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# BIg Ideas

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## Rachel Seib

on 5 November 2014

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#### Transcript of BIg Ideas

Big Ideas
For grades 2-4 Number Strand Specific Outcomes
Specific outcomes
Numbers:
Describe a quantity (how many or how much). This includes whole numbers, decimal numbers, fractions and percents. Each real number can be associated with a point on a number line.

BIG Ideas
At a Glance
these are the 9 "Big Ideas" for the Number Strand in the AB Program of Studies
Base Ten Numeration System:
This system allows the recording of numbers using the digits 0-9, groups of ten, and place value. There are many patterns inherent in this system.

Amelia, Anastasia, Brandon, Alexis & Rachel
EDEL 316 (A03)

Equivalence:
There are different, but equivalent ways of representing a number, measure, numerical expression or equation. (ie. decomposing numbers in a variety of ways, composite numbers…)

Numbers
Base Ten Numeration System
Equivalence

Comparison
Patterns

Properties:
For a given set of numbers there are relationships that are always true, and these are the rules that govern arithmetic and algebra.

(ie. commutative property for addition and multiplication).

Basic Facts and Algorithms:
Basic facts and algorithms use concept of equivalence to make calculations simpler.

Estimation:
Numerical calculations can be approximated by replacing numbers with other numbers that are close and easier to compute mentally.

Operation Meanings and Relationships
Properties
Basic Facts and Algorithms
Estimation
Comparison:
Numbers, expressions, and measures can be compared by their relative measures. Benchmark numbers (5, 10, 100…) are useful for relating and estimating numbers.

Patterns:
Relationships can be described and generalizations made for mathematical situations that have numbers that repeat in predictable ways. (ie. skip counting, multiplication facts…)

Operation Meanings and Relationships:
The same number sentence (ie. 12 – 4 = 8) can be associated with different concrete or real-world situations AND different number sentences can be associated with the same concrete and
real-world situation.

SO6.
Estimate quantities to 100, using referents
[C,ME,PS,R]

SO10.
Apply mental mathematics strategies, such as:

SO2.
Demonstrate if a number (up to 100) is even or odd. [C, CN, PS, R]

SO8.
Demonstrate and explain the effects of adding zero to, or subtracting zero from, any number
[C,R]

SO4.
Represent and describe numbers to 100, concretely, pictorially, and symbolically
[C,CN,V]

SO1.
Say the number sequence 0 to 100 by:
* 2's,5's, and 10's, forward and backward, using starting points that are multiples of 2,5, and 10 respectively
* 10's, using starting points from 1 to 9
* 2's, starting from 1
[C,CN,ME,R]

SO5.
Compare and order numbers up to 100
[C,CN,ME,R,V]

SO9.
Demonstrate an Understanding of addition (limited to 1- and 2-digit numerals) with answers to 100 and the corresponding subtraction by:

SO10.
Relate decimals to fractions and fractions to decimals
[C,CN,R,V]

SO7.
Illustrate, concretely and pictorially, the meaning of place value for numerals to 100. [C,CNR,V]

SO3.
Describe order or relative position, using ordinal number (up to tenth) [C,CN,R]

* Using personal strategies for adding and subtracting with and without the support of manipulatives
* Creating and solving problems that involve addition and subtraction
* Using the commutative property of addition (the order in which numbers are added does not affect the sum)
* Using the associative property of addition (grouping a set of numbers in different ways does not affect the sum)
*Explaining that the order in which numbers are subtracted may affect the difference
[C,CN,ME,PS,R,V]
* Using doubles
* Making 10
* One more, one less
* Two more, two less
* Building on a known
double
* Thinking addition for subtraction for basic
related subtraction
facts to 18.
[C,CN,ME,PS,R,V]
SO3.
Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:

SO2.
Represent and describe numbers to 1000, concretely, pictorially and symbolically. [C, CN, V]

SO9.
Demonstrate an understanding of addition and subtraction of numbers with answers to 1000 (limited to 1-, 2- and 3-digit numerals), concretely, pictorially and symbolically, by:
* using personal strategies for adding and subtracting with and without the support of manipulatives
* creating and solving problems in context that involve addition and subtraction of numbers.
[C, CN, ME, PS, R, V]

