Month: October 2021

Place Value: Part 2 — Base Ten System

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by C. Elkins, OK Math & Reading Lady

Students know how to count orally, can subitize, can count on one more, know ordinal positions, and have cardinality (know that the last number counted is the number of objects). These are usually prerequisites with number sense before introducing the symbolic representation and base ten understanding.  Our system is based on the repeated groupings of ten. This is a follow up from my last post (Place Value Part 1 — counting).

There are two levels of understanding place value symbols.

  • Place value:  In the number 23, the 2 has a place value representing the tens place.
  • Face value:  With 23, the 2’s value is 2 tens or 20.

This shows 23 and 32 are not the same amount even though they use the same digits. Conversely, this same understanding applies to decimals. For example, .4 is the same value as .40.

Here are some ways to help students develop place value competence. Below this section are activities to help teach and practice:

  1. Organize objects into groups when counting.
    • If given a group of 72 objects, do students try to count them one at a time (and probably losing count somewhere along the way)?  Or, can they make specific size groups such as 10’s to see they have 7 groups of 10 and 2 extra?
    • If given a group larger than 100 (like 134), do they note that 13 groups of ten is the same as 130?
  2. Partition numbers into groups based on powers of ten (ones, tens, hundreds).
    • Students learn that 52 = 5 tens, 2 ones = 50 + 2
    • Students learn that 348 = 3 hundreds, 4 tens, 8 ones = 300 + 40 + 8
    • Students learn that .45 = 4 tenths, 5 hundredths = .4 + .05
  3. Realize the relationship among the different places. Using the number 67 . . .
    • Most frequently it is represented as 6 tens, 7 ones.
    • But it can also be represented as 5 tens, 17 ones (which by the way is crucial to understanding the regrouping process for subtraction.  Show your students a problem in which regrouping is needed for 67, with the result of 5 tens 17 ones. Ask them if 5 tens, 17 ones = 67?  How many think no?).
    • 67 can also be represented as 4 tens 27 ones, and so on.

A clear understanding of these three guiding concepts of place value are very important and you will do well to thoroughly help students experience them before embarking on number operations.

Some activities to help with the above:

Organizing:

  • Provide objects for students to count:  beans, cubes, tiles, candy, popcorn kernels, etc. Depending on the size of the objects, give directions such as “take a handful” or a spoonful, or a cupful. Make it a competition with a partner, who has more? who has less?  Provide different containers for students to compare which holds more or less (and thus, working a little on measurement standards and conservation of space understanding).
  • Use base ten ones units.  Give an amount and after making piles of tens, have student trade each pile for a tens rod.  How many tens? How many ones?

Partitioning:

  • Build given numbers with base ten pieces. Example:  “Build 47”
  • Match pictures with expanded form using task cards.
  • I recommend using place value pieces for KG and 1st graders because the pieces show the relationship (a tens rod is 10 ones units stuck together, the 100’s flat shows ten 10s and 100 units). For 2nd and up, it might be more feasible to use the place value disks because these allow students to work with larger amounts. I know I never had enough hundreds’ flats to build larger numbers. Here’s a link to place value disks on Amazon: Place value disks
  • Use place value number strips that layer:  2000 + 500 + 30 + 8 when laid on top of each other shows the number 2,538. See sample here at Amazon: Place value layered strips
  • Work on mental math thinking of adding tens and ones:  10 + 2 = 12, 30 + 5 = 35, 70 + 6 = 76. I often use this technique to show visually what we want our brain to do. These are tens and ones layered strips I made, available here for free: Digit cards 0-10 and 10-100

Relationships:

  • Practice making exchanges with place value pieces. Build a number the standard way (67 = 6 tens, 7 ones). Then exchange a ten for ten ones. Verify that 5 tens 17 ones = 67.
  • How many ways can you show this number? (Give a number and provide manipulative for students to check and list.)
  • Using base ten rods and flats, verify that 15 tens = 1 flat + 5 tens (100 + 50)
  • Show how number changes by changing the ones or the tens.

    Use a 100 chart. Select a number and find what ten more or ten less would be — also 1 more, 1 less. Use the chart to show ways to count by tens other than the traditional 10, 20, 30, etc.  Try 27, 37, 47, 57. Practice counting forward as well as backwards.

Important tip when using base ten manipulatives (from personal experience):

When students are using base ten pieces, don’t allow random placement of the ones on their work mat. Why? This often means a student may not have the correct amount because they miscounted their “mess.” It also means the teacher has to spend more time when circulating the room to count their “mess” to check for correctness.

The ones cubes are organized!

All it takes is noticing the student(s) who likes to organize their ones pieces.  Their 5 pieces look like the dots on a dice or how it might look on a ten frame, or the 9 ones are in 3 rows of 3. As the teacher, you can show others how this is so helpful to you (and them) to arrange them in some kind of order. Knowing they get some choice in the order is key. And noticing how different students show the same amount in different ways is an ego booster to most students.  Finally, it really does help boost students’ understanding of number bonds when they show 5 can be 4 and 1, or 2 and 3, or 2 and 2 and 1. That’s a two for one, right?

Enjoy your place value lessons — and share some you think would also be helpful!

Take care, stay safe!

 

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.