Homework help for sixth grade Connected Mathematics 2 units

*Note: These suggestions are written for the problems and homework in the student books. If teachers choose to send home other work, they may choose to send home additional suggestions for helping your child.

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Prime Time Download the two-page parent reference sheet and letter

Investigation 1

Problem 1.1

Remember that the factors of a number are the numbers that can be evenly divided into it. The factors of 10 are 1, 2, 5, and 10. In this game, the focus is on the proper factors, which are all of the factors of a number except the number itself (1, 2, and 5 in the previous example). The sample game on pages 8-9 should help you to see how the game is played. As you are playing, think about the strategies you are using to gain points.

Problem 1.2

Continue the table that is started for you above the problem. Keep in mind that better moves are ones that gain you a lot of points but do not gain your opponent a lot of points. Note the definitions of prime and composite numbers in the Follow-Up.

Selected ACE questions (not all in order)

1: See the definition of factor in Problem 1.1 above.

2-9: Think: What would I multiply by the given factor to get the number (the product)?

10: This refers to the table in Problem 1.2. Note the definition of prime above.

11-13: Keep in mind that better moves are ones that gain you a lot of points but do not gain your opponent a lot of points. See Problem 1.1 for a reminder of how the game is played (pages 8-9 for a sample game).

14: Are 14 and 15 factors of 84? Why or why not?

15: Look back at Problems 1.1 and 1.2 and the new vocabulary that you know.

16: A digit is one numeral within a written number; the digits in 47 are 4 and 7. A sum is the answer to an addition problem. See above for the definition of factor.

17: All groups must be the same size in each way that you find. Can you find all of the ways?

19, 21: See Problem 1.1 for a reminder of how the game is played (pages 8-9 for a sample game). For 21, see #20 for a reference to perfect numbers.

20: The proper factors are all of the factors of a number except the number itself (1, 2, and 5 in the case of 10).

Investigation 2

Problem 2.1

See page 18 for the rules of the Product Game. Note the definition of multiple on this page also.

Problem 2.2

See page 18 for the rules of the Product Game. Really think about the strategies that you have to use in your game and what numbers make a game harder or easier.

Problem 2.3

Note the definition of factor in Problem 1.1 above and multiple on page 18.

Selected ACE questions

1-4: See page 18 for the rules of the Product Game.

5-8: The product is the number given.

9-10: Think about what factors you would need to get all of these numbers. Note the definition of factor in Problem 1.1 above.

11-12: See Problem 2.3 for a reminder of how to create these Venn diagrams. Note the definition of common multiple in #11. In #12, divisible means can be divided by.

13: Note the definition of multiple on page 18.

14: Note the definitions of these terms in Problem 1.1, on page 18, and above. A divisor is a number by which we divide another number (in the problem 5 divided by 2, 2 is the divisor).

15: Note the hints for #11-12 and 14 above. Note the definition of common factor in this problem.

16: Note the definition of common multiple in #11.

17-18: It might help to make a diagram or two to come up with possible answers.

20: A digit is one numeral within a written number; the digits in 47 are 4 and 7. A sum is the answer to an addition problem.

Investigation 4

Problem 4.1

It might help to list the times at which each person will be at the ground during the rides. For the follow-up, keep in mind that the numbers will be different for the two sisters.

Problem 4.2

It might help to imagine that you are starting at a point when each type of cicada is out and then list the number of years before the next cycle for each type. For the follow-up, the product is what you will get if you multiply the number of years in each cycle by the other(s). Are the answers you came up with in each part (A and B) greater than, less than, or equal to the product for each part?

Problem 4.3

It might help to start with a smaller number and decide whether that number of students could attend the picnic under the given conditions. Then, choose other numbers. You may start to see a pattern that will help you solve the problem. Drawing a diagram might help also.

Selected ACE questions

1-7: The common multiples of two numbers are all of the numbers that are multiples of both (or all) of the original numbers. For instance, one common multiple of 12 and 8 is 24. Another is 48. (A multiple is a number that you would get by skip counting from 0 by the original number.) Least means lowest.

8-13: The common factors of two numbers are all of the numbers that are factors of both (or all) of the given numbers. For instance, one common factor of 12 and 8 is 2. Another is 4.

14: It might help to draw a diagram or to make lists of possible numbers of hot dogs and buns in packages. Can you do it in a more efficient way?

15: It might help to make a list of the number of days that pass before pizza and applesauce are served again. Can you do it in a more efficient way?

16: It might help to make a list of the number of seconds that pass before each sign blinks again. Can you do it in a more efficient way?

17: Assume that at some point, the 12-year locusts do meet their predators.

18: It might help to make a list of the number of years that pass before each planet reaches that position again. Can you do it in a more efficient way?

19-20: Use a pattern to solve parts b and c (do not write it all out!).

21: It might help to draw a diagram of the pens in the drawer. Can you do it in a more efficient way?

22: See the hints above for the definitions of common factor and multiple. A digit is one numeral within a written number; the digits in 47 are 4 and 7.

Investigation 5

Problem 5.1

See the example on page 46 to see how to work on the problem.

Problem 5.2

Read the explanation on pages 48-49 first. Note the definition of exponent on page 49.

Problem 5.3

Read the explanation on page 50 first. Note the definition of relatively prime on page 51. For reminders about other vocabulary terms, see the notes above.

Selected ACE questions, not all in order

1-7, 11: See page 50 for help with prime factorization.

8-10: See the example maze on page 52.

12: A product is the answer to a multiplication problem. Remember that a prime number only has two factors, itself and 1. See Problem 1.1 notes above for the definition of factor.

13-15: Try to use the method shown on page 50 to solve these. The greatest common factor is the largest number that divides evenly into both (or all) given numbers. The least common multiple is the smallest number that is a multiple of both (or all) given numbers, meaning that skip counting from 0 by whole numbers will allow you to reach each number.

17: The common multiples of two numbers are all of the numbers that are multiples of both (or all) of the original numbers. For instance, one common multiple of 12 and 8 is 24. Another is 48. (A multiple is a number that you would get by skip counting from 0 by the original number.)

18: See page 50 for help with prime factorization.

19: See #5 on page 47 for a hint about this.

20: It might help to list some possible numbers of days that they could have worked and find out how much they would have earned. Look for patterns to help you.

21-22: See the notes above for help with these terms, if you need reminders.

24: It might help to draw a diagram of what the books might look like in the box. Can you think about it more efficiently?

Investigation 6

Problem 6.1

This is a fun problem with a very interesting solution and patterns to explore. Take your time in working your way through the lockers so that you have done it correctly and can identify the patterns and the solution.

Selected ACE questions

1-8: For help with any of these terms, see the notes above and/or the lessons in previous investigations.

9-10: Give a general rule that would tell you which numbers would be included in each problem.

12: Be sure to be clear in your explanation – why does the one group have many fewer primes than the other?

13: This refers to Problem 6.1. A square number is a number that can be divided evenly by a number to obtain that number again (e.g., 25, 81).

15: It might help to work backwards on this question.

17: See page 50 for help with prime factorization.

18: List some numbers that have 2 and 6 as factors to see if you can find a rule for what other numbers must also be factors.

19: List some multiples of 12 to see if you can find a rule for other numbers of which they must also be multiples.

20: This means that 10 and 6 are each factors of some other numbers. List some numbers for which this is case to find a general rule for other common factors.

Bits and Pieces I Download the two-page parent reference sheet and letter

Investigation 1

Problem 1.1

To determine if each of these statements could be true, consider half of the thermometer and how much money it would represent; then compare each of the statements to this as a starting point. Keep in mind the amount of money that the whole thermometer represents. You might also consider how to divide the thermometer into fourths, eighths, thirds, or other fractions to relate these to the statements in the problem.

Problem 1.2

The questions in this problem are asking you to think about relationships between the fraction strips. For instance, many people create the one with fourths by folding the one with halves in half again. When you mark the fractions in part E, you will probably want to write the fractions at the end of each piece that each fraction represents (on a fold or at the end), not in the middle. This will make it look a bit like a ruler.

Problem 1.3

You will probably want to use your fraction strips to help with the questions in this problem. However, you might be able to answer some the questions without them, if you can picture the sections of the fraction strips without actually placing them on the page. You can use the blue shaded areas on the page to help you think about how the whole thermometer might be divided into equal fractional parts. For E1, you will probably want to create a fraction and simplify it.

Problem 1.4

Notice that the goals are all different on these thermometers, even though on some of them the goals are the same heights as each other. You will probably want to use your fraction strips to help with the questions in this problem. For the shorter thermometers, think about how you could still use part of one of your fraction strips to determine what fraction of these are shaded. You might be able to answer some the questions without the strips, if you can picture the sections of the fraction strips without actually placing them on the page. You can use the shaded areas on the page to help you think about how the whole thermometers might be divided into equal fractional parts.

ACE questions

For #1: Consider half of the thermometer and how much money it would represent; then write statements that relate to this. Keep in mind the amount of money that the whole thermometer represents. For b, it may help to write a fraction and simplify.

