NVACS: Operations and Algebraic Thinking

Addition, Subtraction, Multiplication, and Division

Concept Overview for 4th Graders

4.OA.1

Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 X 7 as a statement that 35 is 5 times as many as 7 as 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.2

Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with symbols for the unknown number to represent the problem, distinguish multiplicative comparison from additive comparison.

4.OA.3

Solve multistep word problems posed with whole numbers and having while-numbers 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.

4.OA.4

Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

4.OA.4

Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.

In grade 3-5 students need to develop a stronger understanding of the various meanings of different types of multiplication and division problems. These are called Problem Situations. Using algebraic reasoning in grades 3-5 prepares students for algebra in middle school.

**Different Problem Situations for Multiplication and Division**

❏**Multiplication Equal Grouping Problems: **One factor tells the number of things in a group and the other factor tells the number of equal sized groups.

❏**Array Problems: **This type of problem is often known as an area problem. In an array problem the role of the factors is interchangeable.

❏**Multiplicative Comparison:** One number identifies the quantity in one group or set while the other number is the comparison factor.

❏**Division Equal Group Problems:**

❏**Quotative Division:** Sometimes referred to as repeated subtraction, the number of objects in each group is known, but the number of groups is unknown.

❏**Partitive Division:** Dividing a set into a predetermined number of groups

**Examples of Different Problem Situations for Multiplication and Division**

❏**Multiplication Equal Grouping Problems: **

I have 5 friends that need 4 cookies each.

How many cookies to I need to bake? 5 x 4 = 20

❏Array Problems:

How many desks would there be in the classroom if I had 6 rows of 7 desks. 6 x 7 = 35

❏**Multiplicative Comparison:**

Jill wrote ten pages for the group assignment. Sara wrote 5 times as many. How many pages did Sara write?

❏**Division Equal Group Problems:**

**Quotative Division:**

I have 30 popsicles and I want to give 5 to each person. How many people will get popsicles?

**Partitive Division:**

Mark has 24 apples. He wants to share equally among his 4 friends. How many apples will each friend receive?

**Letter representing an unknown quantity**

In earlier grades students solved open sentences with boxes for the unknown or missing value. In the upper grades of elementary students need to transition to variables from the use of boxes. 5 + n = 12. In elementary school students should use relational thinking to find the value of the variable. Context (story problems) can help students develop the meaning of variables.

David ate 23 grapes and Sue ate some, too. The container of grapes had 51 and they were all gone! How many did Sue eat? 23 + g = 51 or 51 - 21 = g

Estimation

Students need to determine when estimation is appropriate, what the level of accuracy is needed, selecting the appropriate method of estimation, and verifying solutions or determining the reasonableness of situations using various estimation strategies.

**Interpreting Remainders**

Students need to understand that a remainder in a division problem can have different meaning based on the context of the problem.

**Examples of word problems to show interpreting remainders for 44 **6 = p

❏Remain as a left over

Maria had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches could she fill and how many pencils would she have left? 44 6 = p; p = 7 r. 2, Maria can fill 7 pouches with 2 left over.

Answer: 7 R. 2

❏Partitioned into fractions or decimals

Maria had 44 pencils and put 6 pencils in each pouch. What fraction represents the number of pouches that Mary filled? 44 6 = p = 7

Answer: 7

❏Discard leaving only the whole number answer

Maria had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches did she fill? 44 6 = p; p = 7 R. 2, Maria can fill 7 pouches completely.

Answer: 7

❏Increase the whole number answer up one

Maria has 44 pencils. Six pencils fit into each of her pencil pouches. What would be the fewest number of pouches she would need in order to hold all of her pencils? 44 6 = p; p = 7 R. 2; Maria needs 8 pouches to hold all the pencils.

Answer: 8

❏Round to the nearest whole number for an approximate result

Maria had 44 pencils. She divided them equally among her friends before giving one of the leftovers to each of her friends. How many pencils could her friends have received? 44 6 = p; p = 7 R. 2; some of her friends received 7 pencils and some of her friends received 8. Answer: 7 or 8

Prime

Prime numbers have exactly 2 factors, the number one and their own number.

Example: 17, Factors of 17 are 1 and 17.

Composite

Composite numbers have more than 2 factors.

Example: 24, Factors of 24 are 1,2,3,4,6,8,12,24.

Factor

Are numbers you multiply together to find the product.

Example: 2 x 3 = 6, two and three are factors of 6.

Factor Pairs

The factor pairs are the two numbers you multiply together to find the product.

Example: Factor pairs for 96 are, 1 and 96, 2 and 48, 3 and 32, 4 and 24, 6 and 16, 8 and 12

**Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit.**

Shape Pattern /

Rule / Each number in the pattern can be described by multiplying the number of the pattern sequence by itself. For the 5th shape in the pattern you multiply 5 by 5 and the answer is 25.

n x n = 25. n= the number in the sequence.

Features / The numbers in this pattern can be used to build arrays that are squares. The dimensions in the pattern match the number of the pattern in the sequence. The 3rd shape in the pattern has 3 as the dimension.

RPDP.net

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Some of the examples in the concept overview were modified and used from the North Carolina Unpacking Document.