Name 
Description 
MAFS.4.G.1.1:  Draw points, lines, line segments, rays, angles (right, acute, obtuse),
and perpendicular and parallel lines. Identify these in twodimensional
figures. 

Related Access Points
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Description 
MAFS.4.G.1.AP.1a:  Identify a point, line and line segment and rays in twodimensional figures.  MAFS.4.G.1.AP.1b:  Identify perpendicular and parallel lines in a twodimensional figure.  MAFS.4.G.1.AP.1c:  Identify an angle in a twodimensional figure. 

MAFS.4.G.1.2:  Classify twodimensional figures based on the presence or absence of
parallel or perpendicular lines, or the presence or absence of angles of
a specified size. Recognize right triangles as a category, and identify
right triangles. 

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MAFS.4.G.1.AP.2a:  Identify and sort objects based on parallelism, perpendicularity, and angle type. 

MAFS.4.G.1.3:  Recognize a line of symmetry for a twodimensional figure as a line
across the figure such that the figure can be folded along the line
into matching parts. Identify linesymmetric figures and draw lines of
symmetry. 

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MAFS.4.MD.1.1:  Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a twocolumn table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ... 

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Description 
MAFS.4.MD.1.AP.1a:  Within a system of measurement, identify the number of smaller units in the next larger unit.  MAFS.4.MD.1.AP.1b:  Complete a conversion table for length and mass within a single system. 

MAFS.4.MD.1.2:  Use the four operations to solve word problems^{1} involving distances, intervals of time, and money, including problems involving simple fractions or decimals^{2}. Represent fractional quantities of distance and intervals of time using linear models. (^{1}See glossary Table 1 and Table 2) (^{2}Computational fluency with fractions and decimals is not the goal for students at this grade level.) 

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MAFS.4.MD.1.AP.2a:  Solve word problems involving distance using line plots. 

MAFS.4.MD.1.3:  Apply the area and perimeter formulas for rectangles in real world and
mathematical problems. For example, find the width of a rectangular
room given the area of the flooring and the length, by viewing the area
formula as a multiplication equation with an unknown factor. 

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Description 
MAFS.4.MD.1.AP.3a:  Solve word problems involving perimeter and area of rectangles using specific visualizations/drawings and numbers. 

MAFS.4.MD.2.4:  Make a line plot to display a data set of measurements in fractions of
a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction
of fractions by using information presented in line plots. For example,
from a line plot find and interpret the difference in length between the
longest and shortest specimens in an insect collection. 

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Description 
MAFS.4.MD.2.AP.4a:  Solve problems involving addition and subtraction of fractions with like denominators (2, 4, and 8) by using information presented in line plots. 

MAFS.4.MD.3.5:  Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
 An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “onedegree angle,” and can be used to measure angles.
 An angle that turns through n onedegree angles is said to have an angle measure of n degrees.


Related Access Points

MAFS.4.MD.3.6:  Measure angles in wholenumber degrees using a protractor. Sketch
angles of specified measure. 

Related Access Points

MAFS.4.MD.3.7:  Recognize angle measure as additive. When an angle is decomposed
into nonoverlapping parts, the angle measure of the whole is the sum
of the angle measures of the parts. Solve addition and subtraction
problems to find unknown angles on a diagram in real world and
mathematical problems, e.g., by using an equation with a symbol for
the unknown angle measure. 

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Description 
MAFS.4.MD.3.AP.7a:  Find sums of angles that show a ray (adjacent angles). 

MAFS.4.NBT.1.1:  Recognize that in a multidigit whole number, a digit in one place
represents ten times what it represents in the place to its right. For
example, recognize that 700 ÷ 70 = 10 by applying concepts of place value
and division. 

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Description 
MAFS.4.NBT.1.AP.1a:  Compare the value of a digit when it is represented in a different place of two threedigit numbers (e.g., The digit 2 in 124 is ten times the digit 2
in 472). 

MAFS.4.NBT.1.2:  Read and write multidigit whole numbers using baseten numerals,
number names, and expanded form. Compare two multidigit numbers
based on meanings of the digits in each place, using >, =, and <
symbols to record the results of comparisons. 

