Category: Teaching Fractions

Developing Number Theory and Fraction Concepts

Many students can begin to feel challenged in math in middle school. Students who have been good at, and have even enjoyed, math suddenly look to their teachers, friends or parents for assistance. Why does this happen? If you look at the concepts that are significant in middle school grades (fractions, decimals and integers), you find that these concepts appear to break all the rules their teachers have told them up to this point.

Typically, students are taught that when you multiply two numbers, the product is always larger. When you divide two numbers, the quotient is always smaller. However, these rules apply to whole numbers, not fractions. When you multiply two fractions, the resulting product may be smaller! When you divide two fractions, the quotient may be larger!  Many students become frustrated, confused and give up on math. As teachers, we need to make sure our students understand concepts, not just memorize rules about them. Students need time to explore and discuss real life examples of the concepts we are teaching. Below are some high-level tasks that allow students to explore number theory and fraction concepts. As with all tasks, students should represent their work in numbers, pictures and/or words. They should have time to communicate their thoughts and findings with others.

Topic: Factors and Multiples

Task: Max is making table favors for a party. The candles come in boxes of 15 and the candleholders come in boxes of 9. Max does not want any leftover candles or holders. What is the fewest number of candles and candleholders he needs without any leftover? How many boxes of each should he buy? Task: At a day camp, there are 12 girls and 18 boys. The camp counselors would like to split the campers into teams. However, they must follow these rules: 1) All campers must be on a team; nobody can be left out, 2) all teams must have the same number of campers, and 3) each team can only have all boys or all girls; no boys and girls can be on the same team. What is the greatest number of camperseach team could have?

Topic: Understanding Fractions

Task: In Penny’s Pet Shop,  of the pets were dogs,  of the pets were cats,  of the pets were birds and the rest were gerbils. There were 48 pets in all. How many of each type of pet were there? Task: Ms. Kinny has  tank of gas in her Volkswagen Beetle. Miss Jamison has  tank of gas in her Ford Mustang. Dr. Beck has  tank of gas in her Honda Accord. Mrs. Hughey has  tank of gas in her Toyota Prius. Without finding common denominators, list the women in order from the person who has the least amount of gas in her car to the person who has the greatest amount of gas in her car.

And for more of our Fun Learning Math Games, you can visit here:

http://www.math-lessons.ca/activities/index.html

http://www.math-lessons.ca/activities/FractionsBoard5.html

http://www.math-lessons.ca/activities/Geometry.html

Ordering and Comparing Fractions

When students are asked to order and compare fractions, they almost always start by finding common denominators. This strategy is based on rote memorization and leads to little or no true understanding of fractions (and can be utterly frustrating!). Students cannot visualize the fractions. This article explains how to help your students compare and order fractions using reasoning skills, not math formulas.

There are three steps outlined below. Each step should be introduced separately, practiced and then combined with the steps learned previously.

Step 1 – Use benchmarks – Using benchmarks of 0, 1 and greater than 1 (improper fractions) help students get a general idea of the size of the fraction.

Example – Put the following fractions in order from least to greatest:

Encourage students to find those fractions that are equivalent to 0, 1 and greater than 1 first. Then identify if any of the fractions are exactly  Compare the numerator and denominators on the remaining fractions to determine if they are less than  or more than  Try to relate the fraction to real life examples. (“If I received 11 out of 12 on a test, did I get more than half the questions correct or fewer than half the questions correct?”)

Provide a simple table for those students who have trouble organizing their work.

Step 2 – Use Common Denominators – Many students think ordering fractions with common denominators is even easier than using benchmarks. Since each fraction will have the same number of parts to make the total, comparing is easy. Again, present fractions in real life situations that allow students to visualize them. For example, if you took a math quiz worth 25 points, who would get more of the quiz correct: the student who gets 24 questions correct (  ) or the student who gets 13 questions correct (  )?

Step 3 – Use Common Numerators – This strategy is a bit more difficult for students to grasp. The use of fraction towers, fraction circles and/or drawings helps students grasp this concept.

When the numerators are the same, you are receiving the same number of pieces of the object. However, since the denominators are different, the whole will be cut into a different amount of pieces. For example, imagine you are eating a candy bar. You receive one piece (the numerator), no matter what. If you are all by yourself, you get the whole candy bar. Now imagine one of your friends comes by. You want to share the candy bar; so you split it into 2 pieces (in half). What happens to the size of your one piece as you share with more and more friends?

