Speaking the Language of Math

It's time for my students to learn the language of math, beginning with vocabulary relating to four basic operations: addition, subtraction, multiplication, and division. After all, it's hard to complete a problem if you don't know what the words mean. For example,

• Name the factors of 24.
• What is the sum of 6 and 4?
• Find the difference between 16 and 6?

Each student will receive this sheet, and I'll test them later this week. This way, I can be sure that they all know the language of math!

Many more math vocabulary lessons and assessments will pop up this year, too: measurement terms, properties, shapes, etc. They need to be able to walk the walk AND talk the talk.

Thematic Teaching with Trees 2

Our tree day was a smashing success!

Students spent more than two hours planning, drafting, editing, and illustrating their personifications of trees. One of the most difficult parts of the assignment was giving a tree human qualities but not making it a human in a tree's body. After extensive planning, some children reverted to writing about a tree as just a tree while others simply made their trees talk. Fortunately, we could look back at the planning sheets and work together to make their choices in figurative language more effective.

Mid-day we explored factor trees and completed a close read about how leaves change in fall. Then after lunch, students cut and mashed leaves (red and green) for our chromotography experiment. I added a bit of isopropyl alcohol, and we let them set for one hour. Next they cut coffee filter strips and awaited the results. This step took longer than anticipated. I thought we'd be able to see the lines clearly after about fifteen minutes. In reality, it took about an hour, but the results were spectacular! Students could really see that chlorophyll leaves the leaf after food production is finished for the summer.

Since my class is learning how to conduct valid scientific investigations, I made sure to control as many variables as possible for this experiment. All leaves came from the same burning bush. We used the same type of cups, and the isopropyl alcohol was the same temperature for both cups. 

Have you tried this experiment with success in your classroom? Please share by commenting! I'd love to know how you did it.

P.S. We decided to leave the filter paper in the liquid over the weekend and see what happened. Guess what?

Wow! With science, sometimes patience really pays off!

Thematic Teaching with Trees

In preparation for the fall ball, the maple shed her green summer frock and donned a magnificent magenta cloak.

Tomorrow, my fourth grade class will have some fun with trees.

Activity #1 - Writing: Personification of a Tree
     Each student will use this organizer to plan three to five sentences that personify a specific tree. He/she will sketch the tree's trunk and branches then print the sentences across the page. Finally, after outlining everything in black, the student will color the trunk and sponge paint leaves onto the picture.

Activity #2 - Reading

     Students in my class will read Why Leaves Change Color (North Carolina Testing Program). This brief one-page piece fits well with the longer personification piece. Other options include Why Leaves Change Color, a one-page text written by Stephen Caney, and Autumn Leaves and Fall Color (posted by Science Made Simple).

Activity #3 - Math

     Factor Trees (MathAids.Com) provide an essential link in learning about multiplication and division.

Activity #4 - Science

     Each group will cut and mash a particular leaf, pour isopropyl alcohol over it, wait one hour, and extrapolate colors using coffee filters. You can see examples of how this works in two videos: Time for Me to Leaf (8:45) and Leaf Color Chromatography (4:24).

Activity #5 - Social Studies

     I will supply each group with projected dates for peak fall colors, as shown below. They will cut out the states and dates, arrange them in similar piles, then create a map (with  a key) to show when peak colors will appear in each state. You can download a free map from Free US and World Maps.com by clicking here.

Want to infuse a little excitement into learning this fall? Try thematic teaching with trees!

Please Excuse My Dear Aunt Sally

Are you looking for some clever ways to teach order of operations? Look no farther!

Once your students get a handle on this mnemonic device, they're ready to rock and roll. Kick off your lesson with a teacher-created song, Order of Operations (1:49), by Mrs. Wagstaff. Then choose expressions with just the right degree of difficulty at Order of Operations: PEMDAS Worksheets from Math Worksheets 4 Kids.

After they've learned the order of operations, you can challenge your students any day (or every day) to form as many equations as possible using the date. For example:

C'mon! It's time to play with some numbers!

Autumnal Equinox

Fall arrives in the northern hemisphere today at 20:44 UTC. I'll say, "Goodbye, summer," at 3:44 this afternoon.

