Oxon Hill Road 2.6 mi
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Here is the beginning of another round: Jump from 0 to 22.9.
You can make two jumps of 10, then one jump of 1, another jump of 1, and then…?
17. Describe the different ways you can jump to the final destination of 22.9.
18. Use Student Activity Sheet 10 to complete the ten rounds as described on the next pages.
Complete Rounds 1 and 2 individually. After each round, write the total number of jumps you made in the box at the right.
Round 1: Go from 0 to 5.3 in the fewest jumps.
Round 2: Go from 0 to 6.9 in the fewest jumps.
Compare your results with a classmate. Score two points for a win and one point for a tie.
Do the following problems individually.
Round 3: Go from 0 to 29.8 in the fewest jumps.
Round 4: Go from 0 to 28.1 in the fewest jumps.
Round 5: Go from 0 to 51.6 in the fewest jumps.
Compare your results with a classmate. Score two points for a win and one point for a tie. Keep track of your total score.
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0 10 20 21 22
Notes
17 Discuss this problem with the whole class before they start to play the game.
32 Models You Can Count On
Reaching All Learners
Intervention
If students have difficulty doing the Jump Jump Game, eliminate the rule that tells them to do the problems on page 32 and 33 in as few jumps as possible. Play more rounds with whole numbers to help them learn how to reduce the number of jumps.
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Hints and Comments
Materials
Student Activity Sheet 10 (one per student)
Overview
Students see another example of jumping on a number line, which they finish. Then they start with the first five rounds of the Jump Jump Game.
About the Mathematics
By moving from one number to another in the fewest jumps and by jumping only 0.1, 1, or 10 spaces at the time, students are learning a method for adding numbers mentally. This method is known as the compensation method. The compensation method uses the idea that it is easier to first work with rounded numbers and then compensate for the changes made when rounding the numbers. For example, the formal expression 1.7 1.9 does not necessarily need to be solved by lining up the tenths, adding 7 and 9, carrying the 1, and so on. A more convenient way to solve 1.7 1.9 might be to round 1.9 up by 0.1 to 2, add 1.7 2, and then subtract 0.1.
Planning
Organize a setting so that students can work in pairs on problem 17 and then work individually on the first five rounds of the game.
Comments About the Solutions
17. Adding 22.9 to a number is the same as adding 23 and then subtracting 0.1. For example, to add 15.8 22.9 mentally, first add 23 to 15.8, and then subtract 0.1.
15.8 22.9 (15.8 10 10 3) 0.1 38.7 (This notation shouldn’t be taught to students here.) More examples of the compensation method are given below for applicable problems.
18. Round 2: Adding 6.9 to a number is the same as adding 10, and then subtracting 3, then
subtracting 0.1. For example, to add 17.8 6.9 mentally, first add 10 to 17.8, and then subtract 3, and then subtract 0.1.
17.8 6.9 [(17.8 10) 3] 0.1 24.7.
Or you can first add 7 to 17.8, and then subtract 0.1.
17.8 6.9 (17.8 7) 0.1 24.7.
Round 4: Adding 28.1 to a number is the same as adding 30 and then subtracting 2, then adding 0.1.
For example, to add 35.6 28.1 mentally, first add 30 to 35.6, and then subtract 2, and then add 0.1.
35.6 28.1 [(35.6 30) 2] 0.1 63.7.
Solutions and Samples
17. Start at 22 and jump nine small jumps of 0.1 to the right to 22.9.
Or take one jump of 1 to the right (to 23) and 0.1 back (22.9).
18. Answers may vary. Students may use other jumps and more jumps, and jumps can be done in different order. Sample (minimal) solutions:
Round 1: 5 jumps of 1 to the right, 3 jumps of 0.1 to the right, make 8 jumps.
Round 2: 1 jump of 10 right, 3 jumps of 1 left, 1 jump of 0.1 left, make 5 jumps.
Round 3: 3 jumps of 10 right, 2 jumps of 0.1 left, make 5 jumps.
Round 4: 3 jumps of 10 right, 2 jumps of 1 left, 1 jump of 0.1 right, make 6 jumps.
Round 5: 5 jumps of 10 right, 2 jumps of 1 right, 4 jumps of 0.1 left, make 11 jumps.
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The next few problems are a little different. Do the problems individually.
Round 6: Go from 5.0 to 26.8 in the fewest jumps.
Round 7: Go from 32.4 to 54.6 in the fewest jumps.
Compare your results with a classmate. Score two points for a win and one point for a tie. Keep track of your total score.
Round 8: Go from 4.5 to 8.4 in the fewest jumps.
Round 9: Go from 5.6 to 17.3 in the fewest jumps.
Round 10: Go from 44.4 to 51.6 in the fewest jumps.
Compare your results with a classmate. Score two points for a win and one point for a tie. Write your total score in the star on Student Activity
Sheet 10 or draw your own star.
19. Make up three additional Jump Jump Game problems.
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Notes
19 Students can pose their problems to the class and have classmates come to the board/overhead to show the fewest number of jumps possible.
19 Have students share some of their problems with the class.
33 Models You Can Count On
Reaching All Learners
Extension
Have students play additional rounds of the game without using number lines. For example, ask students to add 17.3 8.9 mentally.
Advanced Learners
Challenge students to play the Jump Jump Game with fractions.
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Hints and Comments
Materials
Student Activity Sheet 10 (one per student)
Overview
Students continue with the Jump Jump Game. Then they create three Jump Jump Game problems by themselves.
Planning
Students can continue working in the same setting.
Comments About the Solutions
Round 6: Going from 5.0 to 26.8 is the same as adding 22 to 5 and then subtracting 0.2. This is the same as adding 21.8.
To add 5.0 21.8 mentally, first add 5.0 22 and then subtract 0.2.
Round 8: Going from 4.5 to 8.4 is the same as adding 4 to 4.5 and then subtracting 0.1. This is the same as adding 3.9.
To add 4.5 3.9 mentally, first add 4.5 4 and then subtract 0.1.
19. Students can exchange the games they made up and solve each other’s games.
Solutions and Samples
Problem 18 continued:
Round 6: 2 jumps of 10 right (25), 2 jumps of 1 right (27), 2 jumps of 0.1 left, make 6 jumps.
Round 7: 2 jumps of 10 right (52.4), 2 jumps of 1 right (54.4), 2 jumps of 0.1 right, make 6 jumps.
Round 8: 4 jumps of 1 right (8.5), 1 jump of 0.1 left, make 5 jumps.
Round 9: 1 jump of 10 right (15.6), 2 jumps of 1 right (17.6), 3 jumps of 0.1 left, make 6 jumps.
Round 10: 1 jump of 10 right (54.4), 3 jumps of 1 left (51.4), 2 jumps of 0.1 right, make 6 jumps.
19. The problems students design will vary.
Sample student work:
Go from 0 to 10.00 (answer: 1 jump of 10 right) Go from 7.6 to 27.6 (answer: 2 jumps of 10 right) Go from 1 to 52.3 (answer: 5 jumps of 10 right, 1 jump of 1 right, 3 jumps of 0.1 right, for a total of 9 jumps)
Go from 21.6 to 90.5 (answer: 7 jumps of 10 right, 1 jump of 1 left, and 1 jump of 0.1 left, for a total of 9 jumps)
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On the television show “Guess the Price,” contestants attempt to guess the actual prices of various items. The person who guesses the closest to the actual price wins that item and is eligible for the Big Wheel Finale. People watching at home can see a number line that shows the correct price and the guesses of each contestant.
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