For #2: The questions in this problem are asking you to think about relationships between the fraction strips. For instance, many people create the one with fourths by folding the one with halves in half again.

For #3-6: See the directions. You might be able to answer some the questions without fraction strips, if you can picture the sections of the fraction strips without actually placing them on the page. You can use the shaded areas on the page to help you think about how the whole thermometers might be divided into equal fractional parts.

For #7-11: As a starting point to answer these questions, consider half of the dispenser and how much liquid it would hold (and where the gauge would be if the dispenser were half full). Keep in mind the amount of liquid that the whole dispenser holds. You might also consider how to divide the gauge into fourths, eighths, thirds, or other fractions to relate these to the statements in the problem.

For #12: Think about your fraction strips.

For #13-15: See the explanation on page 14.

For #16: Think about snack bars in real life.

For #17: Think about what the thermometer would look like if it were enlarged on a copy machine.

For #18-19: You will probably want to write fractions and simplify them.

For #20-26: A number is divisible by another number when it can be divided evenly by it. A divisor is a number that can be divided evenly into another number. Factor is another word for divisor. For #24, you might want to choose a few examples and test the statements made. For #25-26, list fractional parts other than twelfths (#25) and tenths (#26).

For #27: It might help to look at your fraction strips or draw pictures.

For #28-30: Consider how to divide the bars representing the totals into fourths, eighths, thirds, or other fractions to relate these to the bars representing the orange juice. #30 is asking you to think about the scale on the graph and what it might be (then answer the questions based on this).

For #31: It will help to draw a picture of the pizzas.

For #32-35: See the explanation on page 14 for #13-15.

For #36-37: Consider the denominators of these fractions (whether they are missing or not) to be the number of items in a set. Then write the fractions based on this, thinking about what half of the set would be.

Investigation 2

Problem 2.1

Think about how to divide the 48 inches of rope into different numbers of pieces to share the rope with more and more people. Remember that you cannot change the marks you have already made in each step. For part D, think about your fraction strips, and consider what you know about equivalent fractions. For part E, you might want to use fraction strips or even a different object to figure out what fraction was cut off.

Problem 2.2

For all of these problems, keep in mind the denominators of the fractions – that is, into how many equal parts the whole is divided. If you are working with thirds, you should think about how the line would look if it were divided into three equal parts, and where the marks would be in this case. Always keep in mind that the fraction represents the part of the whole between 0 and 1 (the distance from 0 to the point).

When thinking about equivalent fractions, always think about what the parts would look like and not just what the numbers in the fractions are.

Problem 2.3

For each of the fractions, consider the denominator to be the number of items in a set. Then think about what half of the set would be, and decide whether the numerator is less than the number in half the set, more, equal to it, or even more than the number in the whole set. For parts C and D, it will still be useful to think about these fractions in relation to one-half and one whole (as in parts A and B).

Problem 2.4

You will want to use what you know about equivalent fractions to help you with this problem. It may also help to think about numbers that fall in between whole numbers (3 and 4, 8 and 9, etc.).

Problem 2.5

Whenever you work with mixed numbers and fractions greater than one (referred to in this book as improper fractions), it is very helpful to think about the number of pieces in one whole, the number of wholes in the problem, and any leftover pieces. Pictures can also be helpful. Many students get confused trying to remember shortcuts that they may not have truly learned to begin with (though once you do understand these shortcuts and why they work, they can be helpful).

ACE questions – not all in order

For #1-3: Consider the number of people in each problem and the number of items to be shared. Sometimes the number of items is not as important as the number of people.

For #4-7: You may want to draw pictures first, then decide whether each statement is correct or not. Be sure to keep the wholes in each picture the same size.

For #8-11: Think back to your fraction strips and how they were folded. You can also draw pictures of rectangles if this helps.

For #12-23, 25-33: For each of the fractions, consider the denominator to be the number of items in a set. Then think about what half of the set would be, and decide whether the numerator is less than the number in half the set, more, equal to it, or even more than the number in the whole set. ½ and 1 whole are good benchmarks to use to compare fractions, as are ¼ and ¾.

For #24, 70: You will want to use what you know about equivalent fractions to help you with these problems. It may also help to think about numbers that fall in between whole numbers (3 and 4, 8 and 9, etc.).

For #34, 36, 41-45: Whenever you work with mixed numbers and fractions greater than one (referred to in this book as improper fractions), it is very helpful to think about the number of pieces in one whole, the number of wholes in the problem, and any leftover pieces. Pictures can also be helpful. Many students get confused trying to remember shortcuts that they may not have truly learned to begin with (though once you do understand these shortcuts and why they work, they can be helpful).

For #35, 71: Think about the denominator as an indication of how big each piece of the whole is, and the numerator as how many pieces there are.

For #37-40: Think about where 0 and 1 are on each line, and consider how to divide this distance into equal parts based on the denominators in the problem (which, as you know, indicate how many parts there should be in the whole). You will also need to think about where the whole numbers would be – they should be evenly spaced.

For #46-47: Consider how to divide the whole pencil into fourths, eighths, thirds, or other fractions to relate it to the smaller pencils.

For #48: You will probably want to write fractions and simplify them.

For #49-50: The LCM is the smallest number into which the given numbers all divide evenly.

For #51-58: The GCF is the largest number that divides evenly into the given numbers.

For #59-64, 72: Think about the given fractions and their denominators – what would all of the pieces of 1 whole look like on each number line? Where would 1 whole end?

For #65-69: It might help to draw pictures of these fractions and divide them up more as needed.

Investigation 3

Problem 3.1

For all of these tasks, you will want to keep thinking about 100 and what you know about how it can be divided into equal groups (in different ways). In many of these tasks, you will want to consider how many groups there would be if 100 were divided into certain sizes of groups, like 2 or 5 or 10. This will help you to work with the fractions that are in the tasks.

Problem 3.2

You will want to keep in mind the 100 pieces in the first grid. However, in this problem, these pieces are divided into smaller and smaller parts – remember that the grid shown on page 40 is always the whole (the little squares do not become the whole at any time in this problem).

Problem 3.3

Comparing fractions and decimals can be tricky; however, it is easier if you always keep in mind the equal fractions and decimals that you know, such as 0.5 and ½, or 0.25 and ¼, or 0.75 and ¾. Knowing these can help you with the others. Also, for the fractions for which you do not know the equivalent decimals, consider how the denominator relates to 100, and think about how you could create an equivalent fraction with a denominator of 100. Then you can use this to find the decimal.

Problem 3.4

See parts B and C if you are looking for ways to solve part A.

Problem 3.5

When ordering decimals, it is very important to remember that the digits to the left represent larger numbers than the digits to their right. You need to look at the same place values in each decimal when comparing – not the entire number to the right of the decimal point.

ACE questions – not all in order

For #1-5: Consider the whole to be 100, and think about the part of 100 that each problem refers to.

For #6-7, 9-12: Think about the decimal given in relation to 100, and consider how you would write a fraction equal to this decimal. Then write two other fractions equal to that one.

For #8, 13-18: Think about each denominator and how it relates to 100; use this to write a fraction with a denominator of 100 and then a decimal that is equal.

For #19-21: See the explanation on page 48.

For #22, 43-50, 53: For these tasks, it may be helpful to write all the decimals so that they have the same number of digits after the decimal points. If this is already the case, use what you know about whole numbers and halves (for instance) to find decimals in between the others given.

For #23-24: Refer to the place value chart on page 45 for help. For #24, try to use these place values as the denominators of your fractions.

For #25: For all of these problems, keep in mind the denominators of the fractions – that is, into how many equal parts the whole is divided. If you are working with thirds, you should think about how the line would look if it were divided into three equal parts, and where the marks would be in this case. Always keep in mind that the fraction represents the part of the whole between 0 and 1 (the distance from 0 to the point).

For #26: Use a calculator or (better yet) use the fraction 1/9 and its decimal to help find the others.

For #27: Focus on fifths as parts of a whole, knowing that 0.2 is equal to 1/5.

For #28, 31, 32, 37-41: Comparing fractions and decimals can be tricky; however, it is easier if you always keep in mind the equal fractions and decimals that you know, such as 0.5 and ½, or 0.25 and ¼, or 0.75 and ¾. Knowing these can help you with the others. Also, for the fractions for which you do not know the equivalent decimals, consider how the denominator relates to 100, and think about how you could create an equivalent fraction with a denominator of 100. Then you can use this to find the decimal.

For #29, 30: For each of the fractions, consider the denominator to be the number of items in a set. For #29, think about what half of the set would be, and decide whether the numerator is less than the number in half the set, more, equal to it, or even more than the number in the whole set. ½ and 1 whole are good benchmarks to use to compare fractions, as are ¼ and ¾.

For #33-36, 42: When ordering decimals, it is very important to remember that the digits to the left represent larger numbers than the digits to their right. You need to look at the same place values in each decimal when comparing – not the entire number to the right of the decimal point.

For #51: It will help to draw a picture and to think about how the numbers in the problem relate to 100 (to write the decimal).

For #52: Remember how many parts into which the whole is divided.

