Calculating Unit Costs: It's Complicated

Over the years, the size and weight of various grocery items has shrunk, while the cost has stayed the same or increased. Bags of chips now only contain 10 or 12 ounces instead of 16 ounces (even though the bag itself has stayed large), candy bars have shrunk in size, and breakfast bars now have only 8 to 10 smaller bars per box instead of 12 bars.

One way the 90 calorie bars have reduced the number of calories in each bar is by reducing the size of the bar. It's hard to compare costs when the sizes of the items are no longer the same.

In Excel Math, students learn to add and subtract, use coins and dollars, compare costs, multiply and divide, and solve word problems involving money. Even the most proficient mathematicians might be scratching their heads over the changing size of packaging and food items in our stores today.

Toilet paper rolls have reduced their number of sheets per roll or the size of the square sheets to decrease the total feet of paper while keeping the price the same. So when comparing prices, I now have to decide if I want one or two-ply, larger sheets per roll, more sheets per roll, or a larger total on the roll. Then I need to find out if the larger rolls will even turn on my dispenser at home. Could it get more confusing?


Here are some soaps I bought for our visiting guests this summer. (Yes, I over did it!) The soap came in various packages of 6 or 8 bars, but even the individual soap boxes were completely different sizes and so were the soaps. You really have to do the math to find out which one is the better price when the sizes vary so much.

We'll say that six 3.9 ounce bars of soap cost $5.00. The package lists the total or net weight at 23.28 ounces or 1.45 pounds (which means the actual bar size is closer to 3.88 ounces). Instead, we could buy a single bar of soap at 4.25 ounces for $1.19 per bar.  Which would be most cost effective? Let's do the math! The package of six bars of soap cost 23¢ per ounce ($5.00 ÷ 23.28 = .23). The single bar of soap costs 28¢ per ounce ($1.19 ÷ 4.25 = .28). So in this case we could save 5¢ per ounce or $1.16 on every six bars of soap. (Your students can do this type of problem-solving experiment to find cost per unit of some of their favorite items: snack food, fruit, facial tissue, desserts, movie tickets, etc.)

Cereal and cracker boxes also contain less food than they used to in years past. There are fewer nuts in each can or bag. Making unit cost comparisons is no longer as easy as looking a the price of the item. Read more about the shrinking sizes of grocery items at WallStreetJournal.com.

Ask your students to take a look at their favorite foods to see how the package size has been reduced over the years. You could even take a class survey and make a picture chart of the packaged foods that are favorites in your classroom.

What are some of your favorite shrinking packages? Share your thoughts by clicking on the word "comments" below.
Read more . . .

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New to Excel Math? Preview elementary math lessons that really work for Kindergarten through Sixth Grade on our website: www.excelmath.com. Also find math resources for teachers, parents and students and download a sample packet at excelmath.com.

The unique spiraling system built into Excel Math lessons helps children become confident math students. Read more and see sample lessons at http://excelmath.com/downloads/sample_lessons.html.

Multiplying Fractions, Part I

We've pulled together some activities and worksheets to help your students learn to multiply fractions. Let's start with multiplying whole numbers and fractions using number lines, repeated addition and area models. Giving your students lots of practice will help get these concepts into long-term memory.

In Excel Math, students learn to add and subtract fractions, multiply and divide fractions, and solve word problems involving decimals and fractions. The unique spiraling system built into Excel Math lessons helps children become confident math students.

Many former Excel Math students mention that they first began to develop a love for math during their elementary school years. With Excel Math lessons, students learn practical ways to apply mathematical concepts to their everyday lives.

Preparation: Draw a number line on the board with a range of 0 - 2, showing increments of 1/5:

If you have room, create a life-size number line on the floor or wall of your room using colored tape.

Give each student a Fraction Pieces page and scissors. Download the Fraction Pieces file here.

Excel Math Fraction Pieces Worksheet
Click here to download a copy

Write some multiplication problems on the board: 

4 x 1/2 = 

3 x 2/5 = 

8 x 2/3 = 

5 x 4/6 =

2 x 3/12 = 

Do the first problem together. Show the students how they can use their fraction pieces and repeated addition to calculate the answers: 

4 x 1/2  =  1/2 + 1/2 + 1/2 + 1/2

Have a student write the process and the solution on the board. (The students do not need to simplify their answers, unless you want them to do that extra step.)

Do the second problem together. Have the students use their fraction pieces once again.

Point out that 3 x 2/5 = 6 x 1/5 

Remind your students that multiplication can be thought of as repeated addition:

Use arrows to show this on the number line:

Have a student stand on your floor number line (or point to the wall line) and jump from 0 to 2/5, then to 4/5 and finally to 6/5 as your students say the red fractions together.

