Wednesday, January 30, 2013

Without Math, Art Loses Its Shape!

Renee's TPT Store
We end January with my last guest blogger, Renee Goularte.  (Notice that art is part of her last name.)  She is a retired elementary school teacher/art educator, a working artist, and a writer. Her professional teaching career began in San Jose, CA in 1988 in a third grade classroom. She soon moved to a 2/3 combination class and then spent four years teaching in a Multi-age 1-2-3 program, which she started with a colleague. Renee has also taught a 4/5 combination class in addition to Kindergarten, and has worked specifically with GATE students, ELL students, and at-risk students. Before she retired in 2010, she taught art to Kindergarten, and first and second graders. For her, teaching art has come full circle.  In addition to her work in the classroom, Renee has written articles on teaching, written lessons professionally, maintained a number of websites and blogs, and facilitated math, language, and art workshops for teachers and parents. She also has a Teachers Pay Teachers store where she offers many creative products.  Click under her picture to go there.

Like myself, Renee believes in hands-on, active learning that touches all learning styles. From a personal experience, she knows visual literacy is an important component of the curriculum, and that creativity can be fostered and nurtured.

In many of the art lessons I teach, I use math, especially math vocabulary. Sometimes this is incorporated when having students look at famous art works, especially (and most obviously) some abstract art, but also (and less obvious) art that is more representational. When I ask students to tell me what they notice in works by Wasily, Kandinsky or Pablo Picasso, many of the responses include basic math vocabulary (square, triangle, line), but those responses might also include vocabulary that is more specific, words like acute, obtuse, parallelogram or isosceles.

Many of the lessons I teach to young children (I work mostly with primary-aged children) include the creation, manipulation, and use of different kinds of shapes and lines, whether they are drawing, painting, or making collages. When appropriate, I like to remind the students that they are “doing math” as well as “making art”.


Here are a few of my favorite art-making activities with math connections.

Geometric Shape Collage

This is an activity that I have done with elementary students of all ages. For younger students, the directions are more general and fewer shapes are used. For older students, the directions and requirements are more specific and complex. For example, younger students are asked to use one circle, two lines, three triangles, and four colors (for a little problem solving challenge), while older students are asked to use one circle, two lines, three non-congruent triangles, four different quadrilaterals, and five colors. Directions can be altered to include geometry vocabulary as desired, and there are many ways to extend the art lesson to incorporate more mathematics. For example, one can ask younger students to do a rubbing of their design and label all the shapes, or have the  older students calculate combined areas of the quadrilaterals and find the ratio of that area to the total area.

Geometric People

One of my standard lessons with second grade students, these figures are created entirely from geometric shapes (mostly rectangles). The lesson includes a discussion about our joints, some attention to human body proportion, length and width, and the depiction of movement. When I have time, I begin with an introduction to the art of Keith Haring and a movement activity that requires students to arrange their bodies in different poses.

Kandinsky-Inspired Abstract Design

This is one of my favorite “no prep needed” art activities for incorporating math vocabulary and most of the elements of design. All you need is white paper, assorted colored markers, and crayons. It is adapted from an activity in the book “Drawing With Children” by Mona Brooks. I usually have students look at a Kandinsky print and tell me what they notice. Invariably, math vocabulary bubbles up: acute angles, triangles, parallel lines, etc. Then I lead them through the drawing by having them draw one thing at a time: three dots anywhere, one line that goes off the edge of the paper, another line parallel to the first, a third line that intersects the first two, etc. I have some standard directions for this activity that get varied now and then, according to what students are coming up with. No matter what, the directions use lots of math vocabulary. They are asked to color it however they choose, using only one color per closed shape and leaving part of the composition white. Later, they write about their art work, comparing it to Kandinsky’s work. These are always successful, colorful, and interesting!

 Symmetrical Cityscapes

This is an easy, fun art activity that connects to mathematics with its focus on symmetry, proportion, and a little work with geometric shapes. It gives students the opportunity to be creative while applying their knowledge of bilateral symmetry. I like to use construction paper crayons on black paper, but I’ve also used regular crayons on white paper. An even more artistic version of this lesson has students do a watercolor wash for a sky, onto which the cut out skyline is glued. Students love this art lesson, and it is inherently successful; even if mistakes are made in the symmetry, the end results are always beautiful.

There is no doubt artists use math all the time when creating art. Sometimes that math is obvious in the subject matter, and sometimes it is more subtle in the composition, but it’s nearly always present. To teachers who insist that there is no time for art, I say “think math” and you can kill two birds with one stone!

