Understanding the Learning Trajectory

Big Ideas:

• The value of each digit in a number depends on its position in that number.

• Numbers are based on powers of ten.  The value of each place is 10 times the value of the digit to the right.

• The structure of numbers is based on unitizing amounts into groups of ones, tens, hundreds, etc.

Math Strengths Instructional Video 2.1a↗

Important Assessment Look-fors:   

• Student counts one hundred as a single unit.

• Student composes and decomposes numbers into ones, tens, and hundreds.

• Student knows the number of hundreds that can be made from any group of tens and the number of tens left over.

• Student determines the total value of a group of hundreds, tens, and ones by reorganizing them into all possible hundreds, then all possible tens, with leftover ones.

• Student describes any 3 digit number in terms of its value in hundreds, tens, and ones.

Purposeful Questions:

• How many groups of ones, tens, and hundreds make this number?

• How do the digits in this number relate to the groups of hundreds, tens, and ones in this number?

• How can the hundreds, tens, and ones in this number be regrouped to represent an equivalent value?

Understanding the Learning Progression

Big Ideas:

• The same value may be represented as 10 ones and 1 ten or 10 tens and 1 hundred.

• Ones, tens, and hundreds can be grouped and counted as units.

• The place value structure of hundreds, tens, and ones helps determine 10 more, 100 more, 10 less, and 100 less without counting by ones.

Important Assessment Look-fors: 

• Students moves vertically on a 120s chart to demonstrate 10 more/10 less/100 more/100 less than a number.

• Student counts forward/backward by tens to determine 10 more/10 less than a given number.

• Student adds or removes a ten rod to demonstrate 10 more/10 less of a number.

Purposeful questions:

• Is there a way to show 10 more/10 less in one jump on the 120s chart? Is there a way to show 100 more/100 less in one jump on the 120s chart?

• How can I represent 10 more/10 less, with base ten blocks?

• Is there another model you can use to show 10 more/10 less? 100 more/100 less? 

• Explain to a friend how you would mentally solve 10 more/10 less/100 more/100 less questions.

Understanding the Learning Trajectory

Big Ideas:

• According to the compare learning trajectory, students compare by counting, compare numbers with place value, use knowledge of number relationships and mental number line as well as benchmarks to determine relative size and position when comparing. (Clements & Sarama, 2019)↗ 

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

• Comparing the magnitude of two digit and three digit  numbers uses the understanding that the tens place is greater than the ones place and the hundreds place is greater than the tens place (Common Core Standards Writing Team, 2019).

Important Assessment Look Fors:

• Student chooses numbers that are very different from the original number when comparing quantities.

• Student uses the place value structure of numbers to compare and order different amounts.

• Student uses symbols to represent greater than, less than, and equal to relationships. 

Purposeful Questions: 

• What determines whether a number is greater than, less than, or equal to another number?

• How is understanding place value helpful when comparing and ordering numbers?

• What words and symbols are used to compare and order numbers?

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)

• Rounding should be conceptually understood to allow for flexibility in thinking about distance between numbers. 

Important Assessment Look Fors:

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

• Student can identify the tens, hundreds, thousands place in order to round. 

• Student can identify the tens a number is between. 

• Student determines the closest multiple of ten for the given number. 

Purposeful Questions: 

• How did you know to round ________ to ________? ( 37 to 40)

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

• Explain why you chose _________ to round to ______? (80)

• What numbers would round to 60?  How do you know? 

Understanding the Learning Trajectory

Big Ideas:

• Within the learning trajectory of counting forward and back, this level demonstrates students’ ability to count "counting words" (single sequence or skips counts) in either direction starting at any number. Recognizes that decades sequences mirror single-digit sequences (Clements & Sarama, 2019).

• Students are able to skip counting by saying a sequence of numbers represents a grouping of objects. Organizing objects into groups while skip counting by that quantity is more efficient than skip counting by ones (Richardson, 2012).

• Skip counting can occur at various places in the sequence of numbers. Skip counting  by tens forward leads to place value strategies for addition (Richardson, 2012).

Important Assessment Look Fors:

• Student counts by two or fives rather than by ones when objects are presented in groups of 2 or 5, respectively. 

• Student skip counts by 10s to continue a counting sequence. 

• Student recognizes that a zero in the ones place signals counting by tens. 

• Student applies the pattern of 5s and 0s in the ones place when counting by fives beginning at a multiple of 5.

Purposeful Questions: 

• How are the objects grouped in the pictures? 

• How does the way objects are grouped determine the skip counting pattern?

• How do the numbers you say while skip counting help you determine the pattern? 

• What are some patterns noticed in the specific numbers said when skip counting by a specific amount?

Understanding the Learning Trajectory

Big Ideas:

• Within the learning trajectory of counting forward and back, this level demonstrates students’ ability to count "counting words" (single sequence or skips counts) in either direction starting at any number. Recognizes that decades sequences mirror single-digit sequences (Clements & Sarama, 2019).

