1. Progression of Arrays

2. This makes counting easier. Kindergarten: Module 1 Lesson 7
In kindergarten, when students first start counting objects, they begin to sort objects in vertical columns and horizontal rows.
This makes counting easier.
Kindergarten: Module 1 Lesson 7
Vertical Columns
Horizontal Rows
This sorting of objects
sets the foundation for students creating and
using arrays.

3. Kindergarten: Module 1 Topic E
Students engage in counting numbers above 5 in varied configurations. Students use their knowledge of counting up to 5 to reason about larger numbers in the more difficult linear, array, and scattered configurations.
Students can
easily count in
this array because they
learn to skip
count in pairs

4. My square covered with little squares.
Introduction of Area
Kindergarten: Module 3 Lesson 16
Students are
introduced to area
in kindergarten by
counting how many of different objects they can fit on a larger square or rectangle. These objects may be smaller squares or even beans just as long as they are set up as an organized array.
My square covered with little squares.

5. Count 10 objects within counts of 10 to 20 objects Kindergarten: Module 5 Lesson 2
Students know that this array represents 10 in the same way that they can read a sight word without reading the individual letters.

6. First Grade: Module 2 Lesson 1
Show how to add these objects together (three addends).
Based on their knowledge of a 10 frame or array, students can easily combine the triangles and the square to make a group of ten.
Make a “friendly number”.
Other examples:

7. 30 24 29 34 First Grade: Module 4 Lesson 1
Students will identify groups of ten and what is left over to make a number up to 40. 30 24 29 34

8. Second Grade: Module 1 Lesson 1
Ways to make ten:
Combine ten frame cards to find all the ways to make a ten.
Use the ten frame cards to find the missing addend.
Begin to represent the ways to make ten symbolically using number bonds

9. Second Grade: Module 6 Topic B
Arrays and Equal Groups
Second Grade: Module 6 Topic B
Having students draw the same number of objects in an array of horizontal rows as well as an array of vertical columns helps students understand that the orientation of the objects does not affect their value.

10. Second Grade: Module 6 Lesson 6
Decompose arrays to make a repeated addition sentence
Second Grade: Module 6 Lesson 6
Breaking apart the array into a repeated addition sentence builds the groundwork for moving into multiplication.

11. Even + Even = Even Odd + Even = Odd Odd + Odd = Even
Arrays to investigate odd and even numbers
Second Grade: Module 6 Lesson 20
4 + 4 = 8
Even + Even = Even
7 + 6 = 13
Odd + Even = Odd
3 + 3 = 6
Odd + Odd = Even

12. 3 6 5 3 15 3 Third Grade: Module 1 Lesson 2
Relate multiplication to an array model
Third Grade: Module 1 Lesson 2 3 6
Relate division to an array model
Third Grade: Module 1 Lesson 4 5 3 15 3

13. 2 rows of 9 and 9 rows of 2 are both equal to 18.
Commutative Property of Multiplication
Third Grade: Module 1 Lesson 8
2 rows of 9 and
9 rows of 2 are
both equal to 18.

14. Third Grade: Module 1 Lesson 11
Interpret the quotient as the number of groups or the number of
objects in each group
Third Grade: Module 1 Lesson 11
Finding the number of groups or the number of objects in each group is easier to visualize when it is in an array.
This array is easily translates into a tape diagram. 3 18

15. Third Grade: Module 4
Area model: a model for multiplication that relates rectangular arrays to area 4 3 12
Framing the objects into rectangles that form a larger rectangle
Solid rectangle that students can find the area in square units
Arranging objects into an array of rows and columns

16. Reasoning between arrays and written numerical work
Grade 4 Module 3
If three rows of four is equal to 12, then it is easy to see that three rows of 40 is equal to 120, three rows of 400 is equal to 1,200, and three rows of 4,000 is equal to 12,000.

17. Fourth Grade: Module 3 Lesson 22
Using arrays to find factors of a number
Fourth Grade: Module 3 Lesson 22
When introducing this skill it is helpful to have the students try to make even arrays of 4, 5, 7, and 8.

18. < Fourth Grade: Module 5 Lesson 15
Using arrays to compare fractions
Fourth Grade: Module 5 Lesson 15
<
When using arrays to compare fractions, it is important for the teachers and students to draw one where the fraction is represented as columns and the other one is represented as rows. This will help when students begin to use the arrays to add, subtract, multiply, or divide fractions or find parts of a number.

19. Fifth Grade: Module 2 Overview
Area model: a model for multiplication that relates rectangular arrays to area

20. of 12 Fifth Grade: Module 4 Lesson 6 What is 1/4 of 12? 3
Fractions of a Set (Parts of a Number)
Fifth Grade: Module 4 Lesson 6
of 12 9 6 3
What is 1/4 of 12?
What is 2/4 of 12?
What is 3/4 of 12?

21. X X = Fifth Grade: Module 4 Lesson 13 2 12 Multiplying Fractions
How many parts of the array overlap? 2
Since they have been comparing fractions drawing one in columns and the other in rows, it is a natural progression to draw one fractional representation on top of the other fractional representation to visually see the product of the two fractions.
How many total parts of the array? 12 X =

22. Sixth Grade: Module 1 Lesson 25
Connect percents to fractions
Sixth Grade: Module 1 Lesson 25
In this example, representing the percent as a 5 X 20 array, allows the students to visualize that 80% and 4/5 represent the same portion of the whole.

23. 4 4 3 20 = Sixth Grade: Module 2 Lesson 1
Division of fractions by whole numbers
Sixth Grade: Module 2 Lesson 1 4 3
How many shaded parts are in each group?
How many total parts of the array? 20 4 =

24. Sixth Grade: Module 4 Lesson 8
Commutative Property of Multiplication
Sixth Grade: Module 4 Lesson 8
Students will see that 3 groups of 4 and 4 groups of 3 represent the same value.

25. -12 3 X (-4) -1 -1 -1 -1 -1 -1 -1 -1 Seventh Grade: Module 2 Lesson 10
Multiplication of Integers
Seventh Grade: Module 2 Lesson 10
3 X (-4)
-12

26. Distributive Property
Seventh Grade: Module 3 Lesson 3
4(3 + 2) 3 2 4 12 8
+ 8 = 20
(4 X 3) + (4 X 2)

27. Distributive Property (Cont.)
Seventh Grade: Module 3
Eighth Grade: Module 4
5(8x + 3) 8x 3
x x x x x x x x
x x x x x x x x 5
40x 15
x x x x x x x x
x x x x x x x x
x x x x x x x x
40x + 15

28. Eighth Grade: Module 7 Lesson 15
Pythagorean Theorem
Eighth Grade: Module 7 Lesson 15
a² + b² = c²
3² + 4² = 5²
8.G.B.6: Explain a proof of the Pythagorean Theorem and its converse.

29. Multiplying Binomials and Polynomials
High School Algebra
(a + b)² = (a + b)(a + b)

30. Multiplying Binomials and Polynomials
High School Algebra
(a + b + 1)(b + 1)

31. Multiplying Binomials and Polynomials
High School Algebra
(a + b)(c + d)(e + f + g)