# MATH

## FRACTION IDEAS

### GRADES: 4-8

These are just starter lesson plans, I use the text questions to practice what we have learned.

#### MATERIALS:

- Skittles
- colored paper
- magazines
- newspapers

#### METHOD:

- Skittles on the Overhead. (An Introduction to Fractions) Using 72 Skittles, divide these into 1/2s, 1/4s, 1/8s, 1/16s, 1/3s, 1/6s, and 1/24s. I did this by telling a story to my class. I said I was going to share my Skittles with my teacher’s assistant, so we would each have 36, and what if we shared them with 2 other people, then how many would we have? And so on.
- Fraction Kits. Make kits in groups of four, using different colored papers. Use regular colored paper and show students how to divide into:

`1 whole 6-1/6s`

`2-1/2s 8-1/8s`

`3-1/3s 12-1/12s`

`4-1/4s 16-1/16s`

- Left-shoe Fractions. Using left shoes to make fractions, children sit in a circle and pick out groups of left shoes, and tell what fraction they have made in their groups. For example: if you have 24 students/shoes and 12 of them have blue on them, you will have 12/24 for a fraction, or 6/12 or 3/6 or 1/2. Free Time: Look in magazines, and newspapers for examples of where fractions are used, cut these out and put them into a page of math notebook.
- Fraction Equivalences. Students will already have some experience with fraction equivalences from previous lessons. You re-invent the wheel, have prepared fraction circles and re-state equivalences with the circle fractions. (die. whole, halves, quarters, thirds, sixths, etc.) Also, could show a circle graph, to show where these fractions are commonly used.
*Submitted by,*

**REBECCA FLUET**

OUTREACH SCHOOL

PEMBINA HILLS

*cheyenne@west-teq.net*

## PERCENTS, FRACTIONS, DECIMALS AND RATIOS

### GRADES: 5-9

If you are working with percents, fractions, decimals, and ratios this is a great project for a cooperative group!

#### MATERIALS:

- 12″ x 18″ green construction paper
- lots of die-cut multicolored circles
- die cut stars
- glue
- Ziploc bags

#### METHOD:

- Distribute a bag of multicolored circles to each group. (I have a worksheet that I distribute that has all colors listed.)
- After discussing how percents, decimals, fractions, and ratios are ways to write the same thing, I have each group sort their circles into piles according to color.
- On their worksheet they tally up all the different colors, then they create ratios, fractions, decimals, and percentages of each color to the set.
- Then I have the students cut out trees from the green paper by folding them in half and cutting out the shape.
- They then glue their “balls” onto their tree, add a star to the top and they are ready to display.
- As an extension to this activity I have them create a circle graph. That denotes the percents of colored circles on their tree.
- If you don’t have a die-cut machine, you can use triangles or squares that are easier to cut. Younger children like working with colored food like Fruit Loops, Rainbow Nilla Wafers, M&M’s, Skittles etc.

*Submitted by,***MAIRA MAGUIRE**

CENTENNIAL MIDDLE SCHOOL

MIAMI, FL

*Mywonka@aol.Com*

## WALLPAPERING

### GRADES: 1-4

The children do this activity as a class activity and enjoy the ownership of their joint effort. The cubby becomes their personal space and is coveted for silent reading times and reward times.

#### MATERIALS:

- one A4 photocopy per child.
- colored pencils or crayons.
- maths pads or paper for recording findings (each child)
- empty fridge or stove box that children can keep in the class as a cubby.
- 1 pair of scissors per child.
- wallpaper or other paste suitable for paper or card.
- writing pencils for each child

#### METHOD:

- Either by hand or on the computer, using Publisher, create a large rectangle on a piece of paper that is A4 copy size and divide it into an equal number of squares, rectangles or triangles. Squares are the easiest shape to create. All of the shapes must be identical in size. You can either make all sheets one shape e.g. squares, or make each shape and make an equal number of each so children can select a shape of their choice.
- Photocopy one copy of the shapes for each child in the class.
- The children are instructed to color in each shape on the sheet using 2 colors of their choice. Each alternate square, rectangle or triangle is colored like this i.e. red,blue, red, blue. Whatever 2 colors the child chooses.
- When all of the shapes are colored in, each child counts how many there are and records the number in his pad.
- Each child then cuts out the shapes.
- Taking turns, each child then proceeds to select a space on the cubby (which already has windows and doors cut out with a Stanley knife) and is instructed to glue the pieces alternating each color and recreating the original rectangular shape. The end result will be an approximation of the original uncut rectangle with the shapes glued into the original positions they maintained prior to cutting.
- Each child records their name on their finished piece of wallpapering.
- Prior to the wallpapering, children need to discuss how they will proceed with the project so that all of the children’s work will fit on the cubby. problems are talked through and tentative solutions are explored prior to gluing beginning.
- When all of the children have glued their wallpaper on the cubby, every child counts how many pieces of work have been placed on the cubby. If there are all three shapes, the children can make a graph of how many children used triangles, how many used squares and how many used rectangles. This graph may be done as a whole class graph with children calling out the answers to the teacher. Alternatively, each child can create their own graph in their maths pads or the teacher can make a photocopied model of a bar graph and the children can use rulers to draw in the bars.

