# Teaching Place Value With Arrow Cards

## By Wendy Petti

What are arrow cards and what do you do with them?

Most teachers are familiar with place value charts and base ten blocks for helping to build, reinforce, and extend place value concepts. Teachers in the United Kingdom also use arrow cards to support place value skills and concepts, and this useful tool is beginning to catch on in the United States as well. Let's take a look at how to make, use, and explore with arrow cards.

## WHAT ARE ARROW CARDS?

Arrow cards are a set of place value cards with an "arrow" or point on the right side. Students can organize the cards horizontally or vertically to represent numbers in expanded notation. They can overlap cards and line up the arrows to form multi-digit numbers.

 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Images courtesy of Wendy Petti.

When arrow cards are color-coded by place, they are easier to organize, and the color helps reinforce the concept of place. A yellow "3" looks and feels different from an orange "30" and a red "300." Students in primary grades can work with arrow cards up to the hundreds or thousands, while students in upper elementary grades might benefit from using arrow cards up to 6, 7, 8, or even 9 digits.

Arrow cards form a useful transition between base ten blocks and written numerals. Base ten blocks are very concrete. They look exactly like their value. Using base ten blocks, 300 is represented with 3 hundred squares; each hundred square is the size of 10 tens or 100 ones.

An arrow card representing 300 is larger than arrow cards representing tens or ones, but not in proportion to their value; it is simultaneously concrete and abstract. It is a physical and numerical representation emphasizing both the place and the value of the 3.

## MAKING ARROW CARDS

There are several online sources for arrow cards up to the thousands. Print and photocopy the arrow card templates onto card stock, which is stiffer and more durable than regular copy paper. You might use colored card stock, but another option is to print the arrow cards onto white card stock and have students lightly color the cards with an agreed-on color scheme, using colored pencils or crayons. Another coloring option would be to trace around the numerals with colored markers.

The printable accompanying this article is a multi-page spreadsheet holding arrow card templates up to the hundred millions. The collection includes three place value charts -- to the thousands, the hundred thousands, and the hundred millions -- sized for use with the arrow cards. Cutting out the arrow cards is efficient because they touch each other. Print only the pages you need, of course. Keep it simple at first, even in the upper grades.

OTHER PRINTABLE ARROW CARDS
From Kentucky Mathematics
* Blank Arrow Cards Students can draw their own numerals.
* Arrow Cards to Thousands
* Arrow Card Expansion Set -- 10,000s and 100,000s

Number Downloads from Brenden Is Teaching Includes both colored and black-and-white arrow cards.

Arrow Cards from Tim Whiteford Large and easy to cut; to thousands.

## USING ARROW CARDS

Getting Started

When students first begin working with arrow cards, use them hand-in-hand with base ten blocks so students can see and feel the value of each number.

At the outset, you might use a large set of arrow cards for whole class demonstrations, while pairs of students assemble numbers using smaller sets. You also might like to magnetize your demonstration set to use on a metal chalkboard in expanded notation form; you can purchase rolls of magnetic tape at craft stores. When you layer the cards, you'll need to hold them.

Students should sort the cards into groups according to place -- ones, tens, hundreds, and thousands -- and each group should be a different color.

Work through a series of "show me" activities, in which students at first hold up single arrow cards and then hold up numbers theyve built.
"Show me 8. Show me 40...60.... How many tens are in 60? Show me 700...."

Build a few two-digit numbers. "Show me 11, 12... 46, 47...."
Remind students that when they build 46, the 40 is still there. Its 40 + 6. Forty is 4 tens; 46 is 4 tens plus 6. Set up the same number with base ten blocks.

After introductory work, students should be able to follow instructions with minimal teacher modeling as needed.

Number Pairs

Move to "show me" activities in which student pairs build two related numbers:
"Show me 35 and 53." Both numbers use a 3 and a 5. Discuss the difference in the value of 5 as it is used in each number. Break the numbers apart to look at their components.

