grade 1 mathematics, quarter 3, unit 3.1 partitioning ... · partitioning circles and rectangles...

Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin

33

Grade 1 Mathematics, Quarter 3, Unit 3.1

Partitioning Circles and Rectangles into Halves and Quarters

Overview Number of Instructional Days: 5 (1 day = 45–60 minutes)

Content to be Learned Mathematical Practices to Be Integrated • Partition circles into two and four equal shares.

• Partition rectangles into two and four equal shares.

• Describe the shares using the word halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.

• Describe the whole as two of or four of the shares.

• Understand that decomposing into more equal shares creates smaller shares.

Construct viable arguments and critique the reasoning of others.

• Construct arguments using concrete referents such as objects, drawings, diagrams, and actions.

• Listen to or read the arguments of others.

• Ask useful questions to clarify or improve arguments.

Look for and make use of structure.

• See complicated things as single objects or as being composed of several objects.

• Look closely to discern structure.

Essential Questions

Students are expected to use the mathematical language whole, halves, fourths, quarters, half of, fourth of, quarter of, and equal share in response to the following questions:

• How can you divide a circle/rectangle into two/four equal shares? How can you show it another way?

• How could you fold this piece of paper (circle/rectangle) to show two/four equal shares? How can you fold it another way?

• Which shows half/fourth/quarter of the whole? How do you know?

• How many equal shares (half/fourth) do you need to make one whole?

• Which is the bigger/smaller share? How do you know?

• When you divide a circle/rectangle into more shares, what happens to the size of each share?

Grade 1 Mathematics, Quarter 3, Unit 3.1 Partitioning Circles and Rectangles into Halves and Quarters (5 days)


34

Written Curriculum

Common Core State Standards for Mathematical Content

Geometry 1.G

Reason with shapes and their attributes.

1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Common Core Standards for Mathematical Practice

3 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.

7 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.



35

Clarifying the Standards

Prior Learning

In kindergarten, students composed simple shapes to form larger shapes. They identified shapes as two- or three-dimensional. They also built and drew shapes.

Current Learning

Earlier in grade 1, students compose simple shapes to form larger shapes, so they understand that a whole can be composed of smaller parts. This is taught at the reinforcement level.

In this unit, students build on this concept by recognizing that fractional parts of a whole must be equal in size. They can decompose these shapes into two or four equal shares. Students use mathematical terms to describe the whole and its fractional parts (half of, fourth of, quarter of, halves, fourths, equal share, whole). This is taught at the developmental level.

These skills are not considered a critical area of focus for grade 1.

According to A Research Companion to Principles and Standards for School Mathematics,

a conceptual breakthrough for students is to understand that the magnitude of a quantity (e.g., the whole) is unchanged when the size of the shares changes. (p. 101)

conceptualizing fractions is based on conceiving two quantities as being in a reciprocal relationship of relative size. For example, if a share is one half of the size of the whole, then the whole is twice as large as the share. (p. 107)

Future Learning

In grade 2, students will decompose circles and rectangles into thirds and describe the whole as three of the shares. The shares are equal and smaller.

Students will partition a rectangle into rows and columns of same-size squares and count to find the total number of them. In addition, students will partition circles and rectangles into three equal shares and describe the shares using the words thirds and a third of, and they will describe the whole as three thirds. Finally, they will recognize that equal shares of identical wholes need not have the same shape.

Grade 3 will be the first time students represent equal shares of a figure using fraction notation (e.g., 1/2, 1/3, 1/4, 3/4).

Additional Findings

“Sharing (or partitioning) is the action of distributing an amount of something among a number of recipients so that each recipient receives the same amount. Segmenting (or measuring) is the action of putting an amount into parts of a given size.” Please refer to page 106 to understand how sharing (partitioning) and segmenting (measuring) are highly related, and for examples of each.



36

According to Fostering Children’s Mathematical Power by Arthur Baroody, (pp. 9-7 through 9-11),

Developmental Implications. Even under ideal instructional circumstances, constructing an understanding of such a novel concept as rational numbers is a difficult and lengthy process. Expect children’s whole-number knowledge to interfere (e.g., for them to view 2/3 as two whole numbers, rather than as a relationship between two quantities, or to see 1/3 as larger than 1/2). Only gradually and by actively grappling with rational number situations can children construct the understanding necessary to distinguish common fractions from whole numbers.

