Counting by Twos Number Line & Worksheet Game

"Two, four, six, eight. Who do we appreciate?"

This rhyme is the just the beginning of learning to count by twos (and it may bring back childhood memories). To help your students visualize counting by twos, duplicate the number line below on the board. Have a student circle the number 2. Then have the student skip over 3 and circle the number 4. Do the same until all the even numbers are circled. Then let the students count the numbers by two as another student points to them: 2, 4, 6, 8, etc.

If you have time, enlarge the number line so it can span across the floor of your classroom or hallway (with the numbers spaced about 5 inches apart). Have a student step on the number 2. Then have the student skip over 3 and step on number 4. Do the same as the children count by twos. Have the student step on each number as the class names it. Continue until you reach 20. Then let the students begin at 0, counting the numbers by two as another student steps on them.

Give each student scissors and a copy of the Frogs and Lily Pads Worksheet below from Excel Math Grade 2:

Excel Math Grade 2 Student Worksheet Lesson 70 Manipulative 5
Click here to download a PDF file of this page for your students.

Let each student cut out the first frog on the page so he can jump the frog to the appropriate numbers as you name them. Have the student print his name at the top of the page and on the back of the cut out frog. Have each student place his frog at number 2 as you call out the numbers by twos (beginning with 2). Have the students move their frogs to the numbers as you call them. Stop with number 8. Then ask, "If the frog keeps  jumping in this pattern, which number will he jump on next?" (10)

Continue counting by twos until all the frogs are on number 20. Then have the students move their frogs to start on number 6. Count by twos until all the frogs are on number 14. Then ask, "Is the frog jumping in a pattern?" (yes) "What is the frog's jumping pattern?" (jumping by twos  or adding two for each jump or skipping every other number) "If this is the pattern that the frog has been jumping, what were the previous numbers on which he landed?" (0, 2, 4)

If you can take the students outside, use chalk to draw a number line (from 0 to 20) on the ground. Choose a student to step on a number. Have the other students count by twos until the first child reaches 20. Do this several times, starting on a new number each time and giving each student a turn.

Remind the students that counting by twos means skipping every other number. Have them fill in the missing numbers in each sequence. You may want to do the first one together as a class:

A.  2, 4, 6, ___ 10

B.  9, ___, 5, 3, 1

C.  12, 14, 16 ___

D.  ___, 13, 11, 9, 7

Have the students check their work on the number line. Now let them try filling in the numbers  in each sequence when more than one number is missing:

E.  7, 9, 11, ___, ___

F.  10, 12, 14, ___, ___

G.  20, 18, 16, ___, ___

H.  19, 17, 15, ___, ___

Once again, have the students check their work using the number line. Here are the answers:
A-8; B-7; C-18; D-15; E-13, 15; F-16, 18; G-14, 12; H-13, 11

Download a sample lesson and find out more about Excel Math on our website, excelmath.com. Do you have a favorite teaching tool or lesson you use to help your students count by twos? Tell us about it in the comments section. Give us your email so we can contact you with our thanks.

The Star-Spangled Banner—Long May She Wave

Photo of Ft. McHenry Courtesy of National Park Service
Click here for more photos

During the War of 1812, the British tried to capture the port of Baltimore and Fort McHenry. (Fort McHenry was built prior to the War of 1812. The fort is named for James McHenry, Secretary of War, 1796-1800.)

Francis Scott Key, an American lawyer, watched the battle from a British ship in the harbor. Mr. Key had gone there to ask for the release of a friend who had been taken prisoner.

The battle was fierce and lasted all night. The sounds of fighting suddenly stopped just before morning. Key thought the battle had been lost. However, when the sun came up on September 14, 1814 soldiers at the fort raised a huge American flag to celebrate victory of British forces. Key saw the huge American flag waving high above the fort. He knew then that the American troops had successfully held off the British.

Key was so inspired by those "broad stripes and bright stars," he wrote a poem. He used these new words with a melody from a familiar drinking song to set the poem to music. The poem became the words to the national anthem of the United States, "The Star-Spangled Banner." Most of us know the first verse but might be hard-pressed to remember the others:

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!

