Understanding the Learning Trajectory

Big Ideas:

• The structure of numbers is based on unitizing amounts into groups of ones, tens, and hundreds, etc and these groups can be reorganized so that numbers can be represented in multiple ways. For example 2,560 can be 2 thousands, 5 hundreds, 6 tens, OR 25 hundreds and 6 tens.

• Place value refers to the value of each digit and depends on the position of the digit in the number.

• Numbers are arranged in periods and the places in those periods repeat.

Important Assessment Look Fors:

• Students are able to successfully identify multiple ways to identify a given number.

• Students are able to use values and various representations to create a given number.

• Students are able to place 0 (zero) place holders when applicable and use the comma in the appropriate location.

• Students are able to differentiate between the value and place of a digit.

Purposeful Questions:

• Explain why you wrote that number in the way you did?

• How did you determine the digit in the (ones, tens, hundreds, etc.) place? What is the value of the digit in the (ones, tens, hundreds, etc.) place?

• Read that number out loud. Does that match what is in the question?

• How could you represent this number in a different way?

Understanding the Learning Trajectory

Big Ideas:

• An understanding of 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.

• When rounding to the nearest 10, 100, or 1,000, the goal is to approximate the number by the closest number with no ones, no tens and ones, or no hundreds, tens, and ones (Common Core Standards Writing Team, 2019).

• In mathematics a number line can be used to locate a given number and determine the closest multiples of ten, hundred, or thousand.

Important Assessment Look Fors:

• Students can write the number accurately with the appropriate number of digits.

• Students have an understanding of the base-10 system and are able to identify the tens, hundreds, thousands place in order to round.

• When rounding students use placeholders after the rounded place value.

• Students are able to determine the closest multiple of ten, hundred, or thousand for the given number.

Purposeful Questions:

• How did you know to round ________ to ________? (224 to 220)

• What digit is in the (tens, hundreds, thousands) place? Why did it round to _______?

• Explain why you chose _________ to round to 3,670.

Understanding the Learning Trajectory

Big Ideas:

• Comparing magnitudes of four-digit numbers uses the understanding that 1 thousand is greater than any amount of hundreds, tens, and ones represented by a three-digit number.

• Four-digit numbers are first compared by inspecting the thousands place, then the hundreds place and so on.

• Whole numbers can be compared by analyzing corresponding place values (Charles, 2005, p.14).

• Numbers can be compared by their relative values (Charles, 2005, p. 14). For example, benchmark numbers are important numbers against which other numbers or quantities can be estimated and compared. Benchmark numbers are usually multiples of 10 or 100.

Important Assessment Look Fors:

• Student uses the >, <,=, and symbols correctly.

• Student uses the terms greatest and least correctly.

• Student composes a number that is less than, greater than, or equal to a given number using appropriate place value.

• Student uses place value understanding when comparing numbers.

Purposeful Questions:

• How did you know which number was larger/smaller?

• Why did you order the numbers in that way?

• Why is that number greater than/less than/equal to the given number?

Understanding the Learning Trajectory

Big Ideas:

• Fractions represent equal parts of a whole, part of a group, or part of a length (number line model).

• The denominator is the total number of parts in the whole or group and the numerator is the number of parts being indicated.

• Mixed numbers are written in two parts: a whole number and a proper fraction.

Important Assessment Look Fors:

• Student names and writes numerators and/or denominators to match the model.

• Student represents the denominator as the total number of parts in the whole, group, or on the number line, not the number of parts that are not being indicated.

• Student differentiates between a proper fraction and an improper fraction.

• Student identifies mixed numbers as a whole number and a proper fraction.

Purposeful Questions:

• Explain how you know that your numerator and denominator are correct.

• How did you determine the fraction being represented was a proper fraction or improper fraction/mixed number?

• Can you also write the mixed number as an improper fraction? How do you know they represent the same amount?

Understanding the Learning Trajectory

Big Ideas:

• Fractions are numerical representations for part of a whole or a set.

• Unit fractions are the basic building blocks of fractions and are iterated multiple times to represent other fractions. (Common Core Progressions, p.7)

• Mixed numbers and improper fractions are greater than one whole.

Important Assessment Look Fors:

• Student divides an area model into the appropriate number of equal-sized pieces to represent the denominator of a given fraction.

• Student adds unit fractions to identify and name a larger fraction.

• Student draws more than one whole (of a set or an area model) to represent mixed numbers and improper fractions.

• Student names a model representing more than 1 whole as a mixed number or an improper fraction.

• Student locates improper fractions on the number in a position greater than 1 whole.

Purposeful Questions:

• Why did you represent that fraction in that way?

