When comparing column A to column B, there are times when you need to substitute a number for the variable (x or n or whatever) to think about the circumstances in which the expressions have quite different values. For example,
Column A 5x versus Column B 6x
If x is a positive number, column B will always be greater.
If x is zero, column A will be equal to column C.
If x is a negative number, Column A will be greater than column B.
So, in that case, you must say that you do not have enough information to say which column has a greater value.
But, let's look at a different kind of problem:
Column A x - .24 versus Column B x - .23
We are always subtracting more from x in the Column A expression, because .24 is bigger than .23. So, no matter whether what number is substituted for the variable x, Column B's value is always going to be greater than column A's!!
Sunday, January 29, 2012
7th grade ERB preparation
The math part of thee ERB is divided into 3 sections: Quantitative Reasoning, Mathematics, and Constructed Response. We've already looked at examples of the kinds of quantitative reasoning problems you might get.
The following questions should help you prepare for the Mathematics section. As you will see, you really don't need a calculator to do these. Thinking carefully is the key to success!. Sometimes a problem will use a variable (a letter standing in for an unknown number), just as we've seen in the practice we've done for the Quantitative Reasoning section.
1. x is to 16 as 1 is to 4. What is x?
Hint: A proportion is useful. We used them a lot when we studied percents.
2. If you know the measures of 2 angles of a triangle, can you calculate the measure of the third? _____ If so, how?
(Remember that the 3 angles need to sum to 180 degrees.)
EX: Triangle A has 2 angles measuring 55 and 35 degrees. The degree measure for the third angle will have to be equal to 180 - (55 + 35)
3. I toss a coin 3 times and each time the coin lands with "heads" facing up. What is the probability that I will get heads the next time?
(Remember that each new toss is not affected by outcomes in the past tosses.)
4. You and I each get on a merry-go-round (menage? or carousel) but our merry-go-rounds are making complete turns at different speeds. Mine makes a complete turn every minute and yours makes a complete turn every minute-and-a-half. How many minutes will it be before we are both once again in our original starting positions?
(Hint: Think about multiples! You are doing the same kind of thinking that you do when you find the LCM of two numbers.)
Let's assume that mine makes a complete turn every 60 seconds and yours makes a complete turn every 80 seconds. When will be both be back in our original starting positions?
What would the answer be if mine made a complete turn every 60 seconds but yours made a complete turn every 45 seconds?
5. Each edge of a cube is 3 ft. long. The cube is going to be covered with paper. What is the least amount of paper needed?
If you forget how many faces a cube has, just think of dice: 6! Find the area of one face (3 ft. squared or 3 ft. x 3 ft.) and then multiply that area times 6!!
6. Which of the following numbers is equal to 1/5? a. 0.02 b. 20 c. 0.2
7. Which of the following numbers is equal to 6/12? a. 0.05 b. 0.5 c. 0.6
8. What is the LCM of 2, 6, 8?
Remember to find the multiples of each number. Then look for the lowest shared multiple. Multiples of 6 are 6, 12, 18, 24, 30, 36, etc.
9. What is 100% of 50?
10. What is 10% of 50?
11. What is 20% of 200?
12. I want to balance two items on a scale: one weighs 5 grams and one weighs 100 kg. How many of the smaller items do I need to balance with one of the larger items?
Practice your metric tables. For example, you should know that 1 mm = .1 cm. Know many meters in a km, how many g in a kg, etc.
13. Imagine a clock. What is the measure of the angle formed when the hour hand is at the 12 and the minute hand is at the 10?
14. Imagine a clock, again. What is the measure of the angle formed when the hour hand is at the 12 and the minute hand is at the 11?
Hint for 13 and 14: a complete rotation is equal to 360 degrees.)
15. Solve 2n - 2 = 8 n =
16. The price of a car was reduced from $20,000 to 18,000. By what percent was it reduced?
Hint: To find percent that a price was reduced you first calculate the amount of the price reduction ($200) and then divide the amount of the reduction by the original price and then you multiply that by 100. So: $200/$2,000 x 100
17. A proportion problem:
To estimate species populations, biologists sometimes place tags on a number of animals. Later they take a second sample and count how many in the sample have tags. Then they can use a proportion to estimate the size of the total population.
In a pond, biologists tagged 200 fish and then released them back into the pond. Soon after they took a sample of 500 fish and found that 10 were tagged. What might they estimate the entire population to be?
10/500 = 200/entire population
18. I have 10 cards in a basket. On each card is a letter. If you pick a card, the probability that the letter on the card is an A is 3/5. What is the probability that the card you pick does not have a letter A?