SO1.
Represent and describe whole numbers to 10 000, pictorially and symbolically. [C, CN, V]
SO3.
Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:
* using personal strategies for adding and subtracting
* estimating sums and differences
* solving problems involving addition and subtraction.
[C, CN, ME, PS, R]

SO13.
Demonstrate an understanding of fractions by:
explaining that a fraction represents a part of a whole
describing situations in which fractions are used
comparing fractions of the same whole that have like denominators.
[C, CN, ME, R, V]

SO8.
Demonstrate an understanding of fractions less than or equal to one by using concrete, pictorial and symbolic representations to:
* name and record fractions for the parts of a whole or a set
* compare and order fractions
* model and explain that for different wholes, two identical fractions may not represent the same quantity
* provide examples of where fractions are used.
[C, CN, PS, R, V]

SO5.
Illustrate, concretely and pictorially, the meaning of place value for numerals to 1000. [C, CN, R, V]
SO9.
Represent and describe decimals (tenths and hundredths), concretely, pictorially and symbolically. [C, CN, R, V]

SO3.
Compare and order numbers to 1000. [C, CN, R, V]

SO2.
Compare and order numbers to 10 000. [C, CN, V]
SO11.
Demonstrate an understanding of multiplication to 5x5 by:
* representing and explaining multiplication using equal grouping and arrays
* creating and solving problems in context that involve multiplication
* modeling multiplication using concrete and visual representations, and recording the process symbolically
* relating multiplication to repeated addition
* relating multiplication to division.
[C, CN, PS, R]

SO4.
Apply the properties of 0 and 1 for multiplication and the property of 1 for division. [C, CN, R]

SO6.
Describe and apply mental mathematics strategies for adding two 2-digit numerals, such as:
* adding from left to right
* taking one addend to the nearest multiple of ten and then compensating
* using doubles.
[C, CN, ME, PS, R, V]
SO7.
Describe and apply mental mathematics strategies for subtracting two 2-digit numerals, such as:
* taking the subtrahend to the nearest multiple of ten and then compensating
* using doubles.
[C, CN, ME, PS, R, V]
SO10.
Apply mental mathematics strategies and number properties, such as:
* using doubles
* making 10
* using the commutative property
* using the property of zero
in order to understand and recall basic addition facts and related subtraction facts to 18.
[C, CN, ME, PS, R, V]

SO5.
Describe and apply mental mathematics strategies, such as:
* skip counting from a known fact
* using doubling or halving
* using doubling or halving and adding or subtracting one more group
* using patterns in the 9s facts
using repeated doubling
[C, CN, ME, R]

SO4.
Estimate quantities less than 1000, using referents. [ME, PS, R, V]

SO8.
Apply estimation strategies to predict sums and differences of two 2-digit numerals in a problem-solving context. [C, ME, PS, R]

* Using personal strategies for adding and subtracting
* Estimating sums and differences
* Solving problems involving addition and subtraction.
[C, CN, ME, PS, R]
SO12.
Demonstrate an understanding of division (limited to division related to multiplication facts up to 5x5) by:
* representing and explaining division using equal sharing and equal grouping
* creating and solving problems in context that involve equal sharing and equal grouping
* modeling equal sharing and equal grouping using concrete and visual representations, and
* recording the process symbolically
* relating division to repeated subtraction
* relating division to multiplication.
[C, CN, PS, R]

Justification
Justification
Justification
Justification
Justification
Justification
Justification
Justification
Justification
We chose to include grade two specific outcome three, grade three specific outcome thirteen, and grade four specific outcome eight under the numbers big idea because all of these specific outcomes use whole numbers or fractions to describe a quantity. Due to the fact that these outcomes will use real numbers they could all be placed on a number line. All three outcomes stress comparing and/or ordering numbers which involves taking their quantity into consideration.

There is a clear connection between the grade three and grade four outcomes as they both involve demonstrating an understanding of fractions. The outcome in grade four is an extension of the grade three outcome because it uses higher level thinking. The grade four outcome specifically states that students need to be striving to achieve proficiency with symbolic representations. Although the connection may not be as clear between the grade two and three outcomes, the grade two outcome lays a foundation for future grades to branch from. Without understanding how to order and describe numbers -using correct place value- comparing fractions would be extremely difficult.

