# 4th Grade Fractions: Bringing Order to Disorder

## Discussion/Introduction

Ordering: it’s one of the first math lessons in kindergarten, and it continues to be an important concept up through graduate school. Today, in your fourth grade math lesson, you’re going to be looking at ordering a set of numbers that may seem un-order-able to the uninitiated: dissimilar fractions.

What is the secret to ordering fractions who use mismatching names and don’t want to stand in line? Find out their other, secret names!

You’ll want to do this 4th grade fractions lesson after you’ve introduced equivalent fractions and given your students the tools to recognize and generate equivalent fractions. A free fourth grade lesson plan that covers those topics is available here.

## Objective

Your students will learn to compare two and more fractions with different numerators and different denominators. They’ll practice two different ways of doing this: 1) creating common denominators and numerators, and 2) comparing to a benchmark fraction such as ½.

## Supplies

• Name card holder pockets that can go around children’s necks
• My Name Is Fraction Name Flashcards, Set 1 and 2 (Set 1 has just 16 cards if you have more students than this, print out multiples.)
• Quick Quiz, one copy per student

## Methodology/Procedure

Start on a positive note: today, we get to play a fun new game! Remember, if you have fun with this, your students will too, and there’s no better way to wire the brain for optimal learning than by making learning fun.

Hand out “My name is” fraction cards from your first stack to everyone in the class. Duplicates are not a problem. Have them insert the cards in the transparent pouches and hang them from their neck.. Explain that the fraction on this card is their name for the rest of the class; and that it will be especially important during a new game that is going to be played.

Use your cellphone or MP3 player to turn on some music, and tell the students to mill around or ‘dance’ while the music is on. As soon as the music is off, they need to line up in order of their fractions. If they’re super-fast, there might be a prize.

Play the music, turn it off, and give the students a small amount of time to make sense of their fractions and try to order themselves. Equivalent fractions stand share one place in the line (standing side by side) . You do not need to wait for them to be done before calling a halt.

Observe that it was pretty difficult, and ask why. If no-one brings it up, you can observe that though we’ve learned to order fractions with similar denominators or numerators—fractions where the division size or number of portions was equal– we never learned anything about how to order fractions where both the top and the bottom were different. These fraction names could just as well be in different languages; they simply don’t want to be compared!

If the fraction names are in different languages and so can’t be compared, what is one way we could compare them?

Allow your students discussion time to consider this problem. If they do not arrive at the solution, give it to them: Even if fractions can’t be easily compared in their current state, you can translate them into similar languages so they are comparable! Remind them that one quantity can be described by more than one fraction; and that these different fractions that refer to the same portion are called equivalent fractions. Ask them how they might find other versions of their fraction names.

[Note: if your students came up with the alternative method of ordering dissimilar fractions—comparing to a benchmark—validate their thinking, then go on to discuss creating common denominators and numerators as an alternative way. ]

Summarize their responses on the board, and review any bit that needs reviewing. Then invite them to the table at the front of the room, where you’ve created a big messy draw pile of all the remaining fractions. Give them some time to look through these cards and find their own equivalent fractions. Offer unobtrusive help to anyone who needs it.

When they’ve all got their cards, tell them you’re to play the game again, and ask them whether it will be easier this time. Lead the discussion round to the fact that equivalent fractions, being representations of the same quantity, stand in the same place on the number line. Demonstrate how students can order themselves easily by comparing the fractions, finding pairs with the same numerator or the same denominator, and choosing an order based on that.

Give everyone a chance to compare cards and order themselves, then applaud them on their work and return everyone to their seats. If they’ve made reasonable time, offer them a token prize.

Write your findings on the board: we can compare dissimilar fractions by renaming them as equivalent fractions with common denominators or numerators.