How can I explain this to my students? On September 8th we talked about latitude, longitude, and the tilt of the earth. Now it's time to go a little deeper.

In preparation, I read Equal Day and Equal Night - But Not Quite by Time and Date. This site provided solid background information (to prepare me for questions my students might ask).

Next, I located a five-minute video that clearly describes how the tilt of the earth causes seasons. Cycles of Light and Temperature by Power of 10 Texas will (powerfully) explain this phenomenon to my students. Awesome!

Finally, I want my students to explore with one or more experiments. This lesson, posted by Nauticus, provides a series of experiments that demonstrate the seasons, the tilt of the earth, and day and night.
Tomorrow looks like a sunny day here in northwest Indiana, so I plan on heading outside with thermometers taped to black construction paper. Each group will prop one paper/thermometer set-up at an angle while laying another flat on the ground. Then they'll record the temperature at five, ten, and fifteen minutes. What do you predict will happen?

The autumnal equinox gives us a great reason to explore the seasons, and kids love it!

Writing Fables, Part 3 - Planning for Strong Writing

Good ideas don't always translate into good stories. When writing narratives, I remind my students to do their "BEST WORC."

How does this play out for our first brief narrative of the year, a fable? 

• A fable's beginning is brief and to the point. It sets the stage for the plot by introducing the characters and setting.
• The end of a fable wraps up the plot and exposes the moral.
• "Show, Not Tell" is a complex strategy. For our first attempt, I simply ask students to use dialogue.
• Word choice for this assignment focuses on specific nouns (bulldog instead of dog), active verbs (sauntered instead of walked), and sequence words (at the end of the day, finally, etc.)

In my experience, slowing down for this little bit of extra planning makes a big difference in student writing. This short narrative is the first step in building strong writers.

Common Core State Standards W.3.3, W.3.4, W.3.5, W.4.3, W.4.4, W.4.5, W.5.3, W.5.4, W.5.5

Writing Fables, Part 2 - Planning Story Elements

After reading several fables, my class came up with this definition of a fable: a short narrative with a simple setting, simple characters (who are often animals that act like people), and a moral, or lesson.

Since my students would be required to use animals as characters in their fables, we warmed up with this activity:

The first step in actually planning the fable was to select a moral. We discussed dozens of proverbs, which are age-old lessons, like these:

Students then selected a moral, developed some ideas for the plot, and chose animals that possessed traits necessary to teach the lesson.

Each of my young writers now had a plan and were ready to begin developing their fables.

You are welcome to use these ideas in your classroom. I'd love to hear how they work for you! If you'd like the packaged version, Writing Fables is available in my Teachers pay Teachers store.

Common Core State Standards Addressed: W.3.5, W.4.5, W.5.5

Writing Fables, Part 1 - Dialogue

Short and sweet: what a great way to kick off narrative writing! Since fables are short stories with simple characters, settings, and plots, they work well at the beginning of the year. I'm able to introduce my expectations without overloading my students.

The first expectation is using dialogue effectively. Before we begin planning our fables, my students need to know how to write sentences containing dialogue. After exploring the difference between indirect and direct quotes, I explain that dialogue tags tell who's speaking (noun or pronoun) and the action (said, exclaimed, etc.) Now they're ready for two important rules:

1. The sentence inside the quotation marks retains its original capitalization and punctuation (with few exceptions).
2. The dialogue tag interrupts the normal flow of the sentence; therefore, it is set off by commas.

A PowerPoint presentation explains how to place the dialogue tag at the beginning, middle, or end of a sentence and provides practice.

Now students are ready to practice on their own. This year, my students used two pages in their English workbook for practice. The first page simply asked them to place quotation marks where needed, and the second page asked them to write and punctuate direct quotes. Today we'll practice a bit more with this practice sheet:

For more (fun) practice, try comic strips. Just find a good strip online, print it, and paste it on top of a lined sheet of paper. Ask students to write the dialogue in narrative form. In my experience, it's best to start with strips that go back and forth between two characters then move to strips in which one-word quotes are used or one character continues speaking in a new frame. This will help them build their skills at creating a new paragraph whenever a new character speaks, as well as combining one speaker's quotes into one longer quote.

