Understanding the Learning Progression

Big Ideas:

• Large numbers have the same place value structure as smaller numbers that students have explored in younger grades. Numbers of these quantities can be difficult to conceptualize therefore it is best to connect and explore larger numbers with real-world contexts. (Teaching Student-Centered Mathematics, 2014)

• Students should be able to recognize equivalent relationships of the same number to connect these place value skills to estimation and computation. (Teaching Student-Centered Mathematics, 2014)

• Students should understand the structure of the base-ten system and that 10 of any like units makes a single unit of the next highest place value. Example 10 hundred thousands is the same as 1 million.

• (Common Core Standards Writing Team, 2019).

• The position of the digits determine what they represent and which size group they count. This is the main principle of the place-value system. (Van De Walle et al., 2018).

Important Assessment Look-fors:

• Students can read numbers with varying amounts of digits up to a nine-digit whole number.

• Students can identify the place and value of each digit in a nine-digit whole number.

• Students can write a nine-digit number in standard form.

• Students can describe the base ten structure of place value.

• Students can represent a number in various forms such as standard and written form.

• Students can represent a number in standard form that includes zeros as placeholders (example 23,000,001).

Purposeful questions:

• What is the value of the digit?

• Can you represent this number in multiple ways?

• Can you read this number? What is your strategy for reading big numbers?

• What is this number written in word form?

• Can you identify the standard form of this number written in word form?

Understanding the Learning Progression

Big Ideas:

• When comparing numbers, students need to develop a sense of the relative size of numbers. For example, understanding that 185 is greater than 15, but less than 1,219 and is about the same as 179. (Van de Walle et al., 2018)

• Number sense is flexible thinking about numbers and their relationships. Numbers are related to each other in a variety of ways. The number 67 is more than 50, 3 less than 70, and composed of 60 and 7 as well as 50 and 17. When thinking about the number 67 in a variety of forms, we are able to apply these skills to estimation and computation. (Teaching Student-Centered Mathematics)

• Knowing the value of each place and the period of a number helps students when determining the value of digits in any number and is important to understand when comparing and ordering numbers. (VDOE Grade 4 Curriculum Framework)

Important Assessment Look-fors:

• Students can compare numbers with a variety of different place value digits.

• Students can use the symbols equal (=) and not equal (≠) when comparing numbers in a variety of forms.

• The student can order numbers from least to greatest and/or greatest to least.

• The students can compare two whole numbers using the correct symbols and terms.

• The student can compare and/or order numbers where the digits in the greatest place value is equivalent, requiring students to compare the value of digits in a different place value.

Purposeful questions:

• Can you explain why this number is greater? Can you explain why this number is less than?

• Can you create a number that is greater than the given number?

• What strategies did you use when ordering these numbers from least to greatest? Greatest to least?

• Explain the strategy you used to compare the two given numbers.

• Which symbol can be used when comparing these two numbers to make this statement true?

Understanding the Learning Progression

Big Ideas:

• A strong understanding of place value is important for the development of number sense when exploring problems that involve rounding numbers. (Georgia Standards of Excellence Curriculum Framework, Unit 1, Number and Operations in Base Ten)

• The use of a number line to determine which multiple a number is closer to is a strategy that develops a conceptual understanding of rounding instead of learning rules or mnemonics when rounding to a specific place value. (Georgia Standards of Excellence Curriculum Framework, Unit 1, Number and Operations in Base Ten)

• The concept of estimation and rounding are similar in many ways. Estimation is flexible thinking of friendly numbers to make mental computation easier or to compare it to a reference. When estimating there isn’t always one correct answer, but instead the purpose of creating that friendly number should be considered. Looking at the number 327, an estimation could be 300, 325, 330, or even 350. The estimations 325 and 350 are not the nearest ten or hundred but could be considered when estimating the number 327 depending on the purpose of the estimation. When rounding a number, students identify the closest multiple of a specific place value. Rounding is also used as a way of creating a friendly number and could be used for mental computation. (Elementary and Middle School Mathematics, John Van de Walle)

Important Assessment Look-fors:

• The student can round a number to a specific place value.

• The student can identify various numbers that would round to a specific place value.

• The student can identify a range of numbers that would round to a special place value.

• The student is able to use a model to represent a conceptual understanding of rounding whole numbers to the nearest thousand, ten-thousand, or hundred-thousand.

