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 multidigit arithmetic.
CCSS.MATH.CONTENT.4.NBT.B.5
Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit 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 wholenumber quotients and remainders with up to fourdigit dividends and onedigit 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 wholenumber 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.
Number and Operations in Base Ten
Use place value understanding and properties of operations to perform multidigit arithmetic.
CCSS.MATH.CONTENT.4.NBT.B.5
Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit 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 wholenumber quotients and remainders with up to fourdigit dividends and onedigit 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 wholenumber 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.
Let the investigation begin!!
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 1100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1100 is a multiple of a given onedigit number. Determine whether a given whole number in the range 1100 is prime or composite.
Anchor Investigation: Candy Boxes (136)
Students explore the number system through a rectangular array investigation. As a class project, they make all the different rectangular arrays for the numbers 136. They post the arrays for each number sequentially on the wall (136) 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:
 Centimeter Paper
 T2.Centimeter Grid Paper1.pdf(relink 2cm paper)
 HalfSheets of Construction Paper (cut vertically), with large number written at the top of each sheet (Numbers: 136)
 Scissors
 Glue Sticks
 Chart Paper
 Tape
Investigations Session 1.1: Things Than Come in Arrays
Students consider how examples of arrays from the real world represent multiplication.
Materials:
Students consider how examples of arrays from the real world represent multiplication.
Materials:
 T1.Things that Come in Arrays.pdf
 T1.Things that Come in Arrays SP.pdf
 M1Things_ Arrays.pdf
 M1.pdf.SP Cosas que van en matrices.pdf
 M3M4. Family Letter.pdf
 M3M4.pdf.Carta a la Familia.pdf
 Color Tiles or Connecting Cubes


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.
 T2.Centimeter Grid Paper SP.pdf
 Color Tiles or Connecting Cubes (as needed)
 12 x 18 construction paper (2 sheets per pair, plus extras)
 Scissors (1 per pair)
 Glue Sticks (1 per pair)
 Markers (1 per pair)
 *Session 1.3 also uses: "Ways to Make" posters (see Session 1.2) AND Materials for making arrays (See Session 1.2)
 M5M6.pdf Family Letter.pdf
 M5M6.pdf.SP Carta a la Familia.pdf
 M7M8.pdf Family Letter.pdf
 M7M8.pdf.SP Carta a la Familia.pdf
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.
 T5.pdf Array Cards.pdf
 T5.pdf.SP Tarjetas de Matrices.pdf
 T15.pdfArray Cards.pdf
 T15.pdf.SP Tarjetas de Matrices.pdf
 T18 Array Cards.pdf
 T18.pdf.SP Array Cards.pdf
 T23.pdf Array Cards.pdf
 T23.pdf.SP Array Cards.pdf
 M9M29.pdf Array Cards.pdf
 M9M29.pdf.SP Tarjetas de Matrices.pdf
 M30.pdf Factor Pairs.pdf
 M30.pdf.SP Pares de Factores.pdf
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 1112
Homework: page 10
 M31.pdf Representing 8 X 6 Assessment.pdf
 M31.pdf.SP Evaluación.pdf
 M32.pdf Assessment Checklist.pdf
 M32.pdf.SP Lista de Comprobación de 8 x 6.pdf
 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
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 Tchart from page 63 in T.E.
 C2 Multiplicative Comparison Problems.pdf
 C3 Multiplicative Comparison Problems.pdf
 C4 Multiplicative Comparison Problems.pdf
100 Hungry Ants (another rich investigation with arrays of 100)
100 hungry ants_factor game_silent multiplication1.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.
Quick Images Videos
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.
 T24.pdf Quick Images1.pdf
 T25.pdf Quick Images1.pdf
 T24.pdf SP Miradas Rapidas.pdf
 T25.pdf.SP Miradas Rapidas.pdf
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.
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.
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.
 M35M40.pdf Multiplication Cards.pdf
 M35M40.pdf Tarjetas de Multiplicación.pdf
 M41.pdf Blank Multiplication Cards.pdf
 M41.pdf Tarjetas de Multiplicación en Blanco1.pdf
 M42.pdf Practicing with Multiplication Cards.pdf
 M42.pdf Instrucciones para practicar con tarjetas de multiplicación.pdf
 M43M44.pdf Family Letter.pdf
 M43M44.pdf Carta a la Familia.pdf
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.
 M41.pdf Blank Multiplication Cards1.pdf
 M41.pdf Tarjetas de Multiplicación en Blanco.pdf
 M45.pdf Multiple Turnover Game Instructions.pdf
 M45.pdf.SP Baraja de Multiplos.pdf
 M46M49.pdf Multiple Cards.pdf
 M46M49.pdf.SP Cartas de Múltiplos.pdf
 M50.pdf Multiple Turn Over Recording Sheet.pdf
 M50.pdf.SP baraja de Múltiplos, hoja de anotaciones.pdf
 Chart Paper
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 Notes1.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 onedigit 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.
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
U1 Investigations 2 and 3 CC Alignment Notes1.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 onedigit 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.
 M2.Centimeter Grid Paper.pdf
 M53.pdf Four 100 charts.pdf
 M53.pdf.SP Cuatro Tablas de 100.pdf
 M54.pdf 300 Chart.pdf
 M54.pdf 300 Chart1.pdf
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
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


Session 3.4.: End of Unit assessment
M55M56.pdf End of the Unit Assessment.pdf
M55M56.pdf.SP Evaluación de final de la Unidad.pdf
M55M56.pdf End of the Unit Assessment.pdf
M55M56.pdf.SP Evaluación de final de la Unidad.pdf