Unit 2 — Multi-digit Arithmetic & Fraction Equivalence
Description
Unit 2 deepens understanding of multiplication and division procedures while introducing fractions. Students fluently add and subtract multi-digit whole numbers using the standard algorithm and develop strategies for multiplying multi-digit numbers by one-digit and two-digit numbers using place value and the distributive property. Division with multi-digit dividends and one-digit divisors is explored through strategies and models. Multi-step word problems requiring interpretation of remainders are solved. In the fraction strand, students develop methods for recognizing and generating equivalent fractions, comparing fractions with different denominators using benchmark fractions and common denominators, and decomposing fractions into sums of smaller fractions. Area and perimeter formulas are applied to solve real-world problems.
Essential Questions
- What is a fraction and how can it be represented?
- How can equivalent fractions be identified?
- In what ways can we model equivalent fractions?
- How can we compare and order fractions?
- How are area and perimeter related?
Learning Objectives
- Fluently add and subtract multi-digit whole numbers using the standard algorithm.
- Multiply a whole number of up to four digits by a one-digit number and multiply two two-digit numbers using place value strategies.
- Divide a whole number of up to four digits by a one-digit divisor and explain the calculation using equations, arrays, or area models.
- Solve multi-step word problems involving all four operations, interpret remainders, and assess reasonableness.
- Recognize and generate equivalent fractions using visual models.
- Compare fractions with different numerators and denominators using benchmark fractions and common denominators.
- Decompose fractions into sums of fractions with the same denominator in multiple ways.
- Solve real-world and mathematical problems by finding area and perimeter of rectangles using formulas.
Supplemental Resources
- Fraction strips or area models for equivalence exploration
- Grid paper for area and perimeter activities
- Printed comparison symbols and benchmark fraction cards
- Graphic organizers for decomposing fractions
- Task cards with multi-step word problems
Measurement
Number and Operations in Base Ten
Number and Operations—Fractions
Operations and Algebraic Thinking
Students use reading comprehension skills to decode words, study vocabulary, and solve word problems. Students identify important information and write explanations of their mathematical thinking using mathematical terms. Students connect everyday vocabulary to strengthen understanding of mathematical terms.
Students represent data, discover patterns, and read information to analyze observations. Students use measuring tools to create models and measure results of experiments. Students analyze data to form conclusions and use data to prove theories across life science, physical science, and earth science modules.
Students understand how to read dates properly and interpret historical information.
Formative Assessments
- Exit tickets on regrouping in addition and subtraction
- Task cards for multiplication and division strategy practice
- Fraction model activities using visual representations
- Classwork involving comparison of fractions with unlike denominators
- Whiteboards for quick practice on multi-step word problems
Summative Assessment
Unit benchmark covering fluent arithmetic operations, multi-step problem solving, fraction equivalence and comparison, and area/perimeter; chapter tests; performance tasks with real-world contexts
Benchmark Assessment
Benchmark tests within adopted program; PARCC released items for operations and fractions
Alternative Assessment
Students may demonstrate understanding through manipulatives, visual models (base-ten blocks, area models, fraction strips), or verbal explanation of strategies in place of written computation. Number lines, partially completed algorithms, or choice boards selecting the correct step in a multi-digit operation may be provided as scaffolds.
IEP (Individualized Education Program)
For multi-digit arithmetic and fraction work, provide visual models such as place value charts, fraction strips, and area model templates to support procedural understanding alongside written work. Reduce the number of problems on practice tasks to prioritize depth of understanding over quantity, and allow students to demonstrate reasoning through oral explanation or by pointing to and describing a model rather than writing full solutions. For multi-step word problems, chunk the problem into smaller parts and highlight key operation language to help students identify and sequence the steps needed. During benchmark assessments, allow extended time, provide multiplication reference charts, and permit use of manipulatives unless the task specifically measures recall.
Section 504
Provide extended time on all practice tasks and assessments involving multi-digit computation and fraction comparison, as the volume of procedural steps in this unit can create barriers to demonstrating understanding. Offer a distraction-reduced setting during multi-step problem solving tasks, and supply reference tools such as a number line and fraction benchmark chart to support working memory demands throughout the unit.
ELL / MLL
Pre-teach and consistently reinforce key vocabulary from this unit — such as equivalent, remainder, denominator, and decompose — using visual word walls, labeled diagrams, and concrete examples before and during instruction. Provide directions for multi-step word problems in simplified language with visual representations of the problem context, and allow students to explain their fraction reasoning using drawings or models in addition to verbal or written responses. Where possible, connect multiplication and division contexts to familiar, culturally relevant situations to build comprehension alongside language development.
At Risk (RTI)
Connect new concepts to prior knowledge by anchoring multi-digit addition and subtraction procedures to students' existing understanding of place value before introducing multiplication and division strategies. Offer entry points through hands-on models — such as base-ten blocks for arithmetic and fraction tiles for equivalence — so students can build conceptual understanding before moving to symbolic notation. Provide structured graphic organizers for multi-step word problems that guide students to identify what they know, what they need to find, and which operation to use, building confidence and independence across the unit.
Gifted & Talented
Challenge students to investigate the relationships between multiplication, division, and fraction equivalence at a deeper level — for example, by exploring how the distributive property connects to area models or how equivalent fractions can be understood through multiplicative reasoning rather than procedural rules. Encourage students to construct and solve their own multi-step real-world problems that require interpretation of remainders or comparison of fractional quantities, and to justify their solutions with multiple representations. Students may also explore how area and perimeter formulas extend to more complex or composite figures, connecting arithmetic fluency to geometric reasoning in ways that go beyond grade-level expectations.