Understanding the Learning Trajectory

Big Ideas:

• Rote counting is a prerequisite skill for addition and subtraction (one more, one less).  Each successive counting number describes a quantity that is one more than the quantity that the previous number describes. In a sense, then, counting is adding: Each counting number adds one more to the previous collection.[LT]2 Facilitator Guide: Counting

• Noticing patterns in numbers is helpful when counting. (Clements, D.H., & Sarama, J [2017/2019])

• Understanding groups of ten can help students stay on track when counting.

Math Strength Instructional Video

Important Assessment Look-fors:   

• Student follows number patterns to count forward by 1s.

• Student moves through decade numbers with ease (e.g., 29, 30, 31).

• Student corrects mistakes.

Purposeful Questions:

• What patterns do you notice  with counting? 

• What comes after 8? How can that help you get past 28?

• Think about what you just said. How can that help you move on?

Understanding the Learning Trajectory

Big Ideas:

• Initially, students begin counting at 1, but eventually progress to counting on from any number and keeping track of their counting using patterns (Clements, D.H., & Sarama, J [2017/2019]).

• The patterns in the ones, tens, and hundreds help students count accurately (Clements, D.H., & Sarama, J [2017/2019]).

• Students progress from reading two-digit numbers as individual digits (ex: they read “23” as (two-three) to a beginning understanding of place value (ex: knowing the 2 in “23” represents 2 tens, or twenty, and read the number as “twenty-three”)

Important Assessment Look-fors:   

• Student correctly continues the number pattern.

• Student generally forms numbers correctly and can read their own writing.

• Student uses number patterns to help write missing numbers in a sequence.

• Student recalls and writes a number when it is said orally.

Purposeful Questions:

• What do you notice about the pattern?

• Would counting forward or backward help you?

• Where might there be a decade/tens number to help get you on track?

Understanding the Learning Trajectory

Big Ideas:

• Counting backward by rote lays the foundation for subtraction (VDOE curriculum framework)

• Counting forward and backward leads to the development of counting on and counting back. (VDOE curriculum framework)

• Counting backward by ones helps students in the counting phase move to a transitional phase in which they use base ten representations to aid subtraction. (OGAP)

Important Assessment Look-fors:   

• Student follows number patterns to count backward by 1s.

• Student moves through decade numbers with ease (22, 21, 20, 19)

• Student states number before when given a number out of order. 

• Student corrects mistakes. 

Purposeful Questions:

• What patterns do you notice when counting?

• What comes before 9? How can that help you count backwards from 29?

• Why do you think it is important for us to say “zero” when we finish counting backwards?

Understanding the Learning Trajectory

Big Ideas:

• One-to-one correspondence is crucial to success with counting. Sometimes students can rote count without one-to-one correspondence.

• Skip counting is a way to count a group of objects quickly and efficiently; leads to algebraic thinking (Clements, D.H., & Sarama, J [2017/2019]).

• Skip counting can strengthen estimation skills by practicing repeated addition (Clements, D.H., & Sarama, J [2017/2019]).

Important Assessment Look-fors:   

• Student uses and identifies a pattern to skip count.

• Student accurately skip counts.

• Student attempts a skip counting strategy before counting by ones.

• Student sees efficiency in skip counting and utilizes it in real world situations.

Purposeful Questions:

• What strategies do you know to help skip count?

• Can you get a running start to help you figure out what the next number might be?

• How can you use ones to help you skip count?

Understanding the Learning Trajectory

Big Ideas:

• Each digit in our number systems has a place and a value.

• A collection can be grouped into tens and ones (Clements, D. H., & Sarama, J.[2017/2019]).

• A group of 10 can be counted as one unit of ten. 

Important Assessment Look-fors:   

• Student makes groups of ten.

• Student uses groups of ten to determine the value of a collection rather than counting by ones. 

• Student writes the number corresponding to a collection of objects.

• Student demonstrates understanding of the value of the digit in the tens place as meaning that many groups of ten, and the value of the digit in the ones place as being that many ones. 

Purposeful Questions:

• Did you make groups? Can you tell me how you grouped your cubes?

• What do these leftover cubes represent?

• How does the number that you counted connects to the number of cubes you have?

