Here is a collection of articles written for the weekly Math Corner by Raina Fishbane and Merry Adelfio.
Reading Lists are Everywhere - What about Math Over the Summer?
How to Help Your Children Be Great Mathematicians
Can You Subtract Multi-digit Numbers from Left to Right?
Why Teach So Many Different Strategies?
More Wonderful Math Literature
Computation and Arithmetic - Is there really more to math than that?
Family Math Activities of the Week
READING LISTS ARE EVERYWHERE --WHAT ABOUT MATH OVER THE SUMMER??
It seems that the end of every school year is the same – students get to bring home all of their work from the year, as well as a wonderful reading list to make sure that they keep reading throughout the summer. But where are the summer math lists?
Just like continued reading is important, children should absolutely keep exploring numbers and thinking about math throughout the summer. But to do effective math work is a little more complicated than simply providing a list of exciting books to read.
Instead, parents need to be involved in their child’s summer math work. Here are the goals that we think would be ideal for the summer:
· Children should maintain their number flexibility. Talk to your kids about numbers. Are they able to take them apart and put them back together? For example, can they find 7 different ways to make the number 6? For older students, can they come up with mental strategies for a multi-digit multiplication problem? The more they do it, the easier it becomes!
· If your children have already memorized their “math facts,” they should continue to practice them over the summer so they don’t forget them. But practice should be fun! If your child is working on addition or subtraction facts, play blackjack (21) or cribbage. Does your child love the computer? Have him or her play Math Blaster or another quality program that will help reinforce their math fact knowledge. (Second graders should know their addition and subtraction facts to 20, third graders should also know their multiplication facts, and fourth graders need to keep practicing long division.)
· Play a math related game together every week – Uno, Top It, Othello, Mancala, etc. We have a large list on the Math Adventures website of wonderful games. Go with your child to a toy store together and pick out some new games to play this summer.
· Take out your old wooden blocks. Think your kids are too old? They are not! Dig them out and watch your kids discover again the joy of building with blocks while developing their visual discrimination and geometry skills.
· Try to incorporate “math talk” in your day to day conversations. Count your steps when you are going for a walk, determine in advance the change you’ll get at a store, estimate the number of people in a group, or notice or find shapes and angles on one of your summer outings.
We will be creating a new Summer Math page on the Math Adventures website. Look for it in June and we hope it will be a good resource for math ideas for your summer.
Have a great summer and please let us know if you have any questions or would like any other recommendations!
Merry and Raina.
DA VINCI CODE MATH!!
The best selling novel the Da Vinci Code is about to be released as a movie.
Have you read the book? There was lots of math throughout the book – it is
filled with secret codes and even talks about the Fibonacci sequence and
describes the properties of the Golden Ratio! But was it all true?
The Da Vinci Code describes the secrets of the Golden Ratio – and claims that
its mysterious properties are found throughout nature, as well as ancient art
and architecture.
But what is the Golden Ratio? The Golden Ratio is an irrational number – a
number that cannot be derived by dividing one whole number by another (i.e.,
a/b).
The Golden Ratio can be found in many different ways, including by dividing any
two consecutive numbers in the Fibonacci sequence. The Fibonacci sequence is the
following series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. Can you figure
out the pattern?
Each subsequent term is the sum of the two terms before it (e.g., 1+1=2, 1+2=3,
8+13=21, etc.).
As you divide progressively larger numbers in the Fibonacci sequence, you get
progressively closer to the Golden Ratio, which starts as 1.61803…., and is
often represented by the greek letter Phi. Like the number pi, Phi goes on
infinitely and never repeats.
To get the Golden Ratio exactly, take any line segment and divide it into two
more line segments such that the ratio of the whole segment to the longer of the
two pieces is equal to the ratio between the longer piece and the shorter piece.
On many plants, the petals on the plant’s flowers is a Fibonacci number: 3
petals on a lily or iris, 5 petals on a buttercup, 8 on many delphiniums, and
many daisies have 34, 55, or even 89 petals!
Sunflower petals grow in two spirals – one almost always has 21 or 34 petals
growing clockwise and the other spiral has 34 or 55 petals growing
counterclockwise.
Looking at the arrangements of seeds on the heads of flowers, the seeds grow in
circular patterns with the numbers of seed in each concentric ring also equal to
a Fibonacci number!
Not convinced? Look at a pinecone. Pinecones clearly show the Fibonacci spirals!