How many of the 10 cards have an A on them?____
19. A dog weighs 5 pounds more than 3/4 of its weight. How much does it weigh?
Hint: 1/4 of its weight will be equal to 5 pounds.
20. 4n = 40 x .1 n= _____
21. 3n = 60 x .001 n = ____
22. 5n = 100 x .001 n=____
To do well on questions like 20-22, it is important to practice multiplying numbers by 10, 100, and 1,000 and .1, .01, and .001. Get comfortable moving around the decimal point! Don't just rely on your calculator to tell you this, but see that multiplying by .1 is the same as dividing by 10.
When you divide by 10, it is the same as moving the decimal one place to the left. When you multiply by .01, it is like dividing by 100 or moving the decimal two places to the left. You can practice with these:
40 x .1 = 4 40 x .01= .4 40 x .001= .04 400 x .001 = .4
23. Imagine that there are ten cards in a basket. The numbers 1-10 are written on them. What is the probability that a card picked at random will have a prime number written on it?___ An odd number?____ An even number? A number divisible by 3?_____ A number divisible by 6?______ A multiple of 4?______
A factor of 8?______A factor of 9?_____A single digit number?______
24. Are all rectangles parallelograms?
25. Are all parallelograms rectangles?
FORMULAS
Know how to use each geometry formula!! I'd like each of you to make up a problem for each formula and provide me with the answer. Remember to be careful to get your units right. Problems for area have answers in square units. Problems for volume have answers in cubic units.
Rectangles
Area = l x w (units will be square units such as square in., sq. ft, etc.)
Perimeter = 2l = 2w (units will be linear units such as in, ft. , etc.)
Squares
A = side x side
Perimeter = 4s
Triangles
A = 1/2 (b x h) or A = (b x h): 2
The base can be any side. The height must be perpendicular to base!
Perimeter = side 1 + side 2 + side 3
The math part of thee ERB is divided into 3 sections: Quantitative Reasoning, Mathematics, and Constructed Response. We've already looked at examples of the kinds of quantitative reasoning problems you might get.
The following questions should help you prepare for the Mathematics section. As you will see, you really don't need a calculator to do these. Thinking carefully is the key to success!. Sometimes a problem will use a variable (a letter standing in for an unknown number), just as we've seen in the practice we've done for the Quantitative Reasoning section.
1. x is to 16 as 1 is to 4. What is x?
Hint: A proportion is useful. We used them a lot when we studied percents.
2. If you know the measures of 2 angles of a triangle, can you calculate the measure of the third? _____ If so, how?
(Remember that the 3 angles need to sum to 180 degrees.)
EX: Triangle A has 2 angles measuring 55 and 35 degrees. The degree measure for the third angle will have to be equal to 180 - (55 + 35)
3. I toss a coin 3 times and each time the coin lands with "heads" facing up. What is the probability that I will get heads the next time?
(Remember that each new toss is not affected by outcomes in the past tosses.)
4. You and I each get on a merry-go-round (menage? or carousel) but our merry-go-rounds are making complete turns at different speeds. Mine makes a complete turn every minute and yours makes a complete turn every minute-and-a-half. How many minutes will it be before we are both once again in our original starting positions?
(Hint: Think about multiples! You are doing the same kind of thinking that you do when you find the LCM of two numbers.)
Let's assume that mine makes a complete turn every 60 seconds and yours makes a complete turn every 80 seconds. When will be both be back in our original starting positions?
What would the answer be if mine made a complete turn every 60 seconds but yours made a complete turn every 45 seconds?
5. Each edge of a cube is 3 ft. long. The cube is going to be covered with paper. What is the least amount of paper needed?
If you forget how many faces a cube has, just think of dice: 6! Find the area of one face (3 ft. squared or 3 ft. x 3 ft.) and then multiply that area times 6!!
6. Which of the following numbers is equal to 1/5? a. 0.02 b. 20 c. 0.2
7. Which of the following numbers is equal to 6/12? a. 0.05 b. 0.5 c. 0.6
8. What is the LCM of 2, 6, 8?
Remember to find the multiples of each number. Then look for the lowest shared multiple. Multiples of 6 are 6, 12, 18, 24, 30, 36, etc.
9. What is 100% of 50?
10. What is 10% of 50?
11. What is 20% of 200?
12. I want to balance two items on a scale: one weighs 5 grams and one weighs 100 kg. How many of the smaller items do I need to balance with one of the larger items?
Practice your metric tables. For example, you should know that 1 mm = .1 cm. Know many meters in a km, how many g in a kg, etc.
