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What number does the rectangle represent if each small square represents:
1
0.1
0.01
0.001
Here is a square.
What number does the square represent if each small rectangle represents:
10
0.1
Here are some diagrams that we will use to represent base-ten units.
1 large square labeled "one." 1 medium rectangle labeled “zero point one, or tenth.” 1 medium square labeled “ 0 point 0 one, or hundredth.” 1 tiny rectangle labeled “0 point 0 0 1, or thousandth.” 1 tiny square labeled “0 point 0 0 0 1, or ten-thousandth.”
Here is the diagram that Priya drew to represent 0.13. Draw a different diagram that represents 0.13. Be prepared to explain why both Priya’s and your diagrams represent the same number.
Here is the diagram that Han drew to represent 0.025. Draw a different diagram that represents 0.025. Be prepared to explain why both Han’s and your diagrams represent the same number.
For each number, draw or describe two different diagrams that represent it.
Use diagrams of base-ten units to represent each sum. Try to use as few units as possible to represent each number.
A vertical equation of 0 point 2 6 plus 0 point 0 7 results in 0 point 3 3. A 1 is written above the 2 in the tenths column.
A diagram of a strategy used to calculate addition expression. A base ten diagram. There are 2 large rectangles and 6 small squares indicated. Directly below, the squares are an additional 7 small squares indicated. A dashed circle contains 10 of the small squares with an arrow, labeled compose, pointing to a third large rectangle. The third rectangle is drawn under the other two existing large rectangles.
Discuss with your partner:
Find each sum. The larger square represents 1. The rectangle represents 0.1. The small square represents 0.01.
Two diagrams of base-ten blocks are indicated. The top diagram has 2 large squares, 5 large rectangles, and 9 small squares. The bottom diagram has 3 large rectangles, 1 small square, and 2 small rectangles.
Here are diagrams that represent differences. Removed pieces are marked with Xs. The larger rectangle represents 1 tenth.
For each diagram, write a subtraction expression and find the value of the expression.
Express each subtraction in words.
Find each difference by drawing a diagram and by calculating with numbers. Make sure the answers from both methods match. If not, check your diagram and your numerical calculation.
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
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 0.008+0.013. 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.
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