counting

Place Value: Part 1 (counting)

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by C. Elkins, OK Math and Reading Lady

Place value is such an important math concept from KG and up. It starts with counting and recognizing amounts and in later grades plays a huge part in composing and decomposing numbers, multiplication, division, decimals, and yes . . . even fractions.  Students need a solid understanding of it to estimate and compare numbers as well. Stay tuned for ideas and freebies below.

If you look up the definition of place value in a dictionary or math glossary it’s likely to refer to “the value of the place, or position, of a digit in a number.”  But I think young students start off with a different understanding of numbers and may become confused. Does a child age 2 see their age and a quantity of 2 cookies in the same way?What about these numbers? – – Should I interpret these in terms of ones, tens, hundreds, tenths, hundredths, etc.? Does place value apply to these?

  • telephone number: 123-456-7890
  • address numbers: 1234 Happy Lane
  • zip codes
  • # on a sports jersey
  • identification numbers (on badges, Social Security, etc.)
  • # on a license plate

The examples above are actually referred to as nominal or nonnumeric because they are used for identification purposes and rarely have any meaning associated with place value.  For example, the address above (1234) is most likely NOT the one thousandth plus house on Happy Lane.

So on the way to understanding place value, let’s look at ways numbers are classified and the basic heirarchy even before we expect them to use the written notation for numbers:

  1. Rote counting:  saying numbers in sequence
  2. Counting objects:  using a 1 to 1 correspondence between number and quantity. You may have to teach how to keep track of counting objects like sliding them to the side when counting, or marking pictures with checks or circles as they are counted on paper.
  3. Subitizing:  recognizing a quantity without counting (accomplished using ten frames, dot cards, dice dots, a Rekenrek, tally marks).  See my other blog posts on subitizing for more info and resources.
  4. Cardinality:  associating the last number named when counting as the quantity of the set. After a child counts a set of objects, ask him/her this: “How many ___ are there?” Can they name the amount without recounting?
  5. Naming the next number in the sequence:  Give a child a set to count. After announcing the amount, add one more object to see if they can name it — or do they start over and recount?  Cardinality and naming the next number are needed in order to practice the skill of counting on.
  6. Concept of zero:  To a young child this means “nothing.” With place value it can be a place holder within a larger number.
  7. Ordinal positions:  learning terms such as first, second, third . . . which don’t even sound like the numbers one, two, three, . . .
  8. Part-Whole relationship:  recognizing that quantities can be decomposed different ways. With 5 objects, can student show different combinations such as two and three, four and one, five and zero.  I often refer to this as number bonds.

The message with today’s blog is to make sure young children have a firm understanding of the above before use of number symbols and teaching about “tens and ones.” I relate this to reading:  Students need to develop phonological awareness about the sounds of letters and words before associating with the printed form (which is the study of phonics).

How do you accomplish the above?

  • Lots of exposure to classroom manipulatives
  • Oral counting practice (even in poems and songs)
  • Match objects one to one. Place objects on top of dots on dot cards and count as you go, or Match # of objects from one picture card to objects of another picture card.
  • Make designs. Example:  “Using your color tiles, what design can you make with ten pieces?”
  • Use ten frames and dot cards during Number Talk sessions (flash quickly and discuss how the quantity is seen).  Example — If you show a dot card with 4 which forms a square shape, do you get a variety of responses such as, “I saw two and two.” or “If it makes a square, there are 4.” See some of my Number Talk blog posts for resources.
  • Use class scenarios to help children name the next number.  “There are 3 of you sitting on the carpet with me. If Megan comes to join us, how many would there be then?”
  • Practice counting on with ten frames and Rekenreks.  Ex:  Show a ten frame like this. The top is full so it is 5. Then count on 6, 7.  How many dots? 7
  • Notice ordinal positions regarding lines of students or arranging manipulative objects. Ex. “Put the blue bear first, the yellow bear second, and the red bear third.”
  • Experience part-whole counting by provide number bond activities such as my favorite, On and Off

    4 on and 1 off

  • Share stories about counting. Check out this link from The Measured Mom: The Ultimate List of Counting Books
  • Develop an observation-type informal assessment checklist to track each child’s ability to do the above.  Assess while they are using math centers or during inside recess opportunities. Here’s a FREEBIE checklist you are welcomed to edit, so I kept it in Word format. Counting Fluency Observation Checklist

 

Enjoy your counting lessons with your children or students. Use these to identify gaps in students’ concepts. Stay tuned for more development toward understanding of place value.

Addition and Subtraction Part 1: Numerical Fluency

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by C. Elkins, OK Math and Reading Lady

To be able to add and subtract, students normally pass through several phases as they build readiness for these operations with numbers.  As teachers, we know oral counting does not necessarily indicate an understanding of numbers and sets, just like reciting the alphabet doesn’t necessarily mean a child can recognize letters and sounds. Read ahead for freebies in the Part-Part-Whole section.

Numerical Fluency Continuum:  There are 7 steps to numerical fluency. If a child gets stuck on any of these steps, it may very likely halt their progress. Hopefully children move through these by the end of 2nd grade, but many students beyond that level have a breakdown which is likely because they missed one of these stages. Can you determine which of these stages your students are in?