• Organizing objects into groups of ten while skip counting backward is  more efficient than skip counting backward by ones. 

• Skip counting backwards by tens is “ ten less” than a number. Skip counting backwards by tens supports place value strategies for subtraction.

Important Assessment Look Fors:

• Student skip counts backward by tens without counting backwards by ones.

• Student determines “10 less” than a number.

• Student uses a number line or  hundreds chart to facilitate their skip counting.

Purposeful Questions: 

• How is skip counting backwards by ten more efficient than skip counting backwards by one?

• How is skip counting backwards by ten the same as “10 less” than a number?

• How is skip counting backwards by ten similar and different than skip counting forward by ten? 

Understanding the Learning Trajectory

Big Ideas:

• Building on their own intuitive mathematical knowledge, students display a natural need to organize things by sorting, comparing, ordering, and labeling objects in a variety of collections. (VDOE Curriculum Framework, VDOE, grade 2)

• Odd and even numbers can be explored in different ways (e.g., dividing collections of objects into two equal groups or pairing objects). When pairing objects, the number of objects is even when each object has a pair or partner. When an object is left over, or does not have a pair, then the number is odd. (VDOE Curriculum Framework, Grade 2)

• Determine if a number is even or odd by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. (corestandards.org)

Important Assessment Look Fors:

• Students can organize marbles into two equal groups or group into groups of two

• Students can apply rote counting by twos to coordinate with groups of two

• Students can write an equation to express an even number as a sum of two equal addends or an odd number of two equal addends plus 1.  

Purposeful Questions: 

• How did you use the marbles to help determine if 25 is even or odd?  

• How do you know 25 is even or odd? 

• How might you write an equation to prove 25 is even or odd? 

Understanding the Learning Trajectory

Big Ideas:

• An understanding of the cardinal and ordinal meanings of numbers is necessary to quantify, measure, and identify the order of objects. 

• The ordinal meaning of numbers is developed by identifying and verbalizing the place or position of objects in a set or sequence (VDOE curriculum framework)

• An ordinal counter is someone who can identify and use ordinal numbers from first to tenth. Ordinal counters can also make connections to counting words. For example, if you are fifth in line, you are number 5 in line. (Clements, D. H., & Sarama, J.)

Important Assessment Look Fors:

• Student can identify where to start when counting, indicating they know left, right, top, and bottom. 

• Student can match the correct ordinal position with the cardinal numbers. 

• Student can place the correct drawing or notation in the correct spot indicating they understand ordinal number placement. 

Purposeful Questions: 

• How did you know which shape is fifth from the right? How did you know which shape to put an “x” on?

• How did you know the order of ice cream colors? How did you know to start from the top? bottom?

• How did you know where to find the twelfth shape? Fourteenth? 

Understanding the Learning Trajectory

Big Ideas:

• Fractions are equal shares of a whole or a unit. Therefore, fraction instruction should begin with equal sharing activities that build on whole number knowledge to introduce fraction as quantities(i.e., 2 sandwiches shared with 4 friends). SOL K.5 and 1.4 focus on equal shares and the number of sharers of 2 and 4 and move to other fractional parts like thirds, sixths, eighths, etc. in support of the learning trajectory. 

• A fraction is a numerical way of representing equal parts of a whole region (i.e., an area model), parts of a group (i.e., a set model), or parts of a length (i.e., a measurement model).

• Parts of a region model may not be congruent but maintain an equal value (area).

• The numerator represents the equal parts being considered and the denominator represents the equal parts that create the whole.

Important Assessment Look Fors:

• Student identifies the shaded parts of a region model as the numerator and the total number of parts as the whole or denominator. 

• Student identifies that the total number of objects in a set represent the whole or denominator, and that the shaded objects represent the identified part, or numerator.

• Student compares lengths of sections and their positions relative to whole numbers on a number line to name fractional marks on a number line. 

• Student compares sizes and shapes of fractional parts to determine whether or not a region model is divided into equal parts. 

Purposeful Questions: 

• How do we name fractions? What does the numerator represent? What does the denominator represent?

• How does a fraction represent a part to whole relationship?

• What does a fraction on a number line represent? What makes the whole on the number lines? What is the part being considered? 

• How do you find a fraction of a set? (Set model - discrete objects)

• How must a shape be divided to represent a fraction? (region or area model- continuous)

Understanding the Learning Trajectory

Big Ideas:

• The whole must be defined when working with fractions.

• In a region/area model, the parts must have the same area. 

• In a set model, the set represents the whole and each item represents an equivalent part of the set. 

• In a length model, each length represents an equal part of the whole. 

Important Assessment Look Fors:

• Student divides a whole region into the appropriate number of equal parts to represent the denominator of a fraction.

