Woodcock-Johnson Applied Problems in Factors, Multiples, and Patterns

Woodcock-Johnson applied problems assessment in factors, multiples, and patterns with detailed solutions and visual infographics.

Woodcock-Johnson Applied Problems: Comprehensive Overview

WJ Assessment Focus

Applied Problems: Real-world mathematical reasoning requiring problem-solving skills.

Mathematical Reasoning: Understanding mathematical concepts and their applications.

Quantitative Skills: Ability to perform calculations and apply mathematical operations.

Key Concepts

Factor: A number that divides another number evenly without remainder.

Multiple: The product of a number and any whole number.

Pattern: A sequence that follows a predictable rule.

Assessment Skills

Problem Analysis: Identifying what the problem is asking.

Strategy Selection: Choosing the appropriate mathematical approach.

Calculation: Performing necessary mathematical operations.

WJ Scoring

Standard Score: Average 100, Standard Deviation 15.

Percentile Rank: Shows performance compared to peers.

Age Equivalent: Performance level relative to age norms.

1
Read Carefully
2
Identify Keywords
3
Choose Method
4
Solve Systematically
5
Check Answer
WJ APPLIED PROBLEMS FOCUS: Connect math concepts to real-world situations while demonstrating mathematical reasoning skills!
WJ Applied Problems: Basic Level
1 Basic Factors Easy
WJ ASSESSMENT Example 1
FACTOR PROBLEM Sarah has 24 apples and wants to pack them in boxes with the same number of apples in each box. What are all the possible ways she can pack the apples? Which packing gives her the most boxes?
Definition:

Factors: Numbers that divide evenly into another number. For example, the factors of 24 are numbers that divide 24 without leaving a remainder.

Method to find factors:
  1. Start with 1 and the number itself
  2. Check each number up to the square root of the number
  3. If a number divides evenly, add both the divisor and the quotient
  4. List all unique factors
Problem
Pack 24 apples equally
Find factors of 24
1, 2, 3, 4, 6, 8, 12, 24
Step 1: Find all factors of 24

1×24=24, 2×12=24, 3×8=24, 4×6=24

So factors are: 1, 2, 3, 4, 6, 8, 12, 24

Step 2: Interpret factors as possible packings

Each factor represents a possible number of boxes or apples per box

1 box of 24 apples, 2 boxes of 12 apples, 3 boxes of 8 apples, etc.

Step 3: List all possible packings

8 different ways: 1×24, 2×12, 3×8, 4×6, 6×4, 8×3, 12×2, 24×1

Step 4: Identify packing with most boxes

The packing with the largest number as boxes gives the most boxes

24 boxes of 1 apple each gives the most boxes (24 boxes)

8 different packings possible, 24 boxes of 1 gives most boxes
Final answer:

Sarah can pack her 24 apples in 8 different ways:

1 box of 24, 2 boxes of 12, 3 boxes of 8, 4 boxes of 6, 6 boxes of 4, 8 boxes of 3, 12 boxes of 2, or 24 boxes of 1

The packing with the most boxes is 24 boxes of 1 apple each.

Applied rules:

• Factor identification: Find all numbers that divide 24 evenly

• Application: Each factor pair represents a possible packing

• Optimization: The largest factor as number of boxes gives the most boxes

🍎
Factor Tree
24
4×6
2×2×2×3
📦
Possible Packings
1×24
2×12
3×8
4×6
6×4
2 Basic Multiples Medium
WJ ASSESSMENT Example 2
MULTIPLE PROBLEM A school bus comes every 12 minutes and a city bus comes every 18 minutes. If both buses arrive at the station at 8:00 AM, when will they next arrive together? How many times will each bus have come by that time?
Definition:

Least Common Multiple (LCM): The smallest positive number that is a multiple of two or more numbers.

Problem
School bus every 12 min, City bus every 18 min
Find LCM of 12 and 18
LCM(12, 18) = 36
Next arrival
8:00 AM + 36 min = 8:36 AM
Step 1: List multiples of each number

Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108...