13. Imagine a clock. What is the measure of the angle formed when the hour hand is at the 12 and the minute hand is at the 10?
14. Imagine a clock, again. What is the measure of the angle formed when the hour hand is at the 12 and the minute hand is at the 11?
Hint for 13 and 14: a complete rotation is equal to 360 degrees.)
15. Solve 2n - 2 = 8 n =
16. The price of a car was reduced from $20,000 to 18,000. By what percent was it reduced?
Hint: To find percent that a price was reduced you first calculate the amount of the price reduction ($200) and then divide the amount of the reduction by the original price and then you multiply that by 100. So: $200/$2,000 x 100
17. A proportion problem:
To estimate species populations, biologists sometimes place tags on a number of animals. Later they take a second sample and count how many in the sample have tags. Then they can use a proportion to estimate the size of the total population.
In a pond, biologists tagged 200 fish and then released them back into the pond. Soon after they took a sample of 500 fish and found that 10 were tagged. What might they estimate the entire population to be?
10/500 = 200/entire population
18. I have 10 cards in a basket. On each card is a letter. If you pick a card, the probability that the letter on the card is an A is 3/5. What is the probability that the card you pick does not have a letter A?
How many of the 10 cards have an A on them?____
19. A dog weighs 5 pounds more than 3/4 of its weight. How much does it weigh?
Hint: 1/4 of its weight will be equal to 5 pounds.
20. 4n = 40 x .1 n= _____
21. 3n = 60 x .001 n = ____
22. 5n = 100 x .001 n=____
To do well on questions like 20-22, it is important to practice multiplying numbers by 10, 100, and 1,000 and .1, .01, and .001. Get comfortable moving around the decimal point! Don't just rely on your calculator to tell you this, but see that multiplying by .1 is the same as dividing by 10.
When you divide by 10, it is the same as moving the decimal one place to the left. When you multiply by .01, it is like dividing by 100 or moving the decimal two places to the left. You can practice with these:
40 x .1 = 4 40 x .01= .4 40 x .001= .04 400 x .001 = .4
23. Imagine that there are ten cards in a basket. The numbers 1-10 are written on them. What is the probability that a card picked at random will have a prime number written on it?___ An odd number?____ An even number? A number divisible by 3?_____ A number divisible by 6?______ A multiple of 4?______
A factor of 8?______A factor of 9?_____A single digit number?______
24. Are all rectangles parallelograms?
25. Are all parallelograms rectangles?
FORMULAS
Know how to use each geometry formula!! I'd like each of you to make up a problem for each formula and provide me with the answer. Remember to be careful to get your units right. Problems for area have answers in square units. Problems for volume have answers in cubic units.
Rectangles
Area = l x w (units will be square units such as square in., sq. ft, etc.)
Perimeter = 2l = 2w (units will be linear units such as in, ft. , etc.)
Squares
A = side x side
Perimeter = 4s
Triangles
A = 1/2 (b x h) or A = (b x h): 2
The base can be any side. The height must be perpendicular to base!
Perimeter = side 1 + side 2 + side 3
Quiz Name___________________ No Calculators
1. The baseball team has won 3/4 of its first 12 games. How many must the team win in its next 16 games to maintain the same average?________
You will need to calculate 3/4 of the next 16 games....and then... the original average of 3/4 will not change!
2. Draw a Venn diagram. There are 55 students in the 7th grade. In a survey, 32 of them said that they like ping pong and 21 of them like judo. In addition, one quarter of those who like ping pong also like judo.
Draw a Venn diagram with two overlapping circles. Then show how many like only ping pong and how may like only judo and how many like both. Label your diagram and put the number of students in each of the part of the two overlapping circles. Put a number outside the circles to show how many do not like either one.
_____like only ping pong
_____like both (overlap area)
_____like only judo
_____like neither one
3. Two subtraction problems:
a. What is 11/15 - 1/5?
b. What is 23/15 - 1/5?
State your answer in b as a mixed number in simplest form. In both problems you will need to restate 1/5 as its equivalent fraction in 15ths.
4. Find the LCD for 1/6 and 1/8.
Hint: Remember that the LCM is quite similar to the LCD. So, look at the multiples of 6 and 8 and find the lowest multiple in common. Mult. of 6: 6, 12,18, 24, 30...
5. How many prime numbers are multiples of 5?____of 2?____
Hint: If a number is a multiple of 5, that means 5 is one of its factors.
