Place Value: Part 2 — Base Ten System

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)

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.

Decimals: Part 1 – The Basics (revised)

by C. Elkins, OK Math and Reading Lady

Number sense regarding decimals usually starts with fourth grade and continues with more complex operations involving decimals in fifth grade and beyond. It is this extension of the place value system and then relating them to fractions and percentages that often perplex our students (and the teachers, too)!  Read ahead to get your freebies (Decimal practice notes, anchor charts, and Discovering Decimals Number of the Day / Game activity).  I have revised this previous post and included some more freebies below.

Students must understand  this base-ten value system extends in both directions — between any two values the 10-to-1 ratio remains the same. When using place value blocks in primary grades, students recognize the 100 square as 100, the tens strip as 10, and the units cube as 1.  Then with decimals, we ask them to reverse their thinking as the 100 square represents 1 whole, the tens strip represents a tenth, and the unit cube represents a hundredth.  This may take repeated practice to make the shift in thinking — but don’t leave it out. Remember the progression from concrete (hands-on) to pictorial to abstract is heavily grounded in research. Students will likely gain better understanding of decimals by beginning with concrete and pictorial representations.

I am sharing my decimal practice notes, which highlight some of the basic concepts to consider when teaching. Pronouncing the names for the decimals is not in these notes, but be sure to emphasize correct pronunciation — .34 is not “point three four.” It is “thirty-four hundredths.” Use the word and for the decimal point when combining with a whole number.  Example: 25.34 is pronounced “Twenty-five and thirty-four hundredths.” I know as adults we often use the term “point,” but we need to model correct academic language when teaching. You can get also the pdf version of these notes by clicking here: Decimal practice teaching notes. Continue reading

Ten Frames Part 3: More addition, subtraction, and place value

by C. Elkins, OK Math and Reading Lady

Welcome back to Part 3 of my Ten Frame series. This will continue with some more ideas on using ten frames for addition and place value. Be sure to grab my free set of mini ten frame dot cards and Place value mat with ten frames to use with these activities.

Add 9:

How often do you see students counting their fingers, drawing tally marks, or other figures to add 9? But what if they could visualize and conceptualize adding 9 is almost like adding ten, but one less? This is where the ten frame comes in handy.

  • To be most efficient with adding 9, help students to add 10 (or a multiple of 10) to any single digit.  Example: 10 + 7, 20 + 4, 50 + 8 . . .
  • Show a problem such as 9 + 7 as part of your daily Number Talk. Observe and listen to how students are solving.
  • Introduce this strategy by showing two ten frames – one with 7 and the other with 9. Check for quick recognition (subitizing) of these amounts on each ten frame.
  • Move one counter from the ten frame with 7 to the ten frame with 9. This will complete it to a full ten frame. Then add 10 + 6 mentally.
  • The purpose is for students to visualize that 9 is just one away from 10 and can be a more efficient strategy than using fingers or tally marks.
  • Practice with several more +9 problems.
  • For 3rd and up try mental math problems such as 25 + 9 or 63 + 9.  Then how about problems like 54 + 19 (add 20 and take away one)?
  • Can students now explain this strategy verbally?

Subtract 9:

  • Let’s say you had the problem 14 -9.  Show 2 ten frames, one with 10 and one with 4 to show 14.
  • To subtract 9, focus on the full ten frame and show that removing 9 means almost all of them. Just 1 is left. I have illustrated this by using 2 color counters and turning the 9 over to a different color.
  • Combine the 1 that is left with the 4 on the second ten frame to get the answer of 5.
  • Looking at the number 14, I am moving the 1 left over to the one’s place (4 + 1 = 5). Therefore 14 – 9 = 5

Use of the ten frame provides a concrete method (moving counters around) and then easily moves to a pictorial method (pictures of dot cards). These experiences allow students to better process the abstract (numbers only) problems they will encounter.

Place Value Concepts: Continue reading

Discovering Decimals Part 1: Basic Concepts

by C. Elkins, OK Math and Reading Lady

Number sense regarding decimals usually starts with fourth grade and continues with more complex operations involving decimals in fifth grade and beyond. It is this extension of the place value system and then relating them to fractions and percentages that often perplex our students (and the teachers, too)!  Read ahead to get your freebies (Decimal practice notes, anchor charts, and Discovering Decimals Number of the Day / Game activity).  I have revised this previous post and included some more freebies below.

Students must understand  this base-ten value system extends in both directions — between any two values the 10-to-1 ratio remains the same. When using place value blocks in primary grades, students recognize the 100 square as 100, the tens strip as 10, and the units cube as 1.  Then with decimals, we ask them to reverse their thinking as the 100 square represents 1 whole, the tens strip represents a tenth, and the unit cube represents a hundredth.  This may take repeated practice to make the shift in thinking — but don’t leave it out. Remember the progression from concrete (hands-on) to pictorial to abstract is heavily grounded in research. Students will likely gain better understanding of decimals by beginning with concrete and pictorial representations.

I am sharing my decimal practice notes, which highlight some of the basic concepts to consider when teaching. Pronouncing the names for the decimals is not in these notes, but be sure to emphasize correct pronunciation — .34 is not “point three four.” It is “thirty-four hundredths.” Use the word and for the decimal point when combining with a whole number.  Example: 25.34 is pronounced “Twenty-five and thirty-four hundredths.” I know as adults we often use the term “point,” but we need to model correct academic language when teaching. You can get also the pdf version of these notes by clicking here: Decimal practice teaching notes.

Anchor charts are excellent ways to highlight strategies in pictorial form. Here are some examples of anchor charts to help students relate decimals to fractions, location on a number line, word form, and equivalencies. Get the free pdf version here: Discovering decimals anchor charts. It includes a blank form to create your own.

 

In this model, I chose the 1000 cube to model 356 thousandths. It’s a little tricky – be sure to see that the 300 part is shaded all the way (front and top – picture 3 slices of 100), the 50 part is shaded (front and half the top – picture half of a 100 slice), and the 6 part is just shaded in the front (picture 6 individual parts). The entire cube would represent 1 whole.

Here’s a matching activity / game in which students match decimal to fraction, word form, expanded form, money, and pictorial form. Included is a blank page so you can make your own or have students take notes. Click here for the FREE activity:  Decimal, Fraction, & Money Match

Another resource ($2.50 at TPT from Joanne Miller) to help students relate the decimal to the pictorial form:Decimal 100 grid Scoot

Finally, below is an activity to practice or reinforce decimal concepts. The page showing can be used as a “Number of the Day” practice. I also created a game using this model, and the whole packet is included in this free pdf. Click here: Discovering Decimals number of the day and game

For more teaching help (videos and interactive models) for decimals, check out the following 3 free resources. These are also listed in my resources section of the blog (top black bar):

  1. www.learnzillion.com
  2. www.teachingchannel.org
  3. http://illuminations.nctm.org

As always, you are welcome to share your decimal discovery ideas. Just click the comment box speech bubble at the top of the article or the comment box at the end of the article.

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

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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