• The student is able to justify why the number rounds to the closest multiple of thousand, ten-thousand, or hundred-thousand when rounding to a specific place value.

• When rounding to a specific place value, the student can identify the multiples of thousand, ten-thousand, or hundred thousand that a given number is between and can justify which multiple the given number rounds to.

Purposeful questions:

• Can you create a model, such as a number line, to represent which thousand the number 4,527,093 is closest to when rounding to the nearest thousand? (The underlined word can be interchangeable with other place values such as ten-thousand or hundred thousand).

• Is the number 345,568 closest to 346,000 or 345,000 when rounding to the nearest thousand? Can you identify the range of numbers that would round to 346,000 when rounding to the nearest thousand?

• Can you round this number to the nearest thousand, ten-thousand, and/or hundred thousand?

• Can you identify a number that would have the same answer when rounded to the nearest thousand and the nearest ten-thousand?

Understanding the Learning Progression

Big Ideas:

• Initial fraction ideas begin with creating experiences with a variety of visual representations (area models, number lines, set models, etc.) which can be utilized to compare and order fractions. Over time, reasoning strategies and generalizations are developed through modeling and mental imagery. There may be a transition from creating models to applying those generalizations and reasoning strategies to solve problems.

• A variety of reasoning strategies can be applied to compare fractions. Potential strategies include but are not limited to:

  • reasoning about relative magnitude ;

  • unit fraction reasoning;

  • benchmark reasoning ( i.e. I know 4/8 is ½ so ⅝ is a little more); and

  • equivalence.

• Behr and Post (1992) indicate that “a child’s understanding of the ordering of two fractions needs to be based on an understanding of the ordering of unit fractions” (1992, p.21).

• The reasoning strategy applied is often impacted by number choice. For example, I might use the benchmark of ½ if I am comparing 3/10 (a little less than ½) and ⅝ (a little more than ½).

Important Assessment Look-fors:

• Students recognize and utilize benchmark fractions.

• Students use their understanding of the ordering of unit fractions to ordering of two fractions.

• Students recognize and apply their understanding of equivalent fractions.

• Students recognize fractions equivalent to ½.

• Students represent and work with fractions greater than 1.

• Students organize their thinking in a way that helps them make sense of the problem. Student representations support their thinking/reasoning.

• Student strategies make sense for the given set of numbers.

• It is evident in the strategy used that the student understands the relationship between the numerator and the denominator.

Purposeful questions:

• Can you share where you started and why?

• How might you use your understanding of unit fractions to compare and order these fractions?

• How can benchmark be used to help you understand the value of these fractions?

• What representation might help you visualize these fractions?

• Is that fraction greater than 1? Less than 1? How do you know?

• What do you know about the value of these numbers?

• What relationships do you see?

• I’m looking at your number line and I’m noticing _____. Tell me more about that...

Understanding the Learning Progression

Big Ideas:

• When two fractions are equivalent that means there are two ways of describing the same amount by using different sized fractional parts. (Van de Walle et al, 2019)

• A variety of representations and models can be used to identify different names for equivalent fractions: region/area, set and measurement models. (Students should use area representations, strips of paper, tape diagrams, number lines, counters and other manipulatives to reason about equivalence.)

• Intuitive methods using drawings and manipulatives support student understanding. Students can develop an understanding of equivalent fractions and also develop from that understanding a conceptually based algorithm. Delay sharing “a rule.” (Van de Walle et al, 2019)

Important Assessment Look-fors:

• Evidence that the student is able to identify equivalent fractions modeled using a variety of different models: area, set and/or measurement.

• The student uses strategies like removing lines or adding additional lines to the area and number line models to show how the models are equivalent.

• The student is able to use a variety of strategies and can justify their reasoning as to why fractions are equivalent.

• When the student represents their fraction as models, they represent the whole using the same size and shape model. Click this link for more information about student representations.

Purposeful questions:

• What strategy (or strategies) did you use to determine which fraction models are equivalent?

• Which fraction models are the easiest for you to identify? What made it easy?

• Which fraction models are the hardest for you to identify? What made it difficult?

• What relationships do you see?

• Can you create another equivalent model that is different from ones shown?

