# Operation Meaning and Relationships

**The same number sentence can be associated with different concrete or real-world situations**

**AND**

*different*** number sentences can be associated with the same concrete or real-world situation.**

Example of the idea that *same* number sentence can be associated with *different *concrete or real-world situations:

I have 11 bags of cookies. Each bag has 9 cookies in it. How many cookies do I have all together?

Dan has a bathroom that is 11 ft long and 9 ft wide. How many square feet is his bathroom?

For both situations you could use the equation 11×9 to get the answer 99. Even though they are different situations the same number sentence is used to solve the answer.

Example of the idea that different number sentences can be associated with the same concrete or real-world situation.

I have 11 bags of cookies. Each bag has 9 cookies in it. How many cookies do I have all together?

There are different number sentences that could be used to solve this problem. Some of these number sentences:

11×9=99

11+11+11+11+11+11+11+11+11=99

(9×10)+9=99

### Examples of Mathematical Understandings

**Whole numbers**

Some real-world problems, involving the different problem types, can be solved using addition and others using subtractions.

Example of using same number sentence with different situations

Joining Result Unknown: Suzy had 4 dolls. Lucy gave her 8 more. How many dolls does she have altogether?

Equation: 4+8=12

Separating Start Unknown: Billy had some baseballs. He gave 4 to John. How he has 8 baseballs left. How many baseballs did Billy start with?

Equation: 4+8=16

Example of using different number sentences with the same situation

Part-Part-Whole Part Unknown: Cami had 12 marbles, 4 are red and the rest are yellow. How many yellow marbles does Cami have?

Equations: 12=4+8

12-4=8

Adding x is the inverse of subtracting x

Example of using different number sentences with the same situation

I am on floor 10 of a building. Before I was on a higher floor. If I wouldn’t have taken the elevator 5 floors down what floor would I be on?

Equations: 10+5=15

10-(-5)=15

Any subtraction calculation can be solved by adding up from the subtrahend

Adding quantities greater than zero gives a sum that’s greater than any addend

Subtracting a whole number from another whole number gives a different that’s less than the minuend

Some real-world problems, including the different problem types, can be solved using multiplication and other using division

Example of using same number sentence with different situations

Price Measure Division: A pair of shoes cost $20 each. How many pairs of shoes would cost $120?

Equation: 120÷20=6

Multiplicative Comparison Measurement Division: In Maui the average yearly rainfall is 120″. Last year they only received 20″. How many times greater is the average yearly rainfall compared to last year?

Equation: 120÷20=6

Multiplying by x is the inverse of dividing by x

Multiplying two whole numbers greater than one gives a product greater than either factor

**Rational Numbers (Fractions and Decimals)**

The real-world actions for addition and subtraction of whole numbers are the same operations with fraction and decimals.

Different real-world interpretations can be associated with the product of a whole number and fraction (decimal), a fraction (decimal) and a whole number, and a fraction and fraction (decimal and decimal)

Different real-world interpretations can be associated with division calculations involving fractions (decimals)

The effects of operations for addition and subtraction with fraction and decimals are the same as those with whole numbers

The product of two positive fractions each less than one is less than either factor

Example using same number sentence with different situations

My dad asked me to mow a third of the lawn. I only mowed half of what I was asked to mow. How much of the whole lawn did I mow?

Equation: 1/3 x 1/2 = 1/6

Jill got a third of the pizza. She ate half of that third. How much of the whole pizza did she eat?

Equation: 1/3 x 1/2= 1/6

**Integers:**

The real-world actions for operations with integers are the same for operations with whole numbers

#### The big ideas that the same number sentence can be associated with different concrete or real-world situations and that different number sentences can be associated with the same concrete or real-world situation can be taught naturally through inquiry lessons. Students will solve real-life situations in different ways without the teacher telling them to.

### Activities:

#### Choosing Number Sentences

Present the students with a problem and have them choose which of the number sentences provided could be used to solve the problem. Ask why the number sentences chosen can be used to solve the problem.

Example:

Diego saved $96 this month by doing odd jobs for the neighbors. Last month, he saved $8. How many times as much money did he save this month as last month?

96 × 8 = _ _ = 8 × 96 8 × _= 96

96 ×_ = 8 96 ÷8 = _ 8 ÷96 = _

_÷8 = 96 96 ÷ _= 8 8 ÷ _ = 96

**Similarities and Differences**

Provide the students with two problems using the same numbers but different meanings for multiplication or division, such as one showing equal grouping and one showing the equal sharing. Ask the students to explain how the problems are the same and how they are different. They may wish to put their explanations in a graphic organizer.

An example of a problem showing equal grouping:

You have 75 pictures put into albums. If each album page holds 4 pictures, how many pages do you need? Explain how you know.

An example of a problem showing equal sharing:

You have 75 marbles to share equally among 4 friends. How many marbles will each friend receive? Explain how you know.

Sue Willis et al. *First Steps in Mathematics: Operation Sense-Operations, Computations, and Patterns and Algebra*. Canadian Edition, page 36

Resources:

Ideas for lessons: http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/structure/st325ip.htm

Big Ideas: http://www4.uwm.edu/Org/mmp/PDFs/Charles-Big%20Ideas_NCSM_Spr05v7(3)p9-24.pdf

Carpenter, T. P. (1999). Children’s mathematics: Cognitively guided instruction. New Hampshire: Heinemann.

Sue Willis et al. *First Steps in Mathematics: Operation Sense-Operations, Computations, and Patterns and Algebra*. Canadian Edition, page 36