For #54: Use your own experience and your own ideas.

For #55-60: Use the suggestions given.

Investigation 4

Problem 4.1

Think about the whole on each percent strip, and think of the strip as being like a number line.

Problem 4.2

Think about the whole in each set of free throws. How does each personπs number of successful free throws compare to the whole?

Problem 4.3

It may help to write fractions for some of the problems and then compare them to fractions with denominators of 100. Also, think about how the size of the whole in each problem relates to 100 – is it larger? Smaller? A factor of 100? A multiple of 100?

Problem 4.4

In part B, the percents should come from the chart above. For parts D and E, remember that a percent is a fraction out of 100, and the first two places after the decimal point in a decimal represent hundredths.

ACE questions – not all in order

For #1: Think about the whole on each percent strip, and think of the strip as being like a number line.

For #2-4: Think about the whole in each problem. How does each part (whether it is people, points, etc.) compare to the whole?

For #5-6: It may help to write fractions for the problems and then compare them to fractions with denominators of 100. Also, think about how the size of the whole in each problem relates to 100 – is it larger? Smaller? A factor of 100? A multiple of 100?

For #7-20: Use the part and the whole in each task to write a fraction. Then, compare them to fractions with denominators of 100. Also, think about how the size of the whole in each problem relates to 100 – is it larger? Smaller? A factor of 100? A multiple of 100? For #10-15, you will need to do some totaling first (from the data) before you write your fractions.

For #21-25, 33: Remember that a percent is a fraction out of 100 (in #21, use the answer for part a to get the others). For #22-23, the first two places after the decimal point in a decimal represent hundredths. For #24, use what you know about finding equivalent fractions easily to help you.

For #26-31: For each of the fractions, consider the denominator to be the number of items in a set. Think about what half of the set would be, and decide whether the numerator is less than the number in half the set, more, equal to it, or even more than the number in the whole set. ½ and 1 whole are good benchmarks to use to compare fractions, as are ¼ and ¾.

For #32: Whenever you work with mixed numbers and fractions greater than one (referred to in this book as improper fractions), it is very helpful to think about the number of pieces in one whole, the number of wholes in the problem, and any leftover pieces. Pictures can also be helpful. Many students get confused trying to remember shortcuts that they may not have truly learned to begin with (though once you do understand these shortcuts and why they work, they can be helpful).

For #34-36: You will want to use what you know about equivalent fractions to help you with these problems. It may also help to think about numbers that fall in between whole numbers (3 and 4, 8 and 9, etc.).

For #37-39: You may want to trace these pictures on your own paper and divide them up further to be able to compare the shaded parts to each other.

For #40: You will probably want to write a fraction using the information in the problem, and then simplify it.

For #41-43: Think of these as regular number lines, and decide how you would figure out what is exactly halfway between the given numbers.

For #44: It might help to use 10% as a way to work with 30% in these problems.

Shapes and Designs Download the two-page parent reference sheet and letter

Homework help for each problem will be posted soon!

Bits and Pieces II Download the two-page parent reference sheet and letter

Investigation 1

Problem 1.1

See the explanation of the game on pages 5-6. When using fraction benchmarks, it is always useful to compare the fractions to common fractions like 1/2, 1/4, and 3/4. Sometimes it helps to draw a picture, or even a mental picture, of the fraction to compare it to one of these fractions. Another option is to think of the denominator as the total number of items in a set and the numerator as the number of items you have; then decide how much of the set you have compared to these fractions (e.g., 7/8 would be larger than 1/2 and is close to 1 whole because 7 out of 8 is more than 4 out of 8 and because 7 is almost all of 8).

Problem 1.2

In parts B and C, it might help to draw a picture, or even a mental picture, of the fractions to get an idea of their size compared to fractions like 1/2, 1/4, 3/4, or 1 whole. Another option is to think of the denominator as the total number of items in a set and the numerator as the number of items you have; then decide how much of the set you have compared to these fractions (e.g., 7/8 would be larger than 1/2 and is close to 1 whole because 7 out of 8 is more than 4 out of 8 and because 7 is almost all of 8).

ACE questions

1-9: Sometimes it helps to draw a picture, or even a mental picture, of the fraction to compare it to one of the numbers in the directions. Another option is to think of the denominator as the total number of items in a set and the numerator as the number of items you have; then decide how much of the set you have compared to these fractions (e.g., 7/8 would be larger than 1/2 and is close to 1 whole because 7 out of 8 is more than 4 out of 8 and because 7 is almost all of 8).

10-19: Sometimes it helps to draw a picture, or even a mental picture, of each fraction to decide about how large it is (it is a good idea to compare it to 0, 1/2, or 1 whole). Then you can think about what the sum of the two fractions would be close to. Another option is to think of the denominator as the total number of items in a set and the numerator as the number of items you have; then decide how much of the set you have compared to these fractions (e.g., 7/8 would be larger than 1/2 and is close to 1 whole because 7 out of 8 is more than 4 out of 8 and because 7 is almost all of 8). When a decimal is involved, it may help to draw a picture (or a mental picture) that is divided into tenths or hundredths, but you can also think about the decimal in terms of money – what part of a dollar would it be? For #16-18, it may also help to use the equivalent decimals that you found for many of these fractions in the last unit.

20-25: Think about (perhaps draw pictures of) the sizes of the numbers given, and think about two smaller fractions that would add to a number between them. A picture might even help you to come up with the two fractions in your answer because you can actually see smaller parts (fractions) within the picture. Another option would be to think about these numbers in terms of money (dollars) and find amounts of money that would add up to a number between them. Then, you would need to write fractions to represent these amounts of money.

26: Think of the denominator as the total number of items in a set and the numerator as the number of items you have; then decide how much of the set you have compared to these fractions (e.g., 7/10 would be larger than 1/2 and is close to 1 whole because 7 out of 10 is more than 5 out of 10 and because 7 is somewhat close to all of 10).

27-29: Sometimes it helps to draw a picture, or even a mental picture, of each fraction to decide about how large it is (it is a good idea to compare it to 0, 1/2, 1/4, 3/4, or 1 whole). Then you can think about what the sum of the two fractions would be close to. Another option is to think of the denominator as the total number of items in a set and the numerator as the number of items you have; then decide how much of the set you have compared to these fractions (e.g., 7/8 would be larger than 1/2 and is close to 1 whole because 7 out of 8 is more than 4 out of 8 and because 7 is almost all of 8).

30: Try to think in terms of quarters or half-dollars (even dimes) – what amounts are the listed amounts close to? This will make it easier to estimate sums.

31: Think: if the rectangle represents 3/4 of the whole, would the whole be larger or smaller than this rectangle? How much larger or smaller? It might help to think about this in terms of quarters (coins). If the rectangle represents 3 quarters, what would a dollar look like?

32: Think: if the rectangle represents 150% of the whole, would the whole (100%) be larger or smaller than this rectangle? How much larger or smaller? It might help to think of each 1% representing something, like a person.

33: Think: if the beans represent 3/5 of the total, would the total be larger or smaller than this set? How much larger or smaller? Try to break apart the set of beans pictured to help you.

34: You found all of these equivalents in the previous unit. You can look back at your notes if you cannot remember all of them (and then try to memorize them). Another option is to view the fraction as a division problem and divide the numerator by the denominator to get a decimal, then change this to a percent by multiplying by 100. For part b, some students might find it easier to mark the fractions first, then the decimals and percents, while others might find it easier to mark the decimals and percents first, then the fractions.

35: When ordering decimals, it is very important to remember that the digits to the left represent larger numbers than the digits to their right. You need to look at the same place values in each decimal when comparing – not the entire number to the right of the decimal point.

36: The question is asking which set of fractions can be rewritten as fractions out of 100 with whole numbers in the numerators. Use what you know about finding equivalent fractions by multiplying the numerator and denominator by the same number.

37-40: Remember that the fourths in each shape must have the same area, but they do not have to be exactly the same shape (though they can be).

41-42: You will want to count the parts of each strip that you can see, write the fractions that are clear because the strips are visible, and then think about how many parts in one strip are equal to one part in the other strip.

43-47: You will probably want to find some fractions, decimals, or percents equal to the fractions on the line to make this an easier task.

Investigation 2

Problem 2.1

The easiest way to work on part A is to divide each section into equal pieces, using the lines that are already there as a start. You will have to think about how small your pieces will have to be in order to make them all equal. For parts B-F, you can use the small pieces that you created in part A to answer the questions, but sometimes you may find that you can simplify your answers so that the denominators are smaller.

Problem 2.2

For each of these tasks, you will want to think carefully about the relationships among the denominators (how the wholes are divided) in each recipe. Pictures may help you to do this, or you may not need them. For instance, in the Spice Parisienne recipe, what is the relationship between 5ths and 10ths? Can you use this to find the totals required in the questions? It will be a bit more difficult to do this in the Garam Marsala recipe, but you can still think about a way that you could divide a whole that would produce halves, thirds, and fourths (think back to the licorice dividing task in the previous unit). Once you have related the fractions within the same whole, you can add (and subtract) them.