And where is that band who so vauntingly swore,
That the havoc of war and the battle’s confusion
A home and a Country should leave us no more?
Their blood has wash’d out their foul footstep’s pollution.
No refuge could save the hireling and slave
From the terror of flight or the gloom of the grave,
And the star-spangled banner in triumph doth wave
O’er the land of the free and the home of the brave. 

O thus be it ever when freemen shall stand
Between their lov’d home and the war’s desolation!
Blest with vict’ry and peace may the heav’n rescued land
Praise the power that hath made and preserv’d us a nation!
Then conquer we must, when our cause it is just,
And this be our motto - “In God is our trust,”
And the star-spangled banner in triumph shall wave
O’er the land of the free and the home of the brave.

Congress proclaimed this song the U.S. National Anthem in 1931. See Key's original manuscript at the Smithsonian website. These words gave new significance to our country's flag. Since then, generations of Americans have revered the flag, assigning to it their own emotions and memories.

On this Memorial Day, we'll see lots of flags flying around our neighborhoods to honor the men and women who have died while serving in the military. Originally known as Decoration Day, Memorial Day began in the years following the Civil Way and became an official federal holiday in 1971, observed on the last Monday of May.

Over the years, our family (just like many others) has celebrated by visiting a local cemetery, holding family barbecues, swimming, and/or attending parades. Unofficially, it marks the beginning of summer. However you celebrate, enjoy your Memorial Day!

Math Multiples & Division Worksheet

A multiple is the result of multiplying two numbers. Some multiples of 2 are:

2, 4, 6, 8, 10 and so on

Some multiples of 5 are 5, 10, 15, 20, 25, and so on.

In Excel Math, we teach students how to multiply and divide, how to find the greatest common factor between two or more numbers, how to identify fact families, and how to recognize multiples.


Help your students identify multiples by writing the first three multiples of 2 on the board. Let your students list two additional multiples of 2. Do the same with multiples of 5. If you have time, continue by listing multiples of 3 and 4. Then have your students complete problems 1-4 on this Grade 3 Excel Math worksheet. Do the first one together:

Excel Math Grade 3 Student worksheet
Click here to download a PDF file of this worksheet

Before they begin number 2, remind the students that 3 will divide evenly into any number whose digits add up to a multiple of 3. So 3 will divide evenly into 189 (the digits of 189 add up to 18, which is evenly divisible by 3). Three will not divide evenly into 17 (the digits of 17 add up to 8, which is not evenly divisible by 3). Let the students finish #2-4 on their own. The answers are shown below.

Arrange students in pairs. Give each student 33 items that can be used for counting (paper clips, beans, pennies, etc.) Point out that factors are all the integers that divide evenly into another integer (whole number). They are "what you get" when you divide. The number 1 always works as a factor. 


Read the instructions for number 5. Ask the students to tell you which numbers are the factors in the problem. (11 and 6) Have the students use their counters to form 6 groups with 11 in each  group. Let them count the total items and print the answer under the line. (66) 

Next, have them take the 66 items and see how many times they can subtract six. Each time they subtract six, ask them what the result is. (The result will be multiples of 6.)


Ask the students how many times they were able to subtract 6. To reinforce the connection between division and subtraction, write on the board: There are 11 groups of 6 in 66.

Explain that another multiplication fact in this fact family would be to simply rearrange the order of the factors: 11 x 6. Have the students write 6 x 11 above the empty line. Have them form 11 groups of 6 with their items. Let them count the total items and print the answer under the line. (66) 

Now have the students take the 66 items and see how many times they can subtract eleven. Each time they subtract 11, ask them what the result is. (The result will be multiples of 11.)

Repeat this process for numbers 6-8. Point out that multiplication and division problems form fact families just as addition and subtraction do. Here are the answers:

Excel Math Grade 3 Lesson 117 Student Worksheet with Answers
Click here to download a PDF file of this worksheet

Since Excel Math uses a unique spiraling strategy, you will need to teach multiple lessons sequentially within each grade level in order to get the concepts into your students' long-term memory. A student's learning of new concepts takes place during Excel Math Lesson Plans and Activities. The concepts are refreshed through Guided Practice, Homework, Create A Problem and Tests.

