Lesson 1

Estimate Quotients

Est. Class Sessions: 2–3

Daily Practice and Problems

Teacher Notes
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TIMS Challenge

This problem can serve as a Problem of the Week. The questions can be quite challenging and interesting to students. Allow students to work through the problems in order and complete as many as they can. Encourage them to draw pictures, make tables, and use patterns to solve the problems. As appropriate, make suggestions to help them start their drawings or tables and identify patterns.

  1. 23/4 yards. Strategies will vary: Students may draw a picture or use a table.
  2. Possible patterns:
    1. After the 1st drop, the distance traveled up and down for each bounce is half the distance traveled up and down on the previous bounce.
    2. Starting with the third bounce, the total distance traveled is a mixed number with 2 yards plus a fraction. That denominator of the fraction doubles on each bounce. The numerator is always 1 less than the denominator.
  3. 215/16 yards. See the table in A.
  4. Students can continue the table or use the pattern in the table in A. The total distance on the 7th bounce is 231/32 yards and on the 8th bounce is 263/64 yards.
  5. Possible response: The ball will never travel 3 yards. The distance traveled up and down gets smaller and smaller each time but never gets to 3 yards.

D. The Ball Bounce

Jessie and John were bouncing a ball on the school playground. John noticed that each time the ball hit the ground it bounced back up 1/2 the distance of the drop. When he dropped the ball from a height of 1 yard, it hit the ground and bounced up 1/2 yard. John and Jessie dropped the ball from a height of 1 yard and let the ball keep bouncing until it stops.

Answer the following questions on a separate piece of paper.

  1. How far will the ball have traveled, up and down, when it hits the ground the fourth time? Write about how you found your answer. Explain all your strategies.
  2. Look for patterns. Describe any that you see.
  3. How far will the ball have traveled, up and down, when it hits the ground for the sixth time?
  4. Show or tell a way to find the distance the ball will have traveled, up and down, after any number of hits.
  5. Will the ball ever bounce a total distance of 3 yards? Explain your thinking.