Related Access Points

MAFS.4.NBT.1.3:  Use place value understanding to round multidigit whole numbers to
any place. 

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Description 
MAFS.4.NBT.1.AP.3a:  Use a hundreds chart or number line to round to any place (i.e., ones, tens, hundreds, thousands). 

MAFS.4.NBT.2.4:  Fluently add and subtract multidigit whole numbers using the
standard algorithm.Clarifications: Fluency Expectations or Examples of Culminating Standards
Students’ work with decimals (4.NF.3.5–3.7) depends to some extent on concepts of fraction  

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Description 
MAFS.4.NBT.2.AP.4a:  Solve multidigit addition and subtraction problems within 1,000. 

MAFS.4.NBT.2.5:  Multiply a whole number of up to four digits by a onedigit whole
number, and multiply two twodigit numbers, using strategies based
on place value and the properties of operations. Illustrate and explain
the calculation by using equations, rectangular arrays, and/or area
models.Clarifications: Examples of Opportunities for InDepth Focus
When students work toward meeting this standard, they combine prior understanding of multiplication with deepening understanding of the baseten system of units to express the product of two multidigit numbers as another multidigit number. This work will continue in grade 5 and culminate in fluency with the standard algorithms in grade 6.  

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MAFS.4.NBT.2.AP.5a:  Solve a twodigit by onedigit whole number multiplication problem using two different strategies. 

MAFS.4.NBT.2.6:  Find wholenumber quotients and remainders with up to fourdigit
dividends and onedigit divisors, using strategies based on place
value, the properties of operations, and/or the relationship between
multiplication and division. Illustrate and explain the calculation by
using equations, rectangular arrays, and/or area models.Clarifications: Examples of Opportunities for InDepth Focus
When students work toward meeting this standard, they combine prior understanding of multiplication and division with deepening understanding of the baseten system of units to find wholenumber quotients and remainders with up to fourdigit dividends and one digit divisors. This work will develop further in grade 5 and culminate in fluency with the standard algorithms in grade 6.  

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Description 
MAFS.4.NBT.2.AP.6a:  Find wholenumber quotients and remainders with up to threedigit dividends and onedigit divisors, using two different strategies. 

MAFS.4.NF.1.1:  Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b)
by using visual fraction models, with attention to how the number and
size of the parts differ even though the two fractions themselves are
the same size. Use this principle to recognize and generate equivalent
fractions.Clarifications: Examples of Opportunities for InDepth Focus
Extending fraction equivalence to the general case is necessary to extend arithmetic from whole numbers to fractions and decimals.  
MAFS.4.NF.1.2:  Compare two fractions with different numerators and different
denominators, e.g., by creating common denominators or numerators,
or by comparing to a benchmark fraction such as 1/2. Recognize that
comparisons are valid only when the two fractions refer to the same
whole. Record the results of comparisons with symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model. 

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MAFS.4.NF.1.AP.2a:  Use =, <, or > to compare two fractions (fractions with a denominator or 10 or less).  MAFS.4.NF.1.AP.2b:  Compare 2 given fractions that have different denominators. 

MAFS.4.NF.2.3:  Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
 Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
 Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
 Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
 Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
Clarifications: Examples of Opportunities for InDepth Focus
This standard represents an important step in the multigrade progression for addition and subtraction of fractions. Students extend their prior understanding of addition and subtraction to add and subtract fractions with like denominators by thinking of adding or
subtracting so many unit fractions.  

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Description 
MAFS.4.NF.2.AP.3a:  Using a representation, decompose a fraction into multiple copies of a unit fraction (e.g., 3/4 = 1/4 + 1/4+ 1/4).  MAFS.4.NF.2.AP.3b:  Add and subtract fractions with like denominators (2, 3, 4 or 8) using representations.  MAFS.4.NF.2.AP.3c:  Solve word problems involving addition and subtraction of fractions with like denominators (2, 3, 4 or 8). 