Visit Excel Math online to find out more about our lesson plans and student worksheets. Excel Math is available for Kindergarten through Grade 6. 

Divide Evenly: Tricks of the Trade

In math, divide evenly means when you divide two numbers, there is no remainder or there is a remainder of zero.

That leads us to the term Factors. Factors are all the integers that divide evenly into another integer (whole number). They are "what you get" when you divide. The number 1 always works as a factor.


In Excel Math, we teach students how to multiply and divide, how to find the greatest common factor between two or more numbers, and how to recognize multiples. For students who need some extra confidence building in division, here are a few ways to help them determine if a number can be divided evenly by another number.


As we said, the number 1 always works as a factor. It will divide evenly into any number.


2 will divide evenly into all even numbers. It will not divide evenly into odd numbers. Two will divide evenly into 2456, which is an even number:

2456 ÷ 2 = 1228

3 will divide evenly into all numbers whose digits add up to a number that's divisible by 3. Three will divide evenly into 2436 since its digits add up to 15 and 15 is divisible by 3:

2436 ÷ 3 = 812

5 will divide evenly into all numbers ending in 0 or 5. Five will divide evenly into 820 since it ends in 0:

820 ÷ 5 = 144

Five will also divide evenly into 375 since it ends in 5:

375 ÷ 5 = 75

9 will divide evenly into all numbers whose digits add up to a number that's divisible by 9. Nine will divide evenly into 2736 since its digits add up to 18 and 18 is divisible by 9:

2736 ÷ 9 = 304

10 will divide evenly into all numbers ending in 0. Ten will divide evenly into 820 since it ends in 0:

820 ÷ 10 = 82

Give each student 20 items that can be used for counting (buttons, tokens, paper clips, etc.). Let the students work in pairs. Write this problem on the board:

20 ÷ 10 = 

Ask your students if 10 will divide evenly into 20. (yes) Ask how they can tell. (20 ends in 0 and 10 will divide evenly into numbers ending in 0)


Have the students divide their 20 counters into ten groups. Ask them how many counters are in each group? (2) 
Let a child write the answer on the board:

20 ÷ 10 = 2

Now have the students divide their 20 counters into two groups. Ask them how many counters are in each group? (10)

Write this problem on the board:

20 ÷ 2 = 

Ask your students if 2 will divide evenly into 20. (yes) Ask how they can tell. (20 ends in 0, which is an even number, and 2 will divide evenly into even numbers)


Let a child write the answer on the board:

20 ÷ 2 = 10

Continue in this way with multiples of 3, 5, 9, etc.

What math "tricks of the trade" do you use to help you remember division facts? Leave a comment with your tip or suggestion.

Oldest Known Mayan Calendar Unearthed

A number of years ago, looters digging at the site of Xultún, once a sprawling Maya city-state in northern Guatemala, exposed part of a painted wall. In March 2010 a team from Boston University discovered a portion of the wall. Little did they know that the wall held never-before-seen paintings and an ancient Maya calendar.

Last week, in a paper published online in Science, William Saturno, an archaeologist at Boston University, and Anthony Aveni of Colgate University in Hamilton, NY, and their colleagues reported uncovering the oldest-known astronomical tables of the Maya. These tables are 500 years older than the astronomical tables previously discovered preserved in fragile bark-paper Maya books, many of which were burned in 1561 and later years by missionaries. Today, only a handful of these readable precolonial books survive.

The astronomical tables discovered in Xultún were incised and painted on the walls of a small room in a 1,200-year-old building. The room may have been a working space for scribes and was built with a stone roof. Human figures, including an elaborately dressed Maya king, also decorate the walls. You can tour the room at National Geographic Daily News.

One wall of this residential room contains a calendar based on phases of the moon, covering about 13 years. Anthony Aveni, an expert on Mayan astronomy, said the calendar would allow scribes to predict the appearance of a full moon years in advance. Record-keeping was a key to Mayan astrology and rituals, and might be used to advise the king on when to go to war or how good the year's crops would be, he said.

