Chapter 1
Counting and Operations with Whole Numbers
Course: Elementary Education Assessment 5006
Subject: Mathematics And Science 5008
Chapter: Counting And Operations With Whole Numbers
This lesson focuses on how students build meaning for the four operations and how they connect models (objects, drawings, diagrams) to expressions and equations. Praxis 5008 expects you to recognize which representation matches a given situation, how to move students from concrete understanding to symbolic work, and how to use properties and number theory to strengthen problem solving.
Representations of Addition, Subtraction, Multiplication, and Division and Connections to Expressions and Equations
Students learn operations most effectively when instruction follows a deliberate progression: concrete → pictorial → abstract. Concrete models help students see and touch quantities, pictorial models help students organize and communicate the relationships, and abstract notation (expressions and equations) allows efficient computation and generalization.
Concrete Representations (Manipulatives and Physical Models)
Concrete representations are physical tools students can manipulate to model quantities and actions. These models clarify what an operation means before students are asked to use symbols.
Addition as Joining or Increasing
Addition represents combining two quantities into a larger quantity or increasing an amount by adding more.
Example: A student has 6 counters and is given 4 more. They physically join the two groups and then count the total: 10. This situation is written as 6 + 4 = 10. The equation matches the action: the first addend is the starting set, the second addend is what is added, and the sum is the total after joining.
Subtraction as Taking Away
Subtraction represents removing a part of a set from a whole, leaving what remains.
Example: A student starts with 12 counters and removes 5. They count what is left: 7. This is written as 12 − 5 = 7. The equation shows the starting whole (12), the removed part (5), and the remainder (7).
Subtraction as Comparing (Finding the Difference)
Subtraction can also represent a comparison between two quantities: how many more or how many fewer.
Example: One student has 14 stickers and another has 9 stickers. Students can line up counters in pairs to compare. After matching 9 pairs, 5 stickers remain unmatched in the larger group, showing a difference of 5. This is represented as 14 − 9 = 5.
Number Lines for Addition and Subtraction
A number line represents addition as movement forward and subtraction as movement backward. This helps students connect operations to distance and change, which later supports integer operations and measurement reasoning.
Example: To model 8 + 6, start at 8 and move 6 steps forward to land on 14. To model 14 − 6, start at 14 and move 6 steps backward to land on 8.
Base-Ten Blocks for Regrouping
Base-ten blocks make place value visible and help students understand why regrouping works in addition and subtraction. Regrouping is not a trick; it is exchanging equivalent amounts.
Example: To compute 47 + 35, students represent 47 as 4 tens and 7 ones, and 35 as 3 tens and 5 ones. Combining ones gives 12 ones, which is exchanged for 1 ten and 2 ones. Now there are 8 tens and 2 ones, giving 82. This connects directly to the written algorithm while preserving meaning.
Pictorial Representations (Drawings and Diagrams)
Pictorial representations are drawings that capture the same relationships as manipulatives. They are critical because they help students move away from physical dependence while still supporting visual reasoning.
Addition and Subtraction Drawings
Drawings can show joining, separating, or comparing without needing physical objects.
Example: A student draws 9 circles to represent 9 apples and crosses out 3 to represent apples eaten. The remaining 6 circles represent 9 − 3 = 6.
Multiplication as Equal Groups
Multiplication represents a total made by having the same number in each group. Students should understand that multiplication describes both the number of groups and the number in each group.
Example: A student draws 5 bags with 4 marbles in each bag. Counting all marbles gives 20. This is represented as 5 × 4 = 20. The first factor (5) indicates the number of groups, and the second factor (4) indicates the amount in each group.
Multiplication as Arrays
Arrays are organized rows and columns that visually show equal groups and connect to area and the distributive property. Arrays also support understanding that multiplication is commutative because rotating an array does not change the total.
Example: An array with 3 rows of 6 dots has 18 dots total, so 3 × 6 = 18. Turning the array shows 6 rows of 3 dots, supporting 6 × 3 = 18.
Area Models for Multiplication
Area models connect multiplication to geometry and are a major bridge to multi-digit multiplication and algebraic expansion.
Example: To compute 14 × 6, students split 14 into 10 and 4. They compute 10 × 6 = 60 and 4 × 6 = 24, then combine for 84. This model supports 14 × 6 = (10 × 6) + (4 × 6) and builds distributive reasoning.
Division as Equal Sharing
Division can be shown as sharing a total equally into a given number of groups. The focus is on fairness and equal amounts.
Example: If 18 cookies are shared among 6 children, a student draws 6 circles (children) and distributes cookies one by one until all 18 are used. Each circle receives 3 cookies, showing 18 ÷ 6 = 3.
Division as Grouping (How Many Groups?)
Division can also represent forming groups of a fixed size and counting how many groups can be made.
Example: If 20 markers are packed into bundles of 4, students form groups of 4 repeatedly and find 5 groups, giving 20 ÷ 4 = 5.
Connecting Representations to Expressions and Equations
Praxis 5165 expects you to help students translate between models and symbols. This translation is essential for building algebra readiness because equations are concise statements about relationships.
Example: A diagram showing 4 groups of 3 objects can be expressed as 3 + 3 + 3 + 3, then written efficiently as 4 × 3, and evaluated as 12. Students should understand that the expression describes the structure, and the equation states the equality: 4 × 3 = 12.