We thought that the grade two specific outcome seven, grade three specific outcome five, and grade four specific outcome nine fit really well under Base Ten Numeration System because they all specifically relate to place value. They also all use the concept of groups of ten, as they involve using numerals to 100 and 1,000, as well as decimals in the tenths and hundredths place.

As stated, the underlying idea in all of these outcomes is place value. They fit well together because they all deal with illustrating and representing numbers, first concretely and pictorially, then progressing to representing them symbolically. There is also a gradual increase in the difficulty of numerals involved (starting with hundreds, then thousandths, and then decimals).

Grade two specific outcome nine, and grade four specific outcomes three and ten all fit under the big idea of equivalence for a number of reasons. Firstly, they portray two or more ways to represent the same number and they also link operations together. Although addition and subtraction are seemingly opposite, through these outcomes we can see that they are corresponding operations.

Outcomes nine and three are clearly related and show a general progression in skill and understanding. In outcome nine, adding is limited to one and two digit numerals whereas outcome three involves three and four digit numbers. Outcome three also adds in the concept of estimating and does not give an option for students to use manipulatives for adding and subtracting. Outcome ten expands off the skills learned in outcomes three and nine by adding in the concepts of fractions and decimals (instead of just using whole numbers). Because students are already familiar with how to solve for numbers using addition and subtraction moving from whole numbers to decimals and fractions is a natural progression.

We decided that grade two specific outcome five; grade three specific outcome three and grade four specific outcome two go hand in hand with the idea of comparison. All of these outcomes require to understand the concept of “comparing”.

The three specific outcomes (5,3,2) work together with the knowledge a learner already possesses to build better understanding of comparing and ordering numbers from zero to ten thousand as the learner progresses through each grade.

Grade two outcome one, Grade three outcome eleven and outcome twelve all fit into Patterns because they require students to understand the use of patterns in order to solve basic math equations. They all require students to represent and explain how the answer can be found, it puts the emphasis on the process rather than memorizing the answers. Thus the students are asked to relate such findings and patterns to different types of math (skip counting, multiplication, division).

At the Grade two level, students are asked to be fairly comfortable with skip counting by 2’s, 5’s and 10’s, forwards and backwards. This will allow them to find patterns, and start preparing them for a strong base in in multiplication, which essentially skip counting and patterns. This is a gradual connection to Grade three, where the students are asked to demonstrate the understanding of multiplication up to 5x5. Here is where the skip counting and patterns from the previous grade comes in handy, they are asked to display a knowledge of the groupings of the equations. In this grade it is also asked to relate multiplication back to addition and division, with understanding the patterns of each type of equation, it is easy to establish a strong basis for these mathematical processes. In Grade three, students are now guided towards understanding division. From learning about multiplication and seeing the patterns there, it is a gradual step up now to find the same corresponding patterns in division. Going back to Grade two where skip counting was needed to be understood forwards and backwards, this is now simply reversing such patterns from multiplication, or seeing them in a different way. There was no gradual transition into a specific outcome in grade four, that directly relates to the previous ones.

Grade two specific outcome four, grade three specific outcome two and grade four specific outcome one all fit well under the big idea of ‘Operations Meanings and Relationships.’ They all require students to represent numbers concretely, pictorially and symbolically, and these representations of numbers would likely be related to real-world situations. By relating numbers to real life and representing them concretely, pictorially, and symbolically, students will be able to give the numbers meaning and make relationships between the numbers and real life. Another two outcomes that fit well under this big idea are grade three outcome nine and grade four outcome three. These outcomes also involve demonstrating an understanding of numbers. However, it is a progression that involves looking at how numbers change with addition and subtraction. These two outcomes require the learner to create and solve problems in context that involve addition and subtraction, and it also instructs learners to use personal strategies. This again gives the learner an opportunity to relate the numbers and problems to real life, and give the numbers meaning and make relationships between the numbers and real life, which is what this big idea is all about.