Understanding the Learning Progression

Big Ideas:

• Fractions have multiple meanings and interpretations. Generally there are five main interpretations: fractions as parts of wholes or parts of sets; fractions as the result of dividing two numbers; fractions as the ratio of two quantities; fractions as operators; and fractions as measures (Behr, Harel, Post, and Lesh 1992; Kieren 1988; Lamon 1999). When a fraction is presented in symbolic form, devoid of context, the intended interpretation of the fraction is not evident. The various interpretations are needed, however, in order to make sense of fraction problems and situations. Students need to explore and understand that fractional parts are equal shares of a whole or set model.

• A fraction can also represent the result obtained when two numbers are divided. This interpretation of fraction is sometimes referred to as the quotient meaning, since the quotient is the answer to a division problem. Chapin and Johnson (2000) give these examples, “the number of gumdrops each child receives when 40 gumdrops are shared among 5 children can be expressed as 40/5 , 8/1 , or 8; when two steaks are shared equally among three people, each person gets 2/ 3 of a steak for dinner. We often express the quotient as a mixed number rather than an improper fraction – 15 feet of rope can be divided to make two jump ropes, each 7 1/ 2 ( 15/2 ) feet long” (p .99– 101).

• When partitioning a whole into more equal shares the parts become smaller. (Teaching Student-Centered Mathematics, Grades 3-5, John Van de Walle)

• When exploring the concept of fractions and connecting it to the division statement, students should be able to identify and recognize that the fraction is the amount that each person would receive when dividing equally.

• Students’ understanding of fractions as division develops through the use of a progression of equal sharing problems, beginning with problems resulting in a whole number (6 cookies shared by 3 children), then problems resulting in a mixed number (5 cookies shared by 2 children), then problems resulting in a unit fraction (1 cookie shared by 4 children), and finally problems resulting in non-unit fractions (2 cookies shared by 3 children) (Empson & Levi, 2011).

Important Assessment Look-fors:

• The student is able to accurately partition models when exploring fair share problems.

• When given context, the student is able to successfully use models to identify the division statement that represents the fraction.

• The student can interpret the fraction and division statement as the amount each person would receive.

• The student can identify the difference between a division statement that represents an improper fraction versus a proper fraction within context. The student can also represent a fair share problem when the numerator is larger or smaller than the denominator.

Purposeful questions:

• Will each person receive more or less than a whole? Explain your reasoning.

• Identify the division statements that represent this fraction as presented in the context.

• How much will each person receive when shared equally?

• Once the students have discovered how much each person will receive, ask the students if each person was to combine their equal share together, what would be the combined total? What do you notice?

• Example: 2 cookies shared with 3 students. Each person would receive ⅔ of a cookie. When combining the equal shares from each of the 3 people, ⅔ + ⅔ + ⅔ = 6/3 or 2. The sum is equivalent to 2 whole cookies or the original amount of cookies that was shared.

• If the dividend is smaller than the divisor, how does this relate to how much each person would receive if the amount was shared equally? Would this be true if the dividend was larger than the divisor?

• Example: 2 pizzas shared with 3 people versus 3 pizzas shared with 2 people.

Understanding the Learning Progression

Big Ideas:

• The structure of the base-ten number system is based upon a simple pattern of tens, where each place is ten times the value of the place to its right. This is known as a ten-to-one place value relationship (Van de Walle et al., 2019)

• Decimals are another form of writing fractions and the connection between the two is important in understanding the concepts of decimals (i.e. connect 1/10 as 0.1 and 1/100 as 0.01 and 1/1000 as 0.001 - Reading the decimal fractions will help students “hear” the connection).

• Understanding of the base-ten system to the relationship between adjacent places and how numbers compare can help support students round for decimals to thousandths. For example, it is important to deepen understanding and fluency with decimals in the different forms, seeing .57 as 5 tenths and 7 hundredths as well as 57 hundredths (Common Core Standards Writing Team, 2019, p. 64). This ability to rename and decompose decimals can help students round to the nearest whole number, tenth or hundredth.

• The decimal point separates the whole from the fractional part. The place value system extends infinitely in both directions of the decimal point, to very large and very small numbers. Connected uses of decimals in real life using money, metric measurements, batting averages can support student understanding.

Important Assessment Look-fors:

• The student uses the identified whole to name a decimal represented by base ten blocks.

• The student can read a decimal in word form and record it in numeric form.

• The student can identify the position and place value of each digit in a decimal.

Purposeful questions:

• Is the decimal represented more or less than a whole? Explain your answer.

• If the whole changed, how would that affect the decimal represented?