Writing dialogue can be easy and fun. Let's get started!

Common Core State Standards: W.3.3b, W.4.3b, L.3.2c, L4.2b, W.5.3b

Conversations on Connotation

Conversations on connotation have begun in my fourth grade class. I'm making a concerted effort to discuss vocabulary in a variety of ways every day.

The main objective of yesterday's reading lesson was to identify elements of fables, but we diverted our attention to vocabulary for a few short minutes.

Part 1: Specific Verbs

Teacher: What is the meaning of perched in this text?

Unsuspecting Student: To sit on a branch.

Teacher: Like a pig?

Students: Haha! What?

Teacher: Does the word perched mean to sit on a branch like a pig would sit on a branch?

Student: Haha. No. It means like a bird sits on a branch.

Teacher: So what exactly does perch mean?

Conversation continued until the class established that perched means "stood on a branch with feet wrapped around it" in this passage, as well as the fact that it's used only for some types of birds (not ducks or penguins, for example). We discussed the importance of using specific verbs in their own writing.

Part 2: More Specific Verbs

We focused on words that mean "walk," such as "roam" and "saunter" in this text. Although we didn't stop for long on this, one day in the future we will actually practice walking in different ways, such as strutting, gliding, stomping, marching, and many other forms of walking. Again, this helps kids see the importance of choosing specific verbs.

Part 3: Connotation

Denotation is definition; connotation is how a word feels. To truly understand a word, you must know what it means, as well as how positive or negative it feels. We looked at the remaining highlighted words on the page (beautiful, attractive, comely, magnificent, and lovely) and determined that they all meant "pretty." I drew a continuum on the board and asked them to help me arrange the words by their degree of "prettiness." It turned out like this:

Taking ten minutes to discuss vocabulary in any subject area can really pay off. I'm trying to squeeze it in wherever I can this year.

4.NBT.B.5 Multiplying Whole Numbers

The Common Core State Standards give us explicit direction in teaching multiplication of multi-digit numbers. 4.NBT.B.5 states: "Multiply a whole digit number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models."

Today I'd like to discuss one path that takes students through "strategies based on place value and the properties of operations" in the quest to multiply multi-digit numbers by one-digit numbers. This is how it's playing out in my fourth grade classroom. (We are focusing our learning on equations instead of arrays or area models.)

Step One
While my class studied numeration, addition, and subtraction, students were also working to master multiplication facts from zero to nine. (This step is not conceptual but necessary for success. I trust that my students' third grade teachers have handled the conceptual side of single-digit multiplication.) As we move into our unit on multiplying multi-digit numbers, I am proud to report that 26 of my 27 students can fluently recall their facts. How can this be achieved? For some ideas, read my blog from August 31st.

Step Two
We learned the commutative (order), associative (grouping), identity (one), and zero properties of multiplication, used the properties to determine unknowns in equations, and identified properties used. This worksheet from Scott Foresman is a simplified version of what I used in my classroom. In searching the Internet, I also found this worksheet from Common Core Sheets.

Step Three
Students practiced multiplying single-digit numbers by multiples of 10 and 100. First we built it out, like this, 7 x 3 = 21, 7 x 30 = 210, and 7 x 300 = 2100. In order to maximize conceptualization, we also decomposed the numbers so students could reason why this is so: 7 x 30 = 7 x (3 x 10) and the associative property allows us to change the grouping to (7 x 3) x 10. Next students practiced using the skill with mixed problems, such as 6 x 800 = n and 40 x 3 = n. You can find some free practice worksheets at Math Worksheets Land and Common Core Sheets.

Step Four
Now it's time to introduce the distributive property and employ knowledge of expanded form. Each year I worry that my students will get stumped at this stage of the game, but they always pull it off! This year was no different. Let's say, for example, we have the problem 6 x 35. We first decompose 35 to 30 + 5, so now we have 6 x (30 + 5). Then we use the distributive property to get (6 x 30) + (6 x 5). After solving for each quantity, we end up with 180 + 30. Easy! 210.  You can find worksheets for practicing the distributive property at Common Core Sheets.