• Student uses the appropriate number of shapes to represent the identified numerator of the fraction.

• Student folds the paper strip (or other model) into equal parts, as represented by the denominator, and shades the parts identified by the numerator.

Purposeful Questions: 

• When creating a model to represent a fraction, what does the numerator represent and what does the denominator represent?

• When making a model to represent a fraction, what must all the pieces have in common? 

Understanding the Learning Trajectory

Big Ideas:

• A whole is composed of equal parts and each part is considered a unit fraction.

• A unit fraction is represented symbolically with a one as the numerator.

• The more equal parts a whole is divided into, the smaller each part becomes.

• The value of a fraction depends on the number of equivalent parts making up the whole and how many parts are being considered.

Important Assessment Look Fors:

• Student creates equal sized parts when representing a fraction in a region model.

• Student compares sizes of unit fractions, of the same size whole, to determine which unit fraction is larger/smaller.

• Student recognizes that parts get smaller as a region is divided into more equal parts, and that a larger denominator indicates a smaller unit fraction.

• Student identifies that the total number of parts in a region is indicated by the denominator and the shaded parts are indicated by the numerator. 

Purposeful Questions: 

• Why is a unit fraction smaller with a larger number in the denominator?

• What happens to the parts of a whole as they are divided into more parts?

• What does the numerator represent in a fraction? What does the denominator represent in a fraction?

• How can skip counting by unit fractions help determine the fraction represented by a model?

• What symbol represents “greater than”, “less than”, and “equal to”? How can we use the symbol to compare two fractions?

Understanding the Learning Trajectory

Big Ideas:

• According to the learning trajectories, children around age 7-8 move from being a “deriver” to a “problem solver”. At the level, deriver, a child can use flexible strategies and related facts/derived combinations (for example, “7 1 7 is 14, so 7 1 8 is 15”) to solve all types of problems. As children develop their addition and subtraction problem solving abilities, they can solve all types of problems by using flexible strategies and many known combinations. For example, when asked, “If I have 13 and you have 9, how could we have the same number?” this child says, “9 and 1 is 10, then 3 more to make 13. 1 and 3 is 4. I need 4 more!” (For more information, go to learning trajectories levels  at https://www.learningtrajectories.org/.)

• Addition and subtraction are related and have an inverse relationship.

• Number relationships provide the foundation for strategies that help students remember basic facts.

• The patterns formed by related facts facilitate the solution of problems involving a missing addend in an addition sentence or a missing part in a subtraction sentence. 

Important Assessment Look Fors:

• Student has an understanding of the word “related.”

• Student models an addition story problem with manipulatives.

• Student creates a number sentence to model the practical problem.

• Student models and writes equations to match various types of word problems.

Purposeful Questions: 

• Is there a way to use a different operation to solve this number story?

• Does the number sentence you wrote match the number story? What could be used to represent what information we are looking for?

• How can using the inverse operation help us? 

Understanding the Learning Trajectory

Big Ideas:

• Flexible methods for computing build computational fluency.

• Composing and decomposing numbers build understanding and flexibility with number.

• Part-whole relationships are foundational to understanding computation.

Important Assessment Look Fors:

• Student identifies the symbols for representing computation and understands the meaning of the symbols for addition and subtraction.

• Student  uses one or more strategies to solve addition and subtraction facts/problems.

• Student uses a manipulative to model a strategy to solve addition and subtraction facts/problems.

Purposeful Questions: 

• What strategy did you use? Why did you choose that strategy?

• Is there another strategy you can use?

• Is there another fact you used to help you solve?

• How can you use a different  operation to check your answer? 

Understanding the Learning Trajectory

Big Ideas:

• Estimation is a valuable time saving skill in practical situations when an exact answer is not needed.

• Estimation helps students complete computations with larger numbers.

• Composing and decomposing numbers can build estimation skills.

Important Assessment Look Fors:

• Student identifies which two tens a number falls between.

• Student breaks apart numbers and finds friendly combinations to solve.

• Student understands when an estimate is needed and not an exact answer.

Purposeful Questions: 

• What strategy did you use to find an estimate? Why did you choose that strategy?

• How did estimation help you determine the solution?

• How can estimation help you decide if your answer is correct?

Understanding the Learning Trajectory

Big Ideas:

• Flexible methods of addition and subtraction involve composing and decomposing numbers in a variety of ways.

• Flexible methods for computation require a strong understanding of the operations including the properties of operations.

Important Assessment Look Fors:

• Student uses estimation strategies before and after solving addition and subtraction problems.

• Student illustrates or acts out a story problems to determine the operation needed to solve.

• Student uses various methods for computing and describes how the method works.

Purposeful Questions: 

• What strategy did you use to find an answer? Why did you choose that strategy?

• Is there another strategy you could try?