Solving Mathematical and Real-World Problems Using the Four Operations, Using Properties, and Checking Reasonableness
Students must solve computation problems and word problems accurately and justify the operation choice. They must also evaluate whether answers make sense in context.
Applying the Four Operations in Context
Students learn to connect key phrases and situations to operations, but Praxis emphasizes reasoning over memorized keywords. Students should identify what is happening: joining, separating, equal groups, or sharing.
Example: “A store sold 46 tickets on Friday and 38 tickets on Saturday. How many in all?” This is joining totals, so 46 + 38 = 84.
Example: “A tank holds 60 gallons. It currently has 27 gallons. How many more gallons are needed to fill it?” This is finding the difference, so 60 − 27 = 33.
Example: “Each student receives 6 stickers. There are 24 students. How many stickers are needed?” This is equal groups, so 24 × 6 = 144.
Example: “144 stickers are shared equally among 24 students. How many stickers per student?” This is fair sharing, so 144 ÷ 24 = 6.
Using Properties of Operations to Solve Efficiently
Properties support flexible thinking. Students should use them to simplify work, especially in mental math and multi-step reasoning.
Commutative Property
This property allows students to change order to make computation easier in addition and multiplication.
Example: To compute 18 + 32 + 2, a student can reorder to 18 + 2 + 32 = 20 + 32 = 52. The total stays the same, but the calculation becomes simpler.
Associative Property
This property allows students to regroup numbers to create friendly combinations.
Example: To compute (6 × 25) × 4, regroup as 6 × (25 × 4). Since 25 × 4 = 100, the product becomes 6 × 100 = 600.
Distributive Property
This property allows students to break apart numbers and combine partial products or partial sums.
Example: To compute 23 × 7, split 23 into 20 and 3: 23 × 7 = (20 × 7) + (3 × 7) = 140 + 21 = 161. Students learn that this is not a shortcut; it is a structural rule that always holds.
Determining Reasonableness
Students must check whether answers fit the context and magnitude of the problem. This is a core Praxis expectation because it reflects mathematical reasoning, not only calculation.
Estimation to Check Magnitude
Rounding gives a quick plausibility check.
Example: For 49 × 8 = 392, estimate 50 × 8 = 400. Since 392 is close to 400, the result is reasonable.
Context Checks
Students must ask whether the answer makes sense given the situation.
Example: If 32 students share 10 pizzas equally, each student cannot receive 5 pizzas. Since 10 ÷ 32 is less than 1, each student receives less than one pizza.
Identifying Properties of Operations and Using Them in Problems
Students must identify properties, state them correctly, and apply them in computation and context.
Commutative Property
Changing the order does not change the result for addition and multiplication.
Example: 9 + 13 = 13 + 9 and 7 × 4 = 4 × 7. This fails for subtraction and division because order changes the result: 9 − 13 ≠ 13 − 9.
Associative Property
Changing the grouping does not change the result for addition and multiplication.
Example: (8 + 5) + 2 = 8 + (5 + 2). A student can group 5 and 2 to make 7, then add: 8 + 7 = 15.
Distributive Property
Multiplication distributes over addition and subtraction.
Example: 6 × (12 − 2) = (6 × 12) − (6 × 2) = 72 − 12 = 60. This property supports mental math and later algebraic simplification.
Real-World Application
Properties help students reason about totals efficiently.
Example: A fundraiser sells 7 tickets for 8 each. Total revenue = (7 × 12) + (7 × 8) = 7 × (12 + 8) = 7 × 20 = 140. This uses distributive reasoning to combine like groups.
Basic Number Theory: Primes, Composites, Factors, and Multiples
Number theory supports divisibility reasoning, fraction simplification, and finding common factors and multiples.
Prime Numbers
A prime number has exactly two positive factors: 1 and itself. Prime numbers cannot be decomposed into smaller factor pairs.
Example: 17 is prime because no whole number other than 1 and 17 divides it evenly. Dividing 17 by 2, 3, 4, or 5 leaves a remainder each time.
Composite Numbers
A composite number has more than two positive factors and can be written as a product of smaller whole numbers.
Example: 21 is composite because it can be written as 3 × 7. Its factors include 1, 3, 7, and 21.
Factors
Factors are whole numbers that divide another number with no remainder. Understanding factors supports area, divisibility, and GCF.
Example: The factors of 30 include 1, 2, 3, 5, 6, 10, 15, 30. Since 5 × 6 = 30, both 5 and 6 are factors.
Multiples
Multiples are numbers produced by multiplying a given number by whole numbers. Multiples support skip counting, LCM, and fraction operations.
Example: Multiples of 6 are 6, 12, 18, 24, 30, 36. Each multiple is 6 × n for a whole number n.
Greatest Common Factor (GCF)
The GCF is the largest factor shared by two numbers. It is used to simplify fractions and solve grouping problems.
Example: For 24 and 36: Factors of 24 include 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18, 36. The greatest common factor is 12.
Least Common Multiple (LCM)
The LCM is the smallest positive number that is a multiple of two numbers. It is essential when adding or subtracting fractions with unlike denominators.
Example: Multiples of 8: 8, 16, 24, 32, 40, 48 Multiples of 12: 12, 24, 36, 48 The least common multiple is 24