Introduce the concept of rational numbers to students concretely and informally with fair-sharing problems—a quotient meaning. (Streefland, 1993). Traditionally, instruction has introduced common fractions in terms of the part-of-a-whole meaning. Yet, given children’s familiarity with fair sharing, it makes sense to introduce children to a quotient interpretation of common fractions first. (Freudenthal, 1983)

Initially, prompt students to estimate the outcome of fair-sharing situations. (Streefland, 1993) Later, encourage them to use objects, manipulatives, or drawings to solve fair-sharing problems informally.


37


Organizing, Representing, and Interpreting Data and Time to the Half-Hour


Content to be Learned Mathematical Practices to Be Integrated • Tell and write time in half-hours using digital

and analog clocks.

• Organize data.

• Represent data.

• Ask questions about total number of data points.

• Tell how many more or less there are in one category than another.

Use appropriate tools strategically.

• Consider the tools needed to solve a problem (tallies, manipulatives, etc.).

• Gain competency in using tools and which tool serves them best.

Attend to precision.

• Communicate precisely, using clear definitions in discussions with others, in speech, and in written symbols.

• Carefully specify the nature and units of quantities in numerical answers and in graphs and diagrams.

Essential Questions • What time does this digital clock show? Show

this time on an analog clock.

• What time does this analog clock show? Write this time as you would see it on a digital clock.

• Given a set of data, how can you organize and represent these data?

• What is the total number of data points represented?

• What comparisons can be made between the categories?

• Using data in two categories, how many more/less are in the first category? (e.g., Using the amounts in our graph, how many more [students like chocolate ice cream] than [strawberry ice cream]? How many less?)

• Looking at these data (bar graph, etc.), what questions could you answer? (e.g., Which flavor is liked the most? How many people took the survey? What is the title of the graph?)

Grade 1 Mathematics, Quarter 3, Unit 3.2 Organizing, Representing, and Interpreting Data and Time to the Half-Hour (10 days)


38

Written Curriculum


Measurement and Data 1.MD

Tell and write time.

1.MD.3. Tell and write time in hours and half-hours using analog and digital clocks.

Represent and interpret data.

1.MD.4. Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.


5 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.

6 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.



39


Prior Learning

In kindergarten, students classified objects into given categories, counted the number of objects in each category, and sorted the categories by count. (Category counts were limited to less than or equal to 20.)

Current Learning

Earlier in grade 1, students tell and write time in hours using analog and digital clocks. They interpret data with up to three categories, ask and answer questions about the total number of data points, how many are in each category, and how many more or less are in one category than in another. These skills are all at the reinforcement level.

In this unit, students tell and write time to the half-hour using analog and digital clocks. They organize and represent data with up to three categories. This is at the developmental level. Students continue to ask and answer questions about the total number of data points, how many are in each category, and how many more or less are in one category than in another. These skills are at the reinforcement level.

Future Learning

In grade 2, students will tell and write time from analog and digital clocks to the nearest five minutes using a.m. and p.m.

Students will draw a picture graph and bar graph (with single-unit scale) to represent a data set with up to four categories. They will solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Additional Findings

According to the K–3, Categorical Data; Grades 2–5, Measurement Data Progression, “If students devise different ways to represent the same data set, then the class might discuss relative strengths and weaknesses of each scheme.” (p. 5) “A student might ask how many specimens there were all together”…. “ask and answer questions leading to addition and subtraction problems such as compare problems or problems involving the addition of three numbers.” (p. 6)



40


41


Counting and Comparing Numbers to 100


Content to be Learned Mathematical Practices to Be Integrated • Read and write numerals to 100.

• Represent a number of objects with a written numeral.

• Compare two 2-digit numbers 10–99 and record the results of comparisons using correct mathematical symbols (<, >, =).

• Add a two-digit number and one-digit number within 100 using concrete models or drawings and strategies based on place value.

• Add a two-digit number and a multiple of 10 using concrete models or drawings and strategies based on place value.

• Mentally find 10 more or 10 less than a two-digit number without having to count, and explain the reasoning used.

Look for and make use of structure.

• Look closely to discern a pattern or structure.

Look for and express regularity in repeated reasoning.

• Look for general methods and shortcuts.

• Continually evaluate the reasonableness of intermediate results.

Essential Questions • What number would you write to represent this

____ ? (amount up to 100)

• What number represents this group of objects? How do you write that number?

• What is this number? (Teacher shows numeral up to 100.)