All the art lessons described here are available in my Teachers Pay Teachers store where you will find lessons and lesson bundles such as Playing With Shapes, Art With Symmetry, Art With Patterns, Exploring Lines and Shapes, GeomARTry, and much more!  I also share many art ideas on my blog entitled: Creating Art With Kids.  I hope you will check it out.


Thursday, January 24, 2013

Jack and His Nines

My guest blogger today is Barbara.  Even though she is not a seller on Teachers Pay Teachers, we have been friends since 6th grade.  She originally lived in England, and I was her pen pal, but then she moved to my home town in Ohio where we became best friends.  She teaches at the University of North Carolina. The college has a drop in "math lab," which is a service to students who would like help with their math assignments. Barbara is one of five instructors who takes turns supervising the math lab and giving help.  She finds it to be a very rewarding job. The courses she mainly helps with are Statistics, Reality Math, Math for Teachers, Pre-Calculus through Calculus, a little Differential Equations, and occasionally some Physics.  I think you will find Barbara's article on the magical nines quite interesting.

As the story goes, one day Jack and Jill were walking up the hill to fetch a pail of water. Jack said to Jill, “Did you bring your calculator?” Jill acknowledged that she never left home without it.

 Jack said, “Okay, Jill, now think of a fairly large whole number and reverse the digits. Then subtract the smaller number from the larger.”  Using her calculator, Jill came up with an answer to which Jack replied, “Now choose one of the digits in the answer and remember it. Then tell me the other digits, in any order, and I will tell you the remaining digit.”

“Okeydokey,” said Jill, “the other digits are 7, 2, 5, and 1.” Jack revealed that the remaining digit was three.  “That’s right!” shouted Jill.  In fact, she was so surprised that she fell down and broke her crown. 

?? How did Jack know Jill's number ??

The secret is in the properties of the number 9.  Did you know if you multiply ANY whole number by 9, the sum of the digits of the answer will also be a multiple of 9?

*A number is a multiple of 9 if its digits add up to 9 or a multiple of 9. If the answer is not a one digit number, continue to add the digits to eventually arrive at just a one digit answer of nine.  (This is sometimes referred to as finding the digital root.)

Example:  175,439,826 is divisible by 9 because when you add all the digits, you get 45. (1 + 7 + 5 + 4 + 4 + 9 + 8 + 2 + 6 = 45)  Adding the two digits of the answer, you get 9.  (4 + 5 = 9)   Nine is the digital root of 175,439,826 so this number is divisible by 9 as well as three since three is a factor of 9.
Example: 53,872,091 is NOT divisible by 9.  Adding the  eight digits you get the sum of 35. Adding the digits of the answer, 3 and 5, you get 8. This tells you that if you divide the original number by 9, you will get an answer with a remainder of 8.
Another interesting property of 9 is that if you choose any whole number, reverse its digits, and subtract the smaller number from the larger, the answer is always a multiple of 9.

Example:  5,132 – 2,315 = 2,817, which is divisible by 9.  (2 + 8 + 1 +7 = 18    1 + 8 = 9)
So Jack knew that when Jill subtracted her numbers, the answer was a multiple of 9. And when she told him the digits 7, 2, 5, and1, he knew that the remaining digit would have to be 3; so, that all the digits would add up to be a multiple of 9. (in this case, 18.)
Beware for there is one condition when this trick might not go as smoothly. If the digits already add up to be a multiple of 9, then the missing digit could be a 0 or a 9. You could then claim that your mental image is coming in fuzzy, and you can’t quite tell if it is a 0 or a 9.

Here is another math trick.  Ask someone to think of any three-digit number (with no repeating digits), reverse the digits, and subtract the smaller number from the larger. When that person has the answer, ask him/her to reverse the digits of the answer, and then add the answer and the reversed answer together.  You can then tell him/her that the final result is 1089. How do you know that? I will leave that for you to figure out! (That's problem solving at its best!)

P.S.  If you need help with the solution, check out the page above entitled Answers to Problems.


Thursday, January 10, 2013

Don't Copy Me!

My guest blogger for this week is Cynthia.  She, too, is a seller on Teachers Pay Teachers.  She has been an educator for 24 years - 16 years teaching kindergarten, one year looping with her students to first grade and seven years in 2nd where she currently teaches.  I hope you enjoy her timely post about limiting paper copies. 

2nd Grade Pad
Hello Everyone!

I am Cynthia from 2nd Grade Pad, the name of my TPT Store.  I am so excited to be doing a guest blog for Vicky!

Has your school limited copies that you can make for your classroom? 
By using three objects everyone has in their classroom, along with training your students in the process, you will find that planning is a snap and MUCH easier than making lots of copies each week.  No matter what your grade-level, implementing is a cinch!