In the DaVinci Code, author Dan Brown states that the proportion of anyone’s
height divided by the distance from their belly button to the floor will equal
Phi. Try it and see whether he was right!
Finally, what does the Golden Ratio have to do with Da Vinci? Look closely at
The Mona Lisa and see whether you can figure it out!
FAMILY MATH
ACTIVITIES
Here are some fun ideas for activities to do with your kids.
For younger children:
Here’s a quick and easy game that incorporates art while teaching about symmetry
and spatial reasoning.
Take a large piece of paper and fold it in half and then open it again so you
see the crease in the middle.
Take out a bunch of buttons, different shapes of macaroni, beans, or cut out
different shapes and different colors from paper.
· Have your child place one shape somewhere on his or her half of the page. Then
you copy his or her move by placing an identical shape on the corresponding part
of your half of the page. By you being the first one to “copy,” you can model
behavior for your child’s turn.
· Now you go first. You place a shape and then your child copies you by placing
an identical shape in the corresponding place on his or her half.
· Keep going until you have made an interesting pattern and have filled much of
the page.
· When you are done, glue everything onto the page and then display your
masterpiece!
· Talk about where else you can find symmetry (a butterfly, a pair of eye
glasses, a flower).
For older elementary aged and middle school aged kids:
Here’s a game based on Euclidian principles that is fun to play while also
giving kids extra practice with multiples, greatest common divisors, and
strengthening logical reasoning.
· With two players, each player secretly chooses any number between 20 and 200.
The players then reveal their numbers and then begin the game. (In order to
illustrate, pretend the 2 numbers picked were 27 and 151).
· The first player subtracts any multiple of the smaller number from the larger
number that produces a difference that is greater than or equal to 0. (If the
first player takes 3 x 27 and gets 81 and subtracts that from 151, the
difference will be 70).
· The second player does the same thing but using the original smaller number
and the new number formed by determining the difference. (So the 2 numbers now
are 27 and 70. Assume the second player takes 27 and multiplies it by 1 and then
subtracts it from 70, the difference is 43 so the 2 remaining numbers for the
next turn are 27 and 43).
· Play continues until one of the players produces a new pair of numbers where
one of the numbers is 0. That player is the winner.
After playing a few times, think about whether it matters who goes first. Is
there a strategy to the initial picks of numbers?
Euclid, who lived around 300 B.C., helped invent a way to determine the greatest
common divisor of two numbers (the largest number that will evenly divide two
given numbers). What is the connection of this game to greatest common divisors?
Are you interested in more math activities to do as a family? The above
activities are all adapted from a wonderful series of books called Family
Math, Family Math for Young Children, Family Math II: Achieving
Success in Mathematics, and Family Math: The Middle School Years that
provide great hands-on experiences for families interested in exploring
mathematics together.
Some of you might remember learning “long” division and not
so fondly. We can assure you that your children will learn to divide at Lower
School; however, they will learn some strategies you might not recognize.
Our favorite way to introduce multi-digit division is by alerting kids to the
fact that they merely need to know how to multiply (and subtract and add) in
order to divide. We teach our students how to use the partial- quotients
method, which is a most forgiving method for division. At each step, the
student finds a partial answer and at the end, these partial answers are added
to find the quotient.
Study the example below delineating how partial quotients can be used to find the answer to 94 ÷ 6.
6 94
Think: How many 6s are in 94?
(At least 10)
The first partial quotient is 10
(10 x 6 = 60)
Subtract 60 from 94
Think: How many 6s in 34?
At least 5 [6s] or 30
The second partial quotient is 5
(5 x 6 =30)
Subtract. Add partial quotients
Total: 15 with a remainder of 4
The partial quotients method works just as well if the divisor is a 2-digit
number. It often helps students to write down some easy facts for the divisor first. For example: In solving a problem such as 400 ÷ 22; some facts for 22 would be
22 x 2 = 44
22 x 5 = 110
22 x 10 = 220
22 400
10 ( 10 [22s] in
5 ( 5 [22s] in 180)
2 (2 [22s] in 70)
1 (1 [22] in 26)
Total 18 remainder 4
The reason this method is easy to use is that the student can choose the numbers he or she feels most comfortable working with. There are different ways to find the partial quotients and yet all these ways lead to the right answer. Study the example below to see the different ways three students approached the problem 371 ÷ 4.
Now try one yourself and discover how much easier “long” division problems can be solved.
HOW TO HELP
YOUR CHILDREN BE GREAT MATHEMATICIANS
We often get asked what parents can do to help their children succeed in math.