A clear progression is seen between grade two specific outcome four, grade three specific outcome two and grade four specific outcome one. Each outcome requires the student to represent numbers concretely, pictorially and symbolically. The progression is seen in the value that is being represented: first 100, followed by 1000 and then 10 000. Also, as students progress, they should be able to represent the numbers pictorially and symbolically without needing to first show them concretely. Grade three specific outcome nine and grade four specific outcome one also involve representing numbers but include processes of addition and subtraction. At first students are allowed to show the values with and without the use of the manipulative, but as they progress, this option is revoked. Also, as they progress, students should be able to estimate values before solving a problem step by step. Another progression is seen when students solve problems. At first they can create and solve problems in context but later should be able to solve the problem without these aids.

The selected outcomes all fit well under the heading of properties as they all present information that is always true, and thus constitute a math rule. The three outcomes then help students to learn the “rules that govern arithmetic and algebra,” which is a prerequisite of being labeled under properties.

Grade two specific outcomes two and eight, and grade four specific outcome four, represent a simple channel of progression that can be seen in terms of properties. The grade two outcomes represent the very basic properties of math, whereas the grade four outcome shows that this same concept can become more complex. Grade two specific outcome two requires that students learn the rules for differentiating between even and odd numbers. This represents a very primary understanding of math rules. This outcome can be measured by the achievement indicator of “us[ing] concrete materials or pictorial representations to determine if a given number is even or odd.” Due to the concrete nature of this indicator, it is clear that outcome two provides an easy starting point in understanding what a math property is. Specific outcome four introduced the concept of adding and subtracting zero, another rule in math that maintains that adding and subtracting zero will cause no change. This constitutes a property, because young children must learn that just because you are adding something, zero, it does not mean the answer must be bigger than the first addend. This outcome can be observed using the achievement indicator of adding or subtracting zero to a given number, and then explaining why the sum or difference is the same as the given number. Because this indicator requires students to not only solve the problem, but to also explain, it starts to become a higher mental process, and allows students to understand the zero property on a deeper level. This greater depth in understanding will lead to an easier grasp of grade four specific outcome four, which builds off of this early knowledge of math rules by extending properties to multiplication and division. Success in understanding the rules of multiplying by zero and one or dividing by one can be achieved more easily if the student first knows numbers can have properties (as seen in outcome 2), and can first apply this understanding to simple operations such as addition and subtraction (as seen in outcome 8). Achievement indicators for this last outcome include, “determine the answer to a given question involving the multiplication of a number by 0 [and one], and explain the answer,” and “determine the answer to a given question involving the division of a number by 1, and explain the answer.” Because both of these achievement indicators go beyond solving and promote conceptual understanding through explanation of process, this will allow a student to confidently approach future progressions.

Grade two specific outcome six and grade three specific outcomes four and eight fit very well under the big idea of estimation. All three outcomes explicitly require the use of estimation and the application of estimation skills.

Learning outcome six (in grade two) and learning outcome four (in grade three) show a clear progression in the child’s skills from one grade to the next. In learning outcome six the students are required to estimate quantities to one hundred (with the use of referents). In learning outcome four, students are required to estimate quantities less than one thousand (with the use of referents). Increasing the maximum quantity in learning outcome four requires students to make connections between their prior knowledge of estimating quantities up to one hundred, and their new task of estimating any quantity less than one thousand. If the students can make those connections they should be able to apply the skills they already have to their new problems, and naturally progress to a deeper level of understanding in estimation. Learning outcome eight takes student’s understanding of estimation one step further. The outcome requires students to bring together all their knowledge of estimation with their knowledge of addition, subtraction, and problem solving. By making connections students will be able to see how the skills they use in estimation can be very useful in addition, subtraction and problem solving. Once connections have been made, students will be able to apply their skills in estimation in a more demanding problems.

Achievement Indicators:
Achievement Indicators:
Achievement Indicators:
Achievement Indicators:
Achievement Indicators:
Achievement Indicators:
Achievement Indicators:
Achievement Indicators:
Achievement Indicators:
All of the following five outcomes fit nicely under the title of basic facts and algorithms, primarily because all of these goals stress the process of mental math in order to solve basic facts. While some of these outcomes go beyond the boundaries of basic facts (operations with only single digits), the mental calculation of basic facts is the basis for all of them.