For a pie version of this, Birmingham Learning Resources shows us:  http://www.bgfl.org/custom/resources_ftp/client_ftp/ks2/maths/fractions/numerators.htm

And: http://www.freemathhelp.com/numerator-denominator.html

And for more of our Fun Learning Math Games, you can visit here:

http://www.math-lessons.ca/activities/index.html

 

 

 

 

 

 

 

 

Fractions for Christmas, Hannukah and Solstice!

This Holiday season, you can have fun both learning about Holiday Celebrations and Learning Fractions in a Fun way!  Here are 3 different Holiday Celebrations that occur in December that have specific special numbers of Days to count for Fractions – Christmas, Hannukah and Winter Solstice.

8 Days of Hannukah

What Fraction is the 3rd Day of The 8 Days of Hannukah?

1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 8/8

Hannukah (sometimes pronounced Chanukah) is known as The Festival of Lights, and this practice is celebrated by the Jewish Peoples of The Earth.

http://www.chabad.org/holidays/chanukah/article_cdo/aid/605036/jewish/Chanukah-FAQs.html.  It is a remembrance of The Miracle of The Holy Olive Oil lasting for 8 days (i.e.) There was only enough oil to last for one day, but through the Miracle of The Holy Spirit during Beit Hamikdesh, it lasted 8 days and 8 nights, for the entirety of The Celebration of Chanukah.  Hannukah is celebrated starting the 25th Day of Kislev (this corresponds to around the 21st-28th of December in the Gregorian calendar).

The typical Menorah consists of 8 branches with an additional raised or lowered branch in the middle to light the candles from, as there is a candle in each branch lit each night of the 8 days.  A fun thing that happens on Hannukah is The Spinning of The Dreidel, a four-sided spinning top that children play with. Each side of The Dreidel is imprinted with a Hebrew letter. http://en.wikipedia.org/wiki/Hanukkah#Dreidel.

 4 Quadrants of The Seasons

Winter Solstice December 21st 

What is The Fraction of The Solstices and Equinox Seasonal Cycles, based on The Gregorian and Celtic Calendar Year?

There are 4 quadrants that are the Seasonal Cycles of The Gravitational Cycles of The Earth, Sun and The Moon.  We call them:

Spring Equinox, Summer Solstice, Fall/Autumn Equinox and Winter Solstice.

All fall around the 21st – 23rd of the corresponding month (in general the 21st, depending on where the moon is, every year.

http://www.timeanddate.com/astronomy/equinox-not-equal.html.

1/4, 2/4, 3/4, 4/4

  1. Spring Equinox:  March 21st                        1/4                           
  2. Summer Solstice:  June 21st                          2/4
  3. Fall/Autumn Equinox: September 21st      3/4
  4. and Winter Solstice: December 21st            4/4

Traditional Song: 12 Days of Christmas

What is the Fraction of the 5th Day of The 12 Days of Christmas?

Here is an example:  If you add together the 3rd Day of Christmas and the 5th Day if Christmas, what Christmas Fraction does this equal? (Answer:  3/12 +5/12 = 8/12)

1/12, 2/12, 3/12, 4/12, 5/12, 6/12, 7/12, 7/12, 8/12, 9/12, 10/12, 11/12, 12/12

Traditional Song:  The 12 Days of Christmas

On the 1st First Day of Christmas,

My true love sent to me

A partridge in a pear tree. (1/12)

On the 2nd Second Day of Christmas,

My true love sent to me

Two turtle doves, (2/12)

And a partridge in a pear tree. (1/12)

On the 3rd Third Day of Christmas,

My true love sent to me

Three French hens, (3/12)

Two turtle doves, (2/12)

And a partridge in a pear tree. (1/12)

On the 4th Fourth day of Christmas,

My true love sent to me

Four calling birds, (4/12)

Three French hens, (3/12)

Two turtle doves, (2/12)

And a partridge in a pear tree. (1/12)

On the 5th Fifth Day of Christmas, (5/12)

My true love sent to me

Five golden rings, (5/12)

Four calling birds, (4/12)

Three French hens, (3/12)

Two turtle doves, (2/12)

And a partridge in a pear tree. (1/12)

On the 6th Sixth Day of Christmas, (6/12)

My true love sent to me

Six geese a-laying, (6/12)

Five golden rings, (5/12)

Four calling birds, (4/12)

Three French hens, (3/12)

Two turtle doves, (2/12)

And a partridge in a pear tree. (1/12)

On the 7th Seventh Day of Christmas, (7/12)

My true love sent to me

Seven swans a-swimming, (7/12)

Six geese a-laying, (6/12)

Five golden rings, (5/12)

Four calling birds, (4/12)