Dry erase boards, dry erase makers, and an eraser are probably the most important three items in my classroom.  I literally, could not teach without them. 
(All of the technology out there, and these are my MUST HAVES???  YES!!)
Each day in math, I work with a small group at my needs-based table.  Each student has these three items.  I work with them doing a spiral review that includes measurement,
adding, subtracting, money, problem of the day, time, and place value.
Above is an example from November.  We go over one problem at a time. 
 The students write down their answer.  When they are finished, they put the lid on the marker and place it on the table.  This helps me to know when they are finished.   
Here are some of the items we use.
If you aren't familiar with the Judy clock, (on the right) when you move the minute hand, it is attached to gears so that it automatically moves
the hour hand.

 Next, all of the students show their answers, and we discuss them.
On the right is a sample of how they show the big number of the day.
In years' past, I have shown this on my big screen. The students would each write their answers on the spiral math board which I would laminate (included in the monthly units).  The students would keep these in their desk to use daily.
This option is also a paperless method. 

If you like what you see, you can purchase the Daily Math Reviews by the month, or you can purchase the entire year at a discounted price here.  My unit is for 2nd grade, but it can easily be used for first or third.  However, the spiral concept and work problems can certainly be used in any grade.  I hope these ideas save lots and lots of paper copies! 
I Spy Something Fishy

                 Cynthia's Blog

On your right is a free resource you might enjoy.  My store also contains many other freebies if you would like to take a look.

Thursday, January 3, 2013

Math Food for Thought

Because I had surgery on my right hand, some of my fellow bloggers graciously agreed to be guest bloggers this month.  I hope you enjoy reading math articles from other teachers who just might give you a different perspective on how to teach math.


Brian's Blog

Hi!  I am Brian from Hopkins' Hoppin' Happenings.  I taught Kindergarten for three years, 2nd grade for five years, and did a short term in 5th grade.  I am currently a substitute teacher. Today, I would like to talk about how to make learning math fun in Kindergarten.

When I taught Kindergarten, I requested that each parent or guardian bring in food that we could use for math.  Examples included Cheez-Its, Cheerios, Trix, Skittles, colored Goldfish, etc.  Every Friday, I taught or reviewed a math skill using the food!  Two of many of the math skills covered were sorting and graphing. Let's say I was using colored Goldfish for the math lesson.  I generally placed the same number in each student baggie with the same colors of each so everyone would get the same answer. This also made it easy for me to check for accuracy. First, the students were asked to sort the Goldfish according to color. Then they would line the different colors up on a graphing grid I had made.  The students then removed one Goldfish at a time and colored in the space where it had been. Next, I asked the whole class questions such as, “Which color of Goldfish are there the most of?  Least of?  How many green Goldfish are there? How many green and yellow goldfish are there in all?” Afterwards the children were allowed to eat their goldfish!  I have also had students graph and sort different kinds of cereal, Skittles, etc.

Another important math skill to use with food is patterning. Give students different kinds of food of different colors and have them create patterns with it. I give them a piece of paper on which they glue down a couple of their patterns and then they are permitted to eat the rest.  The patterns can be simple or difficult depending on the age of the children.

Food also works well for identifying numbers. Show the students a flash card with a numeral written on it. The students must count out that many skittles, Trix, etc.

Food also works well to teach addition and subtraction facts. For example: 3 + 1. The student counts out three Cheez-Its (or whatever food you are using) and adds one more. Have the student solve the problem and write down how many there are all together.  Here is another example, this time subtraction: 4 – 2.  The student counts out four Cheez-Its but then eats two of them. This time the student answers the question, “How many are left?”
You can also use food for math in 1st or 2nd grade and do the same graphing activity, but then ask higher level questions such as:  “How many more orange life savers are than than green?”  You can also use stick pretzels and Cheerios to teach place value and two digit addition or subtraction. Let’s assume the Cheerios are the ones and the pretzels represent the tens.  When adding 18 + 13, the students would trade 10 Cheerios for a pretzel or if subtracting, trade a pretzel for 10 cheerios.

For older students, food can be used to make groups or arrays for multiplication or to figure out how many are in each group for division. You can also give the students two colors of some food to practice fractions.

Try using food in the classroom. Your students are sure to love learning and quickly grasp the math concepts because they are having fun! 


Free Graphing Activity

Brian is offering an exclusive freebie for all of you who read my blog.  It is entitled Goldfish Cracker Graphing Activity. Just click under the goldfish to download it.

Also, be sure to check out Brian's Store on Teachers Pay Teachers for more primary resources.  While you are there, check out his other 37 freebies.