• What is another way you could represent that fraction?

• Can you create an addition sentence that would be equivalent to your fraction representation?

Understanding the Learning Trajectory

Big Ideas:

• Two fractions that have the same denominator have unit fractions that are the same size, so the fraction with the greater numerator is greater because it is made of more unit fractions (Common Core Progressions, p.9).

• For unit fractions, the one with the larger denominator is smaller, by reasoning, for example, that in order for more (identical) pieces to make the same whole, the pieces must be smaller. For two fractions that have the same numerator, the fraction with the smaller denominator is greater because the pieces are larger (Common Core Progressions, p.9).

• Fractions can be compared by reasoning about the relative size of the fractions, assuming the same size whole.

• Benchmark fractions of 0, 1, and ½ are used in comparing fractions.

Important Assessment Look Fors:

• Student use words (greater than, less than or equal to) and symbols (>, <, =) appropriately.

• Student draws models that are the same size when comparing fractions.

• Student attends to both the numerator and denominator when comparing fractions, not just comparing numerators (ex. 1/4 < 2/12 because 1 is less than 2).

• Student uses the benchmarks 0, ½, and 1 when comparing fractions.

• Student uses to identify other fractions that are equivalent to 1/2 (3/6, 4/8, etc.).

Purposeful Questions:

• How do you know this fraction is greater than/less than the other fraction? (i.e., how is 2/5 less than 4/10?)

• Where on a number line would you put these fractions? How can you use the number line to help you compare the fractions?

• How does this fraction compare to the benchmark of 1/2 (or 0 or 1)?

• Can you draw a model to compare this fraction to 1/2?

Understanding the Learning Trajectory

Big Ideas:

• Properties of operations allow for multi-digit numbers to be broken down into single digit numbers (place value) for computation (addition and/or subtraction). (Common Core Progressions, pg. 3)

• Concrete models, drawings, and symbolic representations may be used to find sums and differences.

• Estimation is a form of rounding. Rounding addends before finding a sum or difference allows students to estimate an answer and determine the reasonableness of the final answer to their computation. (Prince William County, Grade 3 Unit 1 guide).

Important Assessment Look Fors:

• Student uses place value to break numbers down for computing with multi-digit numbers.

• Student represents multi-digit numbers with concrete items and/or abstract drawings.

• Student makes reasonable estimations.

• Student finds the estimate or determines the sum or difference of two multi-digit numbers.

Purposeful Questions:

• What steps are needed to provide an estimate? What strategy did you use?

• How does place value help you when determining the sum or difference?

• What strategy did you use to determine the sum or difference? How will you prove your answer?

Understanding the Learning Trajectory

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).

• Making sense of the problem requires an emphasis on thinking and reasoning rather than on key words.

• Exposure and opportunity to engage with a variety of problem types will strengthen students’ ability to solve new problems. For more information about addition/subtraction problem types see the Grade 3 VDOE Standards of Learning Document p. 15.

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

• Estimation should be used to determine if an answer is reasonable.

Math Strength Instructional Video 3.3b↗

Important Assessment Look Fors:

• Student makes sense of the problem and is able to identify the unknown.

• Student uses an appropriate operation to solve the problem.

• Student uses an efficient strategy to solve the problem and is able to explain their reasoning.

• Student determines if their answer is reasonable.

Purposeful Questions:

• What is the unknown?

• How did you know what operation to use to solve the problem?

• Is there another way to solve this problem?

• How can you justify or prove your answer is correct?

Understanding the Learning Trajectory

Big Ideas:

• Early counting and addition strategies, such as skip counting and using doubles, provide a foundation for solving multiplication problems. Multiplication and division are inverse operations. Multiplication and division can be represented in a variety of different approaches and models, such as equal-sized groups, arrays, length model (number line), commutative property.

• The equal-sets or equal-groups model lends itself to sorting a variety of concrete objects into equal groups and reinforces the concept of multiplication as a way to find the total number of items in a collection of groups, with the same amount in each group, and the total number of items can be found by repeated addition or skip counting. The array model, consisting of rows and columns (e.g., four rows of six columns for a 4-by 6 array), helps build an understanding of the commutative property.

• In multiplication, one factor represents the number of equal groups and the other factor represents the number in or size of each group. The product is the total number in all of the groups. Models of multiplication may include repeated addition and collections of like sets, partial products, and area or array models.