And we know there are many other parents who want to support their children’s
mathematics at home but aren’t sure what to do. So, here are some ideas that we
have:
Help your children grow to love math:
If we could give just one piece of advice, we would say help your children grow
to find math fun and exciting. Math is fun and exciting and children are so much
more likely to “do well” when they enjoy it and consider it play. One of our
main hopes for students at Lower School is that they will not grow up to be
“math phobes” like so many adults of our generation.
Keep it subtle and low-key:
Math is everywhere. Talk about math and help expand your child’s mathematical
understanding and reasoning, it is much more effective to point out the
mathematics in the world rather than to drill math facts for hours on end.
Keep the building blocks out in your house long past the toddler stage. Children
throughout elementary school age love building with blocks and learn so much
mathematics each time they do.
Point out the shapes you see on your walks outside, ask your child to guess how
many grapes are in her bowl and then together count them out, compare lengths or
weights of everyday things (kids love nothing more than to weigh produce on the
scales while grocery shopping!), or show your child an interesting graph in the
newspaper and together analyze the data.
Children need to learn their math facts but there are much more effective ways
to help your child learn them than workbooks. Play blackjack and watch how
quickly your child learns math facts to 20 (actually 21). Play cribbage and
learn all the ways to 15. Play scrabble or a similar board game and let him or
her keep score and soon they are adding numbers into the hundreds without ever
realizing that they are doing math!
Let kids have their own “aha” moments:
Children learn so much more when they are allowed to make their own discoveries,
rather than simply being told the answers. If your child is struggling with a
math problem, help him or her to figure out the solution on his or her own.
Instead, of telling them that “No, 3+6 = 9,” try to say “Well, let’s take out
some blocks and put out 3 and 6 and then we will count them all up and figure
out how many are there.” Simply correcting them and telling them the right
answer puts additional pressure on your child. We know it takes patience but
allowing your children the time and space to solve problems on their own is a
wonderful gift and an incredibly effective way to bolster their ultimate math
prowess.
Remember that there is much more to math than computations or all the work
calculators were designed to do:
To succeed in math, children do need to know their basic math facts, but they
need so much more as well! Calculations are just a fraction of the math
universe. As you talk about math, don’t forget about the breadth of math topics
that kids will be exposed to in school and later in life.
Want to know more? We highly recommend all the books written by Theoni Pappas on
mathematics: The Joy of Mathematics being a great place to start
Multiple Ways
to Multiply
Your children will learn multiple ways to multiply at lower school. One way
involves using partial products. They will also learn the algorithm many of you
were taught growing up, but don’t be surprised if they prefer another method. We
thought you might be interested in learning more about these different
approaches to multiplying double-digit numbers.
Using Partial Products
Partial products are the parts of a multiplication problem that add up to the
total product. Within a double-digit multiplication problem (multiplying another
double-digit number), four products can be generated.
Let’s consider 25 x 46:
This equation is the same as ( 20 + 5) x ( 40 + 6). Therefore, the four partial
products of this problem are:
20 x 40 or 800
20 x 6 or 120
40 x 5 or 200
And 5 x 6 or 30
The sum of those parts is the PRODUCT
1,150
Partial products can be shown clearly with geometric models as well. Here is 25
x 46 again shown with the parts labeled as “zones.” The combined areas of all
the zones equals the total area or the product of dimensions.
Can You Subtract Multi-digit Numbers from Left to Right?
Can you subtract multi-digit numbers from left to right?
Absolutely.
Were you taught to start with the ones on the right?
Probably.
But it isn’t necessary. Watch how some of our third graders
subtract moving from left to right.
569
-482
Starting with the hundreds
(569 – 400) 169
Going next to the tens
(169 -80) 89
Finishing with the ones
(89- 2) 87
Want to try it one more time?
723
- 486
(723-400) 323
(323-80) 243
(243-6) 237
Basically, moving left to right, we are removing the parts of the smaller number
bit by bit, first the hundreds, then the tens, and then the ones to arrive at
our answer.
Another way of subtracting left to right is by using partial differences. At the
beginning of fourth grade, many students learn this strategy.
746
-263
(700-200) 500
( 40 -60) -20
(6 -3) 3
Now to find the total difference
we add up those partial differences
(500-20+3) 483
Your children are exposed to each of these different strategies so that
ultimately they can use whichever strategy works best for them. After learning
each, they come to choose their favorites to include in their repertoire of
strategies by the end of fourth grade.