Multiples of 18: 18, 36, 54, 72, 90, 108, 126...

Step 2: Identify the least common multiple

The smallest number that appears in both lists is 36

So LCM(12, 18) = 36

Step 3: Calculate the next simultaneous arrival

They will both arrive together again in 36 minutes

8:00 AM + 36 minutes = 8:36 AM

Step 4: Calculate number of arrivals by that time

School bus: 36 ÷ 12 = 3 times (but we started at 0, so 3 additional arrivals)

City bus: 36 ÷ 18 = 2 times (but we started at 0, so 2 additional arrivals)

Total: School bus arrives 4 times (including 8:00), City bus arrives 3 times (including 8:00)

Both will arrive together at 8:36 AM
Final answer:

The buses will next arrive together at 8:36 AM.

By that time, the school bus will have arrived 4 times (at 8:00, 8:12, 8:24, 8:36) and the city bus will have arrived 3 times (at 8:00, 8:18, 8:36).

Applied rules:

• LCM concept: Find the smallest common multiple of given numbers

• Time calculation: Add LCM minutes to the starting time

• Counting principle: Divide total time by interval to count occurrences

🚌
School Bus Schedule
8:00
8:12
8:24
8:36
8:48
🚍
City Bus Schedule
8:00
8:18
8:36
8:54
9:12
3 Basic Patterns Easy
WJ ASSESSMENT Example 3
PATTERN PROBLEM A pattern of circles follows this sequence: 6, 12, 18, 24, 30, ... If the pattern continues, what will be the 18th number in the sequence? How many circles are needed for the first 7 figures?
Definition:

Arithmetic sequence: A sequence where each term increases by a constant amount called the common difference.

Pattern
6, 12, 18, 24, 30, ...
Rule
6n
18th term
6(18) = 108
Step 1: Identify the pattern rule

6 = 1×6, 12 = 2×6, 18 = 3×6, 24 = 4×6, 30 = 5×6

The nth term is 6n

Step 2: Find the 18th term

Using the rule: 6(18) = 108

Step 3: Find sum of first 7 terms

First 7 terms: 6, 12, 18, 24, 30, 36, 42

Sum = 6 + 12 + 18 + 24 + 30 + 36 + 42 = 168

18th term: 108, Sum of first 7: 168
Final answer:

The 18th number in the sequence is 108.

The first 7 figures require 168 circles total.

Applied rules:

• Pattern recognition: Identify the rule that generates the sequence

• Formula application: Use the rule to find any term in the sequence

• Series summation: Add consecutive terms to find totals

⭕
Sequence Pattern
6
12
18
24
30

Rule: 6n

🎯
Term Calculation
nth term
6n
6(18) = 108
WJ Applied Problems: Intermediate Level
Factor × Factor = Product
Factors Relationship
Factor Identification
1, 2, 3, 4, 6, 8, 12, 24
Factors of 24
Multiples
12, 24, 36, 48...
Multiples of 12
Pattern Rule
6n
nth term
Key definitions:

Factors: Numbers that divide evenly into another number

Multiples: Products of a number and any integer

Patterns: Sequences that follow a predictable rule

Problem-solving methodology:
  1. Read carefully: Understand what the problem is asking
  2. Identify key words: Look for terms like "arrange," "together," "sequence," "pattern"
  3. Choose method: Determine if it's a factor, multiple, or pattern problem
  4. Solve systematically: Apply the appropriate mathematical operation
  5. Check answer: Verify your solution makes sense
Tip 1: When finding factors, work systematically from 1 up to the square root.
Tip 2: For LCM problems, list multiples until you find a common one.
Tip 3: For patterns, look for the relationship between position and value.

Tip 4: Always verify your answer by substituting back into the original problem.
WJ Applied Problems: Advanced Level
4 Greatest Common Factor Medium
WJ ASSESSMENT Example 4
GCF PROBLEM A teacher has 36 markers and 48 colored pencils. She wants to make identical art sets with the same number of markers and colored pencils in each set, using all supplies. What is the greatest number of art sets she can make? How many markers and colored pencils will be in each set?
Definition:

Greatest Common Factor (GCF): The largest number that divides two or more numbers without leaving a remainder.