Within the five outcomes, there are two separate sub-strands in which the divided outcomes best fit. While grade two outcome ten is the basis for both of these paths, this diverges into one progression following grade three specific outcomes six and seven, and the other branching off into grade three specific outcome ten and grade four specific outcome five. Grade two outcome ten provides a strong base for both progressions as it introduces strategies for mentally calculating basic facts, such as making ten, adding one, subtracting one, etc. This specific outcome can be measured with the achievement indicator “explain or demonstrate the mental mathematics strategy that could be used to determine a basic fact.” Because the indicator requires students to explain and demonstrate, it requires greater cognitive effort than simple solving, and therefore contributes to a greater concept of basic fact manipulations. Grade three specific outcome six and seven then build off this concept to two digit numerals. This progression allows students to use previous knowledge by keeping the same main idea, but working with greater values. Specific outcome six can be assessed using the achievement indicator of “add[ing] two given 2-digit numerals, using a mental mathematics strategy, and explain or illustrate the strategy used,” and similarly, specific outcome seven can addressed using the achievement indicator of “subtract[ing] two given 2-digit numerals, using a mental mathematics strategy, and explain or model the strategy used.” The second pathway, while also having a progression like the first, is a bit more straight forward. Grade three specific outcome ten further develops the ideas of grade two specific outcome ten. The grade three version is almost identical the the lower grade one, with the exception of two additional subpoints: using the commutative property, and the zero property. The achievement indicators for this outcome are the same as for grade two outcome ten but add “provide a rule for determining answers when adding and subtracting zero.” Grade four specific outcome five then further upgrades the progression by introducing more complex processes like “using doubling or halving,” and “using patterns in the 9’s facts.” Despite this increase in difficulty, the outcome is still in line with the early components of the progression, as it still uses mental math strategies. This last outcome can be measured using the achievement indicator of “provid[ing] examples for applying mental mathematics strategies,” such as “doubling,” “halving,” etc.
Indicate a position of a specific object in a sequence by using ordinal numbers up to tenth. Compare the ordinal position of a specific object in two different given sequences.

Identify common characteristics of a given set of fractions.
Describe everyday situations where fractions are used.
Cut or fold a whole into equal parts, or draw a whole in equal parts; demonstrate that the parts are equal; and name the parts.
Sort a given set of shaded regions into those that represent equal parts and those that do not, and explain the sorting.
Represent a given fraction concretely or pictorially. Name and record the fraction represented by the shaded and non-shaded parts of a given
region. Compare given fractions with the same denominator, using models. Identify the numerator and denominator for a given fraction. Model and explain the meaning of numerator and denominator.

Represent a given fraction, using a region, object or set. Identify a fraction from its given concrete representation. Name and record the shaded and non-shaded parts of a given set. Name and record the shaded and non-shaded parts of a given whole region, object or set. Represent a given fraction pictorially by shading parts of a given set. Represent a given fraction pictorially by shading parts of a given whole region, object or set. Explain how denominators can be used to compare two given unit fractions with a numerator
of 1. Order a given set of fractions that have the same numerator, and explain the ordering. Order a given set of fractions that have the same denominator, and explain the ordering. Identify which of the benchmarks 0, 12 or 1 is closer to a given fraction. Name fractions between two given benchmarks on a number line. Order a given set of fractions by placing them on a number line with given benchmarks. Provide examples of when two identical fractions may not represent the same quantity; e.g.,
half of a large apple is not equivalent to half of a small apple, half of ten Saskatoon berries is
not equivalent to half of sixteen Saskatoon berries.
Provide, from everyday contexts, an example of a fraction that represents part of a set and an example of a fraction that represents part of a whole.

Explain and show with counters the meaning of each digit for a given 2-digit numeral with both digits the same; e.g., for the numeral 22, the first digit represents two tens (twenty counters) and the second digit represents two ones (two counters).
Count the number of objects in a given set, using groups of 10s and 1s, and record the result as a 2-digit numeral under the headings 10s and 1s.
Describe a given 2-digit numeral in at least two ways; e.g., 24 as two 10s and four 1s, twenty and four, two groups of ten and four left over, and twenty-four ones.
Illustrate, using ten frames and diagrams, that a given numeral consists of a certain number of groups of ten and a certain number of ones.
Illustrate, using base 10 materials, that a given numeral consists of a certain number of tens and a certain number of ones.
Explain why the value of a digit depends on its placement within a numeral.