Problem 2.3

For A-C, see the explanation of fact families on page 21. For part B, you will want to think carefully about the relationships among the denominators (how the wholes are divided) in each problem. Pictures may help you to do this, or you may not need them. For instance, in #3, what is the relationship between 4ths and 12ths? It will be a bit more difficult to do this in #1, but you can still think about a way that you could divide a whole that would produce thirds and fifths (think back to the licorice dividing task in the previous unit). Once you have related the fractions within the same whole, you can add (and subtract) them. For part D, you will probably need to think about a way to divide a whole into eighths, fourths, and thirds, and then use this to find values for M and N.

Problem 2.4

When you are adding and subtracting fractions, it is very important to consider the relationships among the denominators of these fractions (that is, how the whole is divided in each case). For instance, if you have thirds and ninths in the denominators, what is the relationship between thirds and ninths, and how can you use this to add the two fractions easily? Pictures may help, at least at first. Sometimes, when there is no obvious relationship between the denominators, you may have to think about a way that you could divide a whole that would produce both of them (think back to the licorice dividing task in the previous unit). Once you have related the fractions within the same whole, you can add (and subtract) them.

ACE questions, not all in order

1: The easiest way to work on part a is to divide the garden into equal pieces, using the lines that are already there as a start. You will have to think about how small your pieces will have to be in order to make them all equal. For parts b-f, you can use the small pieces that you created in part a to answer the questions, but sometimes you may find that you can simplify your answers so that the denominators are smaller.

2: There are a couple of ways to approach these problems. One way is to divide the whole into equal fractional parts (using the lines that are already there as a start) and then to re-name each part that is already drawn with another fraction. Another way is to consider the relationship between halves, fourths, eighths, and sixteenths and to use this to find equivalent fractions and answer the questions.

3: Consider the relationship between fourths, eighths, and sixteenths, and use this to find equivalent fractions and answer the questions. You will also want to think about how not to add all of the individual parts of pages listed but to write fractions that mean the same thing as the amounts listed (e.g., to what part of a whole are three 1/4-page ads equal?).

4-6: Consider the relationship between the denominators in each task (that is, how the wholes are divided in each case), and use this to find equivalent fractions and answer the questions. Pictures may help you to do this.

7-13, 20-27: Consider the relationship between the denominators in each task (that is, how the wholes are divided in each case), and use this to find equivalent fractions (if needed) and solve the problems. Pictures may help you to do this. Sometimes, when there is no obvious relationship between the denominators, you may have to think about a way that you could divide a whole that would produce both of them (think back to the licorice dividing task in the previous unit). Once you have related the fractions within the same whole, you can add (and subtract) them. Remember that whole numbers are added or subtracted as they always are – now, they also have fractions attached. In #27, a picture may be especially helpful.

14-17: First, consider the relationship between the denominators in each pair (that is, how the wholes are divided in each case), and use this to find equivalent fractions and solve the problems. Pictures may help you to do this. Sometimes, when there is no obvious relationship between the denominators, you may have to think about a way that you could divide a whole that would produce both of them (think back to the licorice dividing task in the previous unit). Once you have related the fractions within the same whole, you can add (and subtract) them. Then, to decide which is larger, you will probably need to find the difference between the two answers (subtract), using the same process described here.

18-19: For 18, see the explanation of fact families on page 21. For 18-19, you will want to think carefully about the relationships among the denominators (how the wholes are divided) in each problem. Pictures may help you to do this, or you may not need them. You will need to think about a way that you could divide a whole that would produce the denominators in each task (think back to the licorice dividing task in the previous unit). Once you have related the fractions within the same whole, you can add (and subtract) them.

28-29: For finding the sums, consider the relationships between the denominators (how the wholes are divided), and use these as you find your answers. Pictures may help.

30-35: Use what you know about finding equivalent fractions by multiplying or dividing the numerator and denominator by the same number.

36-38: You will want to count the parts of each strip that you can see, write the fractions that are clear because the strips are visible, and then think about how many parts in one strip are equal to one part in the other strip.

39-40: For 39, think about the fractions that you used in Problem 2.1 and how you would write a fraction that represented 100%. Then decide which answer choice is accurate. For 40, think about what you know about .25 and what fraction this represents. (Do not forget the whole, though – 1.25 sections).

41-44: When ordering decimals, it is very important to remember that the digits to the left represent larger numbers than the digits to their right. You need to look at the same place values in each decimal when comparing – not the entire number to the right of the decimal point.

45: For part a, try to show this with just numbers and then with just pictures. For part b, your fractions do not have to have the same denominators. You will want to find fractions equal to those given.

46: If the model is 1/3 of a whole, first draw a picture of what the whole would look like. This will make it easier to name the fractions represented in parts a and b, but be sure to keep in mind what the whole is and how many parts it is divided into.

47: If the model represents 1 whole, think about what thirds and sixths of the whole would look like, and actually draw pictures to divide it into these pieces so that you can more easily draw pictures to solve the problems.

48: Consider $160 to be the whole, and determine how you would find the given fractions of this whole (how much they would cost). Use these to answer all of the questions.

49: This problem is challenging, but you will want to think carefully about the relationships among the denominators that you choose. Try to find a simple example with small numbers first, and then this may help you to find others.

50: Making a table or even drawing a diagram will probably help you solve this problem. Remember that you really need to check to see if your answers make sense – sometimes in these types of problems, our answers can be the opposites of what we are supposed to get if we are not careful!

51: For parts a-d, it is basically asking you to read and put points on the number line based on the information in the problem. In part e, you need to look at where they ended and determine what fraction of their goal they reached (remember that you can only count from $0 on).

Investigation 3

Problem 3.1

If you carefully read and follow each part of the directions, you should be able to draw the pictures required and answer the questions. Remember that you always need to think first about how many total pieces you need (the denominator) and how many are actually a part of the problem (the numerator).

Problem 3.2

Remember that the multiplication sign can be interpreted as of (for example, 1/3 x 1/2 is the same thing as 1/3 of 1/2). Note that it asks you to use pictures in part A2. In A4, think about real-life situations that are better modeled by the brownie pan and situations that are better modeled by the number line. For part B, using a picture model may help. A number sentence is an equation using the numbers in the problem and the answer. For part D, it is probably good to assume that Libby is talking about two fractions that are each less than 1 whole, but it would be interesting to consider the question with fractions that are not.

Problem 3.3

Remember that the multiplication sign can be interpreted as of (for example, 1/3 x 1/2 is the same thing as 1/3 of 1/2). Note that it asks you to use models or diagrams to find the exact answers. A number sentence is an equation using the numbers in the problem and the answer. In your pictures, it will help to think first about how many total pieces you need (the denominator) and how many are actually a part of the problem (the numerator). You will probably need to do this twice for each problem since there are two fractions involved with the same whole.

Problem 3.4

In all of these problems, you need to keep in mind the sizes of the fractions involved, not just the numbers you see on the page. That is, you will want to picture what the fractions look like, whether you use a picture on paper or an image in your mind. Remember that the multiplication sign can be interpreted as of (for example, 1/3 x 1/2 is the same thing as 1/3 of 1/2).

Problem 3.5

For A4, your algorithm may have to include steps for mixed numbers or whole numbers that are not necessary for fractions that just have numerators and denominators.

Selected ACE questions, not all in order

1: The intent of this question is that Greg buys 2/5 of a pan of brownies (maybe they are leftovers!), but only 7/10 of the brownies that would fill this 2/5 of the pan are there. How much is still there? Your picture should show the 2/5 and then 7/10 of this 2/5. It often helps to use both horizontal and vertical lines to divide the whole in two different ways.

2: Draw what she has first; then adapt your picture to show 2/3 of what she has. A number sentence is an equation with the numbers in the problem and the answer.

3: It will help to remember that the multiplication sign can be interpreted as of (for example, 1/3 x 1/2 is the same thing as 1/3 of 1/2). You might also use pictures to illustrate any of these problems if you are not sure of the answer. Remember that the denominators will tell you how to divide the whole pictures.

4: This task is asking you to draw two pictures, one with 2/3 of 3/4 shown, and one with 3/4 of 2/3 shown. Are the final amounts the same?

5-10, 15: Use the algorithm you developed in Problem 3.5. You can also use pictures if it helps.

11: It will help to remember that the multiplication sign can be interpreted as of (for example, 1/3 x 1/2 is the same thing as 1/3 of 1/2). Pictures may also help. It is important to think about the size of the numbers involved and not just the numbers that you see on the page.

12-14: A model or picture may help with these problems. A number sentence is an equation with the numbers in the problem and the answer.

16: A number sentence is an equation with the numbers in the problem and the answer. A product is the answer to a multiplication problem. This is a challenging task; one thing you can do to help is to look for examples of these in other work that you have done. Beyond that, you may want to try some simple numbers to start and then adjust them until you find what you need for the task. For instance, you could try 1/3 x 8 and see if it fits any of the situations in parts a-d.

