*Modules are not necessarily taught in this order.

Module 1a: Addition and Subtraction of Whole Numbers

Module 1a begins with a concentration on place value understanding. The students then apply this place value understanding and properties of operations to perform multi-digit arithmetic. The students will also solve word problems involving addition and subtraction.

By the end of module 1a, the students should be able to:

• Round two- and three-digit whole numbers to the nearest ten or hundred, respectively.

• Add two- and three-digit whole numbers (limit sums from 100 through 1,000) and/or subtract two- and three-digit numbers from three-digit whole numbers.

• Order a set of whole numbers from least to greatest or greatest to least (up through 9,999, and limit sets to no more than four numbers).

Module 1b: Multiplication and Division with Factors 0-5 and 10

This module builds upon the foundation of multiplicative thinking with units started in grade 2. First, students concentrate on the meaning of multiplication and division and begin developing fluency for learning products and representing and solving problems involving multiplication and division involving factors of 2, 3, 4, 5, 9, and 10. The restricted set of facts keeps learning manageable, and also provides enough examples to do one- and two-step word problems and to start measurement problems involving weight, money, length, capacities, and time in the second module. The students will solve word problems involving multiplication and division.

By the end of module 1b, the students should be able to:

• Interpret and/or describe products of whole numbers (up to and including 10 × 10).

• Interpret and/or describe whole-number quotients of whole numbers (limit dividends through 50 and limit divisors and quotients through 10).

• Use multiplication (up to and including 10 × 10) and/or division (limit dividends through 50 and limit divisors and quotients through 10) to solve word problems in situations involving equal groups, arrays, and/or measurement quantities.

• Determine the unknown whole number in a multiplication (up to and including 10 × 10) or division (limit dividends through 50 and limit divisors and quotients through 10) equation relating three whole numbers.

• Apply the commutative property of multiplication (not identification or definition of the property).

• Apply the associative property of multiplication (not identification or definition of the property).

• Interpret and/or model division as a multiplication equation with an unknown factor.

• Solve two-step word problems using the four operations (expressions are not explicitly stated). Limit to problems with whole numbers and having whole-number answers.

• Represent two-step word problems using equations with a symbol standing for the unknown quantity. Limit to problems with whole numbers and having whole-number answers.

• Assess the reasonableness of answers. Limit problems posed with whole numbers and having whole-number answers.

• Solve two-step equations using order of operations (equation is explicitly stated with no grouping symbols).

• Identify arithmetic patterns (including patterns in the addition table or multiplication table) and/or explain them using properties of operations.

• Create or match a story to a given combination of symbols (+, –, ×, ÷, , and =) and numbers.

• Identify the missing symbol (+, –, ×, ÷, , and =) that makes a number sentence true.

Module 2: Fractions as Numbers on the Number Line

● The goal of Module 2 is for students to transition from thinking of fractions as parts of a figure to points on a number line. To make that jump, students think of fractions as being constructed out of unit fractions: “1 fourth” is the length of a segment on the number line such that the length of four concatenated fourth segments on the line equals 1 (the whole). ● Once the unit “1 fourth” has been established, counting them is as easy as counting whole numbers: 1 fourth, 2 fourths, 3 fourths, 4 fourths, 5 fourths, etc. Students also compare fractions, find equivalent fractions in special cases, and solve problems that involve comparing fractions.

By the end of module 2, the students will be able to:

• Demonstrate that when a whole or set is partitioned into y equal parts, the fraction 1/y represents 1 part of the whole and/or the fraction x/y represents x equal parts of the whole (limit the denominators to 2,3,4,6,and 8; limit numerators to whole numbers less than the denominator; no simplification necessary).

• Represent fractions on a number line (limit the denominators to 2,3,4,6, and 8; limit numerators to whole numbers less than the denominator; no simplification necessary).

• Recognize and generate simple equivalent fractions (limit the denominators to 1,2,3,4,6, and 8; limit numerators to whole numbers less than the denominator).

• Express whole numbers as fractions, and/or generate fractions that are equivalent to whole numbers (limit the denominators to 1,2,3,4,6,and 8; limit numerators to whole numbers less than the denominator).

• Compare two fractions with the same denominator (limit the denominators to 1,2,3,4,6,and 8), using the symbols >, =, or <, and/or justify the conclusions.

Module #3: Problem Solving with Mass, Time, Capacity, Length, and Money

● Module 3, which focuses on measurement, again provides students with internalization time for learning the 2, 3, 4, 5, and 10 facts as part of their fluency activities. Students can also take this time to work with place value, comparison and rounding concepts. ● The goal is to develop students’ number sense well enough that they can build proportional bar diagrams used in solving word problems in Grade 3 and beyond (e.g., “If this bar represents 62 kg, then a bar representing 35 kg needs to be slightly longer than half the 62 kg bar…”).