  1. One-to-one correspondence: The ability to count objects so each object counted is matched with one number word.
  2. Inclusion of set: Does a child realize that the last number counted names the number of objects in the set? A child counts 5 objects.  When you ask how many, can they state “5.” If you mix them up after they just counted them, do they realize there are still 5?
  3. Counting on: If a child counts 5 objects and the teacher then puts 2 more objects for the child to count, do they start all over or continue counting from 5?  5 . . . 6, 7.
  4. Subitizing:Recognize an amount without physically counting (ie on dice, dot cards, fingers).
  5. More Than / Less Than / Equal To:  Can a child look at two sets of objects and tell whether the second set is more, less, or equal to the first set. Can a child build a second set with one more, one less, or equal to the first set?
  6. Part / Part / Whole: Compose and decompose sets by looking at the whole and the parts that make up the whole.
  7. Unitizing: The child is able to move from counting by ones to count by sets / groups: fives, tens, etc.

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Daily Math Meeting Part 5: Using the 100 Chart and “Guess My Number”

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by C. Elkins, OK Math and Reading Lady

This post will focus on ways to use a 100 chart to teach or review several math standards in the number sense and number operations strands (all grade levels). Each of these strategies can be completed in just a few minutes, making them perfect for your daily math meeting. Choose from counting, number recognition, number order, less/greater than, odd/even, addition, subtraction, multiplication, number patterns, skip counting, mental math, 1 more/less, 10 more/less, etc.

You can use a 1-100 chart poster on the smartboard, in poster form, or as a pocket chart. The pocket chart is the most versatile. See an example here: enasco.com pocket chart   Here is also a link to little colored transparent pieces that can be placed in the pockets to highlight chosen numbers: enasco.com pocket chart transparent inserts   I often show students that a 100 chart is actually just a giant number line all squished together instead of spread out across the room. To do this, I print off a chart, cut it into rows, tape the rows together, then highlight each multiple of 10. Second concept is that the lower numbers are at the top, and the higher numbers are at the bottom.

 Counting, Number Order, and Place Value

  • Instead of starting with a full 100 chart, start with an empty chart. Add 1 number per day in order, building toward the 100th day of school. This would be suggested for KG level. 
  • For other grade levels: Start with the numbers 1-10, 20, 30, 40, 50, 60, 70, 80, 90, and 100. Put the rest of the number pieces in a jar, baggy, or container. Draw one or more numbers at random each day and assist students in placing the number where it belongs. Example:  If you draw out 45, let’s look at the one’s place (5) and know that it belongs in the same column as the 5. Let’s look at the ten’s place. We know it is greater than 40, but less than 50 so this helps us know which row it belongs in. As you progress, start using the currently placed numbers to help locate the new numbers. “I need to place 67.  I see 57 is already on our chart and know that 67 is ten more, so I place it directly underneath.”
  • Number Thief Game:  After your chart is filled, try this game. After the children have left for the day, remove a few of the pieces. Then during your math meeting the next day, the children try to identify the missing numbers. Read how this blogger describes it:  “Swiper” at petersons-pad.blogspot.com
  • Number locating: Just practice locating numbers quickly. If asked to find 62, does the student start at 1 and look and look until they find it? Or can they go right to the 60s row?
  • Place Value Pictures:  You can’t do this on your hundred chart at meeting time, but there are dozens of picture-making worksheets available for free on TPT in which students follow coloring directions to reveal a hidden picture. Students get much better with locating numbers quickly with this type of practice.

Guess My Number: This is great for reviewing various number concepts. Here are a variations of guessing games. You can use with 1-100 chart, or 100-200, etc.

  1. Teacher writes a number secretly on a piece of paper (ex: 84). The teacher gives a single clue about the number, such as: “My number is greater than 50.” Then let 2-3 students guess the number. Confirm that they at least guessed a number greater than 50. Redirect if not. If you have the little colored inserts, place one in each of the incorrect numbers so students will know what was already guessed. If you don’t have those, just write the guessed numbers somewhere where students can see.  Give a new clue after every 2-3 guesses until someone guesses the number.  After guessing correctly, I always show the students the number I had originally written down so they will know I was on-the-level. Here are some example clues for the secret number 84: My number is even.  In my number, the one’s place is less than the ten’s place.  My number is less than 90. My number is greater than 70. If you add the 2 digits together, you get 12.  The one’s digit is half of the ten’s digit. Again, affirm good guesses because at first there may be several numbers that fit your clue.
  2. Yes or No:  This is almost a backward version of Guess My Number. Try this one after students are well-versed with the above game. It starts out the same though. Teacher selects a number. Then students have 10 tries to guess the number. They ask you questions, which can only be answered “yes” or “no.” Keep track on a chart paper of their questions and your answers. Some sample questions students could ask:  Is your number even? Is your number greater than 50? Is your number in the sixties? Is your number less than 90? Are both of the digits even? Some higher level questions could deal with multiples (Is your # a multiple of 2? Is your number divisible by 4?)

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