• Division is the operation of making equal groups or shares. When the original amount and the number of shares are known, divide to determine the size of each share. When the original amount and the size of each share are known, divide to determine the number of shares. Both situations may be modeled with base-ten manipulatives. Division is the inverse of multiplication. Terms used in division are dividend, divisor, and quotient. Students benefit from experiences with various methods of division, such as repeated subtraction and partial quotients.
  Math Strength Instructional Video 3.4a↗

Important Assessment Look Fors:

• Student relates skip counting to multiplication.

• Student uses repeated addition/subtraction and relates it to multiplication/division.

• Student solves problems in different ways that show the same idea.

• Student uses manipulatives to represent a problem and translate that into an accurate picture.

Purposeful Questions:

• Tell me about your answer. What does your answer represent?

• How does your drawing represent the problem? Explain your reasoning for selecting that representation.

• Is there another representation you could use? What would it look like on a number line?

• Which number is the whole (product/dividend) and which numbers are the parts (factors/divisor/quotient) in your related facts?

• What expression (or equation) would match this problem? (e.., Sue has 5 pencil boxes with 8 pencils in each box. How many pencils does she have?).

• Does it matter what order the factors are in? Why or why not?

Understanding the Learning Trajectory

Big Ideas:

• Some basic multiplication facts can be found by breaking apart the unknown fact into known facts. Then the answers to the known facts are combined to give the final value (Charles pg. 22).

• Some real-world problems involving joining equal groups, separating equal groups, comparison, or combinations can be solved using multiplication; others can be solved using division (Charles pg. 21).

• There are several different types of multiplication and division problems. There are three main categories of problems: equal group problems, multiplicative comparison problems, and array problems.

Important Assessment Look Fors:

• Student writes an appropriate expression or equation to represent the problem

• Student accurately solves the equation that they created to represent the problem.

• Student uses a strategy that works best for them to determine the correct product or quotient.

• Student understands what the word problem is asking and is able to identify what they are solving for (result, starting quantity, etc.).

Purposeful Questions:

• How did you know to multiply/divide for that problem?

• How did you get that product/quotient when you solved the problem?

• What strategy did you use to solve the problem?

Understanding the Learning Trajectory

Big Ideas:

• Some basic multiplication facts can be found by breaking apart the unknown fact into known facts. Then the answers to the known facts are combined to give the final value (Charles pg. 22).

• There are patterns and relationships that exist in these facts and those relationships can be used to learn and retain the facts.

• By studying patterns and relationships, a foundation for fluency with multiplication facts and the corresponding division facts can be built.

Math Strength Instructional Video 3.4c↗

Important Assessment Look Fors:

• Student uses a variety of strategies to solve facts (arrays, equal groups, skip counting, etc.).

• Student applies the identity property of multiplication that states that any number multiplied by 1 is that same number.

• Student applies the zero property of multiplication.

• Student uses the commutative property for multiplication when appropriate to do so.

Purposeful Questions:

• What strategy did you use to find the product of _________?

• How do you know ______ is the product of ________?

• Is there another fact that will help you with this one?

• What did you do to solve this problem?

Understanding the Learning Trajectory

Big Ideas:

• Some basic multiplication facts can be found by breaking apart the unknown fact into known facts. Then the answers to the known facts are combined to give the final value (Charles, 2005).

• Students should explore and apply the properties of multiplication and addition as strategies for solving multiplication and division problems using a variety of representations (e.g., manipulatives, diagrams, and symbols). (VDOE Grade 3 Curriculum Framework).

• The properties of the operations are “rules” about how numbers work and how they relate to one another. Students at this level do not need to use the formal terms for these properties but should utilize these properties to further develop flexibility and fluency in solving problems. (VDOE Grade 3 Curriculum Framework).

• Strategies for solving problems that involve multiplication or division may include mental strategies, partial products, the standard algorithm, and the commutative, associative, and distributive properties. (VDOE Grade 3 Curriculum Framework)

• Students should experience a variety of problem types related to multiplication and division. (VDOE Grade 3 Curriculum Framework)

Important Assessment Look Fors:

• The student can explain how to find the product or quotient.

• The student is able to use a variety of strategies and representations.

• The student’s work shows that they understand the context given in the problem and can use it to determine the operation needed in order to solve.

Purposeful Questions:

• How did you represent your thinking?

• How do you know your answer is correct?

• Can you show your thinking using a different strategy?

• How can you use what you know to find a product with a two-digit number as one of its factors?

Understanding the Learning Trajectory

Big Ideas:

• The understanding of addition as putting together allows students to see the way fractions are composed of unit fractions.

• Prior knowledge of addition and subtraction of whole numbers allows for composing and decomposing fractions with the same denominator.

• Whole numbers can be represented as an equivalent fraction, thereby supporting addition and subtraction computations with whole numbers and fractions.

