the things that children do as they play that help them learn math skills. We can encourage children to do things that develop good “math brains” without making them sit down and look at flash cards or do worksheets (I know a lot of kids love flash cards and worksheets, but we need to expand our horizons a little!)

One very important math concept and skill is that of sorting. In a previous post about combining like terms, I talked about how combining like terms was a lot like sorting laundry or marbles, or anything else than can be sorted and grouped.

Children like to sort and group naturally. My son will sort his toy cars by color, my daughter loves to help put the dishes in their correct places.

When our children practice sorting during play, or while helping mom and dad around the house, they are learning valuable math skills that will play a part in whether or not they are easily able to grasp the concept of things like combining like terms later in school.

Another part of sorting is being able to classify objects. In the picture above where my children are sorting laundry, they wouldn’t be able to sort the laundry unless they knew what set of classifications we were using to sort. Are we sorting kid’s clothes and grown-up’s clothes? Are we sorting every color? In our case, we were sorting according to “light” colors, “dark” and “ bright” colors, and “whites” – which meant that I had to explain what each of those classifiers meant.

When my children understood the classifiers, they were able to sort. Being able to classify objects (especially in different ways – like I said above) is also very significant and an important ability for children to develop so they will be successful in math skills later on. They have to be able to classify math “objects” such as numbers, coefficients, variables, exponents, etc.

Another game to play is “One of these things is not like the others” where you have a group of objects, and one object is not like the others. In this game, you are classifying objects, but you are also teaching the concept of “one” (which might seem like an easy concept, but trust me – you probably don’t actually understand “one”, even as an adult. I didn’t fully understand what “one” meant until I was finished with my undergraduate studies).

Try to find ways to help your children classify and sort objects. Try to make it a fun game that has to do with your life (sorting silverware or laundry, classifying cars when you are driving “Can you see a red car? How about a red truck? What about a blue truck?”). As your children learn to classify and sort objects, and as they do it on a daily basis, their brains will be gearing up to understand much more complex mathematical situations as they grow up.

What ways do you teach your children how to classify and sort? Do you feel like you are able to teach your children math concepts?

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Comparing Decimals

Today, Student A had a worksheet for comparing decimals. She was supposed to say which decimal in each pair was greater than the other, or if they were equal.

Because she originally completed the assignment by guessed (a pretty usual occurrence for A), I decided to give her some steps to follow in comparing decimals before we even began correcting the assignment.

First, we talked about how decimals are just like fractions – meaning they are just part of a number, instead of a whole number. With that in mind, we went on to our steps for comparing decimals:

Step 1. If there is a whole number (the part of the number that is to the left of the decimal point), compare the whole numbers first. If the whole numbers are different, then all you have to do is tell which whole number is greater. (for example, 12.3 and 14.72 – 14.72 is greater because 14 > 12, and it doesn’t matter how much “extra” you tack on to the 12, 12 is still smaller than 14). Otherwise, move on to step 2.
Step 2. Since the whole number parts are the same, we will need to compare the decimals. Remember that decimals are like fractions, so .1 = 1/10, and .01 = 1/100. Well, it’s hard to compare fractions when the bottoms are different (that is, it’s not very intuitive) and it is equally hard to compare decimals when we don’t have the same number of places, such as in 4.35 and 4.3.
To make things easier to “see” (instead of having to think too hard – mathematicians never like to think harder than they absolutely have to – that’s why we have “tricks” for everything) we are going to make sure each decimal has the same number of “places”. To do this, all we have to do is add zeros on the right hand side of the number. Going back to our example of .35 and .3, since .35 has two places and .3 only has one, we will add a zero to .3 and have .30. Now we are comparing .30 and .35 – well, that is just like comparing 30 and 35. It is clear that 35 > 30, so .35 > .30.
If you had .30 and .3, when you add a zero to .3 you will get .30 and .30 – in this case, the numbers are equal.

I was really impressed with Student A’s internalizing this concept. After five or six problems she was able to figure them out quickly, and explain her answers just as fast. She really understood the concept, and that was a huge success for us.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Times on the Table

One of the things I worked with a lot with student A over the summer was TIMES TABLES.

While I am not a supporter of a lot of rote repetition when it comes to learning math, I definitely support repetition when it comes to remembering math. But only after the initial learning has taken place.