Step Five
I often hear teachers say, "I hate partial products. What's the point?" The point is that this is the next conceptual link in the chain of events that lead to students' understanding of the multiplication algorithm. After students see how the distributive property works, they can understand that 35 x 6 = 30 + 180. The order has changed somewhat, but clearly 6 x 5 = 30 and 6 x 30 = 180. When added together, we get 210.

Step Six
Finally we reach the common algorithm for multiplying a two-digit number by a one-digit number. It's the same as the partial products method, but we take a shortcut. Instead of writing 30 for the answer to 6 x 5, we simply write the number of ones (0) in the ones place and move the number of tens (3) above to be added after we've multiplied 3 tens by 6. It sounds confusing as I write it here, but the kids understand it because it's the next logical step in this conceptual progression.

While you may think this is a bunch of unnecessary hullaballoo, it's not. Sure, you could skip properties of multiplication, decomposing numbers, using the distributive property, and partial products. After all, your students can learn to use the common algorithm without it. But kids deserve more. To succeed at math in the future, they need deeper understanding of mathematical operations and how they relate to the Base-10 number system. I know I sound preachy, but I'm not the only one: the Common Core State Standards are up on the soap box with me when they say, "Multiply a whole digit number of up to four digits by a one-digit whole number . . . using strategies based on place value and the properties of operations."

Math Is Like a Video Game

Math is like a video game.

In a video game, you learn skills and strategies to move you to the next level, right? It's the same in math.

At the intermediate level (grades 3-5), students learn skills such as adding, subtracting, multiplying and dividing. They lay foundations for algebra and geometry too. Knowing these things allows a student to move along in the video game we call math.

How can a child become an expert at this game? That's more difficult. He or she must really understand what the game is about, which translates to deep understanding of the place value system and how it's manipulated for each operation. Learning strategies, such as properties of addition and multiplication and decomposition of numbers, takes each student one notch closer to becoming a math whiz.

Even with these phenomenal skills, a student cannot master the game. He or she needs to learn (and have permission) to look at problems from different angles, solve them in a variety of ways, and apply skills and strategies in new and different scenarios. With this, the player has mastered the game.

Math is like a video game. Let's help our students conquer it.

Vocabulary in Informational Text

Every morning, before I crawl out of bed, I consider challenges that await me in my classroom. Today, for some reason, vocabulary was on my mind. How can I encourage my students to pay attention to new vocabulary in informational text?

Kids must learn vocabulary associated with so many subjects each day. For example, in math, we're using words like factor, product, dividend, divisor, and quotient. Today's geology lesson requires them to discriminate between humus, silt, clay, and sand. Social studies vocabulary of the day includes marsh, moraine, till, and cavern. If you're a teacher, you'll understand that's just the tip of the iceberg!

I want my students to be detectives as they read informational text, looking for clues to vocabulary meaning in visual aids, definitions, appositional phrases, context clues, word parts, and glossaries. My newest tool can be found below. I'll be trying it out in my classroom today. Feel free to snatch it so your students, too, can become great deTEXTives!

P.S. This addresses Standard 4 for Reading: Informational Text in the Common Core State Standards.

Latitude's Like a Tomato, Longitude's Like an Orange

Latitude, Flatitude; Longitude, Long-itude. There are many different tricks and sayings for teaching this tricky grid-work to children, but I have my own. I think latitude is just like a tomato, while longitude's like an orange.

I reinforce this using my arm while saying, "Latitude tells how far north or south of the equator, and longitude tells how far west or east of the Prime Meridian."

If you're new to teaching this set of concepts (or just need a refresher course), head to Latitude 34 North for concise yet thorough discussions of latitude and longitude, as well as important named circles of latitude (equator, Tropic of Cancer, Tropic of Capricorn, Arctic Circle, and Antarctic Circle), the Prime Meridian and International Date Line. In my experience, it's important to bone up on specific information before teaching so you can answer questions like this one that popped up in my class: "Is it named Topic of Cancer because you're more likely to get cancer when you cross it?" (Great thinking, but no.)

I like linking the reason for seasons to this discussion. My decrepit globe comes out of the closet, and the path around my room becomes the path of Earth around the sun each year. The lucky student in the center of the room becomes the sun. This 30-second You Tube video is a shortened version of the movement and discussion that then transpire in my classroom. We review this three times during our school year: at autumnal equinox, winter solstice, and vernal equinox. Repetition helps students to remember and conceptualize how Earth's tilted axis causes seasons.