For years, some people had speculated that the Maya people thought the world would end in 2012 since the oldest Maya calendar discovered had ended at 2012. Others pointed out that our calendars also show a limited period of time, not because we think the world won't last beyond then but just for the efficiency of producing the calendar. With the discovery of this new calendar, those speculations are largely put to rest. The markings on this wall suggest dates thousands of years in the future.

"Why would they go into those numbers if the world is going to come to an end this year?" observed Aveni. "You could say a number that big at least suggests that time marches on."

"I think we are all astonished by this find," says Stephen Houston, an archaeologist at Brown University (not part of the team). "What you are looking at here is a fundamental human drive to make sense of the disparate patterns of nature, and somehow harmonize them into an elegant mathematical structure."

In Excel Math, we teach students math concepts they will use in everyday life. Students learn how to read a calendar, the days of week, the months of year, and how to calculate dates in the future. They also learn how to find a date in the past. In second grade, students learn how to calculate a date within a week. Take a look at this Excel Math student worksheet from Grade 2:

Excel Math Grade 2 Lesson 84 Student Worksheet

To help your students find the date within a week, going backward, have them write the first letter of the day you give them. (S=Sunday) Then write the days in reverse order, right to left, ending with the day they are trying to find:

T  F  S  S

 Next, put the given date under the given day and count backward. If it is exactly one week, subtract 7 from the given date since there are 7 days in one week.

Day you are trying to find = T  F  S  S = Given day is Sunday. 

Date you are looking for = 11 12 13 14 = Given date is 14.

For example:
If today is Sunday, May 14. Last Thursday was May _____


Day you are trying to find = T  F  S  S = Given day is Sunday.
Date you are looking for = 11 12 13 14 = Given date is 14.
So last Thursday was May 11.


Try another one:
If today is Friday, July 19. Last Saturday was July _____


Day you are trying to find = S   S   M  T  W Th  F = Given day is Saturday.
Date you are looking for =  13 14 15 16 17 18 19 = Given date is 13.
So last Saturday was July 13.


Let your students try circled #2 from the worksheet above in pairs or as a class, just to make sure everyone understands how to solve the problems.


Then let them solve the two problems in #1 on their own and use the CheckAnswer to make sure their work is correct. If the CheckAnswer is wrong, they can go back and try the problems again to fix their calculations.


This unique self-assessment enables students to confirm their work. The CheckAnswer process helps students develop confidence and good work strategies. Students solve a set of 3-5 problems (A-C), add the answers, and see if the sum matches the CheckAnswer (D). If it matches, they move on. If not, they go back and re-check their calculations:

Read more about the Excel Math Student Lesson Sheets at excelmath.com. Watch a video of Excel Math in action at excelmath.com. 

National Fitness Month: Get Moving with Math

Since 1983, May has been observed as National Physical Fitness and Sports Month. You can teach your students the value of physical activity and help them understand how exercise can improve their mental and physical health. This is just one way to encourage students to lead happier, healthier and more productive lives.

Download a free guide to getting fit for students ages 6-17: Get Fit and Be Active! It suggests various exercises for kids along with some fun ways to get in shape for the President's Youth Physical Fitness Awards program offered by many schools. Visit www.fitness.gov/council_pubs.htm for this guide and lots of others on healthy eating and exercise for people of all ages.

In Excel Math, we help students develop strong minds as they learn to think independently, build confidence, combine math with literacy, and increase their test scores. We also believe in helping students become well-rounded and healthy members of society. Many of our math story problems include examples of exercise and healthy eating. Teachers even call to thank us for removing cookies and candy from our math word problems. (Even so, I'm still partial to a good cookie every now and then.)

May 18 is "Bike to Work" day. In many parts of the country, May is a beautiful time of year to exercise outdoors. Here in San Diego, the trees are budding and flowers are blooming. The water's warming up enough for a swim. It's gorgeous weather for riding a bike. Let's take a look at one of our Fourth Grade Excel Math Create A Problem worksheets about biking and calculating distance:

Excel Math Grade 4 Student  Worksheet Create A Problem 18
Click here to download a PDF file of this worksheet

In this activity, fourth grade students fill in the graph and label it using the information given in the paragraphs. They then write their own word problem to go with the information shown on the graph. Create A Problem exercises let students express their own understanding of story problems, merging math with literacy. We start with simple stories and give students a chance to observe what is happening in the story. They then use those observations and the accompanying charts and graphs to solve problems.

