Colorado Academic Standards Online
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clear Content Area: Mathematics - 2019 // Grade Level: Fourth Grade // Standard Category: 1. Number and Quantity
Mathematics - 2019
Fourth Grade, Standard 1. Number and Quantity
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Prepared Graduates:
- MP7. Look for and make use of structure.
Grade Level Expectation:
4.NBT.A. Number & Operations in Base Ten: Generalize place value understanding for multi-digit whole numbers.
Evidence Outcomes:
Students Can:
- Explain that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that \(700 \div 70 = 10\) by applying concepts of place value and division. (CCSS: 4.NBT.A.1)
- Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using \( \gt \), \( = \), and \( \lt \) symbols to record the results of comparisons. (CCSS: 4.NBT.A.2)
- Use place value understanding to round multi-digit whole numbers to any place. (CCSS: 4.NBT.A.3)
Academic Contexts and Connections:
Colorado Essential Skills and Mathematical Practices:
- Write multi-digit whole numbers in different forms to support claims and justify reasoning. (Entrepreneurial Skills: Literacy/Writing)
- Use the structure of the base-ten number system to read, write, compare, and round multi-digit numbers. (MP7)
Inquiry Questions:
- How do base ten area pieces or representations help with understanding multiplying by \(10\) or a multiple of \(10\)? How can base ten area pieces be used to represent multiplying by \(10\) or a multiple of \(10\)?
- Imagine two four-digit numbers written on paper and some of the digits were smeared. If you saw just \(325\blacksquare\) and \(331\blacksquare\), could you determine which number was larger?
- When is it helpful to use a rounded number instead of the exact number?
Coherence Connections:
- This expectation represents major work of the grade.
- In Grade 3, students use place value understanding and properties of operations to perform multi-digit arithmetic.
- In Grade 4, this expectation connects to using the four operations with multi-digit whole numbers to solve measurement and other problems.
- In Grade 5, students extend their understanding of place value to decimals, and read, write, and compare decimals to thousandths.
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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:
4.NBT.B. Number & Operations in Base Ten: Use place value understanding and properties of operations to perform multi-digit arithmetic.
Evidence Outcomes:
Students Can:
- Fluently add and subtract multi-digit whole numbers using the standard algorithm. (CCSS: 4.NBT.B.4)
- Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. (CCSS: 4.NBT.B.5)
- Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. (CCSS: 4.NBT.B.6)
Academic Contexts and Connections:
Colorado Essential Skills and Mathematical Practices:
- Solve multi-digit arithmetic problems. (Entrepreneurial Skills: Critical Thinking/Problem Solving)
- Explain the process and result of multi-digit arithmetic. (MP3)
- Precisely and efficiently add and subtract multi-digit numbers. (MP6)
- Use the structure of place value to support the organization of mental and written multi-digit arithmetic strategies. (MP7)
Inquiry Questions:
- How can a visual model be used to demonstrate the relationship between multiplication and division?
Coherence Connections:
- This expectation represents major work of the grade.
- In Grade 3, students use place value understanding and properties of operations to add and subtract within \(1000\) and to multiply and divide within \(100\).
- In Grade 4, this expectation connects to using the four operations with whole numbers to solve problems.
- In Grade 5, students understand the place value of decimals and perform operations with multi-digit whole numbers and with decimals to hundredths.
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Prepared Graduates:
- MP3. Construct viable arguments and critique the reasoning of others.
- MP5. Use appropriate tools strategically.
- MP6. Attend to precision.
- MP7. Look for and make use of structure.
Grade Level Expectation:
4.NF.A. Number & Operations—Fractions: Extend understanding of fraction equivalence and ordering.
Evidence Outcomes:
Students Can:
- Explain why a fraction \(\frac{a}{b}\) is equivalent to a fraction \(\frac{n \times a}{n \times b}\) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. (CCSS: 4.NF.A.1)
- Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as \(\frac{1}{2}\). Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols \( \gt \), \( = \), or \( \lt \), and justify the conclusions, e.g., by using a visual fraction model. (CCSS: 4.NF.A.2)
Academic Contexts and Connections:
Colorado Essential Skills and Mathematical Practices:
- Explain the equivalence of fractions. (MP3)
- Use visual models and benchmark fractions as tools to aid in fraction comparison. (MP5)
- Precisely refer to numerators, denominators, parts, and wholes when explaining fraction equivalence and comparing fractions. (MP6)
- Use \(1\), the multiplicative identity, to create equivalent fractions by structuring \(1\) in the fraction form \(\frac{n}{n}\). (MP7)
Inquiry Questions:
- Why does it work to compare fractions either by finding common numerators or by finding common denominators?
- How can you be sure that multiplying a fraction by \(\frac{n}{n}\) does not change the fraction’s value?
Coherence Connections:
- This expectation represents major work of the grade.
- In Grade 3, students develop an understanding of fractions as numbers and the meaning of the denominator of a unit fraction.
- In Grade 4, this expectation connects to building fractions from unit fractions, using decimal notation and comparing decimal fractions, and using the four operations with whole numbers to solve problems.
- In Grade 5, students use equivalent fractions as a strategy to add and subtract fractions with unlike denominators and apply and extend previous understandings of multiplication and division to fractions.
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Prepared Graduates:
- MP7. Look for and make use of structure.