Now help your students create a visual fraction model:

Let your students use their fraction pieces and repeated addition to solve the remaining problems. Have them represent the problems on a number line and with visual models. Talk about the different ways to find answers to these problems.

You may want to suggest that your students replace the “x” in a multiplication problem with the word “of”: 

1/2 x 6, 

1/2 times 6, and 

1/2 of 6 

all mean the same thing.

If your students seem to catch on to this concept, try a word problem with fractions:

In a relay race, each runner runs 1/2 of a mile. If there are 6 team members how many miles long is the race? (3 miles)

Have your students use a number line and a visual fraction model to solve the word problem. If you have time, let them make up a fraction word problem of their own.

What techniques do you use to help students multiply fractions and whole numbers? Share your suggestions by clicking on the word "comments" below.

Read more . . .

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New to Excel Math? Preview elementary math lessons that really work for Kindergarten through Sixth Grade on our website: www.excelmath.com. Also find math resources for teachers, parents and students and download a sample packet at excelmath.com.

Happy Spoonerism Day!

William Archibald Spooner
1844-2930

Today marks the birthday of Reverend William Archibald Spooner. He was born in London in 1844, and is especially remembered for the famous and often funny slip of the tongue named after him. A spoonerism was originally the accidental switch of the initial consonants or vowels of two or more words, but has also since become a popular wordplay game. Read more about Spoonerism Day at Mentalfloss.com

Here are a few Spoonerisms:
"tons of soil" for "sons of toil"
"I'll sew you to your sheet" for "I'll show you to your seat"
"It is kisstomary to cuss the bride" for "It is customary to kiss the bride"

William Archibald Spooner studied at Oxford and became an Anglican priest. For decades he was a respected member of the faculty at Oxford, lecturing on Christianity, philosophy, and ancient history, but he is mostly remembered for unintentionally transposing letters or syllables as he spoke.

The term 'spoonerism' comes from Spooner's name, but actually,  spoonerisms were rare. His fame for such comments was fueled mostly by students at the college. These students amused each other by creating many intentional spoonerisms that were undoubtedly funnier than the Professor's occasional accidents. Read more at www.nndb.com

Laughter is one way to engage students in the classroom. Excel Math lessons are another. With Excel Math, students learn practical ways to apply mathematical concepts to their everyday lives. The unique spiraling system and spaced repetition in Excel Math helps children become confident math students. Read the glowing reports from teachers and administrators who have seen it in action.

Do you have a favorite Spoonerism? Click on the word "Comment" below to share it with us. You may want to have your students make up a few of their own.

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The Link Between Bedtime and Learning

"Early to bed, early to rise
Makes a man healthy, wealthy and wise."

This is a quote from Benjamin Franklin, inventor, writer, and one of the writers of the Declaration of Independence. For a quick biography of Benjamin Franklin, visit USHistory.org. New research shows he just may have been on to something.

Consistent bedtimes for young children may actually help their performance in school as they grow older.

Researchers from University College London have taken a look at data on bedtimes and cognitive tests for reading, math, and spatial abilities for 11,178 seven-year-old children and found that consistent bedtimes during early childhood were linked to markers of cognitive performance.

Their findings showed that irregular bedtimes at age three were independently associated with lower cognitive test scores in reading, math, and spatial skills.

Read more . . .

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What techniques do you use to keep students alert and engaged? Share your suggestions by clicking on the word "comments" below.

Disney Math

Disneyland Entrance

On July 17, 1955 Disneyland officially opened in Anaheim, California. The $17 million theme park was built on 160 acres of former orange groves.

Legend has it that Walt Disney had his workers mark the trees that were to be cut down with a certain color of colored ribbons. Those trees that were to be spared were tied with a different ribbon color. Somehow, the instructions did not translate properly (or the person coordinating the cutting was color blind) and most of the trees Walt Disney wanted spared were the ones that got cut down!

Today Disneyland hosts more than 14 million visitors a year, who spend close to $3 billion. Read more at ThisDayInHistory.com.

Cinderella's Castle at Disneyland 

In Excel Math, students learn to recognize coins, calculate costs and solve word problems involving money and decimals. The unique spiraling system built into Excel Math lessons helps children become confident math students. 

Many former Excel Math students mention that they first began to develop a love for math in their elementary school years. 

With Excel Math, students learn practical ways to apply mathematical concepts to their everyday lives.

If you're looking for ways to save on your next Disney vacation, Read More . . .

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New to Excel Math? Preview elementary math lessons that really work for Kindergarten through Sixth Grade on our website: www.excelmath.com.

You can find math resources for teachers, parents and students and take a walk through the curriculum at excelmath.com/tour/tour01.html.

