Objectives: Use differences to identify patterns in number sequences. Make predictions by using patterns in number sequences. Standards Addressed: 2.4.8A:

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Objectives: Use differences to identify patterns in number sequences. Make predictions by using patterns in number sequences. Standards Addressed: 2.4.8A: Make conjectures on logical reasoning and test conjectures by using counterexamples A: Invent, select, use and justify the appropriate methods, materials and strategies to solve problems.

 A number sequence is a string of numbers, or terms, in a certain order.  Often there will be three dots after the last given number. The three dots (…) indicate that there are more terms that are not listed.  If the difference from one term to the next in a number sequence is always the same, the difference is called a constant difference.

 a. 1, 3, 5, 7, 9, …  2, 2, 2, 2  1 st differences are constant  Notice that the terms increase by 2. The 1 st difference is +2. Add 2 from the previous terms to find each new term.  9+2= = =15  The next three terms are 11, 13, 15.

 b. 80, 73, 66, 59, 52, …

 c. 1, 4, 7, 10, 13, add 3 16, 19, 22  d. 30, 25, 20, 15, 10, … subtract 5 5, 0, -5

 a. 1, 4, 9, 16, 25, …

 b. 37, 41, 48, 58, 71, …  4, 7, 10, 13  1 st difference are NOT constant  3, 3, 3  2 nd difference are constant  Use the constant 2 nd difference to find the next 3 1 st differences, which are 16, 19, 22.  Use these three 1 st differences to find the next three terms.  The next three terms of the sequence are 87, 106, and 128.

 c. 2, 6, 12, 20, 30, …  4, 6, 8, 10  1 st difference are NOT constant  2, 2, 2  2 nd difference are constant  The next three terms of the sequence are 42, 56, 72.  d. 8, 20, 30, 38, 44, …  12, 10, 8, 6  1 st difference is NOT constant.  2, 2, 2  2 nd difference is constant.  The next three terms of the sequence are 48, 50, 50.

 A conjecture is a statement about observations that is believed to be true. When mathematicians make a conjecture, they try to either prove that the conjecture is true or find a counterexample to show that the conjecture is not true.

 A. The table below shows the relationship between temperature in Celsius and temperature in Fahrenheit. Use the method of constant differences to find the Fahrenheit temperatures that correspond to the Celsius temperatures of 50, 60, 70.  Constant Difference of +18.  50  = 122  60  = 140  70  =158 Celsius Fahrenheit

 B. A projectile is launched from ground level. The data in the table below provides its height above ground during the first 4 seconds immediately after the launch. After 10 seconds the projectile hits the ground. Use the method of constant differences to find the maximum height of the projectile.  112, 80, 48, 16  1 st difference are NOT constant  32, 32, 32  2 nd difference are constant  5, 400 Time Height

 solving a simpler problem, making a table or chart, or looking for a pattern.

 Suppose that 10 friends have just returned to school. Each friend has exactly one conversation with each of the other friends to talk about what they did during summer break. Use problem-solving strategies to determine how many conversations there will be.