• Will your strategy work for other numbers? How do you know?

• How can estimation help you decide if your answer is correct? 

Understanding the Learning Trajectory

Big Ideas:

• Addition and subtraction are used to solve problems that arise in everyday life.

• Addition and subtraction have an inverse relationship.

• Different representations  can be used to model a variety of problem types. 

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

• Using related combinations can be helpful in solving problems that contain larger numbers (i.e., 9 + 7 can be thought of as 9 broken up into 2 and 7; using doubles, 7 + 7 = 14; 14 + 2 = 16 or 7 broken up into 1 and 6; making a ten, 1 + 9 = 10; 10 + 6 = 16).

Important Assessment Look Fors:

• Student creates a representation that models the problem and supports their understanding.

• Student interprets the context described in the problem and is able to solve for different unknowns (i.e., problem types).

• Student chooses an efficient strategy that makes sense to them and can be used to solve the problem.

• Student records the problem and their answer(s) using pictures, words, and/or symbols.

• Student estimates to check the reasonableness of each solution.

Purposeful Questions: 

• Describe your problem solving process. Tell me what you know and what you did.

• Is there another way you can represent/model your thinking?

• What strategies are most efficient for this problem and how do you know?

• Is your answer reasonable? How do you know?

Understanding the Learning Trajectory

Big Ideas:

• Coins have unique values and physical attributes.  The size of a coin does not indicate its value. The value of some coins and bills can be represented as a combination of other coins.

• Skip counting can be used to determine the value of a set of like coins and can serve to build the foundation for multiplication.

• The ability to switch between different types of skip counting (i.e., 1, 5, 10, and 25) allows students to see patterns in numbers. 

Important Assessment Look Fors:

• Student identifies and describes attributes of a penny, nickel, dime and quarter.

• Student identifies the number of pennies equivalent to a nickel, dime and quarter.

• Student determines the value of a collection of coins using the skill of skip counting.

• Student determines the value of a collection of coins by counting the highest valued coin first, adding the values of the individual coins, or other efficient strategies (i.e., combining two or more coins by utilizing benchmark numbers).

• Student compares the values of two sets of coins and a dollar bill using the terms greater than, less than or equal to.

Purposeful Questions: 

• Can you tell me what coins you have? 

• What’s the value of each coin? Of this collection of coins? 

• Are there any coins that you can use to make 25? Are there other ways to make that same amount?

• What would be an effective way to count a combination of coins?

• Can you use skip counting to help you could these coins?

• Which combination  of coins has the greatest value/least value?

Understanding the Learning Trajectory

Big Ideas:

• The value of some coins and bills can be represented as a combination of other coins.

• Combinations of coins and bills can be represented in different ways without changing the value.

• The cent symbol, dollar symbol, and decimal point are used to write a value of money.

Important Assessment Look Fors:

• Student can use two or more sets of coins to make the same value.

• Student uses the cent symbol to write the value of a combination of coins.

• Student uses the cent, dollar symbol and decimal point to write the value of a combination of coins and one dollar bill.

Purposeful Questions: 

• How many ways could you receive change at the store if change is given in pennies or other coins.

• How many ways could you show this same amount with like coins?

• Can you write the value of this money combination using the cent and dollar symbol?

Understanding the Learning Trajectory

Big Ideas:

• Measurement is a process of comparing a unit to the object being measured.

• Objects can be described, compared and ordered by length.

• Same size units must be used when measuring with nonstandard units (i.e., same size paperclips, etc.).

• There are standard measurement tools for measuring length.

Important Assessment Look Fors:

• Student describes, compares, and orders objects by length.  

• Student demonstrates the appropriate use of nonstandard measurement tools (i.e., same size paperclips).

• Student identifies and chooses the appropriate measuring tool when measuring length.

Purposeful Questions: 

• How would you describe and compare the lengths of these two objects? (longer or shorter)

• What nonstandard measuring tool could be used to measure the length of this object?

• Which measuring tool would be the most effective to measure the length of this object?

• Is the measurement closer to __ inches or __ inches?

Understanding the Learning Trajectory

Big Ideas:

• Measurement is a process of comparing a unit to the object being measured.

• Objects can be described, compared and ordered by weight.

• Same size units must be used when measuring with nonstandard units (Example: same blocks, cubes, etc.).

• There are standard measurement tools for measuring weight.

Important Assessment Look Fors:

• Student describes, compares, and orders objects by weight.  

• Student demonstrates the appropriate use of nonstandard measurement tools (i.e., same blocks, cubes, etc.).

• Student identifies and chooses the appropriate measuring tool when measuring weight.

Purposeful Questions: 

• How would you describe and compare the weights of these two objects? (lighter or heavier)

• What nonstandard measuring tool could you use to measure the weight of this object?

• Which measuring tool would be the most effective to measure the weight of this object?