# Content Standards Alignment

Math Pathways & Pitfalls (MPP) staff at WestEd have collaborated with other WestEd programs such as the Comprehensive School Assistance Program (CSAP) around a professional development and instructional intervention model that addresses the needs of districts and teachers to align both instructional materials and instructional practices with the Common Core State Standards for Mathematics.

Math Pathways & Pitfalls has developed K-8 intervention materials and lessons in Mathematics that are aligned to Common Core content standards in the domains of Operations & Algebraic thinking and Numbers and Operations. While the curriculum does not address all the standards, it does address the critical standards at each grade level. More importantly, the lessons provide a model for teaching and learning for a robust conceptual understanding of any mathematics domain. The table below shows the alignment for grade 3.

## Alignment with Grade 3 Common Core State Standards

Common Core State Standards – OVERVIEW for Grade 3 MATH PATHWAYS & PITFALLS
Related Units & Lessons
Operations and Algebraic Thinking
Represent and solve problems involving multiplication and division. Unit #3, Lessons #10, #11
Understand properties of multiplication and the relationship between multiplication and division. Unit #3, Lesson #11
Note: This lesson really does deal with the properties of multiplication, but It doesn’t really deal with division.
Multiply and divide within 100. Unit #3, Lesson #10
Note: Again, it doesn’t deal with division.
Number and Operations in Base Ten
Use place value understanding and properties of operations to perform multi-digit arithmetic. Unit #3, Lessons #1, #2, #3, #4, #5, #6, #7, #8, #9
Number and Operations—Fractions
Develop understanding of fractions as numbers. Unit #4, Lessons #1, #2, #3, #5, #6

### Grade 3 Areas of Focus

In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.

Common Core State Standards – FOCUS for Grade 3 MATH PATHWAYS & PITFALLS
Related Units & Lessons
(1) Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division. Unit #3, Lessons #10, #11
Note: These lessons do not deal with division.
(2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators. Unit #4, Lessons #1, #2, #3, #5, #6

### Grade 3 Standards and Clusters

Common Core State Standards – CLUSTERS for Grade 3 MATH PATHWAYS & PITFALLS
Related Units & Lessons
Operations and Algebraic Thinking 3.OA
Represent and solve problems involving multiplication and division.
1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. Unit #3, Lessons #10, #11
3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.  Unit #3, Lessons #10, #11
These lessons do not deal with division.
4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ? ÷ 3, 6 × 6 = ?. Unit #3, Lesson #10
Understand properties of multiplication and the relationship between multiplication and division.
5. Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) Unit #3, Lesson #11
Multiply and divide within 100.
7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. Unit #3, Lesson 11
Number and Operations in Base Ten 3.NBT
Use place value understanding and properties of operations to perform multi-digit arithmetic.
1. Use place value understanding to round whole numbers to the nearest 10 or 100. Unit #3, Lessons #1, #2, #4, #6
Note: The lessons do not use the word “round” but the students are introduced to benchmarks, which cause them to round so I left them in.
2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. Unit #3, Lessons #3, #4, #5, #7, #9
Note: We could also include #8 but it’s not within 1000.
Number and Operations—Fractions 3.NF
Develop understanding of fractions as numbers.
1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. Unit #4, Lessons #1, #2,
2. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. Unit #4, Lesson #3
3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Unit #4, Lesson #5
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. Unit #4, Lessons #1, #2
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Unit #4, Lessons #5, #6