It doesn’t do a student any good to memorize the times tables if the student doesn’t even understand what the times tables are. Such was the case with A. We spent a lot of time just learning about numbers in general, and then specifically multiplication, before we moved on to memorizing the times tables.

As part of “figuring out” the times tables, we learned a lot of tricks, and used some games, songs, and pictures to learn the times tables. We looked at patterns in the numbers, talked about sets, and played a lot with Cuisenaire rods to get a physical grasp of the concept.

Only then did we move on to drills.

Here are the posts describing how we learned each set of times tables.

Twos
Threes – Three is a Magic Number
Fours
Fives
Sixes
Sevens
Eights
Nines – The Great Nines
Tens
Elevens – Some Tricky 11’s – in each place
Twelves

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Combining Like Terms – Sorting Letters

Did you ever play sorting games as a little kid? If you fold your own laundry, then you should be a pro at sorting – pants, shirts, socks, sweaters.

Combing like terms is just like sorting laundry, or marbles (as illustrated in my sorry attempt at drawing marbles).

When we have “letters” (also known as variables) in an expression we often want to combine them so that we have an easier expression to look at.

Think of it like having a pile full of different color marbles (like in the illustration above). Let’s look at an example:

2x + 4y – 3xy + 7y – 5x + 2xy

Let’s look at the different “color” marbles we have (that is, the different “terms”)

We have x’s, y’s, and xy’s (the xy’s are their own type of term, because xy is not the same as x or y).

Now we combine the things that are “like” – We have 2x and –5x, so that is –3x. We have 4y and 7y, so that is 11y. Finally, we have –3xy and 2xy, which is –xy.

Our “simplified” expression is –3x +11y –xy.

Because of the commutative property of addition, we can write the expression in any order we want to.

-3x – xy + 11y
11y – 3x – xy

etc, etc

Here are some practice problems for combining like terms (also known as “simplifying expressions”)

Simplify the Expressions by combining like terms

1.

2.

3.

4.

5.

(Pre-Math Tip! Play sorting games with your preschooler. You may not realize it now when you are sorting blocks and dishes and beads, but your preschooler is building the foundation for basic algebra skills like being able to combine like terms!)

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Three is a Magic Number

You might be wondering what our pattern was for the 3’s times tables. Well, instead of looking for a pattern (there isn’t a really obvious one) we just learned Three is a Magic Number from Schoolhouse Rock. I wasn’t really convinced that it would work for A, but it did! So, here the video of the song. Good luck with threes!

PS – you’ll notice that the song leaves out 3 * 11 and 3 * 12 – well, 3 * 11 is easy, right? We talked about that yesterday – it’s just “3 in the tens place, and 3 in the ones place.” 12 is easy, too - “3, and then 3 doubled” 3 * 12 = 36. We’ll talk more about the 12’s when we get to them, but the first few 12 times tables are pretty straightforward.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Some Tricky 11’s – in each place

Remember that A and I have been talking about patterns in multiplication? Well, the easiest of the multiplication tables (besides 10s) are 11s – just double the number, right? Well, what happens when you get to 11 * 11? It isn’t as hard as you might think – you just have to figure out the right pattern.

Think about 11 * 9:
        We don’t even really have to think about it as “90 + 9” because we can just think of it as “write 9 twice”. But now when we move on to 11 * 10, looking at it this way will help when we do 11 * 11 and 11 * 12:

We can think of the number 11 as “one in the tens place, and one in the ones place” – which when extended to multiplication (for example, 11 * 5) becomes “5 in the tens place and 5 in the ones place” (where you replace “5” with whatever number you are multiplying by.

For 9, that is easy, for 10, it isn’t quite as straightforward – because what does “10 in the tens place” mean? Well, in the illustration above, it means “100” – because a 10 in the tens place means we need to put another zero on the end, and a 10 in the ones place is really just 10. So that’s how we get 11 * 10 = 110.

Now we can move on to 11 * 11. We think the same thing; “11 in the tens place, and 11 in the ones place.” Let’s write it out like we did with 9 and 10:

So you can see that when we write “11 in the tens place” what we get is 110, and when we put “11 in the ones place” we get 11. So the answer is 11 * 11 = 121.