Extend the idea. "Show me all the 3-digit numbers you can make with a 3, 5, and 7 in any place." (How many sets of arrow cards will a pair of students need for this activity, in order to display all of the arrangements?)
"Show me all the 2-digit and 3-digit numbers you can make using 3, 5, or 7 in any place."
After building each number, students should keep a written record so they can reuse the same cards.

Zero

After students show a few 3-digit and 4-digit numbers, ask them to show a number with zeroes.
"Show me 104. Show me 6,018. Show the year you were born."
In the number 104, the 0 means an absence of tens. We don't need a "zero tens" card to show this. We can show 104 with 100 + 4; when we put them together, we will see a zero in the tens place, from the 100 card.

"Show me 10 more than 13...20 more than 13." If students arrange the tens cards in sequence, they can pull away each ten to reveal the next, while the 3 remains stable in the ones place. Model the same numbers with base ten blocks.

"Show me 10 less than 97...20 less...."
"Show me 100 more than...100 less than...1000 more than...1000 less than...."
You can transition to the language, "Build a number that is...."
"Build a number that is 40 more than 39....Build a number that is 300 less than 415...."

For these activities, using a place value chart and two sets of cards, student pairs can arrange the original number and the new number. Roam the room to check for understanding; students cannot hold up the place value chart and two numbers.

Regrouping

Discuss with students, "How can we build a number that is 60 more than 278?" If they count by tens, what happens after 288, 298....? Students should be able to see they will need to trade to the 300 arrow card in order to continue. Building on that idea, what is 70 + 60? (Is it easier to think in terms of 70 + 30 + 30?) What is 270 + 60? What is 278 + 60? Students should be able to use arrow cards and mental math rather than paper and pencil calculations. Using this same strategy, but taking things step-by-step, what is 278 + 64?

How can we build a number that is 50 less than 432? What is 30 less than 430? How can we adjust the number from there? What is 20 less than 400? What is 20 less than 402?

## WORKING WITH LARGER NUMBERS

When students build numbers beyond the thousands, it might help to paperclip the arrow cards together to keep them aligned. The place value chart will be very helpful in reinforcing the value of each place. In working with larger numbers, more of the questions are likely to do with multiples of 10: What is 100 times greater than 320? What number, when multiplied by 1,000, becomes 960,007,000? What is the value of the 6? It can be a challenge even to say those large numbers; give students plenty of practice.

## GOOD QUESTIONS EXTEND THINKING

Students need to be able to understand and apply place value skills and concepts to respond to most of the "show me" and "build a number" tasks. Stretch students' thinking skills even more with questions having more than one response.

"Build a number whose digits add up to 21."
"Build a 3-digit number in which the tens digit is two less than the hundreds digit and the ones digit is two less than the tens digit. What is the highest number you can build? The lowest number?"
"What numbers can you make smaller than 100 that have 7 in the tens place?"
"Can you find several ways to build 1,436 as the sum of smaller numbers?" (For example: 1,000 + 200 + 230 + 6...)
"I'm a number between 600 and 900 with one zero. What number could I be?"
"Build a 5-digit number that is a multiple of 10."
"Build a number that is 100 times larger (or 1,000 times larger) than the last number you built.",br> "What is the largest 6-digit number you can build that has 3 zeroes, but is not a multiple of 1,000? What is the smallest?"

Finally, get your students' creative juices flowing with this one!
"Can you create a new arrow card activity and share it with the class?"

Acknowledgements

Some of the arrow card activities are adapted from an unpublished manuscript and private communication with Tim Whiteford, PhD, who has collected some outstanding elementary math resources and links at: Math Education

Some of the place value extension questions were inspired by the place value questions in Good Questions for Math Teaching: Why Ask Them and What to Ask (K 6) by Peter Sullivan and Pat Lilburn, pages 33-36.

Wendy Petti is the creator of the award-winning Math Cats Web site, author of Exploring Math with MicroWorlds EX (LCSI, 2005), and a frequent presenter at regional and national math and technology conferences. She teaches grades 4 at Washington International School.

Article by Wendy Petti
Education WorldÂ®