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**Problem H1
**The number line shown below has seven points labeled with numbers or letters. The line is not drawn to scale.

Name the lettered point or points that could possibly represent the following:

**a. **c • d

**b. **d c

**c. **c – d

**d. **c + d

**Problem H2
**Given three rational numbers, a, b, and c, you know that:

a > 1

0 < b < 1

0 < c < 2

Fill in the blanks with the symbols <, =, >, or ? so that each sentence will be true. Use ? to indicate that you do not have enough information to ascertain the relationship.

** **

**Problem H3**

A clearance sale offers an additional 50% off items that are already reduced by 20%. Explain why this is not the same as 70% off the original price.

**Tip:** Pick a specific dollar amount that is easy to calculate; for example, $100.

**Fibonacci Bracelets**

You can make “bracelets” using Fibonacci-like sequences of numbers. Here’s how:

Choose any pair of one-digit numbers. Make a Fibonacci-like sequence by recording only the units digit of the sum of these numbers and subsequent number pairs.

For example, the sequence starting (1,3) makes the following pattern:

**1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9, 2, 1, 3, . . .**

Eventually the sequence repeats (in the example above, after the number 2). At this point, attach the last digit in the sequence to the first digit, thus making a bracelet of digits.

**<— 1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9, 2, —>**

Note that sequences starting with any clockwise consecutive pair of numbers in the circle will make the same bracelet. Thus (3,4), (7,1), (1,8), etc. will result in the same bracelet. You should note, however, that although the pair (1,3) is in this bracelet, the pair (3,1) is not.

**Problem H3
**A clearance sale offers an additional 50% off items that are already reduced by 20%. Explain why this is not the same as 70% off the original price.

Pick a specific dollar amount that is easy to calculate; for example, $100

**TAKE IT FURTHER
Problem H4
**Assuming that you can start the sequence with any two one-digit numbers, how many different bracelets are possible?

This activity requires careful organization of information. The method of organization you choose is directly related to the amount of time required to answer the question.

**Suggested Reading:**

For more about the Fibonacci series in nature, see *Renaissance: Numbers in Nature* at www.learner.org/exhibits/renaissance/fibonacci/.

**Kilpatrick, J.; Swafford, J.; and Findell, B., ed. (2001). Adding It Up: Helping Children Learn Mathematics. A Report of the National Research Council. Washington, D.C.: National Academy Press.** Reproduced with permission from the publisher. © 2001 by National Academy Press. All rights reserved.

Download PDF File:

Adding It Up: Helping Children Learn Mathematics

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**Problem H1
a. **The product must be positive but smaller than either c or d; therefore, c • d = b.

**Problem H2**

**a. **The product is smaller, because b is less than 1 and will therefore produce a smaller result: a • b < a.

**b. **There is not enough information, because c may be less than 1 or greater than 1: b • c ? b.

**c. **There is not enough information, because the product of a and c may be less than 1 or greater than 1: a • b • c ? b.

**d. **The quotient is larger, because b is less than 1 (see the area model for division in Problem A3 for more details): ab > a.

**e. **There is not enough information, because c may be less than 1 (yielding a larger quotient) or greater than 1 (yielding a smaller quotient): ac ? a.

**f. **As in question (e), there is not enough information, because c may be less than 1 (yielding a larger quotient) or greater than 1 (yielding a smaller quotient): bc ? b.

**g. **The answer to bb will always be 1, and we are told that b is less than 1. Therefore, bb > b.

**h. **As in question (a), the product of b^{2} will be smaller than b, since b is less than 1: b^{2}< b.

**Problem H3
**This is not the same, because the additional 50% off is coming from the sale price rather than the original price. The sale price is only 0.8 times as large as the original, so the additional discount is really 40% (0.8 • 50%) of the original price.

For example, if an item cost $100 originally, it would be $80 with the sale price. With the additional sale, 50% is taken from the $80 sale price, not $100, and the final price is $40.

Put another way, the resulting price is 0.5 • 0.8 = 0.40 of the original, which is a 60% discount, not a 70% discount.

**Problem H4
**Since we can choose any two one-digit numbers as our starting pair, there are 100 possible two-digit combinations, so the total number of digits in the bracelets will be exactly 100. There are six total bracelets. Here they are:

112358314594370774156178538190998752796516730336954932572910 [60 digits]

134718976392 [12 digits]

22460662808864044820 [20 digits]

2684 [4 digits]

550 [3 digits]

0 [1 digit: 0, 0, 0, . . .]

There are a lot of patterns in the 60-digit bracelet. See what you can find!