• When modeling with base ten blocks, what would the cube represent if the whole is equal to a rod? Explain your answer.

• How can you represent this decimal in a variety of ways, such as number line, money, or 10-by-10 grid?

Understanding the Learning Progression

Big Ideas:

• The base-ten place value system extends infinitely in two directions, to very tiny values to large values. (Van de Walle et al., 2018)

• Students extend their understanding of the base-ten system to the relationship between adjacent places, how numbers compare, and how numbers round for decimals to thousandths. (Common Core Standards Writing Team, 2019).

• A strong understanding of decimal place value is important for the development of number sense when exploring problems that involve rounding numbers.

• The use of a number line to determine which whole number a given decimal is closer to is a strategy that develops conceptual understanding of rounding instead of learning rules or mnemonics when rounding. (VDOE Quick Checks SOL 4.3b)

• Strategies for rounding whole numbers can be applied when rounding decimals expressed through thousandths. (VDOE Curriculum Framework, Grade 4)

Important Assessment Look-fors:

• The student is able to use a model, such as a number line, to represent a conceptual understanding of rounding a decimal to the nearest whole number.

• The student is able to justify the closest whole number when rounding a decimal.

• The student is able to round a decimal expressed through thousandths to the nearest whole number.

• The student is able to round a number with multiple places values to the nearest whole (example: 34.987 rounds to 35 when rounding to the nearest whole number.)

• The student is able to identify decimals that would round to a given whole number.

Purposeful questions:

• What strategy did you use when rounding the decimal to the nearest whole number?

• Can you create a model, such as a number line, to represent which two whole numbers the given decimals round to?

• Can you identify a decimal that would round to the given whole number? Is there more than one answer?

• Which digit can be placed in the tenth place so that the given decimal will round to the next whole number? Is there more than one answer?

Understanding the Learning Progression

Big Ideas:

• This standard builds upon the work students did in previous grades in understanding place value and comparing and ordering whole numbers. In grade 4, in addition to comparing greater numbers, students began relating decimal fractions and decimal numbers and comparing decimals using visual models. The place value understanding that supports the ability to compare decimals also supports the understanding of rounding decimals, which is introduced in grade 5 as SOL 5.1.

• Concepts of whole numbers, fractions, and decimals are connected and applied when comparing and ordering.

• Using manipulatives to construct decimals helps students develop an understanding of the relative size of decimal numbers for comparing and ordering.

• It is important for students to connect decimal number sense concepts such as representations, decimals benchmarks, and/or fractions when comparing and ordering decimals.

Important Assessment Look-fors:

• The student can compare decimals with different amounts of digits. (Example 0.9 and 0.234)

• The student can justify which decimal is larger or smaller using a variety of strategies that focus on number sense such as models, decimal benchmarks and/or identifying the value of the greatest place value.

• The student can order decimals least to greatest or greatest to least.

• The student can apply a variety of strategies when ordering decimals with similar digits and/or different amounts of digits. (Example: 0.9; 0.901; 0.09; 0.009)

Purposeful questions:

• How did you determine an equivalent decimal?

• What strategy did you use to determine which decimal is the greatest and which one is the least? Explain your thinking.

• Which decimal(s) can be placed in the space provided so that the decimals are in order from least to greatest? (Example: 0.142; 0.45 ______; 0.8)

• Compare the following decimals using two different strategies to justify which one is greater or least.

Understanding the Learning Progression

Big Ideas:

• In mathematics, any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value. Decimal and fraction numbers can be named in an infinite number of different but equivalent forms (e.g., 3/10= 0.3 = 0.30 = 0.10 + 0.20) (Charles, 2005).

• We can relate fractions to decimals using a variety of representations including 10 by 10 grids, number lines, decimal squares, money, rational number wheel and decimal grids. These representations build an understanding of equivalency. (VDOE Curriculum Framework, 2016)

• An understanding of money can be applied to fractions and decimals by considering parts of a whole dollar that can be represented as equivalent fractions and decimals. In mathematics, these relationships are especially evident with the connection to dimes (tenths) and the connection to pennies (hundredths).

Important Assessment Look-fors:

• The student models and names equivalent fractions and decimals.

• The student identifies equivalent relationships between tenths and hundredths.

• The student names the same value in a variety of ways.

• The student may partition the same model in a variety of ways to notice and name equivalent relationships.

Purposeful questions:

• Tell me about your model. Why did you…?

