classroom.jpgUnit 8 reviews and builds on children's knowledge of tens and ones as they explore regrouping. As children engage in activities using coins, the packaging of objects, and sticks and circles, children begin to understand the concept behind what they do when they add and regroup numerically. Children are first encouraged to develop their own solution methods for adding 2-digit numbers with regrouping before being introduced to proven numeric methods.

Learning to use algorithms for computation with multi-digit numbers is an important part of developing proficiency with numbers. Algorithms are procedures that can be executed in the same way to solve a variety of problems arising from different situations and involving different numbers. This feature has three important implications. First, it means that algorithms are useful tools--different procedures do not need to be invented for each problem. Second, algorithms illustrate a significant feature of mathematics: The structure of problems can be abstracted from their immediate context and compared to whether different-looking problems can be solved in similar ways. Finally, the process of developing fluency with arithmetic algorithms in elementary school can contribute to progress in developing the other strands of proficiency.

Children begin Unit 8 by showing money amounts with dimes, nickels, and pennies by exploring equivalent coin combinations. They learn to exchange pennies for nickels or dimes. When combining two amounts this results in a new ten being made. This experience with grouping not only helps children become more adept with the money system, but also paves the way for regrouping when they add two 2-digit numbers.
Children will use what they learn about unknown addends to make change. They solve these problems either by counting on with sticks and circles or by counting on with actual (or drawn) pennies and dimes. Buying and selling scenarios are used to give children practice with these methods.

Unknown Addends
At the end of Unit 8, children will turn their attention to finding unknown addends in 2-digit addition exercises. Once they can solve problems of this nature, they apply this knowledge to finding change from a dollar. See example:

Regrouping Ones to Make a New Ten
Children group ones as tens in a concrete way through the use of an orchard scenario. They are asked to pack apples in boxes of ten and leave the extras on the side. In the process of adding the apples picked by two people, they can clearly see that there are enough extra apples for another box of ten. They are then grouped and converted into a box of ten.
Then the class moves to pencil and paper representations. At this stage, children are encouraged to generate their own techniques for 2-digit addition problems. Some children will create methods of counting on with tens and ones, but most will rely on the sticks and circles notation that they already know. Children figure out a way to adapt this notation to the new requirements of grouping. See the attachment for some common representations for 27 + 49. Any method that makes us of tens and ones is an acceptable method at this point. Children are asked to share and explain their methods so that many methods are available for children to try. From the beginning, children link a step in their drawing to a step in their numeric method. This helps the numeric methods take on quantitative meaning. We will review some of the common regrouping methods.

Partial Sums or Show All Totals Method
This method enables children to see the tens and the ones on each number and work from left to right as they do when reading. This method produces two subtotals that are added for the final total. It can also be done right to left.

New Ten Below
This method is like the common method except that the 1 for the regrouped 10 is written below the tens column rather than above. This placement makes addition easier since the 1 ten is added last so children can add the 2 numbers they see and then increase that total by 1.
New Ten Above
Some children may be familiar with the common method of writing the regrouped ten above the tens column. If they prefer this method, they can use it as long as they can explain it. This method requires children to add the 1 to the 2, hold that number mentally, and add it to 4.
When trying pencil and paper methods, children are encouraged to check their work by drawing pictures with sticks and circles and relating steps in their numeric method to steps in the drawing.

Fuson, Karen, Math Expressions, Unit 8 Overview