Here are some diagrams that we will use to represent base-ten units.

• A large square represents 1 one.
• A medium rectangle represents 1 tenth.
• A medium square represents 1 hundredth.
• A small rectangle represents 1 thousandth.
• A small square represents 1 ten-thousandth.

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.”

1. 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.

   

2. 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.

   

3. For each number, draw or describe two different diagrams that represent it.

   1. 0.1
   2. 0.02
   3. 0.004

4. Use diagrams of base-ten units to represent each sum. Try to use as few units as possible to represent each number.

1. Here are two ways to calculate the value of . In the diagram, each rectangle represents 0.1 and each square represents 0.01.

   

   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:

   1. Why can 10 ten squares be composed into a rectangle?
   2. How is this composition represented in the vertical calculation?
2. Find the value of by using base-ten blocks or a diagram. Can you find the sum without composing a larger unit? Would it be useful to compose some pieces? Be prepared to explain your reasoning.
3. Calculate . Check if the sum is the same as the value of the base-ten blocks or diagram you used earlier.

4. Find each sum. The larger square represents 1. The rectangle represents 0.1. The small square represents 0.01.

   1.  

      

      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.

   2.