Colorado Academic Standards Online
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clear Content Area: Mathematics - 2019 // Grade Level: Third Grade // Standard Category: 2. Algebra and Functions
Mathematics - 2019
Third Grade, Standard 2. Algebra and Functions
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Prepared Graduates:
- MP1. Make sense of problems and persevere in solving them.
- MP2. Reason abstractly and quantitatively.
- MP4. Model with mathematics.
Grade Level Expectation:
3.OA.A. Operations & Algebraic Thinking: Represent and solve problems involving multiplication and division.
Evidence Outcomes:
Students Can:
- Interpret products of whole numbers, e.g., interpret \(5 \times 7\) as the total number of objects in \(5\) groups of \(7\) objects each. For example, describe a context in which a total number of objects can be expressed as \(5 \times 7\). (CCSS: 3.OA.A.1)
- Interpret whole-number quotients of whole numbers, e.g., interpret \(56 \div 8\) as the number of objects in each share when \(56\) objects are partitioned equally into \(8\) shares, or as a number of shares when \(56\) objects are partitioned into equal shares of \(8\) objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as \(56 \div 8\). (CCSS: 3.OA.A.2)
- Use multiplication and division within \(100\) to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (see Appendix, Table 2) (CCSS: 3.OA.A.3)
- Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations \(8 \times \mbox{?} = 48\), \(5 =\mbox{_} \div 3\), \(6 \times 6 = \mbox{?}\) (CCSS: 3.OA.A.4)
Academic Contexts and Connections:
Colorado Essential Skills and Mathematical Practices:
- Solve problems involving multiples and parts using multiplication and division. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
- Make sense of missing numbers in equations by using the relationship between multiplication and division. (MP1)
- Reason abstractly about numbers of groups and the size of groups to make meaning of the quantities involved in multiplication and division. (MP2)
- Use arrays to represent whole-number multiplication and division problems. (MP4)
Inquiry Questions:
- How can an array be decomposed in a way that connects it to known multiplication facts? How can arrays be used to write and solve multiplication problems?
- How can the area and one side of a rectangle be used to write and solve a division problem?
- How could the number of dots in an array be counted without counting them one by one?
Coherence Connections:
- This expectation represents major work of the grade.
- In Grade 2, students work with equal groups of objects to gain foundations for multiplication.
- In Grade 3, this expectation connects to understanding properties of multiplication, the relationship between multiplication and division, and to fluently multiplying and dividing within 100.
- In Grade 4, students (a) use the four operations with whole numbers to solve problems, (b) build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers, and (c) solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. In Grade 5, students apply and extend previous understandings of multiplication and division to multiply and divide fractions.
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Prepared Graduates:
- MP3. Construct viable arguments and critique the reasoning of others.
- MP6. Attend to precision.
- MP7. Look for and make use of structure.
Grade Level Expectation:
3.OA.B. Operations & Algebraic Thinking: Apply properties of multiplication and the relationship between multiplication and division.
Evidence Outcomes:
Students Can:
- Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If \(6 \times 4 = 24\) is known, then \(4 \times 6 = 24\) is also known. (Commutative property of multiplication.) \(3 \times 5 \times 2\) can be found by \(3 \times 5 = 15\), then \(15 \times 2 = 30\), or by \(5 \times 2 = 10\), then \(3 \times 10 = 30\). (Associative property of multiplication.) Knowing that \(8 \times 5 = 40\) and \(8 \times 2 = 16\), one can find \(8 \times 7\) as \(8 \times \left(5 + 2\right) = \left(8 \times 5\right) + \left(8 \times 2\right) = 40 + 16 = 56\). (Distributive property.) (CCSS: 3.OA.B.5)
- Interpret division as an unknown-factor problem. For example, find \(32 \div 8\) by finding the number that makes \(32\) when multiplied by \(8\). (CCSS: 3.OA.B.6)
Academic Contexts and Connections:
Colorado Essential Skills and Mathematical Practices:
- Flexibly work with different but related arrangements of factors and products or dividends, divisors, and quotients. (Personal Skills: Adaptability/Flexibility)
- Use properties of operations to argue for or against the equivalence of different expressions. (MP3)
- Be specific with explanations and symbols when describing operations using multiplication and division. (MP6)
- Use the relationship between multiplication and division to rewrite division problems as multiplication. (MP7)
Inquiry Questions:
- What are all of the equations that can be written to represent the relationship between the area of a (specific) rectangle and its side lengths?