Now write 1/3 and ¾ on the board, and ask them if they can tell you quickly which is larger. Talk them through using ½ as a benchmark fraction—1/3 is smaller than ½ because 3 is larger than 2, and ¾ is larger than 2/4=1/2 because 3 is larger than 2. So since 1/3 is smaller than ½ and ¾ is larger, 1/3 is smaller than ¾. Ask which symbol you should put between the two fractions, and if necessary, offer a short review of >, =, and <.

Now collect all the flashcards—the pile of originals and the pile of new names—distribute original fractions again, and give the students another chance to collect their new names from the draw pile. Ordering should be faster this time. Be available to help anyone who is struggling.

Ask your students to take the Quick Quiz out and time themselves to see how fast they are able to order the fractions on that page.

Hand out “My name is” fraction cards from your first stack to everyone in the class. Duplicates are not a problem. Have them hang it from their necks in the name card holder pockets. Explain that the fraction on this card is their name for the rest of the class, and especially, for a new game that is going to be played

Use your cellphone or mp3 player to turn on some music, and tell the students to mill around or ‘dance’ while the music is on. As soon as the music is off, they need to line up in order of their fractions. If they’re super-fast, there might be a prize.

Play the music, turn it off, and give the students a small amount of time to make sense of their fractions and try to order themselves. You do not need to wait for them to be done before calling a halt.

Observe that it was pretty difficult, and ask why. If no-one brings it up, you can observe that though we’ve learned to order fractions with similar denominators or numerators—tops or bottoms that are the same—we never learned anything about how to order fractions where both the top and the bottom were different. These fractions speak different languages, almost, and don’t want to be compared!

If the fraction names are in different languages and so can’t be compared, what is one way we could compare them?

Translate them into similar languages so they are comparable! Remind them that one quantity can be described by more than one fraction; and that these different fractions that refer to the same portion are called equivalent fractions. Ask them how they might find other versions of their fraction names.

Summarize their responses on the board, and review any bit that needs reviewing. Then invite them to the table at the front of the room, where you’ve created a big messy draw pile of all the remaining fractions. Give them some time to look through these cards and find their own equivalent fractions. Offer unobtrusive help to anyone who needs it.

When they’ve all got their cards, tell them you’re to play the game again, and ask them whether it will be easier this time. Lead the discussion round to the fact that equivalent fractions, being the same number, stand in the same place on the number line. Demonstrate how students can order themselves easily by comparing the fractions, finding pairs with the same numerator or the same denominator, and choosing an order based on that.

Give everyone a chance to compare cards and order themselves, then applaud them on their work and return everyone to their seats. If they’ve made reasonable time, offer them a token prize.

Write your findings on the board: we can compare dissimilar fractions by renaming them as equivalent fractions with common denominators or numerators.

Now write 1/3 and ¾ on the board, and ask them if they can tell you quickly which is larger. Talk them through using ½ as a benchmark fraction—1/3 is smaller than ½ because 3 is larger than 2, and ¾ is larger than 2/4=1/2 because 3 is larger than 2. So since 1/3 is smaller than ½ and ¾ is larger, 1/3 is smaller than ¾. Ask which symbol you should put between the two fractions, and if necessary, offer a short review of >, =, and <.

Now collect all the flashcards—the pile of originals and the pile of new names—distribute original fractions again, and give the students another chance to collect their new names from the draw pile. Ordering should be faster this time. Be available to help anyone who is struggling.

Ask your students to take the Quick Quiz out and time themselves to see how fast they are able to order the fractions on that page.

## Common Core Standards

In Fourth Grade Number and Operations—Fractions item 2, the Common Core State Standards for Mathematics reads:

4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

## Web Resources/Further Exploration

This lesson plan (available here as a pdf download ) is only one in a series of engaging, fun math lesson plans coordinated to the Common Core and easy to use in the classroom. Browse through the other lessons at http://www.mathwarehouse.com/topic, and enjoy the wealth of other math resources available at Mathwarehouse.com

Other Resources

Fraction Visualizer

Simplify Fractions Calculator

Fraction Games

Simplify Fraction calculator

Fraction Calculator

# 4th Grade Lesson: Playing With Parts

## Discussion/Introduction

What do your students need to grasp to really understand fractions? Is there a short list of formulas they should memorize? Does it all come down to basic familiarity with algebra, and being able to simplify down? Not really.