Record, in more than one way, the number represented by given proportional materials (e.g., base- ten materials) and non-proportional materials (e.g., money).
Represent a given number in different ways, using proportional and non-proportional materials, and explain how the representations are equivalent; e.g., 351 can be represented as three 100s, five 10s and one 1; or two 100s, fifteen 10s and one 1; or three 100s, four 10s and eleven 1s.
Explain and show, with counters, the meaning of each digit for a given 3-digit numeral with all digits the same; e.g., for the numeral 222, the first digit represents two hundreds (two hundred counters) the second digit represents two tens (twenty counters) and the third digit represents two ones (two counters).
Explain, using concrete materials, the meaning of zero as a place holder in a given number.

Write the decimal for a given concrete or pictorial representation of part of a set, part of a region or part of a unit of measure.
Represent a given decimal, using concrete materials or a pictorial representation. Explain the meaning of each digit in a given decimal with all digits the same. Represent a given decimal, using money values (dimes and pennies). Record a given money value, using decimals.
Provide examples of everyday contexts in which tenths and hundredths are used.
Model, using manipulatives or pictures, that a given tenth can be expressed as a hundredth; e.g., 0.9 is equivalent to 0.90, or 9 dimes is equivalent to 90 pennies.

Order a given set of numbers in ascending or descending order, and verify the result, using a hundred chart, number line, ten frames or by making references to place value.
Identify and explain errors in a given ordered sequence. Identify missing numbers in a given hundred chart. Identify errors in a given hundred chart.

Place a given set of numbers in ascending or descending order, and verify the result by using a hundred chart (e.g., a one hundred chart, a two hundred chart, a three hundred chart), a number line or by making references to place value.
Create as many different 3-digit numerals as possible, given three different digits. Place the numbers in ascending or descending order.
Identify and explain errors in a given ordered sequence. Identify missing numbers in parts of a given hundred chart. Identify errors in a given hundred chart.

Order a given set of numbers in ascending or descending order, and explain the order by making references to place value.
Create and order three different 4-digit numerals. Identify the missing numbers in an ordered sequence or on a number line. Identify incorrectly placed numbers in an ordered sequence or on a number line.
Add zero to a given number, and explain why the sum is the same as the given number.
Subtract zero from a given number, and explain why the difference is the same as the given number.

(Students investigate a variety of strategies and become proficient in at least one appropriate and efficient strategy that they understand.)
Explain how to keep track of digits that have the same place value when adding numbers, limited to 3- and 4-digit numerals.
Explain how to keep track of digits that have the same place value when subtracting numbers, limited to 3- and 4-digit numerals.
Describe a situation in which an estimate rather than an exact answer is sufficient. Estimate sums and differences, using different strategies; e.g., front-end estimation and
compensation. Refine personal strategies to increase their efficiency. Solve problems that involve addition and subtraction of more than 2 numbers.

Express, orally and in written form, a given fraction with a denominator of 10 or 100 as a decimal.
Read decimals as fractions; e.g., 0.5 is zero and five tenths. Express, orally and in written form, a given decimal in fraction form. Express a given pictorial or concrete representation as a fraction or decimal; e.g., 15 shaded
squares on a hundredth grid can be expressed as 0.15 or 15 . 100
Express, orally and in written form, the decimal equivalent for a given fraction; e.g., 50 can
be expressed as 0.50.

Extend a given skip counting sequence (by 2s, 5s or 10s) forward and backward. Skip count by 10s, given any number from 1 to 9 as a starting point. Identify and correct errors and omissions in a given skip counting sequence. Count a given sum of money with pennies, nickels or dimes (to 100¢).
Count quantity, using groups of 2, 5 or 10 and counting on.

Identify events from experience that can be described as multiplication. Represent a given story problem, using manipulatives or diagrams, and record the problem in a
number sentence. Represent a given multiplication expression as repeated addition. Represent a given repeated addition as multiplication. Create and illustrate a story problem for a given number sentence; e.g., 2 × 3 = 6. Represent, concretely or pictorially, equal groups for a given number sentence. Represent a given multiplication expression, using an array. Create an array to model the commutative property of multiplication. Relate multiplication to division by using arrays and writing related number sentences. Solve a given multiplication problem. Demonstrate understanding and recall/memorization of multiplication facts to 5 × 5.