17: You may want to draw a picture to help visualize how much of each ingredient is needed; this will help you think about what numbers to use to solve the problems. It will also help to look for patterns in your answers – that is, things that stay the same as you calculate the answer for each ingredient.

18: A picture will help you solve this problem. Start by drawing the whole sandwich.

19: Think about how many batches of 3/4 pound there are, and use this to write a number sentence and solve. If you are stuck, a picture may help.

20, 35: A picture may help, as will your algorithm from Problem 3.5.

21-29: As it says, use your algorithm from Problem 3.5.

30-32, 34: Use your algorithm from Problem 3.5.

33: If you are having trouble considering these numbers in your head, draw a picture of a face clock, and think about how many minutes are in 1 hour and in various parts of an hour.

37: If you are not sure, it might help you to try this situation with several different amounts of money (do not just try one set of dollar amounts for Lea and Roshaun).

38: Note that both mowed these parts of the whole lawn. You will want to think about how to divide the lawn into both twelfths and fifths in order to answer this question.

39: Note that you are trying to figure out how much is left to be picked. You will want to find a way to deal with fifths and sixths at the same time in order to answer this question.

40-45: Look back at the algorithm you developed in Problem 2.4, but also be sure that you are getting answers that are reasonable (not too large or small, given the numbers in the problems).

47: A picture will probably be most helpful with this problem.

48-49: Use your algorithm from Problem 3.5. A picture may help in #49.

Investigation 4

Problem 4.1

The introduction on page 49 will be helpful to read. It is important to think of a fraction division problem in one of two ways (some problems fit one way better than the other): how many ___s will fit into ____, or (when dividing by a whole number) how much will be in each part when I split ____ into ____ parts? Pictures are a must with all of the tasks in this lesson. Follow the directions carefully, and you should be able to answer the questions. Questions B5 and C7 are asking you to think about what the number means in the problem situation. Try to base your answer for part E on the work that you have done up to that point; look for patterns in your answers.

Problem 4.2

Be sure to show either pictures or use written explanations along with your number sentences. Again, when dividing a fraction by a whole number, it is helpful to think of the problem as How much will be in each part when I split ____ (the fraction) into ____ (the whole number) parts? It is important to think about the size of the numbers involved and not just the numbers that you see on the page.

Problem 4.3

For parts A-C, you will want to think about a way to divide each whole in two different ways, one that makes sense for each fraction in the problem. That is, you will want to look at the denominator in each fraction and try to divide the whole in both of these ways (ex: thirds and fifths for B1). This will help you to determine your answers. Be very careful to think about what any remainders represent – for instance, if the remainder appears to be 2 pieces in your picture, it is not 2 pieces of the whole; it is 2 pieces of the fraction by which you are dividing, and you will need to decide what fractional part of this fraction it is. See the middle of page 49 for a picture that may help to make this a bit clearer.

Problem 4.4

It is important to think of a fraction division problem in one of two ways (some problems fit one way better than the other): how many ___s will fit into ____, or (when dividing by a whole number) how much will be in each part when I split ____ into ____ parts? Looking for patterns in your answers is important when deriving this algorithm. It is important to think about the size of the numbers involved and not just the numbers that you see on the page.

ACE questions, not all in order

1-2, 4: Draw pictures to help you, at least until you think you have found a way of finding answers without using the pictures. It is important to think of a fraction division problem in one of two ways (some problems fit one way better than the other): how many ___s will fit into ____, or (when dividing by a whole number) how much will be in each part when I split ____ into ____ parts?

3: Be sure to show either pictures or number sentences with your solutions. The question about the remainder is asking you to think about what the number means in the problem situation.

5: For part a, think of splitting up the 5 1/3 gallons among four trips. For part b, use your answer from part a.

6-8, 10: Pictures may help. When dividing a fraction by a whole number, it is helpful to think of the problem as How much will be in each part when I split ____ (the fraction) into ____ (the whole number) parts? It is important to think about the size of the numbers involved and not just the numbers that you see on the page.

9, 12: Think of this question as How many ___s will fit into ____?

11, 13-14: You will want to think about a way to divide each whole in two different ways, one that makes sense for each fraction in the problem. That is, you will want to look at the denominator in each fraction and try to divide the whole in both of these ways (ex: thirds and ninths for #11a). This will help you to determine your answers. Be very careful to think about what any remainders represent – for instance, if the remainder appears to be 2 pieces in your picture, it is not 2 pieces of the whole; it is 2 pieces of the fraction by which you are dividing, and you will need to decide what fractional part of this fraction it is. See the middle of page 49 for a picture that may help to make this a bit clearer.

15-20: Use your algorithm from Problem 4.4.

21: It is important to think of a fraction division problem in one of two ways (some problems fit one way better than the other): how many ___s will fit into ____, or (when dividing by a whole number) how much will be in each part when I split ____ into ____ parts?

22-23: See page 45 (#36) for an example of a fact family.

24: Remember that you are looking for the difference between the two distances. Use your work from previous investigations. If you are unsure, think about a way to relate fifths and halves in the same whole so that you can easily work with the fractions.

25: Draw a picture if you are unsure.

26-29: In each task, find a way to relate the denominators of the two fractions in the same whole (how can you relate tenths and fifths, for instance?). Also use your work from previous investigations.

30: Remember that you can multiply or divide the numerator and denominator by the same number to get an equivalent fraction.

31-34: Use your algorithm from Problem 3.5.

35: Think about each of these in terms of money, even if you have to think about parts of cents.

36: See the hint at the bottom of page 39. Also think about a way you might use equivalent fractions to prove your answers.

37: It might help you to solve the multiplication problem in each pair first and then think about the division problem. This might take a bit of trial and error at first, but you should look for a rule (a pattern) that works every time.

38: Use a picture to help.

39: The easiest way to solve this problem will probably be to actually try each way with a few numbers and see what happens.

40: Use your algorithms from Problems 3.5 and 4.4, and as it says, consider what you know about the relationship between multiplication and division (how can you turn the problems around to make them easier to solve?).

41: It may help to draw pictures if you are having trouble keeping the numbers and measurements straight.

Covering and Surrounding Download the two-page parent reference sheet and letter

Investigation 1

Problem 1.1

Parts A and B: See page 5 for a picture and explanation of this task. It is probably easier to start with the 36 square meters of floor space and then work on the rail sections (the perimeter). Part C: For perimeter, be sure you are counting the number of units of distance around the outside of the figure.

Problem 1.2

The area is the number of square tiles needed to cover the design. The perimeter is the number of rail sections (units of distance) around the outside of each design. See page 6 for the costs. Part C: your rectangles should not have a hole in the center; it should be made of full rows and columns.

Problem 1.3

Be sure to give your answers in meters (perimeter) and square meters (area). The area is the number of square units needed to cover the design. The perimeter is the number of units of distance around the outside of each design. Part B: it may help to imagine square tiles that cover this design (to find the area). Part C: it may help you to draw a picture. Part D: the rules should look like formulas (including the variables given and an equal sign).

ACE questions – not all in order

Note: The area is the number of square units needed to cover a figure. The perimeter is the number of units of distance around the outside of a figure.

1: It is probably easier to start with the 24 square meters of floor space and then work on the rail sections (the perimeter). You may want to use grid paper. See the above note about area and perimeter.

2-8, 13-20: See the above note about area and perimeter. For #4-6, it may be easier to start with the area and then work on the perimeter. For #17-19, it may help to imagine square tiles that cover each figure (to find the area). For #20, see your work from Problem 1.3, part D.

9-12: Be careful to keep in mind that each unit of distance on the design is 12 feet, so when you calculate each area, you will want to use this fact. That is, each square is not 12 square feet (use your answer for #9 to help with the others). See the above note about area and perimeter.

21: It will probably help to draw the rectangles first using the lengths and widths given; then fill in the table. Dimensions are length and width. See the above note about area and perimeter.

22-24: You will probably want to divide each figure into several pieces that are easier to manage. Then, (#22-23) be sure you have found all of the missing side lengths on each figure; you can do this by looking at the opposite side lengths and deciding what the missing lengths must be in order for the given numbers to make sense. Then, you can find the area of each piece of each figure and the perimeter of the whole figure. For #24, since no numbers are given, you should describe what you would do to find the cost of the carpet and the molding if there were numbers given.

25: Draw a picture first. It may help to imagine or draw in square units that cover this figure (to find the area). However, remember that some of these will be half units because one dimension is 8 1/2 feet. For part c, you have now found the perimeter, and you know that the walls (which go around the perimeter) are 6 feet high. So, how much paint will they need for 6-foot walls around the entire perimeter?

26: Draw a picture first. Find the area, and use this to find the total cost and how many cars this ride can hold.

27: It is asking whether both formulas could be correct. It might help to replace the variables with numbers to check, or it might help to draw a picture to decide.

28: Draw a picture first – how many feet long and wide would a square yard be?

29: A square foot is a square that measures 1 foot on each side, and a square yard is a square that measures a yard on each side.

30: See #29 for part a. For c: 100 cm = 1 meter. For e: 1 cm = 10 mm. For f: a meter is about 39 inches long; how many inches long is a yard?

