Factors, Multiples, and Arrays


Curriculum Resources for this Unit
(Investigations Unit 1/Math Expressions Unit   )
                                                      Common Core Standards
Number and Operations in Base Ten
​Use place value understanding and properties of operations to perform multi-digit arithmetic.
CCSS.MATH.CONTENT.4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
CCSS.MATH.CONTENT.4.NBT.B.6
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

1 Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Operations and Algebraic Thinking
​Use the four operations with whole numbers to solve problems.
CCSS.MATH.CONTENT.4.OA.A.1
Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
CCSS.MATH.CONTENT.4.OA.A.2
Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.1
CCSS.MATH.CONTENT.4.OA.A.3
Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
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Let the investigation begin!!
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​Math Expressions: Unit 2 - Multiplication with Whole Numbers

Investigations Unit 1: Factors, Multiples, and Arrays

​U1 Investigation 1 CC Alignment Notes.pdf (upside down, we need to fix)

Representing Multiplication with Arrays  (5 to 7 Days) 


​​ 4.OA.B.4 (Links to an external site.)Links to an external site.
Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.


Anchor Investigation: Candy Boxes (1-36)
Students explore the number system through a rectangular array investigation.  As a class project, they make all the different rectangular arrays for the numbers 1-36.  They post the arrays for each number sequentially on the wall (1-36) and look for patterns as well as make observations.  They write conjectures about different types of numbers.  For example:  Prime numbers have only one rectangle.  Square numbers always have a perfect square array. etc. *Tip:  Have them glue the the rectangles horizontally from a common starting point on the strip of paper and in order from least to greatest length.  It will help them to discover structure in the number system. 

Candy Boxes Investigation.pptx

Student Activity Book: 3
Homework: Student activity book page 4 
Tips: Use 2cm grid paper. Use two light colors of construction paper. One for prime numbers and one for composite numbers. Model with the class number 16 and how to write the expression before. Assign students as quality control to check the arrays and expressions for accuracy.   
Recommend: Laminate and display numbers all year long.
Conjectures: are something you think will always be true and you can prove it.
Additional Candy Boxes Exploration.docx
Materials: VIDEO: Understanding Arrays: Rows and Columns
     
​Investigations Session 1.1: Things Than Come in Arrays
Students consider how examples of arrays from the real world represent multiplication.
Materials:Real World examples of Arrays! Can you think of other examples?

Math Expressions: Lesson 1 - Arrays and Area Models
Investigations Session 1.2. and Session 1.3.: Making Arrays
Students use what they know about multiplication to find all the arrays for the given multiplication. Students come up with conjectures and discuss special features of some numbers, including prime and squared numbers. 
Student Activity Book: 3
Homework: page 4
Recommendations:
  • Have student use ipads to take picture of arrays around the classroom/school and share them with the class.
  • Laminate one set of array cards for teacher.
Materials:  
VIDEO LINK Prime and Composite Numbers
    


(possibly include photos of chart)


Session 1.4: Which combinations Do I Know?
In the Array Card Game Factor Pair, Students build on known multiplication combination to find the product of more difficult combinations to 12 by 12. 
Student Activity Book: page 7
Homework: page 8
Recommendations: Give partners two post it notes, one for "know," and one for "don't know yet." Use 2 quart size bags so students can keep track of these categories for the next day. Students work with 20 arrays at a time.  
​Finding Factor Pairs
      
Investigations Session 1.5.: Playing Factor Pairs
Continue playing Factor Pairs and solving array problems 
Number Talk: Activity Book page 9
Student Activity Book: pages 11-12
Homework: page 10 After the M31-32 assessment, sort the story problems by type of problem: Array, area and groups of and discuss the differences between them. Illustrating them.
  • Exit Slip: Pick one of the array cards that you don't know yet and show some strategies how find the total on the grid paper.


Math Expressions: Lesson 2 - Connecting Place Value and Multiplication
Investigations Session 1.5A: Place Value Understanding
Session 1.5A Place Value Understanding.pdf (this is in the wrong unit)


Flipped Classroom: Understanding Place Value
      
  • Session 1.6A: Multiplicative Comparison
Session 1.6A Multiplicative Comparison.pdf
Materials: Factor pair game work: Have students identify their five most difficult array combinations they are working on and chose one to work on with strategies you could use to find the product. Make a T-chart from page 63 in T.E.

100 Hungry Ants (another rich investigation with arrays of 100)
100 hungry ants_factor game_silent multiplication-1.pptx (modify for students as this PPT is meant for PD with teachers)
100 Hungry Ants.docx
Read aloud online click below:
100 Hungry Ants Read Aloud (Links to an external site.)Links to an external site.

Multiplication Combinations
Investigations Unit 1: Multiplication, Factors, Multiples, and Arrays
U1 Investigations 2 and 3 CC Alignment Notes.pdf
4.OA1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

​Investigations Session 2.1.: Quick images
Students are introduced to the ten minute activity Quick images, in which they describe multiplication situations and discuss strategies of approach.  Seeing Numbers (Cut Outs on card stock)
Quick Images Videos
Chart Paper; Divide into two columns, and label them "Combinations we are working with" and "Start With".

Math Expressions: Lesson 3 - Mental Math and Multiplication
Investigations Session 2.2.: Multiplication Cards
Students create a set of Multiplication Cards to practice and become fluent with all multiplication combinations to 12 x 12.
Investigations Session 2.3.: Multiple Turn Over
Students review the terms factor and multiple and play a game Multiple Turn Over that provides experience with factors and multiples.Investigations Session 2.4.: Multiplication Combination
Students discuss strategies for identifying factors and multiples in Multiple Turn Over. They work on three different math workshop activities that focus on becoming fluent with multiplication combinations.
 Note: Use the same resources used in session 2.2 and 2.3 
Investigations Session 2.5.: Assessment; multiplication combinations
Students continue to work on three math workshops activities that focus on becoming fluent with multiplication combinations. They also complete an assessment activity that focuses on multiplication combination through 12 x 12.
(You can find these lessons in the TA of Investigations Unit 1: Factors, Multiples, and Arrays)

U1 Investigations 2 and 3 CC Alignment Notes-1.pdf
Finding Factors
4.OA.4 Gain familiarity with factors and multiples: Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
Session 3.1.: Factors of 100
Students are introduce to an activity Counting Around the Class in which they generate the multiples of a given number. they consider factors of 100.  Chart Paper
Session 3.2 Factors of the Multiples of 100. (Actually this comes from Investigations Unit 1, Session 3.2)
Students find the factors of 200 and 300 and other multiples of 100 by reasoning of the factors of 100.
​Note: Use the same resources masters as Session 3.1
Determining Multiples
Session 3.3: Factors of Related Numbers
Students identify and represent the Factors of 16 and 48 to explore the concept that the factors of a number are also factors of a multiple of that number.
Use Centimeter Grid Paper (see session 3.1)
Factor Finding _ Using the Rainbow method
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​Session 3.4.: End of Unit assessment 
M55-M56.pdf End of the Unit Assessment.pdf
M55-M56.pdf.SP Evaluación de final de la Unidad.pdf