We can do the exact same thing for 11 * 12, and we get 120 + 12 = 132.

Now you can probably figure out 11 * 13, 11 * 14, and 11 * 15! And after a while, you can probably see a new pattern as you go.

So when thinking about the 11’s times tables, you can just think “the number in the tens place, and the number in the ones place.” This rule works for all the times tables – from 1 to infinity!

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

An Update on A and The Great 9’s

I posted a while ago about A, a 3rd grade student I have been tutoring.

We have been working really hard on multiplication, and memorizing times tables. A still has some gaps in her fundamental understanding of numbers, so we will be working with that a lot more in the future.

But in the past few weeks, we have been learning “tricks” for figuring out multiplication without having to count, or even having to think too hard about the numbers. We’re working really hard on recognizing patterns, using facts we know to find out things we don’t know, and of course, memorizing the times tables.

The great 9’s

After we learned fact families of 10, and practiced adding numbers to 10, we then started adding numbers to 9. This led to a discussion of 9 fact families, which I thought was a perfect way to start talking about the nine times tables, since the digits of every product of 9 adds up to 9. We started by listing the numbers 0-9 in a row down our paper:

Then we wrote the numbers 0-9 going back up the paper.

And then, we had the 9 multiplication table right in front of us. We had been talking about “subtracting to add” numbers, specifically nines. This means that instead of taking 9 + 5 and thinking 9, 10, 11, 12, 13, 14 we take one from the 5 and add it to the 9, making the problem 10 + 4 which we can much more easily see is 14. The same principle applies to multiplying by nine. The answer to 9 * 5 is found when we subtract one from 5, and we get 4, and then we just think about the “fact family” that includes 4. That gives us 45, and we’re done.

A got pretty good at thinking “minus one, then fact family” – meaning when I said “Nine times six” she would think “Five, four – fifty four.” She got really fast at figuring out the nine times tables, and now they are by far her best.

Now, the “minus one, then fact family” trick only works for 1-10. But 11 times tables are easy up to 9, because you just repeat the numbers – so 9 * 11 = 99. This is where we started learning to use something we already know to figure out a new problem. Instead of simply trying to memorize 9 * 12 = 108 arbitrarily, we think about “subtracting to add” again. We know that 9 * 11 = 99, and 9 * 12 is just adding one more group of nine, and when we add things to 9 (or 99) we can subtract one to add. So 9 * 12 = 9 * 11 + 9 = 99 + 9 = 100 + 8 = 108.

So the theme we are learning with the number 9 is “subtracting to add” – and we are learning that 9 and 1 are pretty much inseparable.

Tomorrow – Some Tricky 11’s

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

A and the Fact Families

I have started tutoring a recent 3rd grade graduate (A) who is struggling in math. Today was our first session, so I tried a few problems out on her to see where the struggle really lies.

Her mom told me she’s had problems with multiplication, which upon investigation was clear – she struggles with multiplication. But I had a feeling that wasn’t where the real problem was, so I talked to her about addition and had her practice some (relatively) simple math facts – like 10 + 20 = 30, and 10 + 2 = 12. Unfortunately, she couldn’t easily and quickly give me the answer – she struggled with these concepts, which are usually understood before a child even begins to learn multiplication.

Addition War

So I decided to play a little game with her I called “addition war.” For those unfamiliar with the card game “war”, it goes like this – each player starts out with a stack of playing cards. At the same time, the two flip over the card, and whoever flips over the higher card wins the “battle” and keeps the two cards. The person who ends up with all the cards is the winner of the war. For “addition war” we took out all the face cards and then we would each flip over a card. Whoever could add the two numbers together first would win the cards. I knew that I would be fast at this game than a third grader (although it’s great practice for me, too!) so I gave her a few seconds head start. I figured that should be all she needed, but it soon was evident that her problem isn’t just with multiplication – she lacks some fundamental concepts for adding in her head, which will cause problems when she tries to do multiplication, and when she tries to memorize times tables (which will have no concrete meaning to her).

Fact Families of 10

With this in mind, we went back to the basics – addition fact families for the number 10. After A masters addition fact families for 10, we’ll move on to what I call “adding to ten” – when you have 7+5, instead of counting up from 7, you add 7 + 3 + 2 = 10 + 2 = 12. This is a much faster way to add numbers, but you have to be really solid on your fact families (especially for 10 – most of the others are relatively easy to remember, and A seems to know most of the smaller fact families, but we will probably need to work on those, too).