6. Find the LCD for 1/8 and 1/12.
7. Find the LCD for 1/3 and 1/4.
8. Use the distributive property in the following:
a. 2 (5 - x) =
b. 2 (3 + x) =
c. 3 (5 + x) =
9. Multiplying with decimals or with multiples of tens: a good mental math exercise
a. 20 x 20 =
20 x 2 =
20 x 0.2 =
b. 25 x 40 =
25 x 4 =
25 x 0.4 =
10. Give the equivalent mixed number for each improper fraction:
5/3 = 5/4 = 8/5 = 15/4 =
The mixed number equal to 15/2 is 7 1/2.
11. Which of these numbers could be the area of a rectangle, if the length of each side is a whole number > 1?
a. 3 b. 5 c. 17 d. 25
12. Range is the difference between the highest and lowest number is a data set.
Example: Seven students got the following scores on a quiz:
3, 5, 5, 7, 8, 17, 18 (the data set)
The range is 15. You find the range by subtracting the lowest score from the highest. So, 18 - 3 = 15.
Mode: most repeated number in a data set (In this example, it is the number 5)
Median: the number that occurs in the middle of a data set when the data is arranged from least to greatest. In this example, it is 7.
Mean: the average. To find it here, you add up all the scores and then divide by the number of scores. (3 + 5 + 5 + 7 + 8 + 17 + 18) : 7 = 63 : 7 = 9
So, the average or mean score is 9.
Find the range, mode, median, and mean for this data set for 7 students' scores:
6, 6, 6, 8, 14, 18, 19
Range is:
Mode is:
Median is:
Mean is:
13. What is an equation that represents the statement: "Four times a number N is equal to twenty"?
14. Draw a trapezoid with two bases (8 in. and 5 in.) and a height of 3 in.
BONUS: What is the next number in the pattern:
1, 2, 6, 15, 31, 56,____
1. The baseball team has won 3/4 of its first 12 games. How many must the team win in its next 16 games to maintain the same average?________
You will need to calculate 3/4 of the next 16 games....and then... the original average of 3/4 will not change!
2. Draw a Venn diagram. There are 55 students in the 7th grade. In a survey, 32 of them said that they like ping pong and 21 of them like judo. In addition, one quarter of those who like ping pong also like judo.
Draw a Venn diagram with two overlapping circles. Then show how many like only ping pong and how may like only judo and how many like both. Label your diagram and put the number of students in each of the part of the two overlapping circles. Put a number outside the circles to show how many do not like either one.
_____like only ping pong
_____like both (overlap area)
_____like only judo
_____like neither one
3. Two subtraction problems:
a. What is 11/15 - 1/5?
b. What is 23/15 - 1/5?
State your answer in b as a mixed number in simplest form. In both problems you will need to restate 1/5 as its equivalent fraction in 15ths.
4. Find the LCD for 1/6 and 1/8.
Hint: Remember that the LCM is quite similar to the LCD. So, look at the multiples of 6 and 8 and find the lowest multiple in common. Mult. of 6: 6, 12,18, 24, 30...
5. How many prime numbers are multiples of 5?____of 2?____
Hint: If a number is a multiple of 5, that means 5 is one of its factors.
6. Find the LCD for 1/8 and 1/12.
7. Find the LCD for 1/3 and 1/4.
8. Use the distributive property in the following:
a. 2 (5 - x) =
b. 2 (3 + x) =
c. 3 (5 + x) =
9. Multiplying with decimals or with multiples of tens: a good mental math exercise
a. 20 x 20 =
20 x 2 =
20 x 0.2 =
b. 25 x 40 =
25 x 4 =
25 x 0.4 =
10. Give the equivalent mixed number for each improper fraction:
5/3 = 5/4 = 8/5 = 15/4 =
The mixed number equal to 15/2 is 7 1/2.
11. Which of these numbers could be the area of a rectangle, if the length of each side is a whole number > 1?
a. 3 b. 5 c. 17 d. 25
12. Range is the difference between the highest and lowest number is a data set.
Example: Seven students got the following scores on a quiz:
3, 5, 5, 7, 8, 17, 18 (the data set)
The range is 15. You find the range by subtracting the lowest score from the highest. So, 18 - 3 = 15.
Mode: most repeated number in a data set (In this example, it is the number 5)
Median: the number that occurs in the middle of a data set when the data is arranged from least to greatest. In this example, it is 7.
Mean: the average. To find it here, you add up all the scores and then divide by the number of scores. (3 + 5 + 5 + 7 + 8 + 17 + 18) : 7 = 63 : 7 = 9
So, the average or mean score is 9.