Lighting Up Students for Success

Light Up Poster from Excel Math

Albert Einstein once said, "A student is not a container you have to fill, but a torch you have to light up."

With Excel Math, students light up as they learn practical ways to apply mathematical concepts to their everyday lives.

The unique spiraling system in Excel Math helps children become confident math students.

Many former students mention that it was during their time studying Excel Math that they grew to develop a love for mathematics.

We created this poster as a download you can use with your own class or print to send home to parents. Read more and get the link . . .

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New to Excel Math? Preview elementary math lessons that really work for Kindergarten through Sixth Grade on our website: www.excelmath.com. 

You can find math resources for teachers, parents and students and take a walk through the curriculum at excelmath.com/tour/tour01.html.

Celebrating the Hoover Dam

Hoover Dam
April 2012

On July 7, 1930, construction of the Hoover Dam began. Originally known as the Boulder Dam, it was renamed in honor of our 31st president, Herbert Hoover in 1947.

Here are some photos my daughter took on her visit there last spring.

The concrete arch structure of the dam was built to take advantage of gravity. It was intended to prevent flooding as well as to provide irrigation and hydroelectric power to some of the driest regions of states such as California and Arizona.

The dam stands 726 feet high and is 1,244 feet long. Hoover Dam was one of the largest man-made structures in the world at the time of its construction.

In Excel Math, students learn to measure items in yards, feet and inches as well as in meters and kilometers. The unique spiraling system helps children become confident math students. Many former Excel Math students mention that they grew to develop a love for math. With Excel Math, students learn practical ways to apply mathematical concepts to their everyday lives.

Arthur Powell Davis drew up plans for this ambitious dam-building project in 1922. Davis was head of the U.S. Bureau of Reclamation (the federal agency given responsibility for irrigation in the West).

Out of two prospective places for the dam, Black Canyon was chosen over Boulder Canyon. Even so, for some reason the planners continued to call the project Boulder Dam.

Several western states all competed for claims on the Black Canyon River. At the time, Herbert Hoover was U.S. Secretary of Commerce. Even with Hoover's exuberant backing and a regional consensus around the need to build the dam, Congressional approval and individual state cooperation were slow in coming.

Hoover finally  negotiated the Colorado River Compact, which divided the river basin into two regions — lower (Arizona, Nevada and California) and upper (Wyoming, Utah, New Mexico and Colorado) — that would help to resolve the political claims on the river and make the building of the dam possible.

Hoover Dam

Here are some additional statistics:

• More than 200 engineers worked to design the dam.
• It was the largest building project that the federal government had ever undertaken.
• 21,000 men worked on building the dam over the course of its construction (around 5,000 at any one time).
• The dam weighs more than 6.6 million tons.
• At its base, the maximum water pressure is 45,000 pounds per square foot.
• The building of the dam created Lake Mead, which extends for 550 miles of shoreline. 
• Lake Mead is 247 miles in area, and is one of the largest man-made lakes in the world.
• A National Historic Landmark, Hoover Dam draws over 7 million tourists a year.
• 10 million visitors head to Lake Mead each year for boating, sailing, fishing, etc.

Read more about the Hoover Dam at www.history.com and This Day in History.

Have you visited Lake Mead or the Hoover Dam?

Tell us about your experience and your impressions of the area by clicking on the word "Comment" below.

New to Excel Math? Preview elementary math lessons that really work for Kindergarten through Sixth Grade on our website: www.excelmath.com. 

Also find math resources for teachers, parents and students and take a walk through the curriculum at excelmath.com/tour/tour01.html.

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Fourth of July Puzzle It Out with Math

Here's a fun flag puzzle your students can use for Basic Math Fact Practice and to learn more about the birthday of the United States of America on July 4.

Download the puzzle and copy it for each student.

If it's easier for you to copy it as two pages instead of back to back, let your students glue the sheets back to back. If you prefer, print only the front of the puzzle and let your students draw their own pieces on the back.

Give each student a flag puzzle, crayons or markers, an envelope and scissors. Let each student color the flag, then turn it over and print a math equation and his or her initials on each puzzle piece on the back.

Back of Puzzle

Front of Puzzle

Make sure they number (or letter) each piece. Have them color a border or frame around the equations, if they wish. Then let them cut out the puzzle pieces and place them in an envelope. Have each student print her name on her envelope.

As they work, talk about the meaning of the 13 stripes and 50 stars. Explain that the flag is also called "Old Glory." Talk about the flag's history and how the design changed over the years as stars were added. Point out that the flag pattern does not have the correct number of stars and stripes. Read more about the history of the U.S. flag at http://www.pbs.org/capitolfourth/flag.html.