To review – the addition fact families for 10 are
10
1,9
2,8
3,7
4,6
5,5

This means that each pair of numbers add up to 10. In order to really drill these “families” in, we used those same face cards (from addition war) as flash cards. I would flip over a card, and A would tell me the other member of the fact family. We wrote all the fact families on a piece of paper for a “cheat sheet” at first. We set a timer for 2 minutes so she would have a reason to be fast, and then I bribed her with chocolate. She gets a fun size candy bar if she can reach our goal of 75 cards in 2 minutes. Today her best score was 52 in 2 minutes, but that was pretty good, since she is still taking a few minutes to name the missing fact family member. We played four or five times today, and it seemed to really help.

Math Manipulatives – Cuisenaire Rods

After we had used the flash cards for a while, we used the Cuisenaire rods to show the fact families. The thing I love about Cuisenaire rods (and really any math manipulative) is that when the student gets a chance to touch math, they are a lot better able to understand it. Math concepts can be abstract, and making them concrete helps students (especially children who developmentally aren’t abstract thinkers) understand. A was able to see the fact families, and get to know them a little better. We talked about how as one of the fact family members get bigger, the other gets smaller (see the stair step pattern?) and that is because we always want to have ten – so we are moving one “unit” at a time from one side of the fact family to the other. This will take a lot of looking at and playing with until A really understands it, but I can tell that she already understands it better than when we started today.

Another game we started to play was “speed” with the fact families. We laid out a bunch of cards and then matched them with their fact families and as she matched them, she turned them over and I would put more cards down for her to match with the fact family. This worked okay, but I think the flash cards was more what she needed today.

 

Goals and Progress

My goal with A is to get her to the point where she can add pairs of one digit numbers without thinking about it for too long. I think we made great progress. She and I will probably be meeting every weekday this summer. Hopefully by the end of the summer we will have multiplication mastered as well. But math concepts have to be built on a strong foundation, so the foundation is what we’re working on right now.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Sequences Part II – The Future is in the Past

In Sequences Part I, we talked about what a sequence is, and then we talked about straight up sequences. If you need a refresher, just visit the post by clicking here.

Today, we’re going to talk about recursive functions for sequences. That is, ones that look like this:

This is a recursive sequence – or one that uses previous terms to define the next term. This particular sequences is called the Fibonacci Sequence. Mostly it’s just a regular sequence that gets a special name because it’s pretty important to mathematics – but we won’t really get in to much of that in this post.

Remember the cups we talked about? Let’s go back to those and see if we can figure out the Fibonacci sequence. Now, the thing about recursive sequences is that you usually have to take at least one of the cups before the cup you’re working on, and use those numbers in your function. Let’s just take a look at it.

With this recursive sequence, we have the first two terms defined for us.

So, now that we have our first two cups filled up, we can start with our second cup, which would be . The sequence says , so basically what that means is that we fill up each cup with everything that was in the last two cups (because we’re adding – and adding is just like putting things together in a mixing bowl when you’re cooking).

So if n=2 (), then that means we need to take and (or the two cups right before cup “2”) and dump them into ‘s cup. =1 and =1, so when you dump those into a cup, you get 2, right? So =2. (I’ve made some illustrations for what’s happening.

           

Now for the third cup, , we dump in the last two cups, which are now cups 1 and 2. =1, and =2, so = 2 + 1 = 3. At first the Fibonacci sequences just looks like , but let’s just keep going.

We need cups “3” and “2” to fill up cup “4”. = 2, and = 3, so when we dump in those two cups we get = 3 + 2 = 5, so =5, and now we don’t have - we have Fibonacci!

When you write it out, it’s pretty easy to see what the next term will be:

Now I’m going to give you a few practice problems. I’ll post the answers tomorrow, to give you a chance to figure them out on your own.

Find the first 5 terms of each sequence.
(write your answers in the comment section, or email them, or post them on Twitter or Facebook. Those who come up with the correct answers before I post them tomorrow will be mentioned in the answers post)

1.

2.

3.

4.

5.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.