MAFS.4.NF.2.4:  Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
 Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
 Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
 Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
Clarifications: Examples of Opportunities for InDepth Focus
This standard represents an important step in the multigrade progression for multiplication and division of fractions. Students extend their developing understanding of multiplication to multiply a fraction by a whole number.  

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Description 
MAFS.4.NF.2.AP.4a:  Multiply a fraction by a whole number using a visual fraction model. 

MAFS.4.NF.3.5:  Express a fraction with denominator 10 as an equivalent fraction with
denominator 100, and use this technique to add two fractions with
respective denominators 10 and 100. For example, express 3/10 as
30/100, and add 3/10 + 4/100 = 34/100.


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MAFS.4.NF.3.AP.5a:  Find the equivalent fraction with denominators that are multiples of 10. 

MAFS.4.NF.3.6:  Use decimal notation for fractions with denominators 10 or 100. For
example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate
0.62 on a number line diagram.


Related Access Points

MAFS.4.NF.3.7:  Compare two decimals to hundredths by reasoning about their size.
Recognize that comparisons are valid only when the two decimals
refer to the same whole. Record the results of comparisons with the
symbols >, =, or <, and justify the conclusions, e.g., by using a visual
model.


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Description 
MAFS.4.NF.3.AP.7a:  Use =, <, or > to compare two decimals (decimals in multiples of .10).  MAFS.4.NF.3.AP.7b:  Compare two decimals expressed to the tenths place with a value of less than 1 using a visual model.  MAFS.4.NF.3.AP.7c:  Compare two decimals expressed to the hundredths place with a value of less than 1 using a visual model. 

MAFS.4.OA.1.1:  Interpret a multiplication equation as a comparison, e.g., interpret 35
= 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as
many as 5. Represent verbal statements of multiplicative comparisons
as multiplication equations. 

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MAFS.4.OA.1.AP.1a:  Use objects to model multiplication involving up to five groups with up to five objects in each and write equations to represent the models. 

MAFS.4.OA.1.2:  Multiply or divide to solve word problems involving multiplicative
comparison, e.g., by using drawings and equations with a symbol
for the unknown number to represent the problem, distinguishing
multiplicative comparison from additive comparison. 

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Description 
MAFS.4.OA.1.AP.2a:  Solve multiplicative comparisons with an unknown using up to twodigit numbers with information presented in a
graph or word problem (e.g., an orange hat costs $3. A purple hat costs two times as much. How much does the purple
hat cost? [3 x 2 = p]).  MAFS.4.OA.1.AP.2b:  Determine the number of sets of whole numbers, ten or less, that equal a dividend. 

MAFS.4.OA.1.3:  Solve multistep word problems posed with whole numbers and having
wholenumber answers using the four operations, including problems
in which remainders must be interpreted. Represent these problems
using equations with a letter standing for the unknown quantity.
Assess the reasonableness of answers using mental computation and
estimation strategies including rounding. 

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MAFS.4.OA.1.AP.3a:  Solve and check one or twostep word problems requiring the four operations within 100. 

MAFS.4.OA.1.a:  Determine whether an equation is true or false by using comparative relational thinking. For example, without adding 60 and 24, determine whether the equation 60 + 24 = 57 + 27 is true or false. 
MAFS.4.OA.1.b:  Determine the unknown whole number in an equation relating four whole numbers using comparative relational thinking. For example, solve 76 + 9 = n + 5 for n by arguing that nine is four more than five, so the unknown number must be four greater than 76. 

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MAFS.4.OA.1.AP.ba:  Find the unknown number in an equation (+,  ) relating four whole numbers. 

MAFS.4.OA.2.4:  Investigate factors and multiples. Find all factor pairs for a whole number in the range 1–100.
 Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given onedigit number.
 Determine whether a given whole number in the range 1–100 is prime or composite.


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MAFS.4.OA.2.AP.4a:  Identify multiples for a whole number (e.g., The multiples of 2 are 2, 4, 6, 8, 10…).  MAFS.4.OA.2.AP.4b:  Identify factors of whole numbers within 30. 