• How do you know the value of __ based on the place that it’s in?

Understanding the Learning Trajectory

Big Ideas:

• Greater than is more, less than is fewer, and equal to means the same as.

• Concrete models, pictorial models, and written numbers can be compared by deciding which has a higher value.

• The digit in the tens place is more important when determining the size of a two digit number (Common Core Writing Team, 2019, pg 6).

Math Strength Instructional Video↗

Important Assessment Look-fors:   

• Student understands and can use the terms less than, greater than, and equal to.

• Student uses the model to determine how many are in each set.

• Student recognizes each tens block as a unit of ten when counting.

• Student determines that the tens place is more important than the ones place when determining which value is greater. 

Purposeful Questions:

• What do you notice about the two sets of blocks?

• How can you decide which set has more?

• Which numbers helped you decide how to compare the two?

Understanding the Learning Trajectory

Big Ideas:

• The digit in the tens place is more important when determining the size of a two digit number (Common Core Core Progressions, 2019).

• Least to greatest means smallest to largest, and greatest to least means largest to smallest. 

• Conceptual subitizing by place value and multiplicative thinking allow students to use the base-10 system to describe quantities and order them. Learning Trajectory Information↗ 

Math Strength Instructional Video ↗

Important Assessment Look-fors:   

• Student determines the value of two images and compares them.

• Student represents numbers with pictures using groups of tens and ones.

• Student uses the tens place to compare and determine the larger or smaller of two numbers.

• Student sees the groups of ten and the group of ones in any number up to 110. 

Purposeful Questions:

• What do you notice about these two pictures?

• Tell me about your representations for each number.

• How did you decide which order you wanted to write the numbers?

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 person was first in line? How did you know which box was first?

• Count the people. What number person in line is third?

• How did you know which box needed a heart?

Understanding the Learning Trajectory

Big Ideas:

• Fractional parts are equal shares or equal-sized portions of a whole or unit. (Van de Walle, 2006)

• Fractional parts have special names that tell how many parts of that size are needed to make the whole. (Van de Walle, 2006)

• Equipartitioning, or equal sharing, leads students to a foundational understanding of fractions, ratios, multiplication, and division. 

• Equipartitioning helps students understand that numbers are made of smaller parts. 

Important Assessment Look-fors:   

• The student can divide the item into equal parts depending on the number of sharers. 

• The student can name the shares as halves or fourths. (Does not need to be written numerically.)

• The student can represent equal sharing with multiple items and two or four sharers. 

Purposeful Questions:

• What does it mean if the cookie is shared fairly?

• How did you divide the circle so that each friend could have their fair share?

• How can you share cookies and brownies so that no cookies or brownies are leftover?

Understanding the Learning Trajectory

Big Ideas:

• Exploring ways to estimate the number of objects in a set, based on appearance, enhances the development of number sense. (VDOE curriculum framework)

• Relationships of numbers to real-world quantities and measures and the use of numbers in simple estimations can help children develop the flexible, intuitive ideas about numbers that are most desired. (Van de Walle, 2006)

Important Assessment Look-fors:   

• The student can select a reasonable order of magnitude.

• The student can base their selection on the number of items rather than the amount of space the objects take up indicating a truer understanding of magnitude. 

• The student can demonstrate an understanding of number conservation by selecting the correct magnitude even when the order of objects seems to move. 

Purposeful Questions:

• How do you choose which number to match to the sunflowers?

• How do you know when an estimate is too small or too big?

• How did you know where to place each card of 5, 50, or 500?

Understanding the Learning Trajectory

Big Ideas:

• Story problems provide an opportunity to act out and reason through what is happening and what needs to be determined (i.e., which part is unknown).

• There are numerous strategies that work together to develop fluency with operations and efficiency in problem solving (i.e., one more/one less, doubles, make ten, using related facts, etc.)

• Various representations may be used to model and solve the same problem: spoken words (act it out), moving objects and manipulatives, drawing pictorial models, and writing number sentences.

• A number sentence is one way of representing the action (or operation) used to solve a problem. Every problem has an action that translates to an operation (Van de Walle et. al., 2018).  

Important Assessment Look-fors:   

• Student represents a story problem with an appropriate number sentence.