• Is there another way you could name that value?

• How do you know __ is equivalent to ___?

• What do you know about money and parts of a dollar that can help you? (For example, if I have 2 dimes, what part of a dollar does that represent?)

• What connections do you see between money, decimals, and fractions?

Understanding the Learning Progression

Big Ideas:

• Computational fluency is the ability to think flexibly in order to choose appropriate strategies to solve problems accurately and efficiently (VDOE Grade 4 Curriculum Framework).

• All of the facts are conceptually related so students can figure out new or unknown facts using what they already know (Van de Walle et al, 2018).

• The development of computational fluency relies on quick access to number facts. There are patterns and relationships that exist in the facts. These relationships can be used to learn and retain the facts (VDOE Grade 4 Curriculum Framework).

• Mastering the basic facts is a developmental process. Students move through phases, starting with counting, then more efficient reasoning strategies, and eventually quick recall and mastery. Instruction must help students through these phases without rushing them to know their facts only through memorization (Van de Walle et al, 2018).

• When students struggle with developing basic fact fluency, they may need to return to foundational ideas. Just providing additional drill will not resolve their challenges and can negatively affect their confidence and success in mathematics (Van de Walle et al, 2018).

• Students use efficient reasoning strategies such as derived facts to find related facts, commutative property, compensation, and break apart factors.

• In order to develop and use strategies to learn the multiplication facts through the twelves table, students should use concrete materials, a hundreds chart, and mental mathematics. Strategies to learn the multiplication facts include an understanding of multiples, properties of zero and one as factors, commutative property, and related facts (VDOE Grade 4 Curriculum Framework).

Important Assessment Look-fors:

• The student knows and is able to apply a variety of strategies (i.e., partial products, using friendly numbers, repeated addition, and/or decomposition strategies, recall, etc.).

• The student demonstrates an understanding of the term “product” and uses a strategy to find a product that leads to a correct answer.

• The student identifies multiple number sentences with the same product.

• Student’s work shows understanding of the inverse relationship between multiplication and division.

Purposeful questions:

• What strategies are most efficient for this fact (or set of facts) and how do you know?

• How can knowing one fact help you with another fact (or a fact that you don’t know)?

• What makes one fact related to another fact? How can knowing related facts be helpful?

• What do you know about the relationship between multiplication and division? How can that relationship help you solve problems?

• What are some ways that you can break apart these numbers to make this problem easier?

Understanding the Learning Progression

Big Ideas:

• Flexible methods of computation for all four operations involve taking apart (decomposing) and combining (composing) numbers in a variety of ways (Van de Walle et al, 2018).

• Students should explore and apply the properties of addition and multiplication as strategies for solving addition, subtraction, multiplication, and division problems using a variety of representations (e.g., manipulatives, diagrams, and symbols). (VDOE Grade 4 Curriculum Framework)

• Flexible methods for computation require deep understanding of the operations and the properties of operations (commutative property, associative property, and the distributive property). How addition and subtraction, as well as multiplication and division, are related as inverse operations is also critical knowledge (Van de Walle et al, 2018).

• Estimation can be used to determine the approximation for and then to verify the reasonableness of sums, differences, products, and quotients of whole numbers. An estimate is a number that lies within a range of the exact solution, and the estimation strategy used in a particular problem determines how close the number is to the exact solution. (VDOE Grade 4 Curriculum Framework). Estimation is a key component of mathematical reasoning, and it requires flexible thinking and number sense.

Important Assessment Look-fors:

• The student applies terms such as estimate, sum, difference and product.

• The student selects a strategy that they understand and can apply successfully.

• The student explains their solution and justifies the reasonableness of their answer.

• The student’s work contains a variety of strategies and representations.

• The student uses estimation to determine the reasonableness of their answer (i.e., a little more than, a little less than, closer to, etc.).

Purposeful questions:

• Tell me about the operation you decided to use and why it makes sense.

• How did you represent your thinking?

• How do you know your answer is correct?

• How does your estimation relate to the actual answer?

• How do you know __ is closest to __?

• How do you know your answer is reasonable?

• Why did you choose that estimate?

Understanding the Learning Progression

Big Ideas:

• Flexible methods of computation involve taking apart (decomposing) and combining (composing) numbers in a variety of ways (Van de Walle et al, 2018).