Find the range, mode, median, and mean for this data set for 7 students' scores:
6, 6, 6, 8, 14, 18, 19
Range is:
Mode is:
Median is:
Mean is:
13. What is an equation that represents the statement: "Four times a number N is equal to twenty"?
14. Draw a trapezoid with two bases (8 in. and 5 in.) and a height of 3 in.
BONUS: What is the next number in the pattern:
1, 2, 6, 15, 31, 56,____
Quantitative Reasoning Questions
Seventh Grade
Let's get ready for some ERB questions in which you have to compare expressions. Remember, expressions are just different ways of expressing possible values. They usually contain some combination of numbers and letters.
1. What's greater: 3a or 2a?
Remember that 3a means 3 times a. The letter a is a variable. It's job in this expression is to stand in for an unknown number. When you have a number in front of a letter, it means you need to multiply the number by the unknown. So, 3a means 3 times an unknown number, which we are calling a.
First, do you think that we know enough about the unknown to answer this question? Try substituting some actual numbers for a.
a. If a is any positive integer, what's greater: 3a or 2a?
b. But if a is zero, what's greater: 3a or 2a?
c. If a is any negative integer, what's greater: 3a or 2a?
So, you should have gotten 3 different answers, depending on what you substituted for a in those 3 different examples above. If not, think again!!
Okay, adding on to that:
2. What's greater: 2(a + 2) or 2a + 3?
First, we have to use the distributive property to multiply 2(a + 2). In other words, we distribute the multiplication to each of the two terms in the parenthesis. So,
2(a + 2) = (2 x a) + (2 x 2) = 2a + 4
Now, let's ask again the same question, now using the restated version of 2(2 + 2)
What's greater: 2a + 4 or 2a + 3?
It is easier to see now that we are comparing apples to apples. The question becomes simpler. Now, does it matter that we don't know whether the unknown a is:
a. a positive integer
b. zero
c. a negative integer?
You should see that the left side of this equation is always going to be greater, because the 2a part of the expression will be equal. So, we are able to say that 4 is always greater than 3!!! This is true no matter what value we substitute for a!!
3. Now let's ask, what's greater if a is an integer: a or - a?
Isn't this a lot like the first question?
4. What is the LCD (lowest common denominator) for 5/6 and 1/9?
Hint: the LCD is the same as the LCM for the two denominators.
Finding the LCM:
multiples of 6: 6, 12, 18, 24 30
multiples of 9: 9, 18, 27, 36
The lowest multiple they have in common is 18! So, the lowest common denominator for 6ths and 9ths will be 18.
We need LCD to add and subtract fractions, although we can use any common denominator that works. So, let's add those two fractions:
5/6 gets restated as 15/18.
1/9 gets restated as 2/18.
Once you have the common denominators, just add the numerators and leave the denominators alone!!
Now, the ERB has a lot of questions on Prime Numbers, so let's do some prime number practice: primes are numbers greater than one with only two factors: themselves and one. List begins: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47........
Now, try not to look at the list and do these problems:
1. What is the number equal to the sum of the first prime number after 20 plus the greatest prime number less than 5?
2. What is the number equal to the sum of the first prime after 30 plus the greatest prime number less than 10?
Translating English to expressions and back again:
1. A and B each brought x marbles to school. C brought 3 more marbles than either of them. Altogether how many marbles were brought to school?
x + x + x + 3 simplifies to 3x + 3
That's an expression that tells you how many marbles they all brought.
2. Try this one yourself: A and B each brought y marbles to school and C brought 200 more than either of them. Altogether, how many marbles were brought to school?
3. Now, let's use multiplication in our expression, remembering what we did at the very beginning for expressing multiplication with a variable.
Joe ate n sandwiches and Jack ate three times as many as Joe. Altogether, how many sandwiches did the two boys eat?
n + 3n which then simplifies to 4n
4. Try this one yourself: Bill ate n grapes, Bob ate twice as many as Bill, and Jack ate 5 times as many as Bill. Altogether, how many grapes did they eat?
5. Now, what if we say, Bill ate n, Bob ate twice what Bill did, and Jack ate 5 times what Bob did. That expression would look different, like this:
n + 2n + 5(2n) which can simplify to n + 2n + 10n and further to 13n
So, 13n would be the expression which tells you how many they ate altogether.
Let's get ready for some ERB questions in which you have to compare expressions. Remember, expressions are just different ways of expressing possible values. They usually contain some combination of numbers and letters.
1. What's greater: 3a or 2a?
Remember that 3a means 3 times a. The letter a is a variable. It's job in this expression is to stand in for an unknown number. When you have a number in front of a letter, it means you need to multiply the number by the unknown. So, 3a means 3 times an unknown number, which we are calling a.