31: For each part, use this number of square units to form as many rectangles as you can (no holes in the center – complete rows and columns). Remember that a square is a special rectangle (it has 4 equal sides). Factors are numbers that you multiply to get to another number.

32: It is usually easier to multiply mixed numbers when they are written as fractions. Change any mixed numbers to fractions, and then use the algorithm you developed in Bits and Pieces II (multiply the numerators; multiply the denominators; simplify if possible).

33: Since 20 is the product of the two numbers, you will need to divide 20 by the given one to find the other one. You will probably want to change the mixed numbers to fractions and then use one of the algorithms that you developed in Bits and Pieces II.

34-35: It will probably help to draw a picture of the pan first; then decide how to cut it into the given number of pieces. Then, think about the length and width of each piece, keeping in mind that they all must have the same area.

36-37: For a and b, it may help to draw a picture and imagine square units that cover the shape. Also, use what you know about finding area and perimeter (see notes above). For c, use your picture and the lengths and widths of the room and the field to figure out how to fit several copies of the classroom neatly into the field.

38: Look carefully for a relationship between the length and width (think about how you can find area of a rectangle with a formula).

39: You may want to try several examples to decide whether you believe this is true or not; then try to make an argument for all rectangles if you think it is true.

40: It will help to start with the 18 square meters and then work on the length and width.

41: Think back to your work in Problem 1.1.

42: It may help to draw this on grid paper and actually try to fit in some tiles that are the size and shape of the one given.

Investigation 2

Note: The area is the number of square units needed to cover a figure. The perimeter is the number of units of distance around the outside of a figure.

Problem 2.1

Parts A-B: Note that the area must stay 24 square meters. As you find more rectangles, add rows to the table and fill them in. See the note about area and perimeter above. Part C: Use (length, perimeter) pairs from the table. To plot a point on the graph, go right from (0, 0) to the length, and then go up to the perimeter and create a point. What patterns did you see in the table – how are these obvious in the graph? Part D: Use what you notice from the table in part A to help you answer this. What type of shape will always have the lowest perimeter for a given area?

Problem 2.2

Part A: See the note about area and perimeter above. Parts B and C: do not answer too quickly; really think about these, and look back at the figures. Part D: think back to your work in Problem 1.1. Part E: start with the 24 square units, and then try to find a longer perimeter. Note that it does not put limits on what length and width you choose.

Problem 2.3

Part A: Note that the perimeter must stay 24 meters. As you find more rectangles, add rows to the table and fill them in. See the note about area and perimeter above. Part B: Use (length, area) pairs from the table. To plot a point on the graph, go right from (0, 0) to the length, and then go up to the area and create a point. What patterns did you see in the table – how are these obvious in the graph? Parts C and D: Use what you notice from the table in part A to help you answer this.

Problem 2.4

See the note about area and perimeter above.

Selected ACE questions – not all in order

1-2: Draw pictures if needed, and look back at your work in Problem 2.1 to help.

3-5: Note that you only need to do steps a and b for #4 and 5 (steps a-c for #3). See the note above for Problem 2.1, and look back at your work from this problem to help. The work here is the same as the work in that lesson.

6: Part a – Read the graph for the perimeter, and once you know the perimeter and the fact that the length is 2 meters, you can find the width (draw a picture if needed). See the note above about area and perimeter if needed. Parts b and c – look for the high and low points on the graph since the perimeter is shown on the y-axis (the vertical one). Describe the length and width of these rectangles once you identify the perimeters. Part d – use the pairs of length and width that you have found in this problem to decide what the area of any of these rectangles is.

7: See the note about area and perimeter above.

8: See your work in Problem 2.3 for help. Think about what shape the frame should be and how you would make this shape with 72 inches of frame.

9: Part a – Read the graph for the area, and once you know the area and the fact that the length is 2 meters, you can find the width (draw a picture if needed). See the note above about area and perimeter if needed. Parts b and c – look for the high and low points on the graph since the area is shown on the y-axis (the vertical one). Describe the length and width of these rectangles once you identify the areas. Part d – use the pairs of length and width that you have found in this problem to decide what the perimeter of any of these rectangles is.

10-12: See the note above for Problem 2.3, and look back at your work from this problem to help. The work here is the same as the work in that lesson.

13: Look back at your work in Problem 2.1.

14, 18-19, 26-27: See the note about area and perimeter above.

15: You will probably want to try these to decide which is not possible. See the note about area and perimeter above.

16-17: Work backward from the area to find the width. For #16, once you have the width and length, you can find the perimeter (see the note above). Note that part c says to estimate; a benchmark is an easy number with which to work.

21: Work backward from the perimeter (it may help to draw a picture).

23: The entire diagram is the field (not just the parts bordered by dark lines). See the note about area and perimeter above. Find the area for part b; then use it to answer the question. For part d, you will have to take out the area covered by the new structures and then find how much seed is needed. Use this answer to help with part e.

24: Look back at your work in Problem 2.1 to help.

25: For each part, use this number of square units to form as many rectangles as you can (no holes in the center – complete rows and columns). Remember that a square is a special rectangle (it has 4 equal sides). Factors are numbers that you multiply to get to another number.

28: See Problem 2.4 for a reminder of what pentominos are.

29: See the note about area and perimeter above.

Investigation 3

Note: The area is the number of square units needed to cover a figure. The perimeter is the number of units of distance around the outside of a figure. You can use a centimeter ruler.

Problem 3.1

Part A: see the note above. Part B: the smallest rectangle should include at least one side of the triangle. Part C: a rule is like a formula.

Problem 3.2

Parts A and B: position each triangle so that one side aligns with a line on the grid paper and one vertex (corner) on that side is at a point on the grid paper. Labeling the base and height means to write their measurements on the triangles. Use your rule from Problem 3.1 to find the area. Part D: is it sometimes easier to turn the triangle one way or another to find the area? Why?

Problem 3.3

A right triangle has a right angle. An isosceles triangle has (at least) 2 equal sides and (at least) 2 equal angles. A scalene triangle has no equal sides or angles. Use your rule from Problem 3.1 to find the area.

Problem 3.4

Use your rule from Problem 3.1 to find the area of each triangle. (or work backward from the area to find the base or height, using your rule) See the note about perimeter above.

ACE questions – not all in order

1-7, 9, 13-20, 22, 33-34: Use your rule from Problem 3.1 to find the area of each triangle. See the note about perimeter above. The area of a rectangle is found by multiplying the length of the base by the height.

8: This is asking whether you could switch the base and height in 7d and still get the same answer (not to use 7 feet for the base AND 7 feet for the height).

10, 21: You may want to draw a picture to show how to answer these questions. Look back to ACE #8 and Problems 3.2 and 3.3. Also see the note for Problem 3.3 above.

11: Remember that multiplying by 1/2 is the same as dividing by 2. Use your rule from Problem 3.1 to find the area of the triangle.

12: Work backward from the area to find the height, using your rule for area.

23-25: Use your rule from Problem 3.1 to find the area of each triangle. (or work backward from the area to find the base or height, using your rule) See the note about perimeter above.

26-31: Use your rule from Problem 3.1 to find the area of each triangle. See the note about perimeter above. The area of a rectangle is found by multiplying the length of the base by the height. You will probably need to divide some of these shapes into two or three shapes and find the area of each to find the total area for that shape.

32: Use the three sails that are outlined in red. The amount of cloth needed is equivalent to the area; use your rule from Problem 3.1.

35-38: Each fractional part is one region. You will probably want to first divide the shape into as many equal pieces as the lowest common denominator of the fractions; then decide how many of these pieces should be in each region.

39: Pay attention to the measurements that are given, and think logically about how these would look folded up.

40: Think about how to divide the hexagon into triangles and how you would find the area of each triangle. See the note about perimeter above.

Investigation 4

Note: The area is the number of square units needed to cover a figure. The perimeter is the number of units of distance around the outside of a figure. You can use a centimeter ruler to help measure distances. See your rule from Problem 3.1 to find the area of any triangle.

Problem 4.1

See the note above.

Problem 4.2

Part A: position each parallelogram so that one side aligns with a line on the grid paper and one vertex (corner) on that side is at a point on the grid paper. Labeling the base and height means to write their measurements on or around the shapes. A3 is asking you to count the number of square units of area and then determine if this is related in any way to the base and height. Part B: your table should have the name of each figure and its area (count the square units), base and height. Part C: a rule is similar to a formula.

Problem 4.3

The area of a rectangle is found by multiplying the length of the base by the height. You may need to work backward from the area to find the base or the height. See the note above about perimeter. Use your rule from Problem 4.1 to find the area of any parallelogram.

Problem 4.4

Parts A and B: It may help to divide the rectangle into thirds in both directions in order to solve A1 and A2. See the above notes about area and perimeter. Part C: use your rule from Problem 4.1.

ACE questions

1-21: Use the notes above and your rules from Problems 3.1 and 4.1 to find each area and perimeter.

22-27: Use the notes above and your rules from Problems 3.1 and 4.1 to find each area and perimeter. You may have to work backward from area or perimeter (using the rules) in order to find the base or height in some cases.