• Students should explore and apply the properties of addition and multiplication as strategies for solving division problems using a variety of representations (e.g., manipulatives, diagrams, and symbols). (VDOE Grade 4 Curriculum Framework)

• Flexible methods for computation require deep understanding of the operations and the properties of operations (commutative property, associative property, and the distributive property). How addition and subtraction, as well as multiplication and division, are related as inverse operations is also critical knowledge (Van de Walle et al, 2018).

• Estimation can be used to determine the approximation for and then to verify the reasonableness of quotients of whole numbers. An estimate is a number that lies within a range of the exact solution, and the estimation strategy used in a particular problem determines how close the number is to the exact solution (VDOE Grade 4 Curriculum Framework).

• Some division situations will produce a remainder, but the remainder will always be less than the divisor. If the remainder is greater than the divisor, that means at least one more can be given to each group (fair sharing) or at least one more group of the given size (the dividend) may be created (Georgia Department of Education Grade 4 Curriculum↗).

Important Assessment Look-fors:

• Students’ use methods they understand and can explain.

• Students' work shows that they understand and can apply the term quotient(s).

• Students’ work demonstrates an understanding of place value and identifying related facts that correlate with the problem.

• Students can explain their solution and the reasonableness of their answer.

• Student’s work contains a variety of strategies and representations.

• The student is able to use estimation to determine the reasonableness of their answer (ie, a little more than, a little less than, closer to, etc.).

Probing questions:

• How did you represent your thinking?

• How do you know your answer is correct?

• How does your estimation relate to the actual answer?

• How do you know __ is closest to __?

• How do you know your answer is reasonable?

• Why did you choose that place value for your estimate?

• What is the meaning of a remainder in a division problem?

• What effect does a remainder have on a quotient?

• How are remainders and divisors related?

Understanding the Learning Progression

Big Ideas:

• Use reasoning and a variety of strategies that the student understands and is able to explain.

• Makes sense of the problem rather than relying on keywords that may not always be helpful. (See VDOE Standards of Learning- Grade 4 Document↗ p.19).

• Inverse operations are related and can flexibly be used to solve problems.

• Estimation, based on number sense and an understanding of place value, can be used to determine if an answer is reasonable.

• Students will be stronger problem solvers given opportunities to engage with a variety of problem types. For more information about addition/subtraction problem types see the Grade 3 VDOE Standards of Learning Document↗ p. 15 and for multiplication/division problem types see the Grade 4 VDOE Standards of Learning Document↗ pp.20-21.

Important Assessment Look-fors:

• Students are able to create a problem using the given information.

• Students are able to correctly represent the problem they created and it matches the way they solved it.

• Student work shows that they understand the problem because they have planned an approach to solve the problem and/or a way to represent their thinking.

• Student work shows that they understand what each number in the problem represents.

• Student work shows that they understand what operations can be used, the meaning behind the operation or how the operations are related.

Purposeful questions:

• How did you represent your thinking? How do you know it matches the story?

• Is there another way that you could solve this problem?

• Does your solution make sense? Explain your answer.

• What equation/number sentence matches your thinking?

• What operations did you decide to use? Why?

Understanding the Learning Progression

Big Ideas:

• Any whole number is a multiple of each of its factors.

• A number can be multiplicatively decomposed into equal groups and expressed as a product of these two factors, called factor pairs (Common Core Progressions, p.30).

• A factor of a whole number is a whole number that divides evenly into that number with no remainder. A factor of a number is a divisor of the number (VDOE Grade 4 Curriculum Framework).

• Common multiples and common factors can be useful when simplifying fractions (VDOE Grade 4 Curriculum Framework).

Important Assessment Look-fors:

• Student lists all factors for a given number or set of numbers.

• Student generates a list of multiples for a given number or set of numbers.

• Student compares factors and/or multiples of a given set of numbers to identify common factors and/or multiples.

• Student identifies the greatest common factor of a given set of numbers and explains why it is the greatest common factor.

• Student identifies the least common multiple of a given set of numbers and explains why it is the least common multiple.

• The student uses an appropriate strategy for finding common factors and/or common multiples.

Purposeful questions:

• How do you know if a factor is the greatest common factor?

• How do you know if a multiple is the least common multiple?

• Why do you think we look for the greatest common factor and not the least common factor?

• Why do you think we look for the least common multiple and not the greatest common multiple?

• What is the difference between a factor and a multiple?

• Can a factor also be a multiple?