MAFS.4.OA.3.5:  Generate a number or shape pattern that follows a given rule. Identify
apparent features of the pattern that were not explicit in the rule itself.
For example, given the rule “Add 3” and the starting number 1, generate
terms in the resulting sequence and observe that the terms appear to
alternate between odd and even numbers. Explain informally why the
numbers will continue to alternate in this way. 

Related Access Points

MAFS.K12.MP.1.1:  Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 
MAFS.K12.MP.2.1:  Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 
MAFS.K12.MP.3.1:  Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 
MAFS.K12.MP.4.1:  Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, twoway tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 
MAFS.K12.MP.5.1:  Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 
MAFS.K12.MP.6.1:  Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 
MAFS.K12.MP.7.1:  Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 
MAFS.K12.MP.8.1:  Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x² + x + 1), and (x – 1)(x³ + x² + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 
LAFS.4.SL.1.1:  Engage effectively in a range of collaborative discussions (oneonone, in groups, and teacherled) with diverse partners on grade 4 topics and texts, building on others’ ideas and expressing their own clearly.
 Come to discussions prepared, having read or studied required material; explicitly draw on that preparation and other information known about the topic to explore ideas under discussion.
 Follow agreedupon rules for discussions and carry out assigned roles.
 Pose and respond to specific questions to clarify or follow up on information, and make comments that contribute to the discussion and link to the remarks of others.
 Review the key ideas expressed and explain their own ideas and understanding in light of the discussion.


Related Access Points
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Description 
LAFS.4.SL.1.AP.1a:  Provide evidence of being prepared for discussions on a topic or text through appropriate statements made during discussion.  LAFS.4.SL.1.AP.1b:  Ask questions to check understanding of information presented in collaborative discussions.  LAFS.4.SL.1.AP.1c:  Make appropriate comments that contribute to a collaborative discussion.  LAFS.4.SL.1.AP.1d:  Review the key ideas expressed within a collaborative discussion. 

LAFS.4.SL.1.2:  Paraphrase portions of a text read aloud or information presented in diverse media and formats, including visually, quantitatively, and orally. 

Related Access Points
Name 
Description 
LAFS.4.SL.1.AP.2a:  Paraphrase portions of a text read aloud or information presented in diverse media and formats, including visually, quantitatively and orally. 

LAFS.4.SL.1.3:  Identify the reasons and evidence a speaker provides to support particular points. 

Related Access Points
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LAFS.4.SL.1.AP.3a:  Identify the reasons and evidence a speaker provides to support particular points. 

LAFS.4.W.1.2:  Write informative/explanatory texts to examine a topic and convey ideas and information clearly.
 Introduce a topic clearly and group related information in paragraphs and sections; include formatting (e.g., headings), illustrations, and multimedia when useful to aiding comprehension.
 Develop the topic with facts, definitions, concrete details, quotations, or other information and examples related to the topic.
 Link ideas within categories of information using words and phrases (e.g., another, for example, also, because).
 Use precise language and domainspecific vocabulary to inform about or explain the topic.
 Provide a concluding statement or section related to the information or explanation presented.


Related Access Points
Name 
Description 
LAFS.4.W.1.AP.2a:  Introduce a topic clearly and group related information in paragraphs and sections.  LAFS.4.W.1.AP.2b:  Develop the topic (add additional information related to the topic) with relevant facts, definitions, concrete details, quotations or other information and examples related to the topic.  LAFS.4.W.1.AP.2c:  Include formatting (e.g., headings), illustrations and multimedia when appropriate to convey information about the topic.  LAFS.4.W.1.AP.2d:  Link ideas within categories of information, appropriately using words and phrases (e.g., another, for example, also, because).  LAFS.4.W.1.AP.2e:  Use increasingly precise language and domainspecific vocabulary over time to inform about or explain a variety of topics.  LAFS.4.W.1.AP.2f:  Provide a concluding statement or section to support the information presented. 

ELD.K12.ELL.MA.1:  English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. 
ELD.K12.ELL.SI.1:  English language learners communicate for social and instructional purposes within the school setting. 