Review for Final Exam

1. (Ch.9) Proportional reasoning-- create a drawing that helps you visualize the relationship and determine an amount prior to increase/decrease
2. (Ch. 8 & Ch.3) Additive and multiplicative comparison--be able to explain and write problems that model these
3. (Ch.9) Proportional Reasoning--write a problem that models this
4. (Ch.7) Subtraction write problems that model “take away” and “comparison”.
5. (Ch.6) Use benchmark numbers to estimate fractional values
6. (Ch. 5) Estimate operations with percents, decimals, and fractions
7. (Ch. 4) Long division--use and explain the scaffolding method, sometimes called the ‘Big Seven’
8. (Ch 4) Division modeled using ‘Equal Share’ or ‘Repeated Subtraction.’
9. (Ch.2 & Ch.14) Working with Bases other than Ten—operations in other bases
10. (Ch.12 & 13) Be able to model position/time using a qualitative graph

CH 13 Test Review

1. Given a speed and distance determine time; given distance and time determine speed.

2. Create a qualitative speed/time graph and answer questions from information given in a story problem

3. Create a position/time and a total distance/time graphs

CH.9, 10, 12 Test Review

1. Create a drawing and give an explanation to illustrate multiplicative reasoning
2. Create a drawing to represent fractional parts
3. Use proportional reasoning to solve word problems
4. Use open (positive) and closed (negative) dots to model addition, subtraction and multiplication
5. Create a graph on a coordinate system
6. Determine slope of a line
7. Explain rate of change in context
8. Explain important points on the graph

Context for division by negative (If you find other examples, please share them by commenting on this entry.)

(1) My friend gave me 12 tickets and said I can owe her the money. The tickets are 4 for a $1. How much do I owe her for the 12 tickets? This could be solved by dividing 12 tickets by -4 to get -3, which means that I owe her $3.

(2) While scuba diving, it takes me 1/2 second to descend one foot. I was descending for 3 seconds. What is my change in depth? This could be solved by dividing 3 by -1/2 to get -6. Which means I have descended 6 feet.

(3) It is now 4pm. A few hours ago I had a dozen cookies. I have been eating the cookies at a constant rate of 4 cookies/hour. At what time did I start eating? This can be solved by dividing 12 by -4 to get -3. So 3 hours have gone by, which means I started eating the cookies at 1pm.

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 simplified 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. Find a fraction between fractions with unlike denominators without converting to decimal numbers or using common denominators. This means understanding how “neighbor numbers” work.
   
5. Illustrate multiplication and division of fractions
   
6. Explain additive and multiplicative comparison and be able to write word problems that illustrate this knowledge
   
7. Illustrate multiplicative relationships (similar to the chocolate bar activity in ch.8) and use that knowledge to solve problems
   

Division by Fractions

Section 7.3
Dividing by a fraction can be thought about as repeated subtraction or sharing equally.

1/2 divided by 3/4 can be read as:

(a) Repeated subtraction: How much of three-fourths are in a half? The answer is 2/3 of 3/4 is in 1/2.
To model this, draw a circle cut in half. Draw the ¾ overlapping one of the halves. You should be able to see that 2/3 of the ¾ are in a half of the circle.

(b) Equal share: If a half is in three-quarters, then how much is in a whole? The answer is 2/3 are in a whole.
To model this, draw a circle cut into quarters. Use a half dozen dots and distribute them equally into three quarters of the circle. You will notice that there are 2 dots in each of the three quarters; therefore using the notion of equal share, there must be 8 dots in a whole circle. Eight dots are 2/3 of a dozen.

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.

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 digits behind the decimal. 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 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 decimals. 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) 123.797979797... = (12379-123)/99 = 12256/99

CH 3, 4, & 5 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.

Story problems that models 3 divided by 4

Partitive (equal share):
There are 3 brownies and 4 children. If each child gets an equal amount of the brownies, how much does each get?

Repeated subtraction:
Asks how much of 4 is in 3.
I have 3 carrots, but I need to put 4 carrots into each bunch to sell them at the farmer’s market. How much of a bunch do I have?

Obviously, the equal share model is much easier to use when the numerator is smaller than the denominator.

Ice Cream Cone Problem

Where N = number of different ice cream flavors and 2 is the number of different cones

The number of two scoop combinations, if scoop one vanilla and scoop two chocolate is considered different from scoop one chocolate and scoop two vanilla: N*N*2.

The number of two scoop combinations, if scoop one vanilla and scoop two chocolate is considered the same as scoop one chocolate and scoop two vanilla: N*N*2 – N * (N – 1)

CH.1-2 Test

You should be able to:

• Use quantitative anaylsis to solve problems (i.e. the trains approaching each other)
  
• 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.)
  
• 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.

Welcome to Math for Elementary School Teachers

Here is where we will post test reviews and other important information and news. You can also use this Blog to share resources with your classmates. Please check the blog on a regular basis.