Unit 1 — Understanding the Place Value System
Description
This unit develops deep understanding of place value for whole numbers and decimals. Students learn that a digit in one place represents 10 times the digit in the place to its right and 1/10 of the digit in the place to its left. Students write, read, and compare decimals to thousandths using base-ten numerals, number names, and expanded form. They use place value understanding to round decimals to any place and to multiply whole numbers using the standard algorithm. Students also evaluate numerical expressions using the order of operations with parentheses, brackets, and braces, and write simple expressions to represent calculations from words and contexts.
Essential Questions
- Why is place value important when working with whole numbers and decimal numbers?
- How does multiplying a whole number by a power of ten affect the product?
- What is the relationship between decimals and fractions?
- How can we read, write, and represent decimal values?
- How does the placement of a digit affect the value of a decimal number?
Learning Objectives
- Evaluate numerical expressions with parentheses, brackets, and braces using order of operations
- Write numerical expressions from word descriptions and interpret expressions without evaluating
- Explain relationships between digits in place value positions of multi-digit numbers
- Explain patterns in products when multiplying by powers of 10 and in decimal point placement
- Read, write, and compare decimals to thousandths in multiple forms
- Round decimals to any place value
- Multiply multi-digit whole numbers using the standard algorithm
- Divide whole numbers with up to four-digit dividends and two-digit divisors
Supplemental Resources
- Printed word lists for place value vocabulary for building mathematical language
- Graphic organizers showing place value relationships for visual organization
- Index cards with numbers and expressions for matching and sorting activities
- Printed passage sets with word problems involving place value for contextual practice
- Colored pencils and markers for creating place value charts and labeled diagrams
Number and Operations in Base Ten
Operations and Algebraic Thinking
Standards for Mathematical Practice
Students read and comprehend informational and literary texts to understand mathematical concepts and solve word problems. Students write explanations of mathematical thinking using precise vocabulary and demonstrate command of conventions including grammar and punctuation. Students engage in collaborative discussions about mathematical strategies and justify their reasoning with evidence.
Students collect, organize, and analyze data to identify patterns and make predictions in scientific investigations. Students use measurement tools and develop understanding of volume and capacity. Students engage in scientific practices including asking questions, developing models, conducting fair tests, and constructing explanations based on evidence.
Formative Assessments
- Daily classwork on place value and operations tasks
- Exit tickets checking understanding of order of operations and decimal comparison
- White board practice on writing and evaluating expressions
- Math journals reflecting on place value relationships
- Quizzes on decimal notation and multiplication strategies
Summative Assessment
Unit 1 test assessing ability to use place value understanding for comparisons, rounding, and multi-digit operations
Benchmark Assessment
— not configured —
Alternative Assessment
Students may demonstrate understanding through oral explanation of place value relationships and order of operations rules, with visual supports such as place value charts or number lines provided. Numerical expressions may be evaluated using manipulatives or diagrams in place of written work, with response options presented verbally or in simplified written format.
IEP (Individualized Education Program)
Students working with IEPs may benefit from visual place value supports such as labeled place value charts and base-ten models to anchor their understanding of digit relationships across whole numbers and decimals. For tasks involving order of operations or multi-step expressions, directions should be broken into numbered steps with key operational symbols highlighted, and students may respond orally or through guided written frames rather than open-ended notation. A reference card showing place value positions, rounding rules, and the order of operations hierarchy can reduce cognitive load during classwork and assessments. Extended time and reduced problem sets should be provided so that instruction targets depth of understanding for key concepts such as decimal comparison and the standard multiplication algorithm rather than volume of practice.
Section 504
Students with 504 plans should have access to extended time on quizzes and the unit assessment, particularly when tasks require reading and interpreting multi-step numerical expressions or comparing decimals in multiple forms. A distraction-reduced environment and preferential seating near instructional displays, such as a posted place value chart, support sustained focus during multi-step computation work. Providing a clean, enlarged workspace with graph paper or pre-formatted place value grids can help students track digit alignment during multiplication and decimal notation tasks.
ELL / MLL
Multilingual learners should have access to a visual vocabulary reference that includes key unit terms — such as place value, thousandths, exponent, expression, and round — paired with visual models, numeric examples, and where possible, translations or cognates in the student's home language. Teachers should use visual demonstrations with place value charts, base-ten drawings, and number lines when introducing decimal relationships and rounding, minimizing reliance on extended verbal explanation alone. Directions for tasks involving order of operations or written expressions should be simplified and delivered in short steps, with visual examples of the expected format before students attempt independent work.
At Risk (RTI)
Students who need additional support should be connected to concrete, visual representations of place value — such as base-ten models and labeled charts — before moving to symbolic notation for decimals and expressions, ensuring a tangible entry point into abstract concepts. Tasks involving the standard multiplication algorithm or decimal comparison can be scaffolded by pre-formatting the workspace and reducing the number of digits or steps required until accuracy builds. Frequent brief check-ins during classwork, along with immediate corrective feedback, help these students catch and address misunderstandings in foundational concepts like place value position relationships before they carry forward into rounding and operations.
Gifted & Talented
Students who have demonstrated mastery of foundational place value and decimal concepts can be extended through tasks that require them to reason about the structure of the place value system — for example, exploring what happens to expressions and products when place value positions shift, or constructing and justifying their own order of operations problems that others must interpret without evaluating. These students can investigate real-world contexts that require precise use of decimals and large-number multiplication, such as scientific measurement or financial data, and explain the mathematical relationships embedded in those contexts. Encouraging students to write about the logic behind the order of operations convention or to explore how expressions with the same numbers but different groupings produce different results deepens conceptual thinking beyond procedural fluency.