• What strategies can be used to find common factors?

• What strategies can be used to find common multiples?

Understanding the Learning Progression

Big Ideas:

• Some real-world problems involving joining, separating, part-part-whole, or comparison can be solved using addition; others can be solved using subtraction.

• The effects of operations for addition and subtraction with fractions and decimals are the same as those with whole numbers (Charles, 2005).

• Estimation keeps the focus on the meaning of the numbers and operations, encourages reflective thinking, and helps build informal number sense with fractions. Students can reason with benchmarks to get an estimate without using an algorithm (VDOE Curriculum Framework).

• A variety of strategies can be utilized to add and subtract fractions including area models, linear models, decomposing fractions, and finding a common denominator.

Important Assessment Look-fors:

• The student recognizes and uses equivalent relationships to add and subtract fractions.

• The student uses benchmark fractions to determine reasonable estimates.

• The student names fractions greater than 1 in multiple ways.

• The student adds and subtracts with or without models.

Purposeful questions:

• How do you know your answer is reasonable?

• How did that benchmark help you?

• What is an equivalent name for that fraction?

• What relationships do you notice? How can those relationships help you?

Understanding the Learning Progression

Big Ideas:

• In mathematics, real-world actions for addition and subtraction of whole numbers are the same for operations with fractions and decimals (Charles, 2005).

• Benchmark fractions like 1/2 (0.5) and 1/4 (0.25) can be used to estimate calculations involving fractions and these estimations can be used to check the reasonableness of exact answers (Charles, 2005).

• Fraction number sense skills, including estimation skills, will be necessary when solving practical problems to determine the reasonableness of an answer. Estimating with fractions is critical to understanding their magnitude or position on the number line (Van De Walle et al., 2018).

• When solving practical word problems with fractions, use the same ideas developed for whole-number computation and apply these ideas to fractional parts instead of whole numbers (Van De Walle et al., 2018).

• In mathematics, when solving practical word problems the focus should be on thinking and reasoning rather than on keywords. Expose students to a variety of different problem types and opportunities to create and solve their own practical problems. For more information about addition/subtraction problem types see the Grade 3 VDOE Curriculum Framework.

• Counting fractional parts should be done just like counting objects like apples or pencils (Van De Walle et al., 2018).

Important Assessment Look-fors:

• The student uses the context of a problem to plan an approach to solve it.

• The student represents their thinking using models and/or manipulatives.

• The student uses estimation strategies when solving practical word problems with fractions to determine the reasonableness of their answer.

• The student solves single step word problems with like and unlike denominators using a variety of strategies.

• The student represents the resulting fraction in simplest form.

Purposeful questions:

• Based on your estimation, does your answer seem reasonable? Explain why or why not.

• What strategy did you use when solving the problem? Is there another way that you could solve it?

• Does your solution make sense? Explain your answer.

• Is the sum or difference more or less than a whole? Explain your answer.

Understanding the Learning Progression

Big Ideas:

• Addition and subtraction with decimals are based on the fundamental concept of adding and subtracting the numbers in like position values and should be connected to a simple extension from whole numbers (Van de Walle et al., 2018).

• Students should explore estimation prior to finding the exact answer.

• In mathematics, real-world actions for addition and subtraction of whole numbers are the same for operations with fractions and decimals (Charles, 2005).

Important Assessment Look-fors:

• Student identifies the sum or difference of decimals when represented by a variety of base-ten models.

• Student finds the sum of difference when one number is a whole number and the other number is represented as decimal.

• Student finds the sum or difference of two numbers with different amounts of digit place values. (Example: find the sum of 9.5 and 4.34)

• Student applies estimation strategies when finding the sum or difference of decimals.

• Student uses estimation to determine the reasonableness of a sum or difference.

Purposeful questions:

• Based on your estimation, does the sum or difference seem reasonable? Explain why or why not.

• What strategy did you use when solving the problem? Is there another way that you could solve it?

• Once students are able to apply a variety of estimation skills, ask students to place decimals points in the given numbers so that the number sentence would be true, Example: Look at the number sentence, where should the decimal be placed in the sum so that the number sentence is true? 4.25 + 2.9 = 715

• (The decimal point should be placed between the digits 7 and 1 because when estimating the sum of 4 and 3, the estimation is 7.)

• Can you represent the number sentence using a variety of models such as base ten blocks, number lines, and/or money to find the sum or difference?