boardwork.jpgUnit 2 develops the concepts of grouping and ungrouping numbers as a key step in performing multi-digit addition and subtraction. The activities in this unit help students gain a practical understanding of addition and subtraction and the relationship between the two operations. Students are expected to apply their understanding of addition and subtraction to numeric calculations and real-world problem solving situations, including those involving tables of data. Estimation and mental math provide students with a method to validate their answers.

Multi-Digit Addition and Subtraction Methods
Math Expressions shows three methods for multi-digit addition: the common algorithm (New Groups Above), plus two methods found to be effective during the research project developing the program, New Groups Below and Show Subtotals. These methods are introduced to help students see and discuss core mathematical ideas about addition and subtraction. When using the New Groups Below method, students record a regrouped digit on the line below the addition exercise, instead of above the addition exercise. This New Groups Below method allows students to see the tens and ones, or hundreds and tens, more closely together than in the New Groups Above method. In the Show Subtotals method, students add in each place, record the total for each place, then add these totals to find the sum.

To subtract multi-digit numbers, we teach students to ungroup all the places before they subtract. This approach reduces errors and helps develop conceptual understanding of multi-digit subtraction. Some students make the common error of consistently subtracting the smaller digit in a place value column from the larger digit, even if the smaller digit is on top. To help students remember to ungroup in subtraction, they are encouraged to draw a "magnifying glass" around the top number to prepare for ungrouping. They "look inside" the magnifying glass to see which places need to be ungrouped.

Place Value
In Math Expressions, students use place-value drawings to help them conceptualize numbers and understand the relative sizes of place values. In the early lessons of this unit, students make these drawings on the dot-array side of their Math Boards in class. Once students have a conceptual understanding of the number of 1s contained inside each place, they move to drawings without dots. Because the drawings do not need to be perfectly scaled, students can make them quickly. (grouping the shapes by fives, makes them easier to count)

Methods of Addition and Subtraction
Most grade 4 students know how to add and subtract ones and tens, perhaps also hundreds. However, some students go through the motions without understanding why they do what they do. In this unit, students explore the conceptual bases for multi-digit addition and subtraction.

For addition, Math Expressions uses the term grouping to refer to the process of combining 10 units of one place value to get 1 unit of the next greater place value (for example, combining 10 ones to make a new ten or combining 10 tens to make a new hundred). This process in often referred to as "carrying" in traditional curricula. The opposite process, referred to as ungrouping, is used for subtraction. Ungrouping involves breaking apart 1 unit of a place value to get 10 units of the next smaller place value (for example, breaking apart 1 hundred to get 10 tens or breaking apart 1 ten to get 10 ones). This process is often referred to as "borrowing" in traditional curricula.

Connecting Addition and Subtraction
Beginning in grade 1, Math Expressions builds understanding of the inverse nature of addition and subtraction. A conceptual support for this idea is a break-apart drawing (referred to as a Math Mountain in earlier grades). A break-apart drawing shows a total at the top and two addends at the bottom. Students learn that they can add the two addends to get the total or subtract either addend from the total to get the other addend.

Fuson, Karen, Dr., Math Expressions, Grade 4, Unit 2 Overview