4: Strengthening Number Sense - Strategies, Algorithms, and Estimation
- Page ID
- 183508
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Understanding and teaching whole-number computation goes beyond memorizing procedures—it involves developing number sense, strategic thinking, and problem-solving skills (NCTM, 2020). As future elementary educators, you will guide students through the journey of making sense of numbers, fostering their ability to reason mathematically, and helping them see computation as a flexible, sense-making process (Boaler, 2022). This chapter explores mental math techniques, estimation strategies, and standard and nontraditional algorithms to equip you with the knowledge to support diverse learners. By integrating hands-on models, place value understanding, and historical perspectives on computation, you will learn how to build a foundation that encourages deep mathematical thinking rather than rote memorization (Devlin, 2012). As you engage with this content, consider how you can cultivate a classroom environment that values multiple solution paths, mathematical reasoning, and a growth mindset—helping students see themselves as capable and competent problem solvers.
Chapter Learning Objectives
SLO 4.1 Students will apply mental math and computational estimation techniques, such as using compatible numbers, properties, and rounding strategies, to solve mathematical problems efficiently and accurately.
SLO 4.2 Students will compare how basic and scientific calculators interpret and solve expressions, distinguishing between arithmetic logic (step-by-step entry) and algebraic logic (order of operations) while using calculator functions effectively (e.g., memory, exponent, and parentheses keys) to computer accurate results.
SLO 4.3 Students will model and justify standard written algorithms for basic operations (addition, subtraction, multiplication, division) using concrete models, place value, and mathematical properties.
SLO 4.4 Students will compute sums, differences, products, and quotients in various bases (2–12) using standard algorithms and nontraditional methods, such as lattice and scaffold techniques.
Essential Questions
- What strategies can be used to develop a deep understanding of number relationships rather than relying on rote memorization?
- How do different estimation techniques (e.g., rounding, compatible numbers) help in real-world problem-solving?
Learning That Transfers
Students will analyze and apply flexible computational strategies in varied mathematical and real-world scenarios so they can begin to think about how they will design instructional methods that help elementary learners develop number sense and mathematical reasoning across disciplines.
Computation Fun
NEED AUDIO FOR STORY
Morgan's Math Adventure: Counting the World Around Us
One bright morning, with the sun shining high, Morgan saw numbers up in the sky! Up ran to Grandpa, a question to share— "Why do numbers dance everywhere?"
Grandpa chuckled, as wise grandpas do, “Numbers help us, through and through! From planning a trip to seeing new sights, Math is what helps us get things just right!”
Off on a road trip, they went with delight, Counting the miles and watching the sights. “We’ve gone 39 miles, the trip is so grand, The whole drive is ninety—where do we stand?”
Grandpa grinned, “Let’s round to make sense, Forty is close, and ninety is dense! About fifty to go, not quite exact, But close enough to keep us on track!”
They stopped at a fruit stand along the way, Morgan saw melons stacked up in display. “If there are seven rows, each with five, How many melons are stacked up high?”
Grandpa smiled, “Multiply and see, Seven times five? That’s thirty-five with me!” Then Morgan asked, “But what if some sell? If ten are bought, how many to tell?”
Grandpa thought, “Thirty-five take away ten, That leaves twenty-five melons again!”
Further ahead, they saw birds in the sky, Some perched on a fence, some soaring high. “If sixty birds sit, but half fly away, How many are left to rest and stay?”
Morgan paused, then started to do, “Half of sixty? That’s thirty left for you!”
The ocean appeared, waves crashing with cheer, Morgan felt proud as they finally drew near. “Numbers are everywhere, big and small, And now I see, I can use them all!”
Grandpa winked, “That’s exactly right! Now keep on counting, all day and night! With estimation, algorithms, and a flexible view, You’ll solve math problems like mathematicians do!”