Grade 5: Alignment to Mathematics Grade-Level Standards

PAGE 2 Session Objective The purpose of these materials is to help develop understanding of the expectations of high-quality summative assessment items. The concepts shown throughout these modules can be useful for classroom questioning and assessment, but the items themselves may need to be slightly modified.

PAGE 3 CCSSO Section C: Align to Standards – Mathematics Criterion C.1: Focusing strongly on the content most needed for success in later mathematics Criterion C.2: Assessing a balance of concepts, procedures, and applications Criterion C.3: Connecting practice to content Criterion C.4: Requiring a range of cognitive demand Criterion C.5: Ensuring high-quality items and a variety of item types

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PAGE 5 Ten Principles of CCSS-Aligned Items 1. Most items aligned to standards in supporting clusters connect to the major work of the grade. 2. Items are designed to address the aspect(s) of rigor (conceptual understanding, procedural skill, and application) evident in the language of the content standards. 3. Items are designed to attend to content limits articulated in the standards. 4. Most items aligned to a single content standard should assess the central concern of the standard. 5. Representations are well suited to the mathematics that students are learning and serve an important purpose within the item itself. 6. Items use mathematically precise language, are free from mathematical errors or ambiguities, and are aligned to the mathematically appropriate standard. 7. The demands of items measuring the Standards for Mathematical Practice are appropriate to the targeted grade level. 8. Item types are chosen to match the item’s purpose and as part of the evidence required by the standards. 9. Most items measuring the Standards for Mathematical Practice are also aligned to content standards representing the major work of the grade. 10. Items written at the cluster or domain level measure key integration points not necessarily articulated in individual standards but plausibly implied directly by what is written.

PAGE 6 Alignment Principle #1 Most items aligned to standards in supporting clusters connect to the major work of the grade.

PAGE 7 Jonah recorded the distance, in miles, that he ran each day for 5 days on the line plot shown. Enter the total distance, in miles, that Jonah ran all 5 days. Most items aligned to standards in supporting clusters connect to the major work of the grade. 5.MD.B.2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.

PAGE 8 Alignment Principle #2 Items are designed to address the aspect(s) of rigor (conceptual understanding, procedural skill, and application) evident in the language of the content standards.

PAGE 9 Items are designed to address the aspect(s) of rigor (conceptual understanding, procedural skill, and application) evident in the language of the content standards. 5.NF.B.5 Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

PAGE 10 Items are designed to address the aspect(s) of rigor (conceptual understanding, procedural skill, and application) evident in the language of the content standards. 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, 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.

PAGE 11 Items are designed to address the aspect(s) of rigor (conceptual understanding, procedural skill, and application) evident in the language of the content standards. 5.NF.B.7 c. Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.

PAGE 12 Alignment Principle #3 Items are designed to attend to the content limits articulated in the standards.

PAGE 13 Items are designed to attend to the content limits articulated in the standards. Alignment 5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real-world problems. Content Limit 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, 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.

PAGE 14 Alignment Principle #4 Most items aligned to a single content standard should assess the central concern of the standard.

PAGE 15 Most items aligned to a single content standard should assess the central concern of the standard. Central Concern Not the Central Concern 5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

PAGE 16 Alignment Principle #5 Representations are well suited to the mathematics that students are learning and serve an important purpose within the item itself.

PAGE 17 Representations are well suited to the mathematics that students are learning and serve an important purpose within the item itself. 5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

PAGE 18 Representations follow the progressions in the standards to highlight how mathematical ideas are developed and connected. Use the “Add tick marks” button to partition the number line. Drag each point to a tick mark that shows its correct location on the number line. 5.NF.B.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a.Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.

PAGE 19 Alignment Principle #6 Items use mathematically precise language, are free from mathematical errors or ambiguities, and are aligned to the mathematically appropriate standard.

PAGE 20 Items use mathematically precise language, are free from mathematical errors or ambiguities, and are aligned to the mathematically appropriate standard. 5.NF.4b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

PAGE 21 Items use mathematically precise language, are free from mathematical errors or ambiguities, and are aligned to the mathematically appropriate standard. 5.NF.4b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

PAGE 22 Alignment Principle #7 The demands of items measuring the Standards for Mathematical Practice are appropriate to the targeted grade level.

PAGE 23 The demands of items measuring the Standards for Mathematical Practice are appropriate to the targeted grade level. 5.NF.A Use equivalent fractions as a strategy to add and subtract fractions.

PAGE 24 Alignment Principle #8 Item types are chosen to match the item’s purpose and as part of the evidence required by the standards.

PAGE 25 Item types are chosen to match the item’s purpose and as part of the evidence required by the standards. 5.NBT.A.3b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. A B C D Which number makes this inequality true? > ☐

PAGE 26 Item types are chosen to match the item’s purpose and as part of the evidence required by the standards. 5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit 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.

PAGE 27 Item types are chosen to match the item’s purpose and as part of the evidence required by the standards. 5.NF.B.5. Interpret multiplication as scaling (resizing), by: b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

PAGE 28 Item types are chosen to match the item’s purpose and as part of the evidence required by the standards. 5.NF.B.5. Interpret multiplication as scaling (resizing), by: b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

PAGE 29 Item types are chosen to match the item’s purpose and as part of the evidence required by the standards. 5.G.B.3 Understand that attributes belonging to a category of two- dimensional figures also belong to all subcategories of that category

PAGE 30 Alignment Principle #9 Most items measuring the Standards for Mathematical Practice are also aligned to content standards representing the major work of the grade.

PAGE 31 Most items measuring the Standards for Mathematical Practice are also aligned to content standards representing the major work of the grade. MP.3 Construct viable arguments and critique the reasoning of others. 5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

PAGE 32 Alignment Principle #10 Items written at the cluster or domain level measure key integration points not necessarily articulated in individual standards but plausibly implied directly by what is written.

PAGE 33 Items written at the cluster or domain level measure key integration points not necessarily articulated in individual standards but plausibly implied directly by what is written. Content Limit: 5.NBT.B.7. Add, subtract, multiply, and divide decimals to hundredths, 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. Primary alignment: NBT Domain

PAGE 34 Ten Principles of CCSS-Aligned Items 1. Most items aligned to standards in supporting clusters connect to the major work of the grade. 2. Items are designed to address the aspect(s) of rigor (conceptual understanding, procedural skill, and application) evident in the language of the content standards. 3. Items are designed to attend to content limits articulated in the standards. 4. Most items aligned to a single content standard should assess the central concern of the standard. 5. Representations are well suited to the mathematics that students are learning and serve an important purpose within the item itself. 6. Items use mathematically precise language, are free from mathematical errors or ambiguities, and are aligned to the mathematically appropriate standard. 7. The demands of items measuring the Standards for Mathematical Practice are appropriate to the targeted grade level. 8. Item types are chosen to match the item’s purpose and as part of the evidence required by the standards. 9. Most items measuring the Standards for Mathematical Practice are also aligned to content standards representing the major work of the grade. 10. Items written at the cluster or domain level measure key integration points not necessarily articulated in individual standards but plausibly implied directly by what is written.

Thank You!