Given
36 markers, 48 colored pencils
Find GCF of 36, 48
GCF(36, 48) = 12
Art sets
12 sets
Step 1: Find factors of 36 and 48

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Step 2: Identify common factors

Common factors: 1, 2, 3, 4, 6, 12

Greatest common factor: 12

Step 3: Calculate items per art set

Markers per set: 36 ÷ 12 = 3

Colored pencils per set: 48 ÷ 12 = 4

Step 4: Verify the solution

12 sets × 3 markers = 36 markers ✓

12 sets × 4 colored pencils = 48 colored pencils ✓

12 sets with 3 markers and 4 colored pencils each
Final answer:

The teacher can make 12 identical art sets.

Each set will contain 3 markers and 4 colored pencils.

Applied rules:

• GCF application: Use GCF to find maximum number of identical groups

• Division: Divide total items by group count to get items per group

• Verification: Multiply back to ensure original quantities are preserved

🖍️
Marker Distribution
36 markers
÷12 sets
3 markers per set
✏️
Pencil Distribution
48 pencils
÷12 sets
4 pencils per set
5 Advanced Pattern Hard
WJ ASSESSMENT Example 5
COMPLEX PATTERN A tile pattern starts with 8 tiles in the first figure, 15 tiles in the second figure, 22 tiles in the third figure. If this pattern continues, how many tiles will be in the 30th figure? Which figure will contain exactly 211 tiles? How many tiles are in the first 12 figures combined?
Definition:

Linear pattern: A pattern where the number of elements increases by a constant amount each time.

Pattern
8, 15, 22, ...
Rule
7n + 1
30th figure
7(30) + 1 = 211
Step 1: Identify the pattern rule

Figure 1: 8 tiles, Figure 2: 15 tiles, Figure 3: 22 tiles

Each figure adds 7 tiles: 8, 8+7, 8+7+7, ...

Pattern rule: 7n + 1 (where n is the figure number)

Step 2: Find tiles in 30th figure

Using the rule: 7(30) + 1 = 210 + 1 = 211 tiles

Step 3: Find which figure has 211 tiles

We already found that the 30th figure has 211 tiles

Let's verify: 7n + 1 = 211 → 7n = 210 → n = 30

Step 4: Find total tiles in first 12 figures

Using the rule for n=1 to n=12:

Sum = Σ(7n+1) from n=1 to 12 = 7×Σn + 12 = 7×78 + 12 = 546 + 12 = 558

Or: 8+15+22+29+36+43+50+57+64+71+78+85 = 558

Step 5: Verify the results

30th figure: 7(30) + 1 = 211 ✓

Figure with 211 tiles: n = 30 ✓

First 12 figures: 558 tiles ✓

30th figure: 211 tiles, 30th figure: 211 tiles, first 12: 558 tiles
Final answer:

The 30th figure will contain 211 tiles.

The 30th figure will contain exactly 211 tiles.

The first 12 figures contain 558 tiles combined.

Applied rules:

• Pattern recognition: Identify the constant difference between terms

• Formula creation: Develop a rule based on the pattern

• Series summation: Use arithmetic series formulas to find totals

🧩
Pattern Rule
Fig 1: 8
Fig 2: 15
Fig 3: 22
Fig 4: 29
Fig 5: 36

Rule: 7n + 1

🔍
Finding Nth Term
7n + 1
7(30) + 1
211 tiles
WJ Applied Problems: Key Concepts and Strategies
GCF(a,b) × LCM(a,b) = a × b
Relationship Formula
Key definitions:

Factor: A number that divides another number evenly

Multiple: The product of a number and an integer

Prime Number: A number with exactly two factors (1 and itself)