Important Assessment Look Fors:

• Student recognizes fractions represented by models.

• Student uses an appropriate operation to solve the problem.

• Student adds/subtracts numerators only while recognizing denominator remains unchanged.

• Student recognizes when a sum is an improper fraction and is able to convert it to a mixed number.

Purposeful Questions:

• Does your answer make sense? How do you know?

• Why did you add/subtract the numerators and not the denominators?

• Why did you add/subtract the numerators and the denominators?

• How did you come up with that fraction for your answer?

Understanding the Learning Trajectory

Big Ideas:

• Prior experiences with coin identification and values support students’ abilities in finding the value of a collection of coins and bills.

• Strategies to determine the value of the collection are counting on, starting with the highest value coin or bill, grouping like coins, or “make compatible combinations.” (Van de Walle, 2019, p.495)

• Properties of operations allow for computation (addition and/or subtraction) of the collection value. (Common Core Progressions, pg. 3)

Important Assessment Look Fors:

• Student groups like coins and uses repeated addition or skip counting to determine the value of a collection.

• Students group coins to allow for the use of benchmark numbers to make counting more efficient (i.e., combines a quarter and a nickel to make 30 cents and is then able to add remaining dimes).

• Student uses counting or addition strategies to find the value of a bill and coin collection.

• Student draws an efficient collection of dollars and coins to represent a value (using combinations other than just pennies).

Purposeful Questions:

• What strategy did you use? Is there another strategy that can be used?

• Which coins might you combine to create benchmark numbers?

• What is the value of the highest coin? Or bill?

• How many quarters, dimes, nickels, pennies, or bills are there?

Understanding the Learning Trajectory

Big Ideas:

• The value of coins and/or bills determine the value of the collection, not the number of coins and/or bills.

• When counting collections of coins and bills, students extend matching strategies used to compare collections of objects in earlier elementary grades.

• The terms is greater than, is less than, and is equal to are used to describe the relation of sets of coins/bills.

Important Assessment Look Fors:

• Student identifies coins/bills and their respective values.

• Student determines the value of each set of coins/bills.

• Student compares their total value OR uses a matching strategy to compare the value of the sets.

• Student provides reasoning for their selection of the term is greater than, is less than, or is equal to.

Purposeful Questions:

• What strategy did you use to find the value of the collection(s)?

• What does “is greater than, is less than, or is equal to” mean? What symbol is used to show “is greater than, is less than, or is equal to?”

• Did you find any equivalent values with the coins and bills? If so, which?

• Can you create a set of coins/bills that has a value “greater than” or “less than” this set?

Understanding the Learning Trajectory

Big Ideas:

• Prior experiences with coin identification supports students’ skill in finding the value of a coin and bill collection.

• Strategies to calculate change include: starting with the total amount purchased and counting up to the amount spent; subtracting the amount purchased from the amount spent; and finding the next whole dollar. A number line and a hundreds chart support students’ learning to make change.

• Properties of operations allow for computation (addition and/or subtraction) of the value of change.

Important Assessment Look Fors:

• Student identifies coins/bills and their respective values.

• Student uses manipulatives or other tactile objects to solve problems involving making change.

• Student uses a variety of strategies to solve problems involving making change (i.e., part-part-whole, counting on, number line, or subtracting).

• Students use the values of the money available and the purchased items to determine an appropriate strategy to make change.

Purposeful Questions:

• What strategy did you use to solve for the change? Can you explain more?

• What coins and/or bills did you choose to make the change?

• Is there another way and/or strategy to solve this problem?

Understanding the Learning Trajectory

Big Ideas:

• Measuring length or distance consists of two aspects, choosing a unit of measure and subdividing (mentally and physically) the object by that unit, placing that unit end to end (iterating) alongside the object (Common Core Progressions, pg. 4).

• The length of the object is the number of units required to iterate from one end of the object to the other, without gaps or overlaps (Common Core Progressions, pg. 4).

• Having real-world “benchmarks” is useful. Prior understanding of concepts of measurement and scale enhances estimation of an object's measurement (Common Core Progressions, pg. 14-15).

Important Assessment Look Fors:

• Student correctly subdivides inches in half.

• Student uses the ruler correctly, placing the zero mark at one end of the object and ending their measurement at the end of the object.

• Student accurately measures the length/width of objects using the correct unit.

• Student uses an appropriate measuring tool for each question.

Purposeful Questions:

• How did you know that measurement is correct?

• Can you show me how you found that measurement? Explain how you know it is correct.

• Where did you place your ruler to measure that item?

• What if we had a broken ruler? Could we still use it to measure objects?