Three French hens, (3/12)

Two turtle doves, (2/12)

And a partridge in a pear tree. (1/12)

On the 8th Eighth day of Christmas,

My true love sent to me

Eight maids a-milking, (8/12)

Seven swans a-swimming, (7/12)

Six geese a-laying, (6/12)

Five golden rings, (5/12)

Four calling birds, (4/12)

Three French hens, (3/12)

Two turtle doves, (2/12)

And a partridge in a pear tree. (1/12)

On the 9th Ninth day of Christmas,

My true love sent to me

Nine ladies dancing, (9/12)

Eight maids a-milking, (8/12)

Seven swans a-swimming, (7/12)

Six geese a-laying, (6/12)

Five golden rings, (5/12)

Four calling birds, (4/12)

Three French hens, (3/12)

Two turtle doves, (2/12)

And a partridge in a pear tree. (1/12)

On the 10th Tenth day of Christmas,

My true love sent to me

Ten lords a-leaping, (10/12)

Nine ladies dancing, (9/12)

Eight maids a-milking, (8/12)

Seven swans a-swimming, (7/12)

Six geese a-laying, (6/12)

Five golden rings, (5/12)

Four calling birds, (4/12)

Three French hens, (3/12)

Two turtle doves, (2/12)

And a partridge in a pear tree. (1/12)

On the 11th Eleventh day of Christmas,

My true love sent to me

Eleven pipers piping, (11/12)

Ten lords a-leaping, (10/12)

Nine ladies dancing, (9/12)

Eight maids a-milking, (8/12)

Seven swans a-swimming, (7/12)

Six geese a-laying, (6/12)

Five golden rings, (5/12)

Four calling birds, (4/12)

Three French hens, (3/12)

Two turtle doves, (2/12)

And a partridge in a pear tree. (1/12)

On the 12th Twelfth Day of Christmas,

My true love sent to me

Twelve drummers drumming, 12/12

Eleven pipers piping, (11/12)

Ten lords a-leaping, (10/12)

Nine ladies dancing, (9/12)

Eight maids a-milking, (8/12)

Seven swans a-swimming, (7/12)

Six geese a-laying, (6/12)

Five golden rings, (5/12)

Four calling birds, (4/12)

Three French hens, (3/12)

Two turtle doves, (2/12)

And a partridge in a pear tree! (1/12)

And for some other of our Fun Learning Math Games, you can visit here:

http://www.math-lessons.ca/activities/HexaRace.html.

http://www.math-lessons.ca/activities/FractionsCards.html.

http://www.math-lessons.ca/activities/fractions-hazard.html.

 

 

Secret Chocolate Fraction Codes

It is Post-Halloween and we are not quite finished making chocolate FUN just yet!  Here is a fun and easy fractions game to organize that is low-cost and easily teachable, any time of the year.

Supplies:

  • Crayola Markers, 3 for each student
  • One Organic Chocolate Bar for each student
  • Piece of Paper

Have everyone bring in a Whole Chocolate Bar (organic if possible – it is healthier!)  –  One that has an equal number of squares in it.  They do not all have to be the same number of squares, but if they are, it is a bit easier for instructions.

Using a non-toxic marker (Crayola is my favorite), have each Learner draw a gridline across the paper on the outside of the bar in their favorite color. http://www.crayola.com/products/list.cfm?categories=MARKERS,BASICS

Step 1:  Counting 1-12 (usually,  this is the number of squares in a bar.)  If it is different, then  ask the learner to count, respectively re their bar, and write down on paper the basic 12 fractions of their bar:

1/12, 2/12, 3/12, 4/12, 5/12, 6/12, 7/12, 8/12, 9/12, 10/12, 11/12, 12/12

Step 2: Then secretly and individually, each student colors a different amount of squares in each bar, using 3 different colored markers.  Encourage Sharing/Trading markers if there is not enough markers to go around.

Step 3: Then, everyone divides into pairs, and one at a time – without  showing each other what they have colored – each student  guesses what 3 numbered fractions the other one has colored.

Eg/  Susan colored 3 squares in Red, 2 squares in Yellow, and 7 squares in Purple.  Therefore, Susan’s Secret Fraction Codes are:

3/12, 2/12 and 7/12.

Bob colored 3 squares in Blue, 6 squares in Green, and 3 Squares in Orange. Therefore, Bob’s Secret Fraction Codes are:

3/12, 6/12 and 3/12.

Step 4:  After successfully guessing the other’s Secret Fractions, each one guesses the 3 respective Colors – of each Fraction Code.

