Critical Foundations, Part 2

  • Critical Foundations, Part 2

    By Francis (Skip) Fennell, Posted January 16, 2015 –   

    Welcome back! As noted in my previous post and worth repeating here, to me, the “signature expectation” of any pre-K–grade 6 mathematics experience is the ongoing nurturing and development of a sense of number, and the ongoing instructional development and understanding of place value and fraction equivalence (I refer to these as pillars of a sense of number sense) are must-haves for all mathematics learners.

    In this blog entry, I offer additional activities and tasks that connect place value to mathematics topics within a continuum that begins with counting and essentially extends counting to understanding numbers greater than 9, then presents the need for a direct focus on place value related to two-digit whole numbers, and extends to comparing and ordering such numbers as well as early work with addition and subtraction. I hope that what you will read below, and in the previous post will generate discussion and a literal swap shop of comments and ideas of what works for you, as well as questions to help all of us in our efforts to deepen place-value understandings.

    Part 2 provides additional activities and tasks and once again seeks your comments, thoughts, and your own activities and ideas. Try these new activities:

    1.   Turn-around numbers. Select any two-digit number (e.g., 67). Ask, “If we turn the digits around, what is the new number?” (76) Have students try 89. Then have them try 54. Ask, “How do you know when the turn-around number will be larger than the start number? How do you know when it will be smaller?”

    2.   True or False? Such questioning promotes early work with estimation. Have students represent the numbers with base ten materials, if appropriate.

    • 98 is close to 80.
    • 98 is less than 100.
    • 98 is almost 100.
    • 98 > 90.
    • 98 < 100.
    • How do you know which is closest to 100: 98 or 101?

     3.   Digit Card Activities. (You will need digit cards with the digits 1–9 on the cards; 0 may be used at times, too.)

    a.  Have students draw a digit card (from digit cards 1–9) and place that number of ones blocks in the ones column on a grid like the one below. Then have them draw another card and place that number of tens rods in the tens column. What is the new number? What is the number if you took a one away? What is the number if you took a ten away?


    b.   Have students draw two digit cards from 1–9, using those two cards to represent ones and tens digits. Ask, “What’s the largest and smallest two-digit numbers that could be created?” Have students represent each number using base-ten materials.

    c.   Have students think about the digit cards 1–9. If they drew two cards, what would the smallest and largest two-digit numbers they could create? What if a student drew three cards—What would be the smallest and largest numbers that could be created?

    Let’s think decimals now. Consider the following.

    4.   What’s Wrong? Have students discuss and show why the following are incorrect:

    • 2.352 > 2.4
    • 2.34 > 2.5
    • 5.47 > 5.632
    • 1.8 = 1.08

     5.   Close to One. Have students determine which of the following is closest to the number 1:

    • 0.9
    • 1.1
    • 0.91
    • 1.09

    6.   Ask students to share their thinking, perhaps comparing the amounts using representations (possible representations: number lines or hundred charts).

    A sampling of place-value–related opportunities would not be complete without some tasks for students to consider. Try the following.

    7.   Task: Which major league baseball stadium holds the largest crowd? The smallest crowd? Have students go to to find out. Then have them complete the following:

    • How many ballparks can seat more than 50,000?
    • How many ballparks can seat < 40,000?
    • How many sellout crowds would be needed for Yankee Stadium to have an attendance of > 3,000,000?

    8.   Task: The number 3 is the smallest whole number greater than 2. But is there a smallest decimal that is greater than 2? If so, what is it? If not, why not? Have students discuss their solutions, which should provide an interesting beginning to work with density of fractions, these particular fractions—decimals.

    Your Turn

    Once again, it’s your turn. An abridged rationale (see Place Value—Part 1 for the full discussion) for the foundational importance of place value and a half dozen +1 sets of activities and tasks has been provided for you to consider. Try them. What do you think? What works? How do you develop place-value understandings? How does what you do connect to developing the level of flexibility with number so necessary for developing and establishing a sense of number? 

    As with the Place Value—Part 1 blog, we want to hear from you! Post your comments below or share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk, or get to me personally on Twitter @SkipFennell; or at Feel free to visit the following websites for information, resources, or just for fun:


    Fennell_Skip-100x140 Francis (Skip) Fennell,, is the L. Stanley Bowlsbey Professor of Education and Graduate and Professional Studies at McDaniel College in Westminster, Maryland, where he directs the Brookhill Foundationsupported Elementary Mathematics Specialists and Teacher Leaders Project ( He is a past president of NCTM and a recipient of NCTM’s Lifetime Achievement Award. Skip is interested in the work of mathematics specialists, implementation of the CCSSM, teacher education, number and fraction sense, and educational policy.   



    Archived Comments

    Come on bloggers and social media types....Thoughts? How important is place value to you - in the classroom? What works for you? What challenges did your students have?

    Equivalent fractions - as a critical foundation - coming soon.
    Posted by: Francis (Skip)F_32412 at 1/17/2015 11:46 AM

    The discussion around digit cards and placing a digit in the tens and ones column has so much potential. I often use a game called "Place it". Students goal is to add 10 numbers up and get as close to 1,000 as possible. Numbers are generated by a number card and can either represent hundreds, tens, or ones. For example a 5 could be used as "5" or as "50" or "500." Students love it and there are ample. conversations for the magnitude of digits and strategies for trying to reach 1,000.
    Posted by: DrewP_77482 at 2/7/2015 6:56 AM


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    Alex Paul - 3/12/2018 12:06:38 AM

    I have been able to understand mathematics much more easily after I started following this website. The way things are explained in here is so easy to understand and I never for once felt that I was reading through some ancient script, which is a very common feeling while learning mathematics.

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    Francis (Skip) Fennell - 2/23/2015 2:22:27 PM
    Thanks Troy. Check out the most recent Blog re: fraction equivalence.

    Troy Whitbread - 2/23/2015 1:14:32 PM
    I enjoyed reading this article. Thank you!