*Submitted by,***SUZANNE SPIERS**

APPLECROSS PRIMARY SCHOOL

PERTH, WESTERN AUSTRALIA, AUSTRALIA

*psyche@iinet.net.au*## FRACTION IDEAS

### GRADES: 4-8

These are just starter lesson plans, I use the text questions to practice what we have learned.

#### MATERIALS:

- Skittles
- colored paper
- magazines
- newspapers

#### METHOD:

- Skittles on the Overhead. (An Introduction to Fractions) Using 72 Skittles, divide these into 1/2s, 1/4s, 1/8s, 1/16s, 1/3s, 1/6s, and 1/24s. I did this by telling a story to my class. I said I was going to share my Skittles with my teacher’s assistant, so we would each have 36, and what if we shared them with 2 other people, then how many would we have? And so on.
- Fraction Kits. Make kits in groups of four, using different colored papers. Use regular colored paper and show students how to divide into:

`1 whole 6-1/6s`

`2-1/2s 8-1/8s`

`3-1/3s 12-1/12s`

`4-1/4s 16-1/16s`

- Left-shoe Fractions. Using left shoes to make fractions, children sit in a circle and pick out groups of left shoes, and tell what fraction they have made in their groups. For example: if you have 24 students/shoes and 12 of them have blue on them, you will have 12/24 for a fraction, or 6/12 or 3/6 or 1/2. Free Time: Look in magazines, and newspapers for examples of where fractions are used, cut these out and put them into a page of math notebook.
- Fraction Equivalences. Students will already have some experience with fraction equivalences from previous lessons. You re-invent the wheel, have prepared fraction circles and re-state equivalences with the circle fractions. (die. whole, halves, quarters, thirds, sixths, etc.) Also, could show a circle graph, to show where these fractions are commonly used.
*Submitted by,*

**REBECCA FLUET**

OUTREACH SCHOOL

PEMBINA HILLS

*cheyenne@west-teq.net*

## PERCENTS, FRACTIONS, DECIMALS AND RATIOS

### GRADES: 5-9

If you are working with percents, fractions, decimals, and ratios this is a great project for a cooperative group!

#### MATERIALS:

- 12″ x 18″ green construction paper
- lots of die-cut multicolored circles
- die cut stars
- glue
- Ziploc bags

#### METHOD:

- Distribute a bag of multicolored circles to each group. (I have a worksheet that I distribute that has all colors listed.)
- After discussing how percents, decimals, fractions, and ratios are ways to write the same thing, I have each group sort their circles into piles according to color.
- On their worksheet they tally up all the different colors, then they create ratios, fractions, decimals, and percentages of each color to the set.
- Then I have the students cut out trees from the green paper by folding them in half and cutting out the shape.
- They then glue their “balls” onto their tree, add a star to the top and they are ready to display.
- As an extension to this activity I have them create a circle graph. That denotes the percents of colored circles on their tree.
- If you don’t have a die-cut machine, you can use triangles or squares that are easier to cut. Younger children like working with colored food like Fruit Loops, Rainbow Nilla Wafers, M&M’s, Skittles etc.

*Submitted by,***MAIRA MAGUIRE**

CENTENNIAL MIDDLE SCHOOL

MIAMI, FL

*Mywonka@aol.Com*

## WALLPAPERING

### GRADES: 1-4

The children do this activity as a class activity and enjoy the ownership of their joint effort. The cubby becomes their personal space and is coveted for silent reading times and reward times.

#### MATERIALS:

- one A4 photocopy per child.
- colored pencils or crayons.
- maths pads or paper for recording findings (each child)
- empty fridge or stove box that children can keep in the class as a cubby.
- 1 pair of scissors per child.
- wallpaper or other paste suitable for paper or card.
- writing pencils for each child