By the end of module 3, the students should be able to:

• Tell, show, and/or write time (analog) to the nearest minute.

• Calculate elapsed time to the minute in a given situation (total elapsed time limited to 60 minutes or less).

• Measure and estimate liquid volumes and masses of objects using standard units and metric units.

• Add, subtract, multiply, and divide to solve one-step word problems involving masses or liquid volumes that are given in the same units.

• Use a ruler to measure lengths to the nearest quarter inch or centimeter.

• Compare total values of combinations of coins and/or dollar bills less than $5.00.

• Make change for an amount up to $5.00 with no more than $2.00 change given.

• Round amounts of money to the nearest dollar.

Module #4: Multiplication and Area

By Module 4, students are ready to investigate area and the formula for the area of a rectangle. They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps. When that shape is a rectangle with whole number side lengths, it is easy to partition the rectangle into squares with equal areas.

By the end of module 4, the students should be able to:

• Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real-world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

• Measure areas by counting unit squares (square cm, square m, square in., square ft., and non-standard square units).

• Explain that shapes in different categories may share attributes, and that the shared attributes can define a larger category.

• Recognize rhombi, rectangles, and squares as examples of quadrilaterals, and/or draw examples of quadrilaterals that do not belong to any of these subcategories.

• Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

• Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, exhibiting rectangles with the same perimeter and different areas, and exhibiting rectangles with the same area and different perimeters. Use the same units throughout the problem.

Module #5: Collecting and Displaying Data

In Module 5, students leave the world of exact measurements behind. By applying their prior knowledge of fractions, they estimate lengths to the nearest halves and fourths of an inch and record that information in bar graphs and line plots. This module also prepares students for the multiplicative comparison problems of grade 4 by asking students “how many more” and “how many less” questions of scaled bar graphs.

By the end of module 5, the students should be able to:

• Complete a scaled pictograph and a scaled bar graph to represent a data set with several categories (scales limited to 1,2, 5, and 10).

• Solve one- and two-step problems using information to interpret data presented in scaled pictographs and scaled bar graphs (scales limited to 1, 2, 5, and 10).

• Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Display the data by making a line plot, where the horizontal scale is marked in appropriate units – whole numbers, halves, or quarters.

• Translate information from one type of display to another. Limit to pictographs, tally charts, bar graphs, and tables.

Module #6: Multiplication and Division with Factors of 6, 7, and 8

Students learn the remaining multiplication and division facts in Module 3 as they continue to develop their understanding of multiplication and division strategies within 100 and use those strategies to solve two-step word problems. The “2, 3, 4, 5 and 10 facts” module (Module 1b) and the “6, 7, 8 and 9 facts” module (Module 6) both provide important, sustained time for work in understanding the structure of rectangular arrays to prepare students for area in Module 4. This work is necessary because students initially find it difficult to distinguish the different squares in a rectangular array area model (the third array in the picture below), count them and recognize that the count is related to multiplication. Modules 1 and 3 slowly build up to a rectangular array area model using hands-on rectangular arrays (i.e., a Rekenrek) and/or pictures of rectangular arrays involving objects only (stars, disks, etc.)—all in the context of learning multiplication and division.

By the end of module 6, the students should be able to:

• Multiply one-digit whole numbers by two-digit multiples of 10.

• Interpret and/or describe products of whole numbers.

• Interpret and/or describe whole-number quotients of whole numbers.

• Use multiplication and/or division to solve word problems in situations involving equal groups, arrays, and/or measurement quantities.

• Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

• Apply the commutative property of multiplication.

• Apply the associate property of multiplication.

• Interpret and/or model division as a multiplication equation with an unknown factor.

• Solve two-step word problems using the four operations.

• Represent two-step word problems using equations with a symbol standing for the unknown quantity.

• Assess the reasonableness of answers.

• Solve two-step equations using order of operations.

• Identify arithmetic patterns and/or explain those using properties of operations. Create or match a story to a given combination of symbols and numbers Identify the missing symbol that makes a number sentence true.

Module #7: Word Problems with Geometry and Measurement

The year rounds out with plenty of time to solve two-step word problems involving the four operations, and to improve fluency for concepts and skills initiated earlier in the year. In Module 7, students also describe, analyze, and compare properties of twodimensional shapes. By now, students have done enough work with both linear and area measurement models to study that there is no relationship in general between the perimeter and area of a figure, one of the concepts of the last module.

By the end of module 7, the students will be able to:

• Solve two-step problems using the four operations (expressions are not explicitly states) Limit to problems with whole numbers and having whole-number answers.

• Solve two-step equations using order of operations (equation is explicitly stated with no grouping symbols).

• Use a ruler to measure lengths to the nearest quarter inch or centimeter.