• How can you use a symbol (<, >, =) to make these statements true. (e.g., 61 ___ 50 + 40; 25 ___ 2 tens and 5 ones; 98 ___89)

• Add these two numbers (e.g., 48 + 7; any two-digit plus one-digit combination). What strategy did you use? Explain your thinking. (Within 100, using concrete models or drawings and strategies based on place value)

• Add these two numbers (e.g., 48 + 30; any two-digit number plus a multiple of 10). What strategy did you use? Explain your thinking. (Within 100, using concrete models or drawings and strategies based on place value)

• What is 10 more than ____? What is 10 less than ____? Explain how you know. What pattern do you notice? (Mental strategies only)

Grade 1 Mathematics, Quarter 3, Unit 3.3 Counting and Comparing Numbers to 100 (10 days)


42

Written Curriculum


Number and Operations in Base Ten 1.NBT

Extend the counting sequence.

1.NBT.1 Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Understand place value.

1.NBT.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

Use place value understanding and properties of operations to add and subtract.

1.NBT.4 Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

1.NBT.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.


7 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.

8 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



43

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.


Prior Learning

In kindergarten, students counted to 100 by ones and tens. They began at any given number within a known sequence. Students wrote and represented numbers 0–20 and represented the number of objects with a written numeral 1–20.

Current Learning

This is a critical area in grade 1.

Reading and writing numerals 0–60 and the concept of equality are taught at the reinforcement level.

Skills taught at the developmental level are as follows:

• Students read and write numbers 61–100 and use the symbols for greater than and less than to compare numbers up to 100.

• Students add within 100, including adding a two-digit number and a one-digit number and a two-digit number and a multiple of 10 (using open number lines, 10 frames, number grid, bundles). They should relate their strategies to a written method and explain the reasoning used.

• Students mentally find 10 more or 10 less without counting, given a two-digit number, and explain the reasoning used.

Future Learning

In grade 2, students will understand that the digits of a three-digit number represent hundreds, tens, and ones. They will count, read, and write numbers up to 1,000 using base-10 numerals and expanded form. Students will compare three-digit numbers using the symbols for greater than, less than, or equal.

Students will add and subtract within 100 using strategies based on place value and properties of operations and/or relationships between addition and subtraction. In addition, they will mentally add and subtract 10 or 100 to a given number 100–900.

Additional Findings

According to K–5, Number and Operations in Base Ten Learning Progressions, “Grade 1 students use their base-ten work to help them recognize that the digit in the tens place is more important for determining the size of a two-digit number. They use this understanding to compare two two-digit numbers, indicating the result with the symbols <, =, and >. Correctly placing the < and > symbols is a challenge for early learners. Accuracy can improve if students think of putting the wide part of the symbol next to the larger number.” (p. 6)



44

According to Fostering Children’s Mathematical Power (Baroody),

“Children with a counting-based concept of numbers do not think in terms of grouped items (larger units like tens, hundreds, and so for the) and single items (units). Thus, they view 24 for instance, as a collection of twenty-four units not as two groups of 10 and four ones. To understand. … multi-digit arithmetic procedures—particularly those involving renaming (carrying and borrowing) —children must discover the importance of grouping quantities into larger and larger units. … They must transcend their informal counting-based concept of number and construct a new understanding of number—based on a grouping concept (e.g., view 24 as two groups of 10 and four single items).” (p. 6-3)

“Children initially do not realize that a digit’s position is crucial to its value. They view a numeral such as 12 either as two distinct single-digit numbers (“one and two”) or as an inseparable whole (‘twelve units”). Children need to learn that each place increases in value by a factor of 10 as we move from right to left, because items are repeatedly grouped by 10 (e.g., 10 units are grouped to make a hundred, 10 hundreds are grouped to make a thousand, and so forth). … These are foreign and relatively abstract ideas for children.” (p. 6-3)


45


Understanding, Representing, and Solving Addition and Subtraction Problems Within 20


Content to be Learned Mathematical Practices to Be Integrated • Use addition and subtraction within 20 to solve

word problems (adding to, taking from, putting together, taking apart, and comparing) by using objects, drawing, and equations with symbols for the unknown number to represent the problem. (See CCSS glossary, Table 1.)

• Solve word problems adding three whole numbers whose sum is less than or equal to 20 by using objects and drawings.

• Apply the Associative Property of Addition as a strategy to add and subtract (e.g., To add 2 + 6 + 4, the second two numbers can be added to make a 10, so 2 + 6 + 4 = 2 + 10 = 12).

• Add within 20, demonstrating fluency for addition within 10 using strategies such as counting on, making 10, decomposing a number leading to a 10, using the relationship between addition and subtraction, and creating equivalent but easier or known sums.

• Understand the meaning of the equal sign and determine if equations involving addition and subtraction are true or false.