• Student uses an appropriate strategy (i.e., picture, number sentence, talk-aloud, fingers, etc.) to make sense of the problem.

• Student identifies and uses an appropriate action to solve the problem.

• Student identifies the unknown part (i.e., start, change, result).

Purposeful Questions:

• What strategy did you use to solve? What are some strategies you know?

• What’s the action in the story? What is happening?

• What part of the story is missing? What are we trying to find out?

Understanding the Learning Trajectory

Big Ideas:

• Subitizing skills and visual recognition of numbers up to 5 facilitate learning combinations to 10.

• A quantity (whole) can be decomposed into equal or unequal parts; the parts can be composed to form the whole (Erikson Institute’s Early Math Collaborative).

• Quantities represented by numbers can be decomposed (or composed) into part-whole relationships (NCTM Number and Numeration, p.25).

Important Assessment Look-fors:   

• Student demonstrates conservation of number by “holding onto” the original number of objects.

• Student uses a strategy (i.e., fingers, counting backwards, adding up, doubles, etc.) to figure out how many counters are hidden.

• Student recognizes with ease the quantity of small groups of objects (up to five).

• Student shows beginning understanding of the inverse relationship between addition and subtraction (i.e.,  6-2=4, 4+2=6).

• Student draws on solutions already made (in prior problems) to answer further questions.

Purposeful Questions:

• What strategy can you use to find the missing number?

• How many did you start with? How has that changed?

• What small group do you see? How is that related to the total?

Understanding the Learning Trajectory

Big Ideas:

• Flexible methods can be used when adding and subtracting.

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

• Part-whole relationships are foundational to developing computational fluency.

• Fluency leads to efficiency by allowing students to recall basic facts when solving more complex problems (Common Core Progressions, p. 4).  

Important Assessment Look-fors:   

• Student sees a total of 10, broken into two parts.

• Student uses flexible strategies to determine the number of dots/counters (5 and 3 more makes 8, counts backward from 10 to make 8).Student writes a complete number sentence to represent a problem.

• Student composes and decomposes numbers to 10 with ease.

Purposeful Questions:

• Tell me about the strategy you are using. How did it help you to solve the problem?

• How can this model help you when counting? 

• When you first look at this card/ten frame, how many do you see? How do you see it? 

• Can you represent that with a number sentence? What would it look like?

Understanding the Learning Trajectory

Big Ideas:

• Counting money is an exercise in unitizatizing, the concept that a group of objects can be counted as one unit (e.g., 10 pennies can be counted as one dime.) (VDOE curriculum framework)

• Counting money helps students gain an awareness of consumer skills and the use of money in everyday life.  (VDOE curriculum framework)

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

Important Assessment Look-fors:   

• The student demonstrates one to one correspondence up to 100 items. 

• The student can make groups of five and/or tens and count the groups by skip counting. 

• The student can recall the value of a penny, nickel, and dime.

Purposeful Questions:

• Why do you think skip counting by groups of 5 or 10 is easier than counting each coin?

• What tools would make it easier for you to count? a hundreds chart? tens frame? How would those tools help you?

• How do you know what a penny, nickel, and dime are worth?

Understanding the Learning Trajectory

Big Ideas:

• Time cannot be seen, but it can still be measured. Time is the duration of an event from its beginning to its end. (Van De Walle, Teaching Student Centered Mathematics, K-3)

• Telling time is related to reading the instrument of a clock rather than measuring time. The numbers on a clock represent the time in that moment. 

• We can communicate our measures of time in units of hours and half hours. 

Important Assessment Look-fors:   

• Student recognizes analog and digital clocks as instruments to tell the current time.

• Student distinguishes between the hour hand and the minute hand on the analog clock and between the two sides of the digital clock. 

• Student uses the vocabulary words “hours” and “o’clock”.

• Student is able to read the time on the clock to the nearest hour and half-hour. 

Purposeful Questions:

• What does it mean to “tell time?”

• Can you explain how to use the arrows/hands on an analog clock to tell time?

• How does the time change if the hour hand is moved?

• What do the numbers on a digital clock mean? How do you know?