First, do you think that we know enough about the unknown to answer this question? Try substituting some actual numbers for a.
a. If a is any positive integer, what's greater: 3a or 2a?
b. But if a is zero, what's greater: 3a or 2a?
c. If a is any negative integer, what's greater: 3a or 2a?
So, you should have gotten 3 different answers, depending on what you substituted for a in those 3 different examples above. If not, think again!!
Okay, adding on to that:
2. What's greater: 2(a + 2) or 2a + 3?
First, we have to use the distributive property to multiply 2(a + 2). In other words, we distribute the multiplication to each of the two terms in the parenthesis. So,
2(a + 2) = (2 x a) + (2 x 2) = 2a + 4
Now, let's ask again the same question, now using the restated version of 2(2 + 2)
What's greater: 2a + 4 or 2a + 3?
It is easier to see now that we are comparing apples to apples. The question becomes simpler. Now, does it matter that we don't know whether the unknown a is:
a. a positive integer
b. zero
c. a negative integer?
You should see that the left side of this equation is always going to be greater, because the 2a part of the expression will be equal. So, we are able to say that 4 is always greater than 3!!! This is true no matter what value we substitute for a!!
3. Now let's ask, what's greater if a is an integer: a or - a?
Isn't this a lot like the first question?
4. What is the LCD (lowest common denominator) for 5/6 and 1/9?
Hint: the LCD is the same as the LCM for the two denominators.
Finding the LCM:
multiples of 6: 6, 12, 18, 24 30
multiples of 9: 9, 18, 27, 36
The lowest multiple they have in common is 18! So, the lowest common denominator for 6ths and 9ths will be 18.
We need LCD to add and subtract fractions, although we can use any common denominator that works. So, let's add those two fractions:
5/6 gets restated as 15/18.
1/9 gets restated as 2/18.
Once you have the common denominators, just add the numerators and leave the denominators alone!!
Now, the ERB has a lot of questions on Prime Numbers, so let's do some prime number practice: primes are numbers greater than one with only two factors: themselves and one. List begins: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47........
Now, try not to look at the list and do these problems:
1. What is the number equal to the sum of the first prime number after 20 plus the greatest prime number less than 5?
2. What is the number equal to the sum of the first prime after 30 plus the greatest prime number less than 10?
Translating English to expressions and back again:
1. A and B each brought x marbles to school. C brought 3 more marbles than either of them. Altogether how many marbles were brought to school?
x + x + x + 3 simplifies to 3x + 3
That's an expression that tells you how many marbles they all brought.
2. Try this one yourself: A and B each brought y marbles to school and C brought 200 more than either of them. Altogether, how many marbles were brought to school?
3. Now, let's use multiplication in our expression, remembering what we did at the very beginning for expressing multiplication with a variable.
Joe ate n sandwiches and Jack ate three times as many as Joe. Altogether, how many sandwiches did the two boys eat?
n + 3n which then simplifies to 4n
4. Try this one yourself: Bill ate n grapes, Bob ate twice as many as Bill, and Jack ate 5 times as many as Bill. Altogether, how many grapes did they eat?
5. Now, what if we say, Bill ate n, Bob ate twice what Bill did, and Jack ate 5 times what Bob did. That expression would look different, like this:
n + 2n + 5(2n) which can simplify to n + 2n + 10n and further to 13n
So, 13n would be the expression which tells you how many they ate altogether.
Constructed Response Answers
ANSWER KEY
7th Grade: CONSTRUCTED RESPONSE ERB practice:
These are the instructions that will be given for this part of the ERB:
There are 8 questions in this test. For each question you must write an answer. It is important that your answer be clear and complete and that you show all of your work, since partial credit may be awarded. You may use drawings, words, or numbers to explain your answers.
In other words, be COMPLETE in your explanations. As we've been practicing in the TEST BEST booklets, you might
1. Restate what you are asked to find.
2. Show your work.
3. Show how you check your answer.
1. I'm going to choose at random one of the letters in the 7-letter word MEASURE. What letter is most likely to be chosen?
E
Explain why.
E is the most likely letter to be chosen because the probability is 2/7 for getting an E and only 1/7 for each of the other letters. You are twice as likely to get an E.
Note: Remember that probability can be expressed as a fraction or as a percent. Probability will never exceed 1.
2. A class is making puppets where the dimensions are 1/3 as big as the actual people being modeled. Find the puppet's dimensions for the example below.
Real Person Puppet
Height 5 feet ______ 20 inches or 1 ft. 8 in.