- MP8. Look for and express regularity in repeated reasoning.
Grade Level Expectation:
4.NF.B. Number & Operations—Fractions: Build fractions from unit fractions.
Evidence Outcomes:
Students Can:
- Understand a fraction \(\frac{a}{b}\) with \(a \gt 1\) as a sum of fractions \(\frac{1}{b}\). (CCSS: 4.NF.B.3)
- Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. (CCSS: 4.NF.B.3.a)
- Decompose a fraction into a sum of fractions with like denominators in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: \(\frac{3}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\); \(\frac{3}{8} = \frac{1}{8} + \frac{2}{8}\); \(2 \frac{1}{8} = 1 + 1 + \frac{1}{8} = \frac{8}{8} + \frac{8}{8} + \frac{1}{8}\). (CCSS: 4.NF.B.3.b)
- Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. (CCSS: 4.NF.B.3.c)
- Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. (CCSS: 4.NF.B.3.d)
- Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. (CCSS: 4.NF.B.4)
- Understand a fraction \(\frac{a}{b}\) as a multiple of \(\frac{1}{b}\). For example, use a visual fraction model to represent \(\frac{5}{4}\) as the product \(5 \times \frac{1}{4}\), recording the conclusion by the equation \(\frac{5}{4} = 5 \times \frac{1}{4}\). (CCSS: 4.NF.B.4.a)
- Understand a multiple of \(\frac{a}{b}\) as a multiple of \(\frac{1}{b}\), and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express \(3 \times \frac{2}{5}\) as \(6 \times \frac{1}{5}\), recognizing this product as \(\frac{6}{5}\). (In general, \(n \times \frac{a}{b} = \frac{n \times a}{b}\).) (CCSS: 4.NF.B.4.b)
- Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat \(\frac{3}{8}\) of a pound of roast beef, and there will be \(5\) people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? (CCSS: 4.NF.B.4.c)
Academic Contexts and Connections:
Colorado Essential Skills and Mathematical Practices:
- Use the structure of fractions to perform operations with fractions and to understand and explain how the operations connect to the structure of fractions. (MP7)
- Recognize the mathematical connections between the indicated operations with fractions and the corresponding operations with whole numbers. (MP8)
Inquiry Questions:
- How is the addition of unit fractions similar to counting whole numbers?
- How does multiplying two whole numbers relate to multiplying a fraction by a whole number?
- (Given two fractions with like denominators, each of which is less than \(\frac{1}{2}\)) Before adding these two fractions, can you predict whether the sum will be greater than or less than \(1\)? How do you know?
Coherence Connections:
- This expectation represents major work of the grade.
- In Grade 3, students develop understanding of fractions as numbers and represent and solve problems involving multiplication and division.
- This expectation connects to other ideas in Grade 4: (a) using decimal notation for fractions and comparing decimal fractions, (b) using the four operations with whole numbers to solve problems, (c) solving problems involving measurement and conversion of measurements from a larger unit to a smaller unit, and (d) representing and interpreting data.
- In Grade 5, students use equivalent fractions as a strategy to add and subtract fractions with unlike denominators and apply and extend previous understandings of multiplication to decimals.
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Prepared Graduates:
- MP1. Make sense of problems and persevere in solving them.
- MP5. Use appropriate tools strategically.
- MP7. Look for and make use of structure.
Grade Level Expectation:
4.NF.C. Number & Operations—Fractions: Use decimal notation for fractions, and compare decimal fractions.
Evidence Outcomes:
Students Can:
- Express a fraction with denominator \(10\) as an equivalent fraction with denominator \(100\), and use this technique to add two fractions with respective denominators \(10\) and \(100\). (Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.) For example, express \(\frac{3}{10}\) as \(\frac{30}{100}\), and add \(\frac{3}{10} + \frac{4}{100} = \frac{34}{100}\). (CCSS: 4.NF.C.5)
- Use decimal notation for fractions with denominators \(10\) or \(100\). For example, rewrite \(0.62\) as \(\frac{62}{100}\); describe a length as \(0.62\) meters; locate \(0.62\) on a number line diagram. (CCSS: 4.NF.C.6)
- Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols \( \gt \), \( = \), or \( \lt \), and justify the conclusions, e.g., by using a visual model. (CCSS: 4.NF.C.7)
Academic Contexts and Connections:
Colorado Essential Skills and Mathematical Practices:
- Approach adding, subtracting, and comparing problems with fractions and decimal fractions by reasoning about their values before or instead of applying an algorithm. (MP1)
- Draw fraction models to reason about and compute with decimal fractions. (MP5)
- Make use of the structure of place value to express and compare decimal numbers in tenths and hundredths. (MP7)
Inquiry Questions:
- How does a fraction with a denominator of \(10\) or \(100\) relate to its decimal quantity?
- How can visual models help to compare two decimal quantities?
- How is locating a decimal on a number line similar to locating a fraction on a number line?
Coherence Connections:
- This expectation represents major work of the grade.
- This expectation connects to several ideas in Grade 4: (a) extending understanding of fraction equivalence and ordering, (b) building fractions from unit fractions, and (c) solving problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
- In Grade 5, students understand the decimal place value system and use it with the four operations.
Need Help? Submit questions or requests for assistance to bruno_j@cde.state.co.us