Equal and Equivalent
5 min
Teacher Prep
Setup
Narrative
In this Warm-up, students use diagrams to explain why two expressions are or are not equal. Students look for and make use of the structure of the diagrams and expressions to explain their reasoning (MP7). Later, they will extend the reasoning used here to expressions with variables.
Monitor for students who use the length of the diagram (in grid units) to represent numbers and align their tape diagrams on one side to make it easier to compare them.
When discussing the connections between the diagrams and expressions, students have opportunities to share the language they have for describing the commutative properties of addition and multiplication. This may be some students’ introduction to the commutative property by name. Students informally used the commutative property in earlier grades as they made sense of operations and explained computation strategies. It is fine for students to continue to use informal language to describe properties of operations throughout the unit.
Launch
Give students 2 minutes of quiet work time, followed by a whole-class discussion.
Student Task
On the grid, draw diagrams that can represent each statement.
- 2+3 equals 3+2.
- 2⋅3 equals 3⋅2.
- 2+3 does not equal 2⋅3.
Sample Response
Sample response:
Activity Synthesis (Teacher Notes)
Select 1–2 students who used lengths to represent the numbers in the expressions to share their responses. Display their diagrams for all to see. Discuss questions such as:
- “How do the diagrams show whether the expressions are or are not equal?”
- “How is multiplication shown in the diagrams?” (3⋅2 is shown as 3 rectangles that are each 2 units in length. 2⋅3 is shown as 2 rectangles that are each 3 units in length.)
Highlight that the following key ideas:
- We can tell that 2+3 and 3+2 are equal because the length of the diagrams represent the value of each expression, and the diagrams are the same length. The same can be said about 2⋅3 and 3⋅2.
- We can tell that 2⋅3 and 3+2 are not equal because the lengths of the diagrams that represent them are not the same.
- 2+3 and 3+2 are examples of expressions that are not identical, but are equal in value.
Explain that the statements and diagrams in this activity demonstrate what students already know about addition and multiplication: that numbers can be added or multiplied in any order without affecting the result. Tell students to keep these ideas in mind later, when they look at whether expressions with variables are or are not equal.
If time permits, consider introducing the formal names of these properties and creating a display with the property names and examples of equations that illustrate them.
- The commutative property of addition states that the order of the addends (numbers being added) does not change the value of the sum, a+b is equal to b+a.
- The commutative property of multiplication states that the order of the factors does not change the value of the product, a⋅b is equal to b⋅a.
Standards
- 6.EE.4·Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). <em>For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.</em>
- 6.EE.A.4·Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). <span>For example, the expressions <span class="math">\(y + y + y\)</span> and <span class="math">\(3y\)</span> are equivalent because they name the same number regardless of which number <span class="math">\(y\)</span> stands for.</span>
20 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Equal and Equivalent
5 min
Teacher Prep
Setup
Narrative
In this Warm-up, students use diagrams to explain why two expressions are or are not equal. Students look for and make use of the structure of the diagrams and expressions to explain their reasoning (MP7). Later, they will extend the reasoning used here to expressions with variables.
Monitor for students who use the length of the diagram (in grid units) to represent numbers and align their tape diagrams on one side to make it easier to compare them.
When discussing the connections between the diagrams and expressions, students have opportunities to share the language they have for describing the commutative properties of addition and multiplication. This may be some students’ introduction to the commutative property by name. Students informally used the commutative property in earlier grades as they made sense of operations and explained computation strategies. It is fine for students to continue to use informal language to describe properties of operations throughout the unit.
Launch
Give students 2 minutes of quiet work time, followed by a whole-class discussion.
Student Task
On the grid, draw diagrams that can represent each statement.
- 2+3 equals 3+2.
- 2⋅3 equals 3⋅2.
- 2+3 does not equal 2⋅3.
Sample Response
Sample response:
Activity Synthesis (Teacher Notes)
Select 1–2 students who used lengths to represent the numbers in the expressions to share their responses. Display their diagrams for all to see. Discuss questions such as:
- “How do the diagrams show whether the expressions are or are not equal?”
- “How is multiplication shown in the diagrams?” (3⋅2 is shown as 3 rectangles that are each 2 units in length. 2⋅3 is shown as 2 rectangles that are each 3 units in length.)
Highlight that the following key ideas:
- We can tell that 2+3 and 3+2 are equal because the length of the diagrams represent the value of each expression, and the diagrams are the same length. The same can be said about 2⋅3 and 3⋅2.
- We can tell that 2⋅3 and 3+2 are not equal because the lengths of the diagrams that represent them are not the same.
- 2+3 and 3+2 are examples of expressions that are not identical, but are equal in value.
Explain that the statements and diagrams in this activity demonstrate what students already know about addition and multiplication: that numbers can be added or multiplied in any order without affecting the result. Tell students to keep these ideas in mind later, when they look at whether expressions with variables are or are not equal.
If time permits, consider introducing the formal names of these properties and creating a display with the property names and examples of equations that illustrate them.
- The commutative property of addition states that the order of the addends (numbers being added) does not change the value of the sum, a+b is equal to b+a.
- The commutative property of multiplication states that the order of the factors does not change the value of the product, a⋅b is equal to b⋅a.
Standards
- 6.EE.4·Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). <em>For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.</em>
- 6.EE.A.4·Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). <span>For example, the expressions <span class="math">\(y + y + y\)</span> and <span class="math">\(3y\)</span> are equivalent because they name the same number regardless of which number <span class="math">\(y\)</span> stands for.</span>