#### METHOD:

- Either by hand or on the computer, using Publisher, create a large rectangle on a piece of paper that is A4 copy size and divide it into an equal number of squares, rectangles or triangles. Squares are the easiest shape to create. All of the shapes must be identical in size. You can either make all sheets one shape e.g. squares, or make each shape and make an equal number of each so children can select a shape of their choice.
- Photocopy one copy of the shapes for each child in the class.
- The children are instructed to color in each shape on the sheet using 2 colors of their choice. Each alternate square, rectangle or triangle is colored like this i.e. red,blue, red, blue. Whatever 2 colors the child chooses.
- When all of the shapes are colored in, each child counts how many there are and records the number in his pad.
- Each child then cuts out the shapes.
- Taking turns, each child then proceeds to select a space on the cubby (which already has windows and doors cut out with a Stanley knife) and is instructed to glue the pieces alternating each color and recreating the original rectangular shape. The end result will be an approximation of the original uncut rectangle with the shapes glued into the original positions they maintained prior to cutting.
- Each child records their name on their finished piece of wallpapering.
- Prior to the wallpapering, children need to discuss how they will proceed with the project so that all of the children’s work will fit on the cubby. problems are talked through and tentative solutions are explored prior to gluing beginning.
- When all of the children have glued their wallpaper on the cubby, every child counts how many pieces of work have been placed on the cubby. If there are all three shapes, the children can make a graph of how many children used triangles, how many used squares and how many used rectangles. This graph may be done as a whole class graph with children calling out the answers to the teacher. Alternatively, each child can create their own graph in their maths pads or the teacher can make a photocopied model of a bar graph and the children can use rulers to draw in the bars.

*Submitted by,***SUZANNE SPIERS**

APPLECROSS PRIMARY SCHOOL

PERTH, WESTERN AUSTRALIA, AUSTRALIA

*psyche@iinet.net.au*

#### PROBLEM SOLVING LESSONS THAT WE RAN LAST YEAR. THESE ARE AN EXCELLENT WAY TO CONDITION YOUR STUDENTS INTO HIGHER LEVEL THINKING SKILLS FROM THE BEGINNING OF THE YEAR!

## TEACHING THE “GUESS AND CHECK” METHOD

### GRADES 3-12

Guess and check is an important critical thinking process that is becoming increasingly prevalent within new math texts. It is usually introduced in some form in third grade, and is used in some form all the way up through senior high.

There are four major steps involved in the “Guess and Check” method:

- Make a plan
- Create a chart or table
- Eliminate possibilities
- Look for a pattern

The following are a number of examples you can use. (Additional examples can be found in virtually any math text book). They are listed in developmental order, less sophisticated to those more sophisticated. Pick those most appropriate to your students. (The numbers can easily be changed to provide additional examples).

With practice your students will develop a self confidence that will enable them to obtain solutions ranging from a variety of correct answers to one correct answer. This will serve as a preparation for high order thinking skills as those used in Algebra, Geometry, etc.

#### EXAMPLE 1

Using pennies, nickles and dimes, how many different combinations can be used to obtain 25 cents? (HINT: there are 12 ways)

Make a chart with pennies, nickles, dimes and “total” as column headings.

TEACHER NOTE: This problem introduces all four of the steps and adherence to ONE CONDITION–the combination must total 25 cents. The students should be able to put these combinations in any order they choose. As they practice this type of problem, they will find that using a particular system or order, (i.e. concentrating on pennies from greatest to least) will emerge as a faster, more accurate method. Initially, in the earlier grades, students should use actual coins and record their findings.

#### EXAMPLE 2

Using nickles, dimes and quarters, how many different combinations (where at least one of each coin is used), can make 50 cents? Before you start, make a prediction. Compare your prediction to your findings.

TEACHER NOTE: There are only 2 combinations. This example introduces TWO CONDITIONS–at least one of each coin AND a total of 50 cents.

#### EXAMPLE 3

Using 17 coins–including AT LEAST ONE NICKLE, DIME AND QUARTER–how many different combinations can be used to make $2.25? Before you start, make a prediction. Compare your prediction to your findings.

TEACHER NOTE: There are only 3 combinations. This example introduces THREE CONDITIONS–at least one of each coin, 17 coins AND a total of $2.25.

#### EXAMPLE 4

Using 17 coins–including AT LEAST ONE NICKLE, DIME AND QUARTER–how many different combinations can be used to make $2.25–WHERE THERE ARE 4 MORE DIMES THAN NICKELS? Before you start, make a prediction. Compare your prediction to your findings.

TEACHER NOTE: There is only 1 combination. This example introduces FOUR CONDITIONS–at least one of each coin, 17 coins, a total of $2.25 AND a relationship of one variable (dimes) to another (nickles).

submitted by

ROB SCHUCK

PACOIMA MIDDLE SCHOOL

LOS ANGELES, CA

rschuck@glendale.edu

## USING A SYSTEMATIC APPROACH TO THE GUESS AND CHECK METHOD

### GRADES 3-12

Last time we traced the developmental stages of guess and check (“Teaching the Guess and Check Method”), utilizing four components. These components involved:

- Making a plan
- Creating a chart or table
- Eliminating possibilities
- Looking for a pattern

The purpose of these components is to demonstrate to the student that through an organized, *systematic* process, answers to seemingly “impossible” problems can be found. The key is the *systematic approach*, because all four components evolve around the system.