Now's the perfect time to explore patterns of sunshine at daylightmap.com. How cool! My class likes to display this all day long. It's interesting and thought-provoking. You can manipulate the calendar to show the equinoxes, solstices, and everything in between.

Teaching latitude and longitude can't be rushed. The preliminary conceptual background discussed above generally takes one full period or longer. Then we move on to locating specific places using latitude and longitude. Each student needs only one resource: an atlas, which can usually be found in his/her social studies book. 

The pointer finger on the right hand scans up and down along the labels for latitude, and the left pointer scans across on the labels for longitude. The teacher calls out the latitude and longitude; students locate them with their pointers and slide along the lines until their fingers come together.

My students respond best when starting with a map of our state. Since we're in the western hemisphere, I have to explain how and why the lines of longitude increase from right to left (instead of from left to right as we normally read a page). We find the latitude and longitude of our city then discuss the range of latitudes and longitudes for our state. Finally, we locate the closest crosshairs for our capital city. 

After exploring our state, we move to a map of our country, the USA. I call out latitudes and longitudes, wait until everyone's hand is in the air, and ask them to shout out the name of the state.

Finally, we work on latitude and longitude in all four quadrants using the world map. In years when I tried teaching with a world map first, my students were confused. This year, however, after using state and country maps first, they were ready to take on the world!

To ensure mastery, several days of full-class practice are necessary. Then students can hone their skills using games like these:

• Latitude and Longitude Map Match Game, KidsGeo.com
• Treasure Hunt, ABCYa.com
• Latitude and Longitude, Purpose Games

This site from the Oswego City School District also provides a great student-centered review of latitude and longitude.

My social studies book provides one page of text, one map, and one worksheet for teaching latitude and longitude. That's just not enough! Do you have a clever way to teach this skill? If so, please share!

Subtracting Across Zeros

The words "subtracting across zeros" can make a third or fourth grade teacher's hair stand on end. All of that crossing out and mess of nines and tens. How can we teach this skill (and the underlying concept) so that our students will truly understand?

Yesterday I tried something new. Each of my students was given four sheets of Kovich Class Cash (thousands, hundreds, tens, and ones), each with ten bills.

As they cut out their funny money, excitement grew. Comments like, "The bigger the bill, the more Mrs. Koviches you get," and "Can we keep it?" and "I'm rich!" could be heard.

Each child organized his/her money on the top of the desk to represent a place value chart.

Then the subtracting began. I explained that the money at the top of the desk was the bank. In the first problem, we began with the number 1000. Each child put a $1000 bill in the middle of his or her desk.

The problem was revealed on the screen in front of the class.

We discussed the fact that taking two hundred dollar bills, seven ten dollar bills, and six one dollar bills away from a single one thousand dollar bill would be impossible. Instead, we would need to go to the bank.

Each child exchanged one thousand dollar bill for ten hundred dollar bills. On the screen I showed how this appears in the common algorithm.

We established that we still had $1000. Ten hundreds equals one thousand. But could we take two hundreds, seven tens, and six ones away from ten hundreds? No. We had to go back to the bank.

The students took one hundred to the bank and exchanged it for ten tens. Then we looked at our algorithm again. 

Now we had nine hundreds and ten tens. Did this still equal 1000? Yes! Could we subtract two hundreds, seven tens, and six ones? No. Back to the bank.

One ten was exchanged for ten ones. Now we had nine hundreds, nine tens, and ten ones. Did this still equal 1000? Yes! And here's how our algorithm looked:

Could we subtract two hundreds, seven tens, and six ones? Yes! Yay! The students busily arranged their bills on their desks, taking 276 away from 1000.

When the flurry of activity was over, seven hundreds, two tens, and four ones were left. We again looked at the algorithm. Yep! 724. That's the same as seven hundreds, two tens, and four ones.

After this, we worked a half a dozen or so more problems, moving faster and faster. Eventually we moved to working the problems without the funny money. When the time came for each student to subtract using paper and pencil on their own, a few used their funny money to get started.