If your students understand fractions visually—if they have a mental image that pops up in their head whenever they see a fraction on the page—they understand fractions. It’s that simple.

The focus of this lesson is a key formula your students need to understand in order to get further in fractions, but when translated into real life terms it is an extremely simple and obvious concept. If you divide a portion of a whole into a certain number (n) pieces, how many of them would you need in order to have the same amount as you did originally? N, of course!

That’s why this discovery-based lesson is focused on visual representations, and why you want to play with real-life fraction like oranges when introducing the concept. In this lesson you’ll also be having your students drawing, coloring, and superimposing so the idea can be really thoroughly cemented in their minds.

## Objective

That students would be able to use visual fraction models, like oranges or drawings on paper, to explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b), and that they would be able to use this idea to make their own lists of equivalent fractions or recognize equivalent fractions in a list.

Vocabulary: partition, equivalence, fractions,  reason, denominator, numerator.

## Supplies

• Index Cards
• Crayons or colored pencils that can be used for shading.
• Scissors
• Orange (unpeeled, or unpeel it during the beginning of the lesson)
• Worksheet (printable here)

## Methodology/Procedure

Start with vocabulary, so you and your students will all be speaking the same language. Write the ‘wordlist’ out on the board, ask the students what each word means, and then summarize each definition after the word.

You’ll want to use these words throughout the lesson, but don’t assume your students have them memorized unless you have a good reason to believe so. The focus of this class isn’t words, and though precise mathematical speech will only be helpful in the future, there’s no reason to confuse students over terminology.

Now pick up your unpeeled orange. Tell your students you are going to partition this orange. What does that mean again? Yes, you’re going to divide it in parts!

Split the orange in half. How many parts did you partition it into? 2!

Ask a student to come and write a fraction describing one of those pieces on the board. He should have no trouble coming up with ½.

Ask him to divide it into half, and then write the fraction denoting one of the resulting pieces on the board. Then, ask him to denote a fraction denoting both of the resulting pieces. Walk him through any part of this he has trouble with. Don’t draw attention to the sameness of the fractions just yet; but if anyone discovers it on their own, acknowledge them and allow them to share their discovery with the class. Ask your demo student to sit down again.

Give each student four index cards. Ask him to divide the first one in half, and shade half. When this is done, go to the next. Ask him to draw the same line and partition it into half and then to make a crosswise line as well to partition once more. How big is each part? One fourth. Walk the students through partitioning the remaining cards into sixths and eighths.

When all the index cards are divided, ask your students to shade half of the first index card. Ask them to shade 2/4th of the second, 3/6th of the third, and 4/8th of the fourth. Write each number down on the board as you request it.

Then ask them to look at all the index cards and tell you if they can see any interesting patterns on them. It won’t be long before someone makes an important observation: they’re all the same!

Are the pieces the same size? No. Are the same number of pieces shaded? No. But the total amount shaded is the same on each card.

Now you want to draw everyone’s attention to the numbers on the board. Are these numbers the same? They don’t look the same. What is the same about them? Give them some time to discuss the problem with their neighbors and consider solutions.

If no-one can think of anything, suggest they dig down deeper by factoring each of the digits. Go through and write all four numbers on the board, factored, with help from the students. Then step back, look at it again, and ask them

“Does this help?”

It does, of course. Let them have as long as they want to discuss and think about it, and it won’t be long before someone discovers that they’ve all got ½ hiding inside of them—masked by some other numbers that – surprise— are equal on top or bottom. This is a very important discovery and should be treated as such. Allow the discoverers to describe the phenomena in their own words, and let them go as far as they feel comfortable with it. When they’ve done, give them this way to condense their findings:

a/b = (n x a) /(n x b) Look, this is what we’ve just discovered. A fraction is always equal to itself multiplied by the same number on both top and bottom.