Identify events from experience that can be described as equal sharing.
Identify events from experience that can be described as equal grouping.
Illustrate, with counters or a diagram, a given story problem, presented orally, that involves equal sharing; and solve the problem.
Illustrate, with counters or a diagram, a given story problem, presented orally, that involves equal grouping; and solve the problem.
Listen to a story problem; represent the numbers, using manipulatives or a sketch; and record the problem with a number sentence.
Create and illustrate, with counters, a story problem for a given number sentence; e.g., 6 ÷ 3 = 2.
Represent a given division expression as repeated subtraction. Represent a given repeated subtraction as a division expression. Relate division to multiplication by using arrays and writing related number sentences. Solve a given problem involving division. Demonstrate understanding and recall/memorization of division facts related to multiplication
facts to 5 × 5.

Estimate a given quantity by comparing it to a referent (known quantity). Estimate the number of groups of ten in a given quantity, using 10 as a referent. Select between two possible estimates for a given quantity, and explain the choice.

Estimate the number of groups of ten in a given quantity, using 10 as a referent (known quantity). Estimate the number of groups of a hundred in a given quantity, using 100 as a referent. Estimate a given quantity by comparing it to a referent. Select an estimate for a given quantity by choosing among three possible choices.
Select and justify a referent for determining an estimate for a given quantity.

Estimate the solution for a given problem involving the sum of two 2-digit numerals; e.g., to estimate the sum of 43 + 56, use 40 + 50 (the sum is close to 90).
Estimate the solution for a given problem involving the difference of two 2-digit numerals; e.g., to estimate the difference of 56 – 23, use 50 – 20 (the difference is close to 30).
Represent a given number, using concrete materials such as ten frames and base ten materials. Represent a given number, using coins (pennies, nickels, dimes and quarters). Represent a given number, using tallies. Represent a given number pictorially.
Represent a given number, using expressions; e.g., 24 + 6, 15 + 15, 40 – 10. Read a given number (0–100) in symbolic or word form. Record a given number (0–20) in words.

Read a given three-digit numeral without using the word and; e.g., 321 is three hundred twenty-one, NOT three hundred AND twenty-one.
Read a given number word (0 to 1000). Represent a given number as an expression; e.g., 300 – 44 or 20 + 236 for 256. Represent a given number, using manipulatives such as base ten materials. Represent a given number pictorially. Write number words for given multiples of ten to 90. Write number words for given multiples of a hundred to 900.

(Students investigate a variety of strategies and become proficient in at least one appropriate and
efficient strategy that they understand.)
 Model the addition of two or more given numbers, using concrete or visual representations,
and record the process symbolically.
 Model the subtraction of two given numbers, using concrete or visual representations, and
record the process symbolically.
 Create an addition or subtraction story problem for a given solution.
 Determine the sum of two given numbers, using a personal strategy; e.g., for 326 + 48, record
300 + 60 + 14.
 Determine the difference of two given numbers, using a personal strategy; e.g., for
127 – 38, record 38 + 2 + 80 + 7 or 127 – 20 – 10 – 8.
 Refine personal strategies to increase their efficiency.
 Solve a given problem involving the sum or difference of two given numbers

Read a given four-digit numeral without using the word and; e.g., 5321 is five thousand three hundred twenty-one, NOT five thousand three hundred AND twenty-one.
Write a given numeral, using proper spacing without commas; e.g., 4567 or 4 567, 10 000. Write a given numeral 0–10 000 in words. Represent a given numeral, using a place value chart or diagrams. Express a given numeral in expanded notation; e.g., 321 = 300 + 20 + 1.
Write the numeral represented by a given expanded notation.
Explain the meaning of each digit in a given 4-digit numeral, including numerals with all digits the same; e.g., for the numeral 2222, the first digit represents two thousands, the second digit two hundreds, the third digit two tens and the fourth digit two ones.