28: Remember that a parallelogram has two pairs of parallel sides. The parallelograms you find may overlap.

29: The main parallelogram is the green shape, and the small ones are the white shapes. Use your rule from Problem 4.1.

30: It may help to draw a picture of this situation.

31: It may help to draw a picture of this situation. Use your rule from Problem 4.1.

32: You can find a decimal equivalent to 2/3 by dividing 2 by 3. When comparing decimals, remember that you must look at the same place values in each number, and the place values to the right represent smaller amounts.

33: Part a – when thinking about the area, keep in mind the rule that you wrote in Problem 4.1, and try to apply it here. Use the note above for perimeter. Part b – what is true about all rectangles that is also true about parallelograms?

34-35: Congruent means the same shape and same size. See the note above about perimeter and area.

36-37: It may help to draw a picture of this situation.

38: Use the notes above and your rules from Problems 3.1 and 4.1 to find each area and perimeter.

39: You will want to divide these shapes into parts and find the area of each part. You can use a ruler to help with perimeter. Your table should include the number of the shape, the area, and the perimeter.

Investigation 5

Note: The area is the number of square units needed to cover a figure. The perimeter is the number of units of distance around the outside of a figure. You can use a centimeter ruler to help measure distances. See your rule from Problem 3.1 to find the area of any triangle and your rule from Problem 4.1 to find the area of any parallelogram.

Problem 5.1

See the note above. Use logic in your answers.

Problem 5.2

See the explanation on page 72 of the terms involved.

Problem 5.3

See the explanation on page 72 of the terms involved. Use your rule from Problem 5.2 to determine the circumference of each circle. For part D, you may want to think about comparing how the area and circumference change as the pizza gets larger.

Problem 5.4

Part A – you can actually cut the squares into pieces as well to be as exact as possible. How can you predict the area of a circle if you know the size of its radius square? You will have to work backward for part D.

Selected ACE questions

1-2: See the note above about area and perimeter.

5-25: See page 72 for the terms involved, and use your rules from Problems 5.2 and 5.4 for circumference and area.

26: It will help to draw a picture of this situation. You will also need to think about how to use your rules for perimeter and circumference to find enough lengths to find the area of each shape.

27: Find the area of the rectangle, then of the half-circle; then add them. For the circle, find the area of the whole circle, and divide by 2.

28-33: Although the directions tell you to estimate, use a ruler to measure whatever distances you can that will be helpful in making good estimates. You can also use your rules from earlier problems for area and circumference.

34-35: See the note above about area and perimeter.

36: Use your rules from Problems 5.2 and 5.4 when you need to find area or circumference. It may help to draw your own picture of this and add to it to show the various items in the problem.

37: Use your rule from Problem 5.4 when you need to find area of the circle. For b, it may help to divide this flag into smaller parts. Be sure to label the 6-foot lengths on your own sketches of these shapes to help you.

38: See page 72 for the terms involved here. Use your rule from Problem 5.4 to find the area of each circle, and you will have to do some subtraction to find the area of just the blue and yellow bands.

39: Possible measurements to describe might be area, perimeter, circumference, radius, diameter, base, height (others?).

40: Use the rule for circumference that you developed in Problem 5.2. To plot points on the graph, go to the right for the diameter and up for the circumference; then plot a point.

41-46: You may need to look back at the problems you have done and the rules you have used to see how each of these might fit one of those problem situations.

47: Irregular shapes are shapes that do not have congruent sides or angles.

48-49: Draw a picture. You will need to use your rule for circumference.

Bits and Pieces III Download the two-page parent reference sheet and letter

Investigation 1

Problem 1.1

Using benchmarks, as suggested in the directions, means to use easy numbers to estimate – in this case, whole dollars, half-dollars, and perhaps the nearest quarter or dime.

Problem 1.2

See page 8 for reminders about place value in decimals. Remember that in adding and subtracting decimals, we must add and subtract the numbers in the same place value columns because these numbers represent the same size pieces of the whole. Another way to look at it: 2 big pizzas and 1 small pizza are not the same as 3 big pizzas.

Problem 1.3

See the Getting Ready section on page 10 for help with writing decimals as fractions. The denominator of the fraction should match the smallest (right-most) place value in the decimal (see page 8 for place value help). To add or subtract the fractions, it will be helpful to use common denominators. For instance, if you need to add 7/10 and 13/100, you can use 70/100 as an equivalent fraction to 7/10 (then everything is in 100ths). In part C, look for fractions that have decimal equivalents that are close to the decimals given. For instance, if you saw 0.23, you could say that it was close to 0.25, or 1/4, and then use this fraction to complete the problem.

Problem 1.4

For part A, look back at your work in the previous three lessons. For part C, remember that you can work backward with addition to solve a subtraction problem and vice versa. For part D1, remember that you can change a fraction to a decimal by dividing the numerator by the denominator. For D2, look back at Problem 1.3 and the note above.

ACE questions (not all in order)

1-6: First, write down the decimal equivalent for 1/2 so that you can use it as a reference. It may help to think of these numbers in terms of money (dollars). Pay attention to the values in the tenths and hundredths places in each decimal.

7: Use benchmarks (easy numbers) to estimate – in this case, whole dollars, half-dollars, and perhaps the nearest quarter or dime.

8-18, 21, 34-36: Remember that in adding and subtracting decimals, we must add and subtract the numbers in the same place value columns because these numbers represent the same size pieces of the whole. Another way to look at it: 2 big pizzas and 1 small pizza are not the same as 3 big pizzas.

19: Using benchmarks, as suggested in the directions, means to use easy numbers to estimate – in this case, whole numbers, halves (0.5), and perhaps the nearest tenth (0.1).

20: It will help to draw a picture of this situation first. To find half of 1.8, think of half of 18, and then relate that answer to what would be half of 1.8. In adding and subtracting decimals, we must add and subtract the numbers in the same place value columns because these numbers represent the same size pieces of the whole.

22: To compare decimals, you must look at similar place values. For instance, 1.3 is larger than 1.15 because 3 tenths is larger than 1 tenth (it does not work to say that 15 is larger than 3). In adding and subtracting decimals, we must add and subtract the numbers in the same place value columns because these numbers represent the same size pieces of the whole.

23-28: To add or subtract fractions, it is usually easiest to find a common denominator, find equivalent fractions for those in the problem, and then add or subtract, keeping the same denominator. You can change a fraction to a decimal by dividing the numerator by the denominator, and we must add and subtract the numbers in the same place value columns because these numbers represent the same size pieces of the whole.

30, 32, 33: In adding and subtracting decimals, we must add and subtract the numbers in the same place value columns because these numbers represent the same size pieces of the whole. See page 12 for an example of a fact family.

31, 49-54: You can change a fraction to a decimal by dividing the numerator by the denominator, and we must add and subtract the numbers in the same place value columns because these numbers represent the same size pieces of the whole.

37-38, 48, 55: To compare decimals, you must look at similar place values. For instance, 1.3 is larger than 1.15 because 3 tenths is larger than 1 tenth (it does not work to say that 15 is larger than 3).

39: Look at a long length that is given to figure out the missing shorter ones. We must add and subtract the numbers in the same place value columns because these numbers represent the same size pieces of the whole.

40-44: To name the figures, think about how many sides there are, and notice if any sides are equal. We must add and subtract the numbers in the same place value columns because these numbers represent the same size pieces of the whole.

45-46: Along a straight line, the angle measures must add up to 180 degrees. This is also true for the angles in a triangle. We must add and subtract the numbers in the same place value columns because these numbers represent the same size pieces of the whole.

47: Think about how many minutes are in an hour and how many inches are in a foot. Are these numbers the same as the place values in the decimal system? Also, when you get the decimal answers in the problem, think about how close to the nearest whole they are. Given the time and length measurements, do these answers make sense?

56-57: You can use a calculator for this problem; it might help to draw diagrams of the quantities in the problems to get a better sense of what you are trying to find out.

58: See the explanation in the problem.

Investigation 2

Problem 2.1

To give an estimate for each part, use benchmarks (easy numbers) to estimate – in this case, whole numbers, halves (0.5), and perhaps the nearest tenth (0.1). See the examples on pages 21-22 to change the decimals to fractions. For part C, see the problems in part A; a factor is one of the numbers being multiplied, and the product is the number we get when the two factors are multiplied.

Problem 2.2

See the Getting Ready section on page 10 for help with writing decimals as fractions. The denominator of the fraction should match the smallest (right-most) place value in the decimal (see page 8 for place value help). To multiply fractions, you can multiply the numerators and then the denominators to get the numerator and denominator in the product. For part B, use your answers in B1 to help find answers for B2-B4. Hint: the digits in the numbers should not change. For part C, use the patterns you found in part B (think about whole number multiplication to help).

Problem 2.3

To give an estimate for each part, use benchmarks (easy numbers) to estimate – in this case, whole numbers and halves (0.5).

