Unit 5, Lesson 3
Adding and Subtracting Decimals with Few Non-Zero Digits
Grade 6
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Andre and Jada drew base-ten diagrams to represent
Andre drew 11 small rectangles.
Jada drew only two figures: a square and a small rectangle.
Here are two calculations of
The calculation on the left adds zero point 2 and zero point zero 5 by aligning the ones units, tenths unit, and hundredths unit. The sum is zero point 2 5.
The calculation on the right adds zero point 2 and zero point zero five by aligning the hundredths unit under the tenths unit. The sum is zero point zero 7.
Diego and Noah drew different diagrams to represent
Diego started by drawing 4 rectangles to represent 0.4. He then replaced 1 rectangle with 10 squares and crossed out 3 squares to represent subtraction of 0.03, leaving 3 rectangles and 7 squares in his diagram.
A base-ten diagram labeled “Diego’s Method.” There are 2 columns for the diagram. The first column header is labeled "tenths" and there are 4 rectangles. The second column header is labeled "hundredths" and there are 10 squares in that column. The last rectangle is circled with a dashed line and an arrow pointing from the rectangle to the column of squares is labeled “decompose.” The last three squares are crossed out.
Noah started by drawing 4 rectangles to represent 0.4. He then crossed out 3 rectangles to represent the subtraction, leaving 1 rectangle in his diagram.
Do you agree that either diagram correctly represents
A base-ten diagram labeled “Elena's Method.” There are 2 columns for the diagram. The first column header is labeled "tenths" and there are 4 rectangles. The second column header is labeled "hundredths" and there are 40 squares in that column. All four rectangles are circled with a dashed line and an arrow pointing from the rectangles to the column of squares is labeled “decompose.” The last three squares are crossed out.
Find each difference. Be prepared to explain your reasoning. If you get stuck, you can use base-ten blocks or diagrams to represent each expression and find its value.
Base-ten diagrams can help us understand subtraction. Suppose we are finding
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
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.
Notice how this representation also shows that a tenth is decomposed into 10 hundredths in order to subtract 7 hundredths.
This works for any decimal place. Suppose we are finding
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.