Divide students into partners. Let them play with one puzzle at a time. Have them lay out the pieces from one envelope picture side up on a table or floor. The first player takes a piece, turns it over, and solves the equation aloud. If he solves the equation correctly, he places the piece in the center of the play area, picture side up, and gets 100 points. The other student draws a puzzle piece, solves the equation out loud, and places the piece approximately where it will go on the puzzle. If the student solves the equation incorrectly, he simply returns the puzzle piece to the pile, flag picture side up. The players receive 100 points for each correct answer.

Play continues alternating in this way until the puzzle is completed.  The player with the most points wins. If you prefer to play without keeping score, have the winner be the player to place the final puzzle piece. Return the puzzle pieces to their envelope, after talking about the flag and its history. If you have time, let the students play again, using the other puzzle.

Alternate way to play:
cand trade envelopes with a classmate. Give each student a paper and pencil. At your signal, have the students solve the math equations, writing the answers on their papers. When the equations are solved, let them put together the puzzle and stand up when it's completed. (Have them yell out "Happy Independence  Day" if you don't mind the noise.) The first person standing wins. Read more about Independence Day at History.com.

Return the puzzles to their rightful owners and trade envelopes once more (with a different partner). The first partner to finish and solve the equations correctly gets 100 points. Also award 100 points for each equation solved correctly.

If you have time, let the students play again, trading puzzles with another classmate.

How do you help students understand the birthday of our country? Leave a note by clicking on the word Comment below.

New to Excel Math? Preview elementary math lessons that really work for Kindergarten through Sixth Grade on our website: www.excelmath.com. Also find math resources for teachers, parents and students and take a walk through the curriculum at excelmath.com/tour/tour01.html.

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Independence Day Math

On July 4 we celebrate our country's birthday with fireworks, picnics, and activities with our families and friends. Some of the streets around our city are lined with United States flags in honor of Independence Day.

Here's a fun craft — Fourth of July rockets. Let your students make some rockets from paper scraps and recyclables (cardboard bathroom tissue tubes, drinking straws, and tissue paper). Then use the festive rockets to decorate your classroom. See complete instructions and photos at http://to.pbs.org/19CWCCv

Have each student print a math equation on the rocket and print the answer on a strip of paper. Let the child fold the paper and slip it into the rocket tube. Have your students trade tubes and see who can solve the equation first.

Also have your students make paper flags from white construction paper. Glue red paper strips for the stripes and a blue square of paper in the top left corner for the star background. Let the students add white star stickers or use chalk or white crayons to draw stars.

Talk about the United States flag as the students look at your class flag. Or show them the picture of the flag shown here. Ask them to explain why there are 13 stripes and 50 stars. See if the class can name all 50 states or the 13 original colonies.

Let your students create math equations that total 50 and 13 using a variety of operations (addition, subtraction, multiplication, division) and print them on the back of their flags. In Excel Math, we help students understand and practice math skills so they become confident and successful in mathematics. Learn more about our unique spiraling system: http://www.excelmath.com/tour/tour03.html.

Your students may not realize that "the star-spangled banner" refers to our flag. Discuss other banners your students may have seen. Point out that our flag is a banner with stars and stripes. In 1814, Francis Scott Key wrote the poem, Defense of Fort McHenry. The poem was later put to the tune of The Anacreontic Song by John Stafford Smith (watch the Georgia Tech Glee Club perform a rousing version of that song below). The song was modified somewhat, Keys words were added, and it was retitled, The Star Spangled Banner. Congress proclaimed The Star Spangled Banner the U.S. National Anthem in 1931.

Like most people, you may be most familiar with the first verse of  The Star Spangled Banner. Did you know it has four verses? Here are the first two:

O say, can you see, by the dawn's early light,

What so proudly we hail'd at the twilight's last gleaming?

Whose broad stripes and bright stars, thro' the perilous fight,

O'er the ramparts we watch'd were so gallantly streaming?

And the rockets' red glare, the bombs bursting in air,

Gave proof thro' the night that our flag was still there,

O say, does that star-spangled banner yet wave

O'er the land of the free and the home of the brave?

On the shore dimly seen through the mists of the deep

Where the foe’s haughty host in dread silence reposes,

What is that which the breeze, o’er the towering steep,

As it fitfully blows, half conceals, half discloses?

Now it catches the gleam of the morning’s first beam,

In full glory reflected now shines in the stream,

’Tis the star-spangled banner - O long may it wave

O’er the land of the free and the home of the brave!

Read the full text to our national anthem and find out more about the Star-Spangled Banner on our previous post: The Star-Spangled Banner—Long May She Wave. How do you help your students understand Independence Day? Leave a comment in the box below to share your suggestions.

New to Excel Math? Preview elementary math lessons that really work for Kindergarten through Sixth Grade on our website: www.excelmath.com.

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