Understanding the Learning Trajectory

Big Ideas:

• Calendars are instruments used to represent units of time.

• Calendars help students understand the concept of a day as a 24-hour period rather than the hours between sunrise and sunset. 

• Reading a calendar is a practical application of counting principles as applied to real world scenarios and the ability to communicate the passage of time or future events to others. 

Important Assessment Look-fors:   

• Students can accurately identify the day of the week.

• Students can distinguish between the day of the week and the date of a day. 

• Students understand that they are to count the boxes holding a number that represents a date when counting the number of Mondays rather than simply counting five boxes underneath the Monday heading. 

Purposeful Questions:

• What is the difference between the day of the week and the date of a day?

• What information is important when naming the date of a day?

• Why do you think there are more Thursdays than Mondays on this calendar?

Understanding the Learning Trajectory

Big Ideas:

• Measurement involves a comparison of an attribute of an item or situation with a unit that has the same attribute. Lengths are compared to units of length, areas to units of area, time to units of time, and so on. Before anything can be measured meaningfully, it is necessary to understand the attribute to be measured. (Van de Walle, 2006)

• Measuring with nonstandard units allows students practice in laying objects end to end without gaps or spacing. This also grow the understanding that the unit of measure must be same size when comparing objects.

• Measuring and comparing volume helps students understand that a space must be filled or occupied.

Important Assessment Look-fors:   

• The student can use a nonstandard unit end to end without overlapping or leaving gaps.

• The student will use the same size unit to measure both objects in each section.

• The student can accurately count the number of unit equal to the length, weight, or volume of the objects demonstrating one to one correspondence. 

Purposeful Questions:

• How do you choose the unit you use to measure objects?

• How does a balance or pan scale help you measure weight?

• How did you decide which container holds more?

Understanding the Learning Trajectory

Big Ideas:

• Building geometric and spatial capabilities fosters enthusiasm for mathematics while providing a context to develop spatial sense.

• What makes shapes alike and different can be determined by an array of geometric properties. (Van de Walle, 2006)

• Shapes and properties includes a study of the properties of shapes in both two and three dimensions, as well as a study of the relationships built on properties. (Van de Walle, 2006)

Important Assessment Look-fors:   

• The student knows and can use the correct names of shapes. 

• The student knows and can use the terms sides, vertices, and angles. 

• The student can describe the circle as curved or round. 

• The student describe and recognize shapes regardless of their orientation.

Purposeful Questions:

• What can you tell me about the angles of different shapes?

• When you are sorting shapes, how do you decide how you will sort them?

• When looking at the street signs what do you look for when naming the shapes?

Understanding the Learning Trajectory

Big Ideas:

• A collection of objects with various attributes can be classified or sorted in different ways. A single object can belong to more than one class. Classification is the first step in the organization of data. (Van de Walle, 2006)

• Data are gathered and organized in order to answer questions about the population from which the data come. With data from only a sample of the population, inferences are made about the population. (Van de Walle, 2006)

• Data collection moves students from seeing only their own data or opinion to viewing how their data fits in with the rest of the collection or the greater whole. 

• Representing data in charts and graphs showcases a student’s natural need to organize things by sorting and counting objects in a collection according to similarities and differences with respect to given criteria. 

Important Assessment Look-fors:   

• The student can accurately tally the data.

• The student can demonstrate understanding that zero is a value on the data chart and represents zero snow days. 

• The student can demonstrate an understanding of classifying or categorizing the data. 

Purposeful Questions:

• What information can we learn from your tally chart?

• Does the data in chart one match the data in chart two in problem two? Why or why not?

• How did you represent the number of snow days?

Understanding the Learning Trajectory

Big Ideas:

• Graphs are read and interpreted in order to compare information that has been gathered.

• Categorical data comes from sorting information into categories.  

• Asking and answering questions about data creates connections to addition and subtraction. 

Important Assessment Look-fors:   

• Student identifies the categories on a graph.

• Student identifies the whole set total.

• Student asks and answers questions based on each graph. 

• Student makes comparisons using the data.

Purposeful Questions:

• Using the lunch orders graph, tell me what you can learn from this graph.

• How do you know which juice is liked the most?

• How can we use the graph to make a comparison?