P.S. Due to popular demand, I have created a PowerPoint presentation and class cash templates that you can use in your classroom. Subtracting Across Zeros with Class Cash is now available in my Teachers pay Teachers store.

Estimating Subtraction Problems

Today I will teach my fourth grade students how to estimate when subtracting. Some of them already understand how to estimate, but most have no clue (as their pretests have shown). And most of those who know to round then compute clearly don't understand the purpose of estimation.

Here's what I've been getting:

Now what's wrong with this picture? (After all, it's the way our textbook teaches it. Shhh! Don't tell, but haven't opened that text yet this year.) The trouble is that the problem is no easier to solve, and that, my friends, is why we estimate. Estimation should make the problem easy enough to do in your head, which will allow you to check if an actual answer is correct or make some other decision in everyday life.

There's no best way to estimate, but some strategies are better than others.  Here are the steps I will take with my students:

1. When estimating with whole numbers, always round each number to the biggest digit possible then compute. This way we can estimate using mental math. (This is today's lesson, which we will practice, practice, practice.)
2. When estimating numbers that include decimals, do the same.
3. Think! If a problem is easy to compute in your head, why estimate? (For example, the answer to 2060 - 1060 is clearly 1000.) If both numbers round to the same number, go to a lower digit. (For example, if we round the problem 7132 - 6854 to the highest digit, we would come up with 7000 - 7000, and our estimate would be zero. That clearly does not make sense, so in this case, we could round to 7100 - 6900 and see that the answer will be around 200.)
4. Continue building strategies, such as using compatible numbers. I found this set of activities online to use with students who already have a decent grasp of estimating.

Good mathematicians can and do estimate every day. It's a skill worth teaching and honing throughout the year.

Assessing Addition & Subtracting for Middle Grade Students

Now that my class has wrapped up numeration, it's time to move on to computation. The first order of business is adding and subtracting. Since I teach fourth grade, our guiding standard is 4.NBT.B.4: "Fluently add and subtract multi-digit whole numbers using the standard algorithm."

Neither adding nor subtracting whole numbers appears in the fifth grade standards, so you know what they say: The buck stops here. I am responsible for mastery of addition and subtraction of any number with and without regrouping. 

Even though I feel that my students should have a good grasp of this concept already, I must pretest to make sure. (And fifth grade teachers should probably do this as well.)

This simple test would show if my students could (1) add without regrouping, (2) add with regrouping, (3) subtract without regrouping, (4) subtract with regrouping, (5) subtract across zeros, and (6) appropriately use estimation as a tool for checking accuracy.

My results? All of my students showed mastery of addition (with and without regrouping) and subtraction without regrouping. About half of the class needs remediation on subtraction with regrouping and/or across zeros. More than three-fourths could not estimate (or estimated in an inappropriate manner). The most common error students made was rounding the answer to their problems in an attempt to estimate.

I've decided to place them in two groups. Those who mastered addition and subtraction (even if they're not too hot on estimation yet) and those who mastered everything will have a quick lesson on the purpose of estimation then move on to adding, subtracting, and estimating decimals. (We reviewed tenths and hundredths last week, so it will segue well.) Students who need help with subtracting with regrouping will get it - - - as well as a lot of practice with estimation.

If you think I'm a bit obsessed with estimation, you are correct. Even though it isn't in the standard, I believe a good mathematician uses estimation to judge an answer's reasonableness all the time. While my fourth graders will be writing their estimates, the ultimate goal is rapid mental calculation. Therefore, except in rare cases, students should round each number to the largest place value when estimating.

Those who peek in my classroom this week will see kids subtracting with regrouping, asking themselves, "Is my answer reasonable?" and estimating like crazy!

P.S. If you're creating computation worksheets for your students, try the font called Courier. All letters, numbers, and spaces are equally justified; therefore, everything stays lined up from row to row.

Look! I Can Work with Large Numbers!

My class wrapped up their study of multi-digit numbers this week. To show off their skills, each student completed this page and hung it on the wall. Now anyone who enters our room knows that all students have mastered 4.NBT.A.2 and 4.NBT.A.3. Yay!

You can use it too. Simply click on the image to download the sheet.