Having already explored the wonderful world of coin problems, the following examples are concerned with consecutive numbers and age problems. Also available are very basic problem solving examples that highlight each of the seven approaches originally addressed two weeks ago (“Setting a Foundation for Problem Solving”).

#### EXAMPLE 1

5 years ago, Jay was seven times older than Mary. In five years, Mary will be half as old as Jay (or Jay will be twice as old as Mary). How old is each now?

Make a horizontal chart with the following headings. (discuss the construction of the heading with the students):

J5YA—M5YA—J7XOLDER?—JNOW—MNOW—JIN5Y—MIN5Y—M1/2J?

KEY:

J5YA (John’s age 5 years ago)

M5YA (Mary’s age 5 years ago)

J7XOLDER? (Is John 7 times Mary’s age?)

JNOW (John’s age now)

MNOW (Mary’s age now)

JIN5Y (John’s age in five years)

MIN5Y (Mary’s age in five years)

M1/2J? (Is Mary half of John’s age?)

TWO POSSIBLE ANSWERS: (Numbers are in order of the columns above)

7—1—YES—12—6—17—11—NO

14—2—YES—19—7—24—12—YES

Therefore, Jay is 19 and Mary is 7

#### EXAMPLE 2

Make a chart similar to the one above.

Let’s make up a consecutive number problem–your choice.

- Using whole numbers: Four consecutive whole numbers have a sum of 14 and a product of 120. What is the second number? (2, 3, 4, 5)
- Using odd, whole numbers: Three consecutive odd, whole numbers have a sum of 9 and a product of 15. What is the third number? (1, 3, 5)
- Using even whole numbers: Three consecutive even, whole numbers have a sum of 12 and a product of 48. What are the numbers? (2, 4, 6)
- Using integers: Three consecutive integers have a sum of 0 and a product of 0. What is the first consecutive integer? (-1, 0, 1)
*submitted by***ROB SCHUCK**

PACOIMA MIDDLE SCHOOL

LOS ANGELES, CA

*rschuck@glendale.edu*

## GUESS AND CHECK–FINAL PROJECT–

## MULTIPLE VARIABLES AND CONDITIONS

### GRADES: 6-12

This is the last of the problem solving contributions that will be submitted, unless there is a sudden outcry for more! more! more! I hope that what has been presented so far has been of use for some of you. So…for the grand finale of problem solving utilizing the guess and check (trial and error) method, I present to you the infamous chickens, pigs, and sheep problem.

**TEACHER NOTE:** Remember that the “guess and check” method utilizes four components. These components involve:

- Making a plan
- Creating a chart or table
- Eliminating possibilities
- Looking for a pattern

The purpose of these components is to demonstrate to the student that through an organized, systematic process, answers to seemingly “impossible” problems can be found. The key is the systematic approach because all four components evolve around the system. Also, by using a systematic approach, it becomes increasingly easier to eliminate possibilities. This is especially true of the problem presented here.

**THE PROBLEM:** You are given $100 to buy 100 farm animals (at least one each of three animals–chickens, pigs, and sheep). If chickens cost 10 cents, pigs cost $2, and sheep cost $5, how many of each animal must you purchase so that the total is 100 animals for $100?

**THE CHART:** There should be five (5) column headings to represent the problem components. You might want to add a few more to make the students check to see which direction they need to make their “guesses”.

CHICKENS (.10)—–PIGS ($2)—–SHEEP ($5)—–100 ANIMALS?—–$100

CHICKENS: 35 ($3.50)

PIGS: 40 ($80)

SHEEP: 25 ($125)

100 ANIMALS?: YES

$100: NO ($208.50)CHICKENS: 50 ($5)

PIGS: 35 ($70)

SHEEP: 15 ($75)

100 ANIMALS?: YES

$100: NO ($150)

**TEACHER NOTE:** These two lines represent a wealth of information. In addition to each column beginning to show a potential pattern of direction for future guesses, a viewer should be able to see the plan I am using. Also, what possibilities have already been eliminated? What other possibilities can be eliminated as a result? If your students become frustrated with their own attempts, you might consider using these two lines (or your own) to help them get back on track.

**THE SOLUTION:** Do you really want me to tell you? Okay, I’ll meet you half way. The number of chickens is a multiple of 10 (Why must this be so?). It is not 50 chickens. The number of pigs feet is almost = the number of chickens. There are less sheep than the other two animals (approximately 1/7 of chickens and 1/2 of pigs).

submitted by

ROB SCHUCK

PACOIMA MIDDLE SCHOOL

LOS ANGELES, CA

rschuck@glendale.edu