Now ask your students if they can show you why this is true using their index cards and a pair of scissors. Make the first half cut on your index card, and observe this is ½. Ask how you can show multiplying the bottom by n (cutting each piece into n pieces). Ask how you can show multiplying on the top by n (taking n of the existing pieces).

If multiplying on the bottom is cutting into n pieces, and multiplying on the top is taking all n, it will be obvious why you get the same amount of material each way. If you have the same total amount—even if the pieces are different sizes, and even if there are a different number of pieces—you have ‘equivalent fractions’.

When they’ve demonstrated they understand the material, hand out the worksheets and give them some time to fill them out. Allow them to discuss the best way to summarize the principal in the first half of the worksheet, if desired.

## Common Core Standards

In Fourth Grade Number and Operations—Fractions item 1, the Common Core State Standards for Mathematics reads:

4.NF. 1 Extend understanding of fraction equivalence and ordering.

Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

## Web Resources/Further Exploration

This lesson plan (available as a pdf download here) is only one in a series of engaging, fun math lesson plans coordinated to the Common Core and easy to use in the classroom. Browse through the other lessons at http://www.mathwarehouse.com/topic, and enjoy the wealth of other math resources available at Mathwarehouse.com

Other Resources

Fraction Visualizer

Simplify Fractions Calculator

Fraction Games

Simplify Fraction calculator

Fraction Calculator

# 4th Grade Christmas Lesson: Measurement Conversion

## Discussion/Introduction

Time for a Christmas lesson! Tis the season to be jolly—and what is jollier than measuring fun with Santa’s Christmas Elves? Measurement is a huge topic in fourth grade. You’ll see your students move from the measuring and recording they did in previous years to something much more involved: manipulating data measurements and switching between units. Is that fun? Well, it can be! Taking your whole class on a journey to the North Pole workrooms is one magically fun way to give your students a first look at how data conversions work in the (almost) real world.

Because it’s important a fourth grade Christmas lesson be engaging. During the days before Christmas break your students’ interest will caught by what is happening outside the classroom, and it’ll be hard to get them to concentrate on dry facts and ordinary problem solving. That’s when a lesson like this one comes in particularly useful—a fun and engaging story lesson that will make even your most listless fourth grader sit up and pay attention.

Math isn’t optional in Santa’s workshops. Here the ability to switch between larger and smaller units describing the same material is essential to bring understanding between the elves who work in the inner courts, mixing and measuring, with the ‘outer regions’ elves who are responsible for bringing in raw material and calculating delivery details.

## Objective

Students will learn to convert from larger measurement data to data in smaller units (hours to minutes, pounds to ounces, ), and be able to make tables of measurement equivalents that give them data which can be read in either direction (large à small or small à large)

## Methodology/Procedure

Get your students’ attention by beginning with a story. Start like this:

Christmas is approaching, and the activity in Santa’s North Pole workrooms ramps up day by day.

Some elves are busy in the inner chambers of Santa’s workshop. They’re doing delicate work that musn’t be interrupted, and during the month of December they’ll work from morning to night. Hazelnuts and ginger nuts, brought in by the North Post doves, are all they get to eat while on the job; and the cold glacial water keeps them awake and alert.

Another group of elves has a very different job: scouring the earth for the raw material that Christmas treats are made of. These elves are nimble on their feet, and they’re able to climb down mountains, shinny down caves, or scrumble through jungles looking for whatever might be needed out in Santa’s workshop. Some even go on galactic missions, with a little help from Santa’s flying reindeer.

There’s only one problem, and that has to do with the language the different groups of elves use. The outer elves measure their raw material with large measurements—kilograms, pounds, meters and feet. If they talk about time, they talk about hours. The inner elves are much more precise. Instead of kilograms, they do their work with grams. They use ounces instead of pounds; and measure with centimeters or inches instead of meters and feet. When they talk time, it’s minutes or seconds.