The stories are designed so your students can observe, analyze and participate in the stories. Several consecutive stories may be related, so they might occasionally need to think back to what they did a week ago.

Here's one possible completed chart and word problem for the Create A Problem worksheet shown above:

Excel Math Grade 4 Student  Worksheet Answer Possibility for Create A Problem 18

Later in the  curriculum we ask students to create a problem or two, and make up a CheckAnswer. Read more about CheckAnswers here. Finally, students are able to finish a story in their own words and write several problems about their story ending. This process demonstrates mastery AND integration. The format of the page allows students to give longer answers along with charts, graphs and other expressions of their solutions. Find out more about Excel Math on our website.

Need a sample packet to decide if Excel Math is right for you and your students? Download one here or leave us a comment with your email address (or your physical address) and we'll send it to you. We won't post your comment so you can contact us without making your contact information public. Take at look at sample lessons for kindergarten through grade 6 here. Try out our math lessons featuring the unique Excel Math spiraling system (read our April 11 blog post to learn about spiraling and spaced repetition) and see how Excel Math can work with your students!

Research Agrees: Spaced Repetition Really Works!

It’s time for finals, MAP testing, SATs, ACTs, and exams of all kinds. According to Dr. Nate Kornell, Assistant Professor of Psychology at Williams College, writing in Psychology Today, “Cumulative exams take advantage of the spacing effect: if you have already studied something, studying it again after a delay can produce a huge amount of learning.” 

The spacing effect is just one component of the spiraling process so carefully built into every Excel Math lesson. Dr. Kornell continues, “If you want to learn something for life, spaced learning is absolutely critical. Research shows that the longer you want to remember something, the more you should space your learning.” Read the full article at http://blogs.edweek.org.

For this reason, Excel Math lessons gradually introduce new concepts, review them with Guided Practice, and refresh the concept over the next few weeks through Homework and Tests. 

This chart shows how concepts in Excel Math are introduced, reinforced, assessed, and reviewed. This unique spiraling system helps produce confident students, and as a result, test scores soar!

Excel Math's unique Spiraling Strategy
Click for a more detailed view

Here’s a chart showing how one concept progresses through the school year: “Selecting the correct symbol for an equation.”

Excel Math Spiraling of One Concept over 75 Lessons
Click for an expanded PDF download

Other topics taught during this sequence of lessons are not shown on the chart so you can easily see how this one concept is taught through spaced repetition. This concept, “Selecting the correct symbol for an equation,” appears 15 times during the 75 lessons (half year) shown.

In Excel Math, teachers gradually introduce concepts to students, use several models to help students explore a subject, then allow the students multiple chances to demonstrate mastery. One of those demonstrations involves testing.


The following graph illustrates Los Angeles Unified School District students from Parmelee Elementary. This school is located in one of the poorest neighborhoods in Los Angeles with a large minority population. In a multi year study students made dramatic gains in their Percentile ranking in year five, which was the first year implementation of Excel Math as a supplement to a district mandated core curriculum. This was the only change made in math instruction that year. Second grade classes experienced nearly 100% gains in math scores after one year of using Excel Math. 

Test Scores at Parmelee Elementary after one year using Excel Math
Click here to download a PDF file

According to Dr. Kornell, tests are not a bad thing. In fact, they can actually help us get facts into long-term memory and keep them there!

Who's in Your Family? Math Fact Families

These are family photos. The people in a family may be related by bloodline or adoption or just by living under one roof and sharing common interests.

Kindlespire Reunion

Family Travels

There are several definitions of the word family:

noun ( pl. families )
1 a group consisting of parents and children living together in a household.
• a group of people related to one another by blood or marriage: friends and family can provide support.
• the children of a person or couple: she has the sole responsibility for a large family.
• a person or people related to one and so to be treated with a special loyalty or intimacy: I could not turn him away, for he was family.
• a group of people united in criminal activity.
• a group of objects united by a significant shared characteristic.