Composite Number: A number with more than two factors

Problem-solving strategies:
  1. Understand: Read the problem carefully and identify what is being asked
  2. Plan: Decide which mathematical concept to apply (factors, multiples, or patterns)
  3. Solve: Apply the appropriate method systematically
  4. Check: Verify your answer makes sense in the context of the problem
Tip 1: To find factors, try dividing by small prime numbers first.
Tip 2: For LCM, multiply the highest powers of all prime factors.
Tip 3: For patterns, look at the differences between consecutive terms.
Tip 4: Always draw a picture or make a table for complex word problems.
Common errors: Forgetting 1 and the number itself as factors, misidentifying pattern rules, miscalculating LCM/GCF.
Key to success: Practice identifying keywords in word problems, master basic multiplication facts, and verify all answers.
Essential formulas and rules:

• Factor pairs: If a×b = n, then a and b are factors of n

• Multiples: Multiples of n are n, 2n, 3n, 4n, ...

• GCF: Greatest common factor is the largest shared factor

• LCM: Least common multiple is the smallest shared multiple

• Pattern rule: Find the relationship between position and value

WJ Visualization: Patterns and Relationships
WJ ASSESSMENT Example 6: Visualizing Patterns
Consider the following pattern sequences:
Sequence A: 4, 9, 14, 19, 24, ... (start at 4, add 5 each time)
Sequence B: 6, 14, 22, 30, 38, ... (start at 6, add 8 each time)
Sequence C: 1, 4, 9, 16, 25, ... (perfect squares)
A:4
A:9
A:14
A:19
A:24
B:6
B:14
B:22
B:30
B:38
C:1
C:4
C:9
C:16
C:25

Analysis: Different sequences follow different rules and grow at different rates.

  • Sequence A: Arithmetic sequence (rule: 5n-1)
  • Sequence B: Arithmetic sequence (rule: 8n-2)
  • Sequence C: Perfect squares (rule: n²)

WJ Questions & Answers

Question: I don't understand how to find all the factors of a number. Can you explain the systematic way to do it?

Answer: Great question! Here's the systematic approach to find all factors of a number:

  1. Start with 1 and the number itself (since 1 × number = the number)
  2. Check 2: Does 2 divide the number evenly? If yes, write down 2 and its partner
  3. Continue checking 3, 4, 5, etc., up to the square root of the number
  4. For each divisor found, write down both the divisor and its partner

Example with 24:

  • 1 × 24 = 24 → factors: 1, 24
  • 2 × 12 = 24 → factors: 2, 12
  • 3 × 8 = 24 → factors: 3, 8
  • 4 × 6 = 24 → factors: 4, 6
  • 5 doesn't divide 24 evenly
  • Since √24 ≈ 4.9, we stop here

All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Question: How do I know when a word problem is about factors versus multiples?

Answer: Look for these key words and situations:

Factor problems usually involve:

  • "Arranging" items into equal groups (like arranging apples in boxes)
  • "Dividing" items equally among people or containers
  • Finding "how many ways" something can be arranged
  • Keywords: arrange, divide, share, distribute equally

Multiple problems usually involve:

  • Events happening at regular intervals (buses, trains)
  • Items bought in packages of certain amounts
  • Finding when things happen "together" or "at the same time"
  • Keywords: every, together, simultaneously, schedule

Remember: Factors divide a number, multiples are created by multiplying a number.

Question: I have trouble seeing patterns in number sequences. Any tips?

Answer: Here are effective strategies for recognizing patterns:

  1. Look at the differences: Subtract consecutive terms to see if there's a consistent change
  2. Check for multiplication: See if each term is multiplied by the same number
  3. Look for geometric patterns: Shapes or visual representations often follow predictable growth
  4. Make a table: Organize position vs. value to see relationships

Example: 6, 12, 18, 24, 30...

  • Differences: 12-6=6, 18-12=6, 24-18=6, 30-24=6
  • Pattern: Add 6 each time
  • Rule: 6n (multiply position by 6)

Practice with simple patterns first, then move to more complex ones!