Remember: The best math thinkers strive to have three ways to solve one problem
rather than only one way to solve three.
We hope these examples will expand your repertoire of subtraction strategies
beyond the “trade-first” (or borrowing) algorithm you might have been taught as
the only way.
WHY TEACH SO MANY DIFFERENT STRATEGIES?
Our classes teach basic computation very differently than classrooms a
generation ago. As many of you by now have likely noticed, one significant
difference is that Sidwell’s teachers expose children to a variety of strategies
for solving addition, subtraction, multiplication, or division problems. Whereas
students in the past were simply taught “carrying” for solving multi-digit
addition problems, today’s students are taught “partial sums,” “using friendly
numbers,” and many other strategies besides “carrying.”
Why expose children to these different strategies? Simply put, children develop
number sense by developing an understanding of the meaning of the operations not
by learning “procedures” or algorithms. Also, kids learn in different ways, and
the way they make sense of mathematics makes the most sense! Most children
naturally come up with the very strategies the LS teachers are highlighting.
Forcing all children to learn and use only one algorithm for problem solving
invariably leads to a significant percentage of capable and bright kids who
struggle with basic computation. In fact, recent studies in the U.S. and abroad
underscore this significant problem – only 60% of U.S. 10 year olds and only 74%
of fifth graders in Japan achieved mastery of multi-digit subtraction problems
using the standard borrowing or regrouping strategy.
Instead, our students are expected to begin addition, subtraction,
multiplication, and division by exploring numerical relationships and developing
a sophisticated understanding of these relationships or basic concepts before
they are ever exposed to paper and pencil algorithms. By exposing kids to the
algorithms too early, kids become less creative, efficient, and flexible in
their problem solving. They try to “remember the procedure” instead of making
sense or understanding the problem and thinking it through. Here is a
particularly illustrative example: The authors of the Everyday Mathematics
curriculum presented the following problem to a large number of third grade
students who had already been taught regrouping:
300
- 1
___
299
The vast majority began to solve this simple subtraction problem by crossing out
the zeros and borrowing! Only a handful understood the problem and were able to
answer 299 without any written computation.
The foundation we want to give your children is one built on making sense and
understanding the relationships among numbers. Instead of merely teaching your
children procedures, we are encouraging them to think and to make meaning of the
operations and how they relate to one another.
Have you heard your child talk about “partial sums” or “partial products” and
haven’t known what they mean? We will explain these strategies in more depth in
future Thursday Letters. You may end up using them yourself in your own
day-to-day mathematics!
MORE WONDERFUL MATH LITERATURE
Here are some more terrific books that combine a child’s passion for books with
a wonder for numbers and mathematics.
For younger children:
Math-Terpieces by Gregory Tang. This wonderful book uses the artwork of
12 famous painters to talk about math concepts such as grouping. Paintings are
recreated in the book and then the text asks math related questions using fun
and inventive verse all based on the paintings.
If You Hopped Like a Frog by David Schwartz. This lively, funny picture
book vividly describes what you could do if you were different animals from a
frog to a dinosaur. For example, if people could jump like a flea, they could
jump over the torch of the Statue of Liberty. The illustrations are fantastic
and the book provides a great and hilarious introduction to the concepts of
ratio and proportion.
For older children:
Chasing Vermeer by Blue Balliett. Chasing Vermeer is a mystery
that was released last year about a stolen Vermeer painting. Two kids try to
figure out where the stolen painting is and throughout the book, they use
pentaminoes to create a wonderful secret code. For any child who loves secret
codes or who to play with pentaminoes or similar geometric games, this book is
wonderful. The painting that is featured in the book is generally on display at
the National Gallery of Art, so you can go see it after you have read the book!
And the sequel to the book will be out soon . . . .
The Librarian Who Measured The Earth by Kathryn Lasky. This book is an
incredible true story of a Greek mathematician two thousand years ago who using
basically just camels and shadows was able to figure out the circumference of
the earth to within 200 miles of its actual distance! It is a fascinating story
about the power of ingenuity and the relevance of mathematics. Although it is in
picture book format, this is a complex book perfect for older readers.
Our school library has an entire section of books related to mathematical
concepts – stop by one morning or afternoon and browse the books and bring some
home to share with your family!
MATH AT HOME
Math is everywhere around us, and there are so many opportunities for parents to
talk about math in ways that are fun and enriching. We will try to share some of
those places where you find math in the real world and share some ideas for
subtly highlighting the math concepts with your children.