The final definition applies to more than just people who are related. It applies to objects, numbers, shapes, and much more. Here is a photos of the Excel Math family of employees (taken a few years ago):

There are families in mathematics, too. We call these fact families.

In Excel Math we teach students to recognize addition and subtraction fact families. Help your students learn to recognize fact families. Begin by listing the following number sentences on the board. These are all part of a "fact family" (but don't tell your students yet):

2 + 3 = 5

3 + 2 = 5

5 – 2 = 3

5 – 3 = 2

Ask your students if they can see how all of these number sentences are related. (They are all considered part of the same "fact family.") By recognizing the relationships in the addition and subtraction fact families, a student can know four different basic facts by memorizing just one fact.

Have your class look at this Student Lesson Sheet from Excel Math Grade 1 Lesson 71:

Excel Math Grade 1 Lesson 71 Student Worksheet
Click here for a PDF file you can download.

You can print this worksheet and use it with your class. Explain that if they know 2 + 1 = 3 they also know:

1 + 2 = 3

3 – 2 = 1

3 – 1 = 2

These number sentences are all part of the same fact family. Poing out that a fact family is a set of related numbers. Talk through the fact families on this worksheet where the answers are given. Have a student describe how the problems in the first box are related. (They include problems using the numbers 1, 2 and 3.) Do the same with the next box. Let your students fill in the missing problems in the next two boxes. Have a student describe each of these fact families.

Give the children a chance to complete the Basic Fact Practice. Go over any of the problems your students have difficulty solving.

If you have time (or for students who finish early), give each student a blank piece of paper and let him continue to create fact families (one or two additional equations) using the problems given for Basic Fact Practice.

If you have children who would like to share one interesting thing about their own families, let them do so. Begin by sharing about your own family.

Since Excel Math uses a unique spiraling strategy, you will need to teach multiple lessons sequentially within each grade level in order to get the concepts into your students' long-term memory. A student's learning of new concepts takes place during Excel Math Lesson Plans and Activities. The concepts are refreshed through Guided Practice, Homework, Create A Problem and Tests.

Visit the Excel Math web store to order these Student Lesson Sheets (155 lesson sheets plus tests) and a Teacher Edition set of 155 Lessons with brainteasers, teaching suggestions, the answer key, and reproducible manipulatives for each grade  level. Excel Math is available for Kindergarten through Grade 6. 

Using Number Lines to Add and Subtract

This is a number line:

It shows the number 243,616. We can show the value of numbers and even compare numbers using a number line. Did you know you can also show addition on a number line? Here's an example:

You can even show subtraction on a number line. Here's an excerpt from Excel Math First Grade Lesson 79:

Excel Math First Grade Student Sheet Lesson 79

We can also help students learn to write an addition number sentence when we give them the number line:

Show your students these examples. Answer any questions they might have. Next, have them try writing a number sentence to match this number line:

Finally, let them answer the problem. Have a child point to the answer on the number line while you call on a child to answer aloud:

Here are some more examples you can use with your class to help them become familiar with using number lines. This math worksheet is from Excel Math First Grade Lesson 92:

Excel Math First Grade Lesson 92 Student Worksheet
Click here to download a PDF file of this worksheet

Since Excel Math uses a unique spiraling strategy, you will need to teach multiple lessons sequentially within each grade level in order to get the concepts into your students' long-term memory. 

Visit our web store to order these Student Lesson Sheets (155 lesson sheets plus tests) and a Teacher Edition set of 155 Lessons with brainteasers, teaching suggestions, the answer key, and reproducible manipulatives for each grade  level. Excel Math is available for Kindergarten through Grade 6. Here's what one teacher had to say about how well it works in the classroom:

“I cannot express how impressed I am with your program. Our STAR results are outstanding and I am convinced without EXCEL we would be struggling to meet our goals. The spiraling piece that is built in…is what makes this so effective. If I ever move schools and my district does not provide this program, I would purchase it with my own money. Thank you for a wonderful program.” — Elementary Teacher, San Juan, California

For more glowing reports about Excel Math from the teachers who use it and to see some of the impressive test results, visit www.excelmath.com.

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Number Line Addition in the Math Classroom

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