Unit 3, Lesson 14
Using Diagrams to Represent Addition and Subtraction
Accelerated 6
Student Lesson Summary
Base-ten diagrams represent collections of base-ten units—tens, ones, tenths, hundredths, and so on. We can use them to help us understand sums of decimals.
Suppose we are finding .
Here is a diagram where a square represents 0.01 and a rectangle (made up of ten squares) represents 0.1.
Base ten diagram. First row, 0 point 0 8. No rods in the tenths column. 8 small squares in the hundredths column. Second row, 0 point 13. 1 rod in the tenths column. 3 small squares in the hundredths column.
To find the sum, we can compose 10 hundredths into 1 tenth.
Base ten diagram. First row, 0 point 0 8. No rods in the tenths column. 8 small squares in the hundredths column. Second row, 0 point 13. 1 rod in the tenths column. 3 small squares in the hundredths column. A square is drawn around the 10 small squares. An arrow is drawn to a tenths rod outside the diagram. The arrow is labeled compose.
We now have 2 tenths and 1 hundredth, so .
We can also use vertical calculation to find .
Notice how this representation also shows 10 hundredths are composed into 1 tenth.
This works for any decimal place.
Suppose we are finding . Here is a diagram in which a small rectangle represents 0.001. We can compose 10 thousandths into 1 hundredth.
Base ten diagram. First row, 0 point 0 0 8. No squares in the hundredths column. 8 small rectangles in the thousandths column. Second row, 0 point 0 1 3. 1 small square in the hundredths column. 3 small rectangles in the thousandths column. A square is drawn around the 10 small rectangles. An arrow is drawn to a small square outside the diagram. The arrow is labeled compose.
The sum is 2 hundredths and 1 thousandth.
Here is a vertical calculation of .
Vertical addition. First line. 0 point 0 13. Second line. Plus 0 point 0 0 8. Horizontal line. Third line. 0 point 0 2 1. Above the 1 in the first line is 1.
Base-ten diagrams can also help us understand subtraction.
Suppose we are finding . Here is a diagram showing 0.23, or 2 tenths and 3 hundredths.
Subtracting 7 hundredths means removing 7 small squares, but we do not have enough to remove. Because 1 tenth is equal to 10 hundredths, we can decompose one of the tenths (1 rectangle) into 10 hundredths (10 small squares).
Base ten diagram. 0 point 23. Two rectangles in the tenths column. 3 small squares in the hundredths column. A dotted rectangle is drawn around one of the rectangles with an arrow to 10 small squares. The arrow is labeled decompose.
We now have 1 tenth and 13 hundredths, from which we can remove 7 hundredths.
Base ten diagram. 0 point 23. One rectangle in the tenths column. 13 small squares in the hundredths column. 7 small squares have an X through them. The words subtract 0 point 0 7 is below the small squares.
We have 1 tenth and 6 hundredths remaining, so .
Here is a vertical calculation of .
Notice how this representation also shows that a tenth is decomposed into 10 hundredths in order to subtract 7 hundredths.
Vertical subtraction. First line. 0 point 23. The 2 is crossed out and has a 1 above it. The 3 is crossed out and has 13 above it. Second line. Minus 0 point 0 0 7. Horizontal line. Third line. 0 point 17.
This works for any decimal place.
Suppose we are finding . Here is a diagram showing 0.023.
We want to remove 7 thousandths (7 small rectangles). We can decompose one of the hundredths into 10 thousandths.
Base 10 diagram. 0 point 0 2 3. Two small squares in the hundredths column. Three small rectangles in the thousandths column. A square is drawn around 1 small square. An arrow is drawn to 10 small rectangles. The arrow is labeled decompose.
Now we can remove 7 thousandths.
Base 10 diagram. 0 point 0 2 3. One small square in the hundredths column. 13 small rectangles in the thousandths column. 7 small rectangles have an X through them. Below the small rectangles are the words subtract 0 point 0 0 7.
We have 1 hundredth and 6 thousandths remaining, so .
Here is a vertical calculation of .
Vertical subtraction. First line. 0 point 0 2 3. The 2 is crossed out and has a 1 above it. The 3 is crossed out and has 13 above it. Second line. Minus 0 point 0 0 7. Horizontal line. Third line. 0 point 0 1 6.
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