• Determine the unknown whole number in an addition or subtraction equation relating three whole numbers (8 + ? = 11).

Make sense of problems and persevere in solving them.

• Understand the problem and look at how to begin to solve the problem.

• Plan a solution instead of jumping right into the problem.

Grade 1 Mathematics, Quarter 3, Unit 3.4 Understanding, Representing, and Solving Addition and Subtraction Problems Within 20 (15 days)


46

Essential Questions • How would you solve this word problem? Can

you solve it another way? (See glossary, Table 1 to pose questions for each problem type, including two and three addend problems).

• What do you notice about these numbers? How could you use what you notice to solve this problem? (e.g., To add 2 + 6 + 4, the second two numbers can be added to make a 10, so 2 + 6 + 4 = 2 + 10 = 12).

• Add these numbers (within 20). What strategy did you use to solve this problem?

• What is ________? (Fluently add within 10. e.g., 2 + 3; 6 + 4; 8 + 1)

• Is this true or false: _____? How do you know? (e.g., 6 = 7 + 2; 2 + 5 = 7 + 1; 4 + 3 = 7; 7 = 8 – 1; 5 – 3 = 2 + 1)

• How can you solve these problems? (e.g., 8 + ? = 11; 5 = ? – 3; 6 + 6 = ?)

Written Curriculum


Operations and Algebraic Thinking 1.OA

Represent and solve problems involving addition and subtraction.

1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.2 2 See Glossary, Table 1.

1.OA.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Understand and apply properties of operations and the relationship between addition and subtraction.

1.OA.3 Apply properties of operations as strategies to add and subtract.3 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) 3 Students need not use formal terms for these properties.



47

Add and subtract within 20.

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Work with addition and subtraction equations.

1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = – 3, 6 + 6 = .


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.


Prior Learning

In kindergarten, students represented addition and subtraction word problems with pictures, sounds and mental images, or equations. They also decomposed numbers 10 and less into pairs more than one way. Students used pictures or objects and recorded the decomposition by a drawing or equation. When given any number 1–9, they found the number that made 10 by using objects or drawings. Students fluently added and subtracted within five.



48

Current Learning

This is a critical area of instruction for grade 1.

Earlier in grade 1, students add within 20. They use subtraction to solve word problems and understand subtraction as an unknown addend problem. Students use strategies such as counting on, making 10, doubles, near doubles, fact families, etc. They understand the meaning of the equal sign involving addition and subtraction. These skills are reinforced in this unit.

In this unit, students solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. (See CCSS glossary, Table 1.) Comparing is taught at the developmental level, and all other strategies are reinforced in this unit.

Students determine if equations involving addition and subtraction are true or false. Also, they will determine the unknown whole number in an addition or subtraction equation relating three whole numbers. These skills are taught at the developmental level.

Students add within 20, demonstrating fluency for addition within 10.

Research suggests using a variety of problem situations (see Table 1) and exploiting everyday situations to introduce and discuss addition and subtraction.

PARCC defines fluency as quickly and accurately. Fluent is not halting, stumbling, or reversing oneself. A key aspect of fluency in this sense is that it is not something that happens all at once in a single grade but requires attention to student understanding along the way.

To understand student difficulties with language in comparison problems refer to K–5 Operations and Algebraic Thinking Learning Progressions, page 12.

Future Learning

In grade 2, students will use addition and subtraction within 100 in all situations to solve one- and two-step word problems with unknowns in all positions. Reference CCSS, page 88, Table 1: Common addition and subtraction situations. Students will fluently add and subtract within 20 using mental strategies. They will know from memory all sums of two 1-digit numbers.

Additional Findings According to Fostering Children’s Mathematical Power (Baroody),

“Children can learn to distinguish among different types of addition and subtraction word problems … and can learn to solve them, albeit, not all at once (Carpenter, Fennema, Peterson, Chiang and Loef, 1989; Fuson, 1992b).” (p. 5-5)

“Because teachers need to take into account developmental readiness when posing problems, they should have some sense of the relative difficulty of problems. Problems in which the outcome, whole, or difference is unknown are relatively easy. Of these situations, equalize and compare problems may be more difficult to model than change and part-part-whole problems … it is even more difficult for children to model a problem in which the change or a part is unknown. Unknown start or unknown first-part problems may be especially confusing, because children are accustomed to knowing the starting amount.” (p. 5-5)

grade 1 mathematics, quarter 3, unit 3.1 partitioning ... · partitioning circles and rectangles...

Documents