Here are diagrams that represent 0.137 and 0.284.

Two base-ten diagrams representing 0 point 1 3 7 and 0 point 2 8 4. First diagram, 1 rectangle labeled, tenths, 3 squares labeled, hundredths, 7 small rectangles labeled, thousandths. Second diagram, 2 rectangles labeled, tenths, 8 squares labeled, hundredths, 4 small rectangles labeled, thousandths.

1. Use the diagram to find the value of \(0.137 + 0.284\). Explain your reasoning.

2. Calculate the sum vertically.

   

3. How was your reasoning about \(0.137 + 0.284\) the same with the two methods? How was it different?

For the first two problems, circle the vertical calculation where digits of the same kind are lined up. Then, finish the calculation and find the sum. For the last two problems, find the sum using vertical calculation.

1. \(3.25+1\)

   

   First calculation, 3 point 2 5 plus 1 point 0, the decimal point for 1 point 0 is shifted one place to the right.

   

   Second calculation, 3 point 2 5 plus 1 point 0, the decimal points are aligned.

   

   Third calculation, 3 point 2 5 plus 1, the 1 is shifted two places to the right. 

2. \(0.5+1.15\)

   

   First calculation, 0 point 5 plus 1 point 1 5, decimal points are aligned.

   

   Second calculation, 0 point 5 plus 1 point 1 5, decimal point of 1 point 1 5 shifted one place to the left.

   

   Third calculation, 0 point 5 0 plus 1 point 1 5 0, decimal point of 1 point 1 5 0 shifted one place to the left.

3. \(10.6+ 1.7\)

4. \(123+0.2\)

Andre is helping a local charity pack envelopes to mail. He can pack 135 envelopes in 90 minutes.

1. If Andre is packing envelopes at a constant rate, how many envelopes can he pack per minute?
2. Noah also works at a constant rate, and he can pack 75 envelopes in 1 hour.  Who is working faster? Explain or show your reasoning.