Coherence Connections:
- This expectation represents major work of the grade.
- In Grade 2, students work with equal groups of objects to gain foundations for multiplication.
- This expectation connects to other ideas in Grade 3: (a) multiplication and division within \(100\), (b) solving problems involving the four operations and identifying and explaining patterns in arithmetic, (c) understanding properties of multiplication and the relationship between multiplication and division, and (d) understanding concepts of area and the relationship to multiplication and division.
- In Grade 4, students use place value understanding and properties of operations to perform multi-digit arithmetic.
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Prepared Graduates:
- MP7. Look for and make use of structure.
Grade Level Expectation:
3.OA.C. Operations & Algebraic Thinking: Multiply and divide within 100.
Evidence Outcomes:
Students Can:
- Fluently multiply and divide within \(100\), using strategies such as the relationship between multiplication and division (e.g., knowing that \(8 \times 5 = 40\), one knows \(40 \div 5 = 8\)) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. (CCSS: 3.OA.C.7)
Academic Contexts and Connections:
Colorado Essential Skills and Mathematical Practices:
- Efficiently solve multiplication and division problems by using facts committed to memory. (Professional Skills: Task/Time Management)
- Recognize the relationship between skip counting and the solutions to problems involving multiplication and division. (MP7)
Inquiry Questions:
- How can I use multiplication facts that I know to solve multiplication problems I do not yet know? (for example, using \(5 \times 4 + 2 \times 4\) to solve \(7 \times 4\))?
- How can I use models and strategies to show what I know about multiplication?
Coherence Connections:
- This expectation represents major work of the grade.
- In Grade 2, students work with equal groups of objects to gain foundations for multiplication.
- In Grade 3, this expectation connects with representing and solving problems involving the four operations.
- In Grade 4, students use place value understanding and properties of operations to perform multi-digit arithmetic, solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit, and gain familiarity with factors and multiples.
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Prepared Graduates:
- MP3. Construct viable arguments and critique the reasoning of others.
- MP4. Model with mathematics.
- MP6. Attend to precision.
Grade Level Expectation:
3.OA.D. Operations & Algebraic Thinking: Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Evidence Outcomes:
Students Can:
- Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (This evidence outcome is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order of operations when there are no parentheses to specify a particular order.) (CCSS: 3.OA.D.8)
- Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations. For example, observe that \(4\) times a number is always even, and explain why \(4\) times a number can be decomposed into two equal addends. (CCSS: 3.OA.D.9)
Academic Contexts and Connections:
Colorado Essential Skills and Mathematical Practices:
- Solve problems involving the four operations. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
- Explain patterns in arithmetic. (MP3)
- Mathematically model changes in quantities described in real-world contexts using the appropriate numbers, operations, symbols, and letters to represent unknowns. (MP4)
- Complement arithmetic strategies with mental computation and estimation to assess answers for accuracy. (MP6)
Inquiry Questions:
- How can a visual model support making sense of and solving word problems?
- How can the patterns in addition and/or multiplication tables help predict probable solutions to a given problem?
Coherence Connections:
- This expectation represents major work of the grade.
- In Grade 2, students represent and solve one- and two-step word problems involving addition and subtraction.
- This expectation connects to several ideas in Grade 3: (a) representing and solving problems involving multiplication and division, (b) multiplying and dividing within \(100\), (c) solving problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects, and (d) using concepts of area and relating area to multiplication and to addition.
- In Grade 4, students use the four operations with whole numbers to solve problems.
Need Help? Submit questions or requests for assistance to bruno_j@cde.state.co.us