Step 5: Once they have successfully guessed the other one’s Secret Fraction Codes, have them, TOGETHER then, add all 3 to make the Whole Number One 1.

Eg/ Susan’s Secret Fraction Codes look like this:

3/12 + 2/12 + 7/12 = 12/12 = 1

Bob’s Secret Fraction Codes look like this:

3/12 + 6/12 + 3/12 = 12/12 = 1

Last Step:  Everyone share their Chocolate Bars with The Teacher! lolololololol

Enjoy!  Yum.

Love The Earth!

Remember to Recycle both the paper and the tinfoil or plastic that the bar was wrapped in!  The more Recycling and Care for The Earth, the more Pretty Colored Feathers (or Stars)you receive from The Teacher!

http://iloveloveearth.weebly.com/enter-the-i-love-earth-competition.html

For another one of our fun and affordable Fraction Games, you can visit here: http://www.math-lessons.ca/activities/chocolate.html

 

 

 

Teaching Fractions with Yummy Pizza Pie

The rate of success for learning fractions in elementary class math depend on the teaching method used.  Some of these methods are outlined here – simple to teach, and simple to apply.  Having real life examples for students to connect with makes it easier for the brain to get into the right lane. Learners then have a good foundation in fractions to better understand how to add, subtract, multiply, and divide.

Most kids LOVE pizza, so make it fun and bring REAL pizza into the classroom, several if parents are okay with chipping in on the cost or if the principal authorizes this as a project.  Or, do what we did, and bring in the ingredients, borrow the home economics room, and cook them right there in the classroom. If neither of these possibilities is viable, then divide the kids into groups of different numbers and ask them to draw a simple picture of a pizza on a piece of paper, with the number of slices drawn equal to the number of kids in their group.

Real-World Activities for Teaching Fractions

For many elementary age children, fractions seem like a foreign language that they likely may never use in real life.  To challenge this thinking, we can use real-world activities that illustrate just how fractions help their parents, and how they will help them as they grow up.

  • Teach them to use the ruler. Rulers are really handy for teaching real-world applications for fractions.  Most of them are already divided into halves, fourths, eighths, even sixteenths.  It’s easy to show them that two halves are the same as one. . . that two fourths are the same as one half, and four fourths are the same as one, and so on.
  • Teach them to use measuring cups and spoons.  Bring lots of these dishes to class and give one measuring spoon and cup to each child.  Ask them to pour one cup of water into a tall glass. Now take guesses from the students:  How many half cups do they think it takes to fill a cup? How many quarter cups will it take?  How

A Quick & Easy Fraction Plan

For some reason, fractions seem to have a reputation for being a lot harder to learn (and teach) than they really are.  They can actually be fun and simple to master, as long as you have a simple plan for teaching them.  Here’s a strategy you could use:

  1. First, remember that the right time to start teaching fractions is right after the children have a good understanding of adding, subtracting, multiplying and dividing.  Then begin with very basic, simple fractions for each operation.

Teaching and Learning Scary Fractions

Materials:

Small Index Size Recipe Cards

Medium sized Pot

Large Stirring Spoon

Imagination

Create a Scary Recipe for a Halloween Witches’ Brew:

Using various fraction amounts of ingredients, all students must add the total amount of liquid for the brew.

Ingredients:

• 4 1/2 cups Mud Water

• 2 1/2 tbls Ghecko Slime

• 1 3/4 cups Chopped Toenails

Pyramid and Fractions

Many of us enjoyed playing with legos and blocks when we were younger. We used these to build what ever our little imaginations at the time seemed to be fascinated with. These building blocks, no pun intended, are the same philosophy applied to very basics of architecture throughout the world.  Many students also enjoy any type of activity that gets them Pyramid.svginvolved as well as allows them to be able to do something with their hands, which makes this next activity ideal for many students. By having the students use their hands to apply basic mathematical methods and apply these techniques into real life situations, their interest is captured in a great way.

Smarties Game for Learning Fractions

Teachers can implement a fun game for students that not only give them a chance to have fun learning fractions, but they can also learn how to use fractions in real life. This game CB058388incorporates many skills that engage the students and motivate them in wanting to learn more. Some of the skills that are used are many of the fundamentals of math – as well as a variety of other subjects. Several of these skills include adding, creating their own fractions, and the opportunity to use their imagination. By using this innovative lesson plan students will have fun interacting with other students, share ideas, as well as practice their ability to understand fractions.

This game is ideal for students in grades 5-7.  Students will learn how to compare fractions with one another.

Materials:

Mini-smarties candy bag