Neighbor Numbers

Neighbor Numbers is a concept we teach children to help them estimate the value of fractions. Neighbor numbers help us determine the approximate value based upon its relationship to a known fraction.

Neighbor numbers are the fractions for which children have a conceptual understanding, such as ¼, ½, ¾, and 1.

A question involving neighbor numbers might be: ‘Which is larger 7/16 or 4/9?’, the child could compare these to ½. The reason they would pick ½ is because they recognize that:

• 7/16 is 1/16 less than 8/16 = ½
• 4/9 is ½/9 less than 4½/9 = ½ and ½/9 = 1/18
• and since 7/16 and 4/9 are both smaller than ½
• and 1/18 is smaller that 1/16
• then 4/9 is bigger than the 7/16
• because 7/16 is farther away from ½.

Neighbor numbers allow us to use logic instead of common denominators to determine the value of fractions.

Neighbor numbers are often used to quickly put a list of fractions in order from smallest to largest. For example, if you are asked to put these (½ 5/7 1/3 15/31) in order, neighbor numbers are the quickest method, as compared to finding common denominators. This is a very common problem for standardized tests.

Division by Fractions

Section 7.3
Dividing by a fraction can be thought about as repeated subtraction (a/b means how many ‘b’s or how much of ‘b’ is in ‘a’) or sharing equally (a/b means how much of ‘a’ does each ‘b’ get).

1/2 divided by 3/4 can be read as: (a) Repeated subtraction: How much of the three-fourths is in a half? To model this, consider the ½ as your whole and ask yourself what part of ¾ is in that whole. We know that two of the fourths is in the half, and those two fourths are 2/3 of the ¾. (b) Equal Share: If three-quarters gets a half, how much would a whole get? To model this you would first consider how much of the half is equally distributed in each of the three quarters (each qtr gets 1/6), then using that equal share you can determine how much is in the whole (4/6 = 2/3).

Here are some sample word problems (from 7.3 #15) that use fractions for division.

Repeated Subtraction:
· If a tortoise is timed traveling an average of 1 2/3 miles per hour, how long would it take the tortoise to travel 6 miles?
· Your holiday cookie recipe uses 1 2/3 cups of flour for each batch of cookies. How many batches of cookies can you make with the 6 cups of flour?
· How many 2 ¾ feet long strips of ribbon can be cut from a ribbon that is 7 ½ feet long?
· If your pea patch is only 7 ½ square yards and the melon plants you want to grow require 2 ¾ square yards each. How many melon plants can you put into your pea patch?

Sharing Equally:
· You have 7 ½ bags of mulch to cover 2 ¾ square yards of garden bed. If you want to distribute the mulch evenly over the garden bed, how many bags will you need to use for each square yard?
· If you want to share 1 7/8 pizza with 3 people, how much pizza would each person get?
· If you have 3 donuts in 1/3 of a box, how many donuts are in a whole box?

Message from Classmate

Feel free to respond:

I think that both Math chapters in each of Rafe Esquith’s books are very helpful! Chapter 5 of his 2007 book “Teach Like your Hair’s On Fire” is awesome and lists websites/TOOLS to assist teachers along with explaining simple and educational math games/warm-ups. He mentions Marcy Cook’s website as one that he uses for materials!
Great stuff!!!!!!!!!!!
Trina Palosaari

CH 6, 7, & 8 Test Review

You should be able to:

1. Provide a pictorial representation of fractions, whether using a discrete whole or a continuous whole.
2. Write a decimal number as a fraction in the form: i.e 2.25 = 225/100 = 9/4, or 2.2525252525... = 223/99
3. Demonstrate understanding of decimal numbers by finding numbers that would be between consecutive decimal numbers. i.e. what decimal numbers are between 0.2 and 0.3
4. Use approximation to find an estimate of the answer when doing addition, subtraction, multiplication or division involving mixed numbers.
5. Find a fraction between fractions with unlike denominators without converting to decimal numbers or using common denominators. This means understanding how “neighbor numbers” work.
6. Illustrate multiplication and division of fractions
7. Explain additive and multiplicative comparison and be able to write word problems that illustrate this knowledge
8. Illustrate multiplicative relationships (similar to the chocolate bar activity in ch.8) and use that knowledge to solve problems

Converting Decimals to Fractions

TERMINATING DECIMALS:
Put the decimal’s digits in the numerator.
In the denominator, the number of zeros equals the number of decimal digits.
Example: (a) 0.079 = 79/1000 (b) 2.13 = 213/100

SIMPLE REPEATING DECIMALS:
Put the decimal’s repeating digits in the numerator.
In the denominator, the number of nines equals the number of repeating decimal digits.
Example: (a) 0.7979797979… = 79/99

COMPLEX REPEATING DECIMALS:
Subtract the non-repeating digits from the combination of non-repeating digits and one set of the repeating digits. Put this number in the numerator.
In the denominator, the number of nines equals the number of repeating decimal digits and the number of zeros equals the number of non-repeating decimal digits.
Example: (a) 0.12379797979… = (12379 - 123) / 99000 = 12256/99000 (which can then be simplified) (b) 12.3797979797... = (12379-123)/990 = 12256/990 (which can then be simplified)

CH 3, 4, & 5 Test Review

You should be able to:

1. Write word problems that use (a) comparison subtraction, (b) take-away model of subtraction, and (c) missing addend.
2. Analyze students' methods for adding, subtracting, multiplying, or dividing. (Analyze means be able to explain the child's procedure or solution method, whether their procedure is reasonable, and if they got the answer correct or not. )
3. Make a sketch that models the multiplication of two numbers (repeated addition, array, area, Fundamental Counting Principle), whether using whole numbers or fractions.
4. Write division word problems that use equal share or repeated subtraction models.
5. Estimate and explain your thinking when dividing very large numbers to determine an approximate percent.
6. Use scientific notation to solve problems with really big numbers or really small numbers and be able to convert those numbers into other units that provide a better understanding of what those numbers represent.

CH.1 & 2 Test

You should be able to:

1. Use quantitative anaylsis to solve problems (i.e. the trains approaching each other)
   
2. Take a number and use place value to think about the number in different ways. (Similar to our discussion about money--$235 can be thought of as 235 ones, 23 1/2 tens or 2.35 hundreds, etc.)
   
3. Work with bases other than ten: (a) take a base ten number and convert it to another base, (b) take a number from another base and convert to base ten, and (c) add and (d) subtract numbers that are in other bases.

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