# Multiplication and Division of Whole Numbers

## Objective

Evaluate numerical expressions involving addition, subtraction, multiplication, division, and grouping symbols.

## Common Core Standards

### Core Standards

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• 5.OA.A.1 — Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

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• 3.OA.D.8

## Criteria for Success

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1. Understand that an expression is a mathematical phrase that contains operators, numbers, and/or letters that represent unknowns, and distinguish it from an equation, which is two expressions that are equal to one another.
2. Understand that a letter can be used to represent an unknown in an equation. (Spiral from Grade 4.)
3. Understand that the order of operations is a convention that says operations should be performed from left to right in the following order (MP.6):
1. Grouping symbols
2. Multiplication and division
4. Understand that grouping symbols are used as a way to indicate when an operation should be performed out of its usual order, e.g., addition before multiplication (which is why parentheses are not necessary when the operations would otherwise be performed in the conventional order).
5. Evaluate an expression using the order of operations, recording the computations as sequential equations that are simplified from one line to the next so that work stays organized.
6. Evaluate an expression that contains a letter as an unknown, given the value of that unknown.

## Tips for Teachers

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• Students may believe the order in which a problem with mixed operations is written is the order in which they are to solve the problem. The use of acronym PEMDAS or GEMDAS can mislead students to always perform multiplication before division and addition before subtraction. At this level, students need opportunities to explore the power of the various operations on numbers and solve equations starting with the operation of greatest power. Because multiplication and division have an equal impact (i.e., they increase or decrease at the same rate), they are performed together. Because they have a greater impact than addition and subtraction (i.e., multiplication and division increase or decrease at a greater rate than adding or subtracting), multiplication and division are performed before addition and subtraction. Thus, the impact an operation has on the solution dictates the order in which they are performed, with grouping symbols used to indicate where one should override that order.
• Students don’t seem to need to know that a value directly before grouping symbols implies multiplication, e.g., $8(9 + 3)$. All released PARCC, SBAC, Illustrative Mathematics, and NYS items include the operator. Thus, all tasks included in this lesson include an operator before parentheses.
• While the Progressions note that “expressions should not contain nested grouping symbols,” multiple examples of nested grouping symbols exist on released PARCC and MCAS tests (OA Progression, p. 32). Thus, it has been included below.
• A supplement to the Problem Set, we recommend the following resources:
• Bowling for Numbers by Illustrative Mathematics
• Twin Puzzles by mathycathy on Desmos (similar to the Illustrative Mathematics game above)
• Similarly, you can buy the game "24 Game" on Amazon. The advantage to this game is that it can be played throughout the year without requiring other materials aside from the game itself. It can also be played when there are just a few minutes to spare, say when a lesson ends early or while waiting for dismissal. You can more directly align this game to today’s objective by having students write down the expressions that they come up with, making sure students include parentheses where necessary.

#### Fishtank Plus

• Problem Set
• Student Handout Editor
• Vocabulary Package

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### Problem 1

Felix and Julysa are computing 3 + 4 x 4. Felix says the answer is 28. Julysa says the answer is 19.

1. How did Felix get his answer?
2. How did Julysa get her answer?
3. Who is correct?

### Problem 2

Evaluate.

a.     ${(10-3) \times 2 + 8}$

b.     ${(1+2) \times (3+4)}$

c.     ${3 \times [20\div(2+3)]}$

### Problem 3

1. Evaluate ${y\times(5-2)}$ when ${y=8}$.
2. Evaluate ${(3+5)\div(h-12)}$ when ${h=14}$.

## Discussion of Problem Set

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• Look at your Problem Set. Did anyone answer #3g without evaluating the expressions? How were you able to do that?
• When evaluating the expression in #5, a student got 18. Can you identify the error in thinking?
• Look at #7. How did you know what to evaluate first with so many sets of grouping symbols?
• Why do we use parentheses in mathematical expressions? When is it important to use parentheses? When are parentheses not necessary?

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### Problem 1

What is the value of this expression?

$100 - [5 \times (3 + 4)]$

#### References

PARCC Released Items Math Spring Operational 2015 Grade 5 End of Year Released ItemsQuestion #25

### Problem 2

What is the value of the expression below when $p=10$?

${(20+30)\div p}$

A.     2

B.     5

C.     23

D.     60

#### References

Massachusetts Department of Elementary and Secondary Education Spring 2013 Grade 5 Mathematics TestQuestion #1

Spring 2013 Grade 5 Mathematics Test is made available by the Massachusetts Department of Elementary and Secondary Education. © 2017 Commonwealth of Massachusetts. Accessed Dec. 5, 2017, 3:51 p.m..

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