Embedded Files
Standard
Standard
Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
Student language:
- "I can find area by using what I know about multiplication and addition."
Explanation
Explanation
3.MD.7a
This standard asks students to tile areas of rectangles, determine the area, record the length and width of the rectangle, investigate the patterns in the numbers, and discover that the area is the length times the width.
For a 4 by 3 rectangle, students tile (section the rectangle into squares), number each square to count 12 square units. The same area could be found by multiplying the length and width (3 x 4 = 12).
3.MD.7b
This standard asks to find the area of a rectangle using multiplication. For example, a rectangle with side lengths 3cm and 4cm, a student could multiply 3 x 4 = 12. So the area is 12 square centimeters.
Note: Students should go back to tiling and counting squares if they forget why they multiply the length and width.
Example
Drew wants to tile the bathroom floor using 1 foot tiles. How many square foot tiles will he need?
3.MD.7c
This standard extends students’ work with the distributive property. For example, in the picture below the area of a 7 x 6 figure can be determined by finding the area of a 5 x 6 and 2 x 6 and adding the two sums.
Example
Joe and John made a poster that was 4ft by 3ft. Melisa and Debbie made a poster that was 4ft by 2ft. They placed their posters on the wall side-by-side so that that there was no space between them. How much area will the two posters cover?
3.MD.7d
This standard asks students to find area of various shapes composed of squares and rectangles. Students decompose a rectilinear figure into different rectangles. They find the area of the figure by adding the areas of each of the rectangles together.
Common Misconception
Students may confuse perimeter and area when they measure the sides of a rectangle and then multiply. They think the attribute they find is length, which is perimeter. Pose problems situations that require students to explain whether they are to find the perimeter or area.
Resources
Resources
Videos
Videos
- Khan Academy
- scroll to 3.MD.7, click on a skill, watch videos on left side in each skill
- Learn Zillion
EngageNY Lessons
EngageNY Lessons
Extra Practice
Extra Practice
- North Carolina Tasks
- Illustrative Math Tasks
- Howard County Tasks
- Click on 3rd grade at bottom, then 3.MD.7
- Common Core Sheets
PARCC
PARCC
Relate area to the operations of multiplication and addition. b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real-world and mathematical problems.
- Products are limited to the 10x10 multiplication table.
- This ES is different from 3.OA.3-1 in the following ways:
- 3.MD.7b-1 emphasizes application/skill while the emphasis of 3.OA.3-1 is on demonstration of understanding of multiplication using not only area but also equal groups and arrays by modeling.
- 3.MD.7b-1 permits mathematical problems while 3.OA.3-1 is restricted to word problems.
- 3.MD.7b-1 allows for factors less than or equal to 5 while the factors used in 3.OA.3-1 are restricted to the harder three quadrants.
Relate area to the operations of multiplication and addition.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
Performance Indicators: 3.MD.7b-1, 3.MD.7d
Performance Indicators: 3.MD.7b-1, 3.MD.7d
Level 5: Exceeds Expectations
- Recognizes area as an attribute of plane figures.
- Understands area is measured using square units. Describes a visual model to show understanding that area that can be found by covering a plane figure without gaps or overlaps with unit squares and counting them.
- Connects counting squares to multiplication when finding area.
- Represents the area of a plane figure as “n” square units.
Level 4: Meets Expectations
- Recognizes area as an attribute of plane figures.
- With a visual model, understands area is measured using square units. Determines area by covering a plane figure without gaps or overlaps with unit squares and counting them.
- Represents the area of a plane figure as “n” square units.
Critiquing Standards
Critiquing Standards
Base explanations/reasoning on the properties of operations.
(Content Scope: Knowledge and skills articulated in 3.MD.7)
- Tasks may include those with and without real-world contexts.
- Students need not use technical terms such as commutative, associative, distributive, or property.
Base explanations/reasoning on concrete referents such as diagrams (whether provided in the prompt or constructed by the student in her response).
(Content Scope: Knowledge and skills articulated in 3.MD.5, 3.MD.6, 3.MD.7)
- Tasks may include those with and without real-world contexts.
- Tasks with a context may present realistic or quasi-realistic images of a contextual situation (e.g., a drawing of a meadow). However, tasks do not provide the sort of abstract drawings that help the student to represent the situation mathematically (e.g., a tiling of the meadow).