(Students investigate a variety of strategies and become proficient in at least one appropriate and
efficient strategy that they understand.)
 Explain how to keep track of digits that have the same place value when adding numbers,
limited to 3- and 4-digit numerals.
 Explain how to keep track of digits that have the same place value when subtracting numbers,
limited to 3- and 4-digit numerals.
 Describe a situation in which an estimate rather than an exact answer is sufficient.
 Estimate sums and differences, using different strategies; e.g., front-end estimation and
compensation.
 Refine personal strategies to increase their efficiency.
 Solve problems that involve addition and subtraction of more than 2 numbers

Use concrete materials or pictorial representations to determine if a given number is even or odd.
Identify even and odd numbers in a given sequence, such as in a hundred chart. Sort a given set of numbers into even and odd.

Add zero to a given number, and explain why the sum is the same as the given number.
Subtract zero from a given number, and explain why the difference is the same as the given number.

Determine the answer to a given question involving the multiplication of a number by 1, and explain the answer.
Determine the answer to a given question involving the multiplication of a number by 0, and explain the answer.
Determine the answer to a given question involving the division of a number by 1, and explain the answer.

Explain or demonstrate the mental mathematics strategy that could be used to determine a basic fact, such as:
• doubles; e.g., for 4 + 6, think 5 + 5
• doubles plus one; e.g., for 4+5, think 4+4+1
• doubles take away one; e.g.,for 4+5, think 5+5–1
• doubles plus two; e.g., for 4+6, think 4+4+2
• doubles take away two; e.g., for 4+6, think 6+6–2
• making 10; e.g., for 7+5, think 7+3+2
• building on a known double; e.g., 6+6=12, so 6+7=12+1=13
• addition for subtraction; e.g., for 7 – 3, think 3 + ? = 7.
Use and describe a mental mathematics strategy for determining a sum to 18 and the related subtraction facts.
Refine mental mathematics strategies to increase their efficiency. Demonstrate understanding and application of strategies for addition and related subtraction
facts to 18. Demonstrate recall/memorization of addition and related subtraction facts to 10.

(Students investigate a variety of strategies and become proficient in at least one appropriate and efficient strategy that they understand.)
Add two given 2-digit numerals, using a mental mathematics strategy, and explain or illustrate the strategy.
Explain how to use the “adding from left to right” strategy; e.g., to determine the sum of 23 + 46, think 20 + 40 and 3 + 6.
Explain how to use the “taking one addend to the nearest multiple of ten and then compensating” strategy; e.g., to determine the sum of 28 + 47, think 30 + 47 – 2 or 50 + 28 – 3.
Explain how to use the “using doubles” strategy; e.g., to determine the sum of 24 + 26, think 25 + 25; to determine the sum of 25 + 26, think 25 + 25 + 1 or doubles plus 1.
Apply a mental mathematics strategy for adding two given 2-digit numerals.

(Students investigate a variety of strategies and become proficient in at least one appropriate and efficient strategy that they understand.)
Subtract two given 2-digit numerals, using a mental mathematics strategy, and explain or model the strategy used.
Explain how to use the “taking the subtrahend to the nearest multiple of ten and then compensating” strategy; e.g., to determine the difference of 48 – 19, think 48 – 20 + 1.
Explain how to use the “adding on” strategy; e.g., to determine the difference of 62 – 45, think 45 + 5, then 50 + 12 and then 5 + 12.
Explain how to use the “using doubles” strategy; e.g., to determine the difference of 24 – 12, think 12 + 12 = 24.
Apply a mental mathematics strategy for subtracting two given 2-digit numerals.

Describe a mental mathematics strategy that could be used to determine a given basic fact, such as:
• doubles; e.g., for 6+8, think 7+7
• doubles plus one; e.g., for 6+7, think 6+6+1
• doubles take away one; e.g., for 6+7,think 7+7–1
• doubles plus two; e.g., for 6+8, think 6+6+2
• doubles take away two; e.g., for 6+8, think 8+8–2
• making 10; e.g., for 6+8, think 6+4+4 or 8+2+4
• commutative property; e.g., for 3 + 9, think 9 + 3
• addition for subtraction; e.g., for 13 – 7, think 7 + ? = 13.
Apply a mental mathematics strategy to provide a solution to a given basic addition or subtraction fact to 18.
Demonstrate understanding, recall/memorization and application of addition and related subtraction facts to 18.