Hands-On Math, KP® Ten-Frame Tiles, Math, Teaching and Learning
We’ve all done it…teaching rules without meaning. For me it was rounding rules. I had found the perfect song. The teacher next door used a baseball metaphor. Our colleague down the hall used the Rounding Mountain. We were all trying help our students remember the rounding rule: “If the digit to the right is 0-4, round down. If the digit to the right is 5-9, round up.” And in each case, of course, we failed to develop any understanding whatsoever.
That’s because rounding is really about “closer to” thinking. The dictionary states that to round is to “alter a number to one less exact and more convenient for calculations.” In other words, rounding is about finding the closer benchmark number for estimation purposes. The benchmark number doesn’t even have to be a multiple of 10!
Here are three ways to develop “closest to” thinking:
1. Use a concrete manipulative such as KP Ten-Frame Tiles. If your students are rounding 68 to the nearest ten, for example, ask them to place 6 tens and 8 extra ones on a large ten-frame. Ask them if the small ten-frame is closer to full or closer to empty? Since only two tiles are missing, they can easily see that the ten-frame is closer to full, and, therefore, closer to 70 than to 60.
2. Use a visual representation such as a number line. If your students are rounding 368 to the nearest hundred, for example, ask them to plot 400 on one side of the number line and 300 on the other side of the number line. Then ask them to mark the halfway point, 350. Finally, ask them to mark 368. Is 368 closer to 300 or closer to 400, using 350 as the halfway point? NOTE: If you’re familiar with “Clothesline Math,” this same visual can be achieved by having students place folded paper strips on a long piece of clothesline rope or string.
3. Write the “closer to” numbers above and below. Simply ask the students to write the possible rounded numbers on above and below the number at hand. If your students are rounding 2.34 to the nearest hundredth, for example, ask them to write 2.3 above and 2.4 below. Then ask them if 2.34 is closer to3 or to 2.4 (note: they may go back to using a number line for help).
Lesson learned – these are now my go-to strategies rather than teaching my favorite rounding song. Know that kids might find these a bit tricky at first, but honestly, they’ll catch on by the second day if you really get them thinking about “closer to” numbers and halfway points.
How about you? What are your thoughts about teaching thinking strategies rather than memorized rules? Please share your stories and comments below.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Hands-On Math, KP® Ten-Frame Tiles, Teaching and Learning
When was the last time you tried to teach a young child to count coins? If you’re like me, you probably found this to be a daunting task, as many math and non-math related issues surface.
As a child, I remember making a deal with my little sister: I offered to trade my four big nickels for her three little dimes. She thought I was being so generous…and we made the trade right there in the backseat of our Ford Pinto.
This goes to show that value-counting coins is a complex skill. It includes
- memorizing coin names and values,
- understanding the difference between counting the number of coins and “counting” their combined values,
- understanding the proportional relationships between the values of various coins,
- skip counting by different denominations,
- adding mentally,
- seeing coin values as fractions as a whole,
- and the list goes on.
It turns out that our traditional method, using coins as the teaching tool, just doesn’t cut it. Coins are concrete objects, but there is nothing about them that represents their proportional values. Today, I’d like to make a suggestion – let’s, instead, use the ten-frame to teach money!
Using the Ten-Frame to Represent Coin Values
The ten-frame, a visual organizer for numbers 0-10, can be extended and partitioned to represent multiple ten-frames and fractional parts. As you can see below, we can partition and/or combine ten-frames to represent coin values – 1, 5, 10, 25, 50, and 100.
Using the largest ten frame to represent $1.00, students can see that $1.00 is composed of 100 small squares – we’ll call those pennies. They can also see that there are 10 small ten frames – we’ll call those dimes. Then it’s a matter of partitioning and combining ten-frames to find the nickels, quarters, and half-dollars.
From Ten Frames to Money Boards
This visual model, pictured below as a KP Money Board and Coin-Value Cards, helps students see the proportional relationships between the various coin values. It also helps them explore the fractional values of the coins.
KP Dollar Board & Coin Cards for download
Using Money Boards and Coin-Value Cards for Problem Solving
Money Boards and Coin-Value Cards also provide a foundation for students to conceptualize a number of important money-related skills:
- Finding coin equivalences for $1.00 (4 quarters, 10 dimes, 20 nickels, 100 pennies)
- Comparing relative values of coins (e.g., 1 dime = 2 nickels; 1 nickel = 5 pennies)
- Value-counting a set of mixed coins
- Making change from $1.00
- Counting past $1.00 by counting in groups of $1.00
- Solving money-based word problems and story problems
One Final Thought – Bills Before Coins
At KP, we take the stance that since whole numbers are mastered before fractions, value-counting bills (whole numbers) should be taught before value-counting coins (fractions of bills). Several years ago, I conducted an action-research project to see if students would learn to value-count coins faster if they first learned to value-count bills. After all, the bills are uniform in size and clearly indicate their values. Guess what I found? Yep – the class that learned to value-count bills first subsequently learned to value-count coins in about half the time it took the class that began with coins. Furthermore, we used $1, $10 and $100 bills to connect to place-value concepts rather than pennies, dimes, and dollars. This was a huge eye-opener to me and my colleagues. Might you give it a try?
We hope you’ll consider using the KP Money Boards and Coin-Value Cards as you plan for your upcoming money units and problem-solving units. Click here for a downloadable version of the KP Money Board and Coin-Value Cards: KP Dollar Board & Coin Cards for download.
What do you think of these ideas? Do they resonate with you? Are they new for you, or have you thought of something similar in the past? If so, will you share with our readers what have you tried and how it has worked? Please join the conversation by leaving your comments below.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