3.C.4-5
3.C.4-5
Distinguish correct explanation/reasoning from that which is flawed, and – if there is a flaw in the argument – present corrected reasoning. (For example, some flawed ‘student’ reasoning is presented and the task is to correct and improve it.)
(Content Scope: Knowledge and skills articulated in 3.MD.7
- Tasks may include those with and without real-world contexts.
Present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equals signs appropriately (for example, rubrics award less than full credit for the presence of nonsense statements such as 1 + 4 = 5 + 7 = 12, even if the final answer is correct), or identify or describe errors in solutions to multi-step problems and present corrected solutions.
(Content Scope: Knowledge and skills articulated in 3.MD.7b, 3.MD.7d)
- Tasks may include those with and without real-world contexts.
- Multi-step problems have at least 3 steps.
Performance Indicators: 3.C.1-3
Performance Indicators: 3.C.1-3
Level 5: Exceeds Expectations
In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a complete written response based on explanations/reasoning using the:
- properties of operations
- relationship between addition and subtraction
- relationship between multiplication and division
- identification of arithmetic patterns
Response may include:
- a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
- an efficient and logical progression of steps with appropriate justification
- precision of calculation
- correct use of grade-level vocabulary, symbols and labels
- justification of a conclusion
- determination of whether an argument or conclusion is generalizable
- evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate). Provides a counterexample where applicable.
Level 4: Meets Expectations
In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a complete written response based on explanations/reasoning using the:
- properties of operations
- relationship between addition and subtraction
- relationship between multiplication and division
- identification of arithmetic patterns
Response may include:
- a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
- a logical progression of steps
- precision of calculation
- correct use of grade-level vocabulary, symbols and labels
- justification of a conclusion
- evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate).
Performance Indicators: 3.C.3-2
Performance Indicators: 3.C.3-2
Level 5: Exceeds Expectations
In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response based on operations using concrete referents such as diagrams- -including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include:
- a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
- an efficient and logical progression of steps with appropriate justification
- precision of calculation
- correct use of grade-level vocabulary, symbols and labels
- justification of a conclusion
- determination of whether an argument or conclusion is generalizable
- evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning, and providing a counterexample where applicable
Level 4: Meets Expectations
In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response based on operations using concrete referents such as diagrams--including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include:
- a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
- a logical progression of steps
- precision of calculation
- correct use of grade-level vocabulary, symbols and labels
- justification of a conclusion
- evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning.
Performance Indicators: 3.C.4-5
Performance Indicators: 3.C.4-5
Level 5: Exceeds Expectations
In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response by:
- presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
- evaluating explanation/reasoning; if there is a flaw in the argument
- presenting and defending corrected reasoning
Response may include:
- a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
- an efficient and logical progression of steps with appropriate justification
- precision of calculation
- correct use of grade-level vocabulary, symbols and labels
- justification of a conclusion evaluation of whether an argument or conclusion is generalizable
- evaluating, interpreting, and critiquing the validity of other’s responses, approaches and reasoning, and providing a counter-example where applicable.
Level 4: Meets Expectations
In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response by:
- presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
- distinguishing correct explanation/reasoning from that which is flawed
- identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems
- presenting corrected reasoning
Response may include:
- a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
- a logical progression of steps
- precision of calculation
- correct use of grade-level vocabulary, symbols and labels
- justification of a conclusion
- evaluating, interpreting and critiquing the validity of other’s responses, approaches and reasoning.
Performance Indicators: 3.C.5-2
Performance Indicators: 3.C.5-2
Level 5: Exceeds Expectations
In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response by:
- presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
- evaluating explanation/reasoning; if there is a flaw in the argument
- presenting and defending corrected reasoning
Response may include:
- a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
- an efficient and logical progression of steps with appropriate justification
- precision of calculation
- correct use of grade-level vocabulary, symbols and labels
- justification of a conclusion evaluation of whether an argument or conclusion is generalizable
- evaluating, interpreting, and critiquing the validity of other’s responses, approaches and reasoning, and providing a counter-example where applicable.
Level 4: Meets Expectations
In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response by:
- presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
- distinguishing correct explanation/reasoning from that which is flawed
- identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems
- presenting corrected reasoning
Response may include:
- a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
- a logical progression of steps
- precision of calculation
- correct use of grade-level vocabulary, symbols and labels
- justification of a conclusion
- evaluating, interpreting and critiquing the validity of other’s responses, approaches and reasoning.
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