Waist 24 inches ______ 8 inches
Foot 12 inches ______ 4 inches
Remember that there are 12 inches in a foot.
Just show your math work. You could say that each 12 inches in real life will equate to 4 inches for a model.
3. Show how the number 5 can be written as the product of a fraction less than 1 and a whole number.
You pretty much have been given this problem already.
Hint: here's just one way you can show the number 8 as the product of a fraction less than 1 and a whole number: 8 = 1/2 x 16. There are many, many more ways to do this, in fact, infinitely many more ways to show 8 as a product. Another would be: 8 = 1/3 x 24 and another would be: 8 = 1/4 x 48. Get the idea?
7th Grade: CONSTRUCTED RESPONSE ERB practice:
These are the instructions that will be given for this part of the ERB:
There are 8 questions in this test. For each question you must write an answer. It is important that your answer be clear and complete and that you show all of your work, since partial credit may be awarded. You may use drawings, words, or numbers to explain your answers.
In other words, be COMPLETE in your explanations. As we've been practicing in the TEST BEST booklets, you might
1. Restate what you are asked to find.
2. Show your work.
3. Show how you check your answer.
1. I'm going to choose at random one of the letters in the 7-letter word MEASURE. What letter is most likely to be chosen?
E
Explain why.
E is the most likely letter to be chosen because the probability is 2/7 for getting an E and only 1/7 for each of the other letters. You are twice as likely to get an E.
Note: Remember that probability can be expressed as a fraction or as a percent. Probability will never exceed 1.
2. A class is making puppets where the dimensions are 1/3 as big as the actual people being modeled. Find the puppet's dimensions for the example below.
Real Person Puppet
Height 5 feet ______ 20 inches or 1 ft. 8 in.
Waist 24 inches ______ 8 inches
Foot 12 inches ______ 4 inches
Remember that there are 12 inches in a foot.
Just show your math work. You could say that each 12 inches in real life will equate to 4 inches for a model.
3. Show how the number 5 can be written as the product of a fraction less than 1 and a whole number.
You pretty much have been given this problem already.
Hint: here's just one way you can show the number 8 as the product of a fraction less than 1 and a whole number: 8 = 1/2 x 16. There are many, many more ways to do this, in fact, infinitely many more ways to show 8 as a product. Another would be: 8 = 1/3 x 24 and another would be: 8 = 1/4 x 48. Get the idea?
Key to Test Best
Test Best Level G ---7th grade key--- partial----math
p. 58 Number relations.
1. c 2. f 3. a 4. 5 degrees C
p. 59 Number theory
1. b 2. h scientific notation can only have one digit BEFORE the decimal.
3. c 4. It is a prime number. It will have no branches if you factor it. Its only factors are itself and one. For the record, here are the first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, etc, Composite numbers are COMPOSED of primes, meaning that they can be the products of two or more prime numbers. For the record, the first few composites are 4, 6, 8, 9, 10, 12, 14, etc....
p. 60 pre algebra
1. c 2. j Distributive property a(b + c) = ab + ac
3. a 4. j (and 5. 15 = 315 ( .10) = $ 46.50
p. 61 Patterns/ functions 1. d 2. Rudy can expect to make 20. 3. h 4. b
p. 62 Probability /statistics 1. b Two 6-sided cubes there are three different ways you can get results of greater than 10 so it's 3/36 or 1/12
2. Average score is the sum of the numbers divided by 12.
3. j 4. c
P. 63 continued
5. f 6. b 7. f 8. b 9. 12 shares x $65/ share = $780
p. 64 Geometry 1. a 2. 600 sq ft. e. h 4. b
p. 65 continue
5. f 6. c. reflection (a translation is just a shift) 7. j 8. d 9. radius
p. 66 measurement
1. d 3. Note that when they say length of one side of a block, they are not using the real math vocabulary. It would be length of one edge. (Side is not a word we use in solid geometry. The vocabulary is face, edge, and vertex.)
4. d
p. 67 continued
5. h 6. b 7. f 8. d 9. 6 yds x 3 ft./yd = 18 ft.
10. 45 minutes
p. 68 Problem Solving
1. d 2. 30 cars 3. g 4. 42, 44, 46, 48, 50, 52
p. 69 Estimation
1. d 2. h 3. c 4. g 5. 3,000 miles (est.)
p. 70 Test
1. b 2. g scientific notation----see note earlier---only 1 digit before decimal point
3. a 4. h 5. c 6. g 7. All are multiples of three. You might also have said that all have three as a common factor, or gcf.