That’s why there are conversion tables posted on the walls in the areas of Santa’s workshop where inner and outer elves meet. These are lists which interpret one measurement in terms of the other. Our job today is to help out the elves by making some of these conversion tables.

The first elf we’re going to help out is a clever little fellow called Crusel. He’s in charge of a candy making room, and he spends his days making and tasting extra-wonderful candy for children that are marked down in Santa’s book as extra-special. Just now he is focusing on candy-canes. Pure white sugar, bright crimson coloring, and delicious peppermint oils are all brought to him by Ranger Elves; and since they measure in pounds and he only knows ounces, he needs a conversion table posted by his door to make ordering his ingredients straightforward and easy.

Does anyone remember how many ounces went in one pound? Yes, 16!

Prepare a two column table on the board; with ‘pounds’ and ‘ounces’ as the headings for the two columns. Write down 1 and 16 as the first row in the table. Then write 2 down underneath 1.

If Morwel, a ranger elf, brings in 2 lbs of crimson color, what will that be in ounces? Wait for an answer; if there is none; explain it:

1 lb was 16, so 2 lb will be 16+16, or twice 16.

Go on to three, four, five, and six, asking the class to provide the answers and the reasoning behind it, and filling in the table as you go. If you feel your students need the practice, you can continue the table up to ten.

Now, if he needs 64 ounces of crimson color, how many pounds must he request?

This should be easy for your students; if not, walk them through the reasoning behind the problem. Then go on:

Orlian, in a little underground chamber, uses fresh milk and butter along with flavorful cocoa powder and crystal sugar to produce delicious Christmas chocolate. He likes to do all his measurements in milliliters; using a set of glass measurement cups and beakers his elf grandmother left him, but the elves that bring him his ingredients don’t like to deal with anything smaller than liters. Can you make a conversion table for his door, so he’ll know how many liters to request when he’s got an order—in milliliters—ready in his head?

Pass out the worksheet, and guide your students into filling out the first two squares. Then give them a few minutes to fill out the rest of the table, walking around the room to help anyone that needs help.

When they’ve filled it out, ask—

• Looking at your table, can you tell me how Orlian will request 4000 milliliters of fresh yellow butter?
• How will Orlian request 5000 milliliters of good new milk?
• If the outside elves brought him 7 liters of crystal sugar, how many milliliters would he have ready to use in his recipes?

There’s one more workshop I’d like us to visit today. Here Kruskru the elf is hard at working making sleighs for little boys and girls to play in the snow with. His ruler is in centimeters and inches, but the elves who bring him his wood and steel only have rulers that measure in feet and meters. Can you fill out this next table to help him convert between feet and inches? Start with one foot—does everyone remember how many inches that is?

Very good, twelve! So one foot in the first box, and twelve inches in the next. Now go to the next line, write ‘2’ under feet, and write down the number of inches in two feet in the next box.

Give your students some time to complete their tables, and, if class size permits, glance over their work when they are finished.

When students have filled out their tables, ask them:

• If Kruskru wants a 24 inch piece of wood, how many feet will he ask for?
• If a woodcutter elf brings Kruskru a seven foot piece of wood, how many inches long will it be?

If you have extra time, stage some play acting. Let half of your students be workshop elves, and the other half rangers. Pair them up, and give them each dimensions to work with: gallons and quarts, ounces to pounds, seconds to minutes. Let the pairs work together to make conversion tables or let them work together in larger groups if there is less time. Then set a few problems to each pair/group.

## Common Core Standards

This lesson is aligned to standard 4.MD.1 in the Common Core State Standards for Mathematics; 4th Grade Measurement and Data Item 1. 4.MD1 reads:

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two- column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …

## Web Resources/Further Exploration

This is just one of many lesson plans available on Mathwarehouse.com, your new favorite website for lesson preparation. You’ll also find a host of other math teaching resources, like our Visual Fraction applet for those fraction lessons or the handy grapher for when you and your students dig into graphs of all times.