This week, try to take some time to cook or bake with your children.
First, shop for the ingredients you will need. While at the store:
Let your child compare prices to determine the least costly things to buy.
Have him or her compare amounts and decide which size ingredients will be needed
for your menu.
Let your child clip coupons and figure out how much you will save if the coupons
are doubled.
Let your child pick out the relevant coins you will use to help pay for the
groceries.
Ask him or her to figure out how much money you will need or to estimate how
much change you may get back.
Then, when cooking:
Pick a recipe that would serve too many or too few people. Ask your child to
help you make the relevant conversions (e.g., “we need to double the amount of
flour. What is twice 1 1/2 cups?”).
Have your child compare amounts of ingredients (whole numbers for younger
children and fractional amounts for older children).
Ask questions throughout the process (for example, if the recipe calls for 2
cups of oil, after you have poured ½ cup into your bowl, stop and ask how much
more you will need to add).
Ask your child to help with equivalencies – if the recipe calls for 2 pints, ask
your child how many cups that would be? Need help with the conversions yourself?
Most cookbooks have a table with conversions or check http://www.onlineconversion.com/cooking.htm
for easy on-line conversions.
Finally, while serving your home cooked math recipe, cut the food into fractions
and compare different pieces as you enjoy your creation!
For an especially hard math challenge, ask your child to figure out the cost per
serving of the food that you cooked.
Need some good recipes? Check out these great websites: www.ciakids.com; http://www.foodnetwork.com/food/lf_kids;
http://familyfun.go.com/recipes.
COMPUTATION and ARITHMETIC –
IS THERE REALLY MORE TO MATH THAN THAT?
Math education has changed dramatically since many of our current parents were
kids. Teachers in the Lower School now teach a broad variety of math concepts in
ways that may seem unfamiliar. Children are expected to understand underlying
math concepts and to be able to use manipulative materials and problem solve in
ways that kids a generation ago were never asked to do. We will use this column
to describe some of these fundamental changes so that they can be more
understandable to parents.
One of the most striking changes in math education has been the move away from a
sole focus on simple arithmetic and the memorization of “math facts.” In fact,
the National Council of Teachers of Mathematics (NCTM) now mandates that
elementary math education cover more than 10 strands – only one of which is
computation! Some of the other strands include geometry, measurement, algebraic
thinking, probability/statistics, and data analysis, all of which are covered in
each of the classrooms at the Lower School.
Kids must still learn their math facts -- generally students are expected to
learn addition and subtraction facts through 10 in first grade, addition and
subtraction facts through 20 in second grade, and their basic multiplication and
division facts in third grade. However, in this day and age of calculators and
computers, they need to accomplish much more than mere memorization of these
facts during these critical years.
Students must develop a deeper understanding of the concepts underlying
addition, subtraction, multiplication, and division and develop an ingrained
number sense to be comfortable manipulating numbers in equations. Furthermore,
they need be able to apply their facts to problem solve. This involves flexible
thinking, facility taking numbers apart, and the ability to see relationships
between numbers.
Last year, the parent of a 1st grade boy called because she was feeling
frustrated with the way her son was doing his math homework. His class was
working on double digit addition and when he would solve the problems on his
homework sheet, he could easily take the numbers apart and put them back
together in ways that would help him solve the problem. (For example, if the
equation was 28+44, he would split the numbers into 20+8+40+4 and then solve the
relatively easy problem of adding the tens. Once he had 60, he knew that 8+4
equaled 12, so he then would add 10 from the 12 to 60 to get 70. Finally, he
would add 2 more for the answer of 72.)
His mother didn’t recognize the way he chose to solve the problems and instead
fell back to the algorithm that she had learned as a child -- add the ones
column first and then carry and then add the tens. Her son could not do the
problem that way and mother and child were both becoming frustrated. When we
described what her son was doing – and explained why it was an elegant and
efficient solution that demonstrated his deep number sense – she laughed that
she had been trying to force on him something that he was not ready for and that
would actually impede his continuing math growth!
Instead, children in the Lower School are expected to develop a rich number
sense that allows them to understand numbers well enough to manipulate them – to
easily see how to take them apart and put them back together. In this way, they
will have the solid understanding they need in order to later do complex
mathematics.
A weekly list of family math ideas and thoughts on math at home is included in our Thursday Letter at school. Below you will find links to the suggestions from those letters.
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