8. c 9. A lot of kids got this wrong: there are five ways to total 6 when you throw 2 dice so the answer is 5/36 or j 10. 40 boards. 11. b use a proportion
12. j 13. d average you can estimate 14. f 15. c 16. f 17. narrow tip is only 35 cents per marker 18. b 19 j 20. c
p. 73
21. h 222. a 23. h 24. ac is the diameter 25. ordered pair is always x before y
or horizontal before vertical: d 26. g it's a rotation and not a translation (shift)
27. d 28. h 29 8:09 a.m. 30. not logical c 31. j 32 set up a proportion to solve: 12/16 = 3/4 He would need 12 pieces. 33. b 34. 45 35. 36 x 5 -8 h
36. c 37. I don't think this problem lists the correct answers. It would be 8 x 8. We might just write to the publishers to complain! 38. a
p. 76 Procedures.
1. e 2. g 3. a 4. f 5. b 6. j
p. 77 Computation
1. a 2. h 3. c 4. h
p. 78. Test
1. b 2. f 3. a 4. h 5. a 6. h 7. c 8. 201,201 9. h 10. c 11. f 12. d 13. h 14. $2.34
Alas, I have not done a key for test 3 which begins on p. 108. It's easy enough to check your work on that with a calculator.
p. 58 Number relations.
1. c 2. f 3. a 4. 5 degrees C
p. 59 Number theory
1. b 2. h scientific notation can only have one digit BEFORE the decimal.
3. c 4. It is a prime number. It will have no branches if you factor it. Its only factors are itself and one. For the record, here are the first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, etc, Composite numbers are COMPOSED of primes, meaning that they can be the products of two or more prime numbers. For the record, the first few composites are 4, 6, 8, 9, 10, 12, 14, etc....
p. 60 pre algebra
1. c 2. j Distributive property a(b + c) = ab + ac
3. a 4. j (and 5. 15 = 315 ( .10) = $ 46.50
p. 61 Patterns/ functions 1. d 2. Rudy can expect to make 20. 3. h 4. b
p. 62 Probability /statistics 1. b Two 6-sided cubes there are three different ways you can get results of greater than 10 so it's 3/36 or 1/12
2. Average score is the sum of the numbers divided by 12.
3. j 4. c
P. 63 continued
5. f 6. b 7. f 8. b 9. 12 shares x $65/ share = $780
p. 64 Geometry 1. a 2. 600 sq ft. e. h 4. b
p. 65 continue
5. f 6. c. reflection (a translation is just a shift) 7. j 8. d 9. radius
p. 66 measurement
1. d 3. Note that when they say length of one side of a block, they are not using the real math vocabulary. It would be length of one edge. (Side is not a word we use in solid geometry. The vocabulary is face, edge, and vertex.)
4. d
p. 67 continued
5. h 6. b 7. f 8. d 9. 6 yds x 3 ft./yd = 18 ft.
10. 45 minutes
p. 68 Problem Solving
1. d 2. 30 cars 3. g 4. 42, 44, 46, 48, 50, 52
p. 69 Estimation
1. d 2. h 3. c 4. g 5. 3,000 miles (est.)
p. 70 Test
1. b 2. g scientific notation----see note earlier---only 1 digit before decimal point
3. a 4. h 5. c 6. g 7. All are multiples of three. You might also have said that all have three as a common factor, or gcf.
8. c 9. A lot of kids got this wrong: there are five ways to total 6 when you throw 2 dice so the answer is 5/36 or j 10. 40 boards. 11. b use a proportion
12. j 13. d average you can estimate 14. f 15. c 16. f 17. narrow tip is only 35 cents per marker 18. b 19 j 20. c
p. 73
21. h 222. a 23. h 24. ac is the diameter 25. ordered pair is always x before y
or horizontal before vertical: d 26. g it's a rotation and not a translation (shift)
27. d 28. h 29 8:09 a.m. 30. not logical c 31. j 32 set up a proportion to solve: 12/16 = 3/4 He would need 12 pieces. 33. b 34. 45 35. 36 x 5 -8 h
36. c 37. I don't think this problem lists the correct answers. It would be 8 x 8. We might just write to the publishers to complain! 38. a
p. 76 Procedures.
1. e 2. g 3. a 4. f 5. b 6. j
p. 77 Computation
1. a 2. h 3. c 4. h
p. 78. Test
1. b 2. f 3. a 4. h 5. a 6. h 7. c 8. 201,201 9. h 10. c 11. f 12. d 13. h 14. $2.34
Alas, I have not done a key for test 3 which begins on p. 108. It's easy enough to check your work on that with a calculator.
Tuesday, November 25, 2008
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