Problem 2.4

Do not use a calculator on these problems because your job is to look for patterns and use what you know to find answers you do not know. In part B, see the examples on pages 21-22 to change the decimals to fractions. To multiply fractions, you can multiply the numerators and then the denominators to get the numerator and denominator in the product. In part C, an algorithm is a mathematical procedure. Use what you found out in parts A-C to complete part D.

Selected ACE questions

1-6: To give an estimate for each part, use benchmarks (easy numbers) to estimate – in this case, whole numbers and halves (0.5).

7-12: To give an estimate for each part, use benchmarks (easy numbers) to estimate – in this case, whole numbers and halves (0.5). See the examples on pages 21-22 to change the decimals to fractions. To multiply fractions, you can multiply the numerators and then the denominators to get the numerator and denominator in the product.

13-20: Use what you learned in Problem 2.4 (your algorithm for decimal multiplication) to solve each of these problems. To compare decimals, you must look at similar place values. For instance, 1.3 is larger than 1.15 because 3 tenths is larger than 1 tenth (it does not work to say that 15 is larger than 3).

21: You are finding 0.75 OF 0.4 in part a; you can change the decimals to fractions and multiply (see note for #7-12) or use your work from Problem 2.4. In part b, use your answer from part a to find this portion OF 8 acres.

24-26: This is asking you to estimate the product (hint: use whole numbers and nearest halves) and then explain (based on this estimate) how you know whether the actual product is greater/less than each of the factors.

27-32: Use the patterns and algorithm you discovered in Problem 2.4. Remember that the digits will not change in most of these problems, but the location of the decimal point will, which changes the value of the number itself.

34-39: To multiply fractions, you can multiply the numerators and then the denominators to get the numerator and denominator in the product. See the examples on pages 21-22 to change the decimals to fractions.

40: To compare decimals, you must look at similar place values. For instance, 1.3 is larger than 1.15 because 3 tenths is larger than 1 tenth (it does not work to say that 15 is larger than 3). To find the area of each carpet, remember to multiply the length by the width. To find the total cost, use the total area of the carpet and the price per square meter.

41-44: The area of a rectangle and a parallelogram is found by multiplying the length of the base by the height. The area of a triangle is found by multiplying the base times the height and dividing by 2. The area of a circle is found by multiplying the radius by itself and then by pi (about 3.14).

45: If these are the areas, work backward to find a length and width that multiply to give each area.

46-48: You can change a fraction to a decimal by dividing the numerator by the denominator.

49: Try multiplying both the numerators and denominators by 10. To multiply fractions, you can multiply the numerators and then the denominators to get the numerator and denominator in the product.

50: The bottom line is the answer to the original problem; this is not two separate problems.

52: Think about how you multiply fractions, say 7/10 and 17/100, and then think about these as decimals and what the product would look like as a decimal.

53: The area of a rectangle is found by multiplying the length of the base by the height.

54: Where could the decimal point go in each factor to give a product that was close to 25?

55: It might help to draw a number line (in tens or hundreds) to solve this problem.

Investigation 3

Problem 3.1

For A1 and A2, if you use a diagram, you could use rectangles divided into ten equal pieces to show tenths and then find a way to show groups of 0.4 in those rectangles. If you explain, you will want to talk about the whole number and tenths in 3.2 and how you know how many groups of 0.4 are in this number. For A3, a number sentence is just an equation (including numbers and an equal sign). For A4, think about how many sandwiches could actually be made and why. For part B, consider a whole number that is close in size to the decimal in the problem, and use this to make an estimate for each answer.

Problem 3.2

For part A, you are asked to estimate first (using whole numbers that are close to the decimals) and then to find exact answers by using fraction notation for the decimals. For example, 4.2 can be written as 4 2/10, or 42/10. When we have common denominators, one easy way to divide fractions (as you learned in Bits and Pieces II) is to divide the first numerator by the second one. This tells you how many groups of the second fraction are in the first one. (e.g., 8/3 ∏ 2/3 = 4 because there are 4 sets of 2/3 in 8/3). If we divided the numerators and got a decimal (non-whole number) answer, this is OK, especially in this case since we are dealing with decimals anyway! For part B, see the explanation here for using fraction notation to divide. The whole number division problem is the problem you do when you divide the numerators only (when you have common denominators). A quotient is the answer to a division problem. For parts C and D, refer to what you did in part B; the process is quite similar.

Problem 3.3

You found in Problem 3.2 that when dividing BY a decimal, you can move the decimal point in this decimal to the right until it is a whole number; then move the decimal point in the dividend (see page 38) to the right the same number of places. This is because you are multiplying each number by either 10, 100, 1000, etc. as you move the decimal point, and when we do that, we can divide the new numbers and get the same answer. (This is like saying that 6 ∏ 1 is equal to 600 ∏ 100; there are 6 groups of 1 in 6, and there are also 6 groups of 100 in 600.) If we are dividing a decimal BY a whole number, we can leave the numbers as is and divide, either using long division or a method such as this (for A2a):

27.5 = 27 5/10 or 275/10. 275/10 ∏ 55 = 5/10 = 1/2 (or 0.5)

For part B, see page 38 for the vocabulary words. In writing the story problem, think about this to help you: 175 ∏ 5 would mean How many groups of 5 are in 175? Think about a real-life situation that would involve the decimals in the problem, and write a problem that involves this type of question. Look back at Problem 3.1 if you need more examples. For part C, a fact family involves four number sentences that show how the numbers are related by multiplication and division. For instance, a fact family could be 8 x 3 = 24, 3 x 8 = 24, 24 ∏ 3 = 8, 24 ∏ 8 = 3. For C2, try to use a fact family to turn these into other problems that are easier to solve. For part D1, think about what happens when many people measure a fairly small object. The mean is found by adding all of the numbers and dividing by the total. D3 is asking you to find the difference between the first week and the second.

Problem 3.4

For A1-5, see page 41 for help. For A6, an example would be whether you can write 2/5 equal to a fraction with a denominator of 10, 100, 1000, etc. Part B2 is referring to B1. For B3, refer to page 8 for decimal place values, and remember that you can write 413/1000, for example, as 0.413. For part C, consider how you say the name of the decimal, like 35 hundredths, and use this wording to write the fraction. (again, see page 8 for help) For parts D and E, see the notes here and page 8 if needed.

ACE questions

1-4: This is asking you to choose addition, subtraction, multiplication, or division, not to actually solve each problem.

5: You may want to use rectangles divided into tenths as the wholes. The quotient is the answer to the problem.

6: Consider whether you are dividing a larger number by a smaller (like 10 ∏ 2) or the other way around (like 6 ∏ 12). In which case will you get a number less than 1 (or greater than 1)?

7-12: To write these as fractions, consider how you say the name of the decimal, like 4 and 5 tenths, and use this wording to write the fraction. (again, see page 8 for help with place value) See the explanation for Problem 3.3 above for further help.

13: See the first part of the explanation for Problem 3.3 above for help (or your work in Problem 3.3). The quotient is the answer to the problem; this is asking you to describe what the quotient tells you about groups of 0.5 in 22.4.

14-21: See the first part of the explanation for Problem 3.3 above for help (or your work in Problem 3.3). The quotient is the answer to the problem.

22: This is asking you to write two other problems with the same digits as 48 and 12 but with decimals instead of whole numbers (e.g., 4.8 divided by _____). The problems you write should have the same answer as 48 ∏ 12. See the first part of the explanation for Problem 3.3 above for help (or your work in Problem 3.3).

23-24: A fact family involves four number sentences that show how the numbers/variables are related by multiplication and division. For instance, a fact family could be 8 x 3 = 24, 3 x 8 = 24, 24 ∏ 3 = 8, 24 ∏ 8 = 3. It might help to write the fact family first and then decide how best to find the value of N.

25-26: As you learned in an earlier unit, you can find the decimal equivalent by dividing the numerator by the denominator. Once you have found the decimal equivalents in each problem, describe what you notice about the decimals and the three different fractions – why do you think this happens?

27: As you learned in an earlier unit, you can find the decimal equivalent by dividing the numerator by the denominator. For part d, look at the decimal part of each number, and think about what fraction that decimal equals (see your table). Then, write a mixed number that is equal to that decimal.

28: For each of these, find the distance between the two marked points; then, think about how to divide that distance among four intervals (the ones that are also marked).

29: Consider whether you are dividing a larger number by a smaller (like 10 ∏ 2) or the other way around (like 6 ∏ 12). In which case will you get a number less than 1 (or greater than 1)?

30: Consecutive means next to each other. An average is found by adding the values and dividing the total by the number of values.

31: A product is the value that we get when we multiply two numbers. Look back to your work in Investigation 2 for reminders about how to multiply decimals.

32: For part a, keep in mind that the intervals are spaced equally. The mean (the average) is found by adding the values and dividing the total by the number of values. Part c is asking you to decide (having done part b) whether this is possible and if so, why, and if not, why not.

33: The formula for circumference is pi times the diameter. The formula for area is pi times the radius squared. You will need to work backward from the area using the formula to find the two answers needed.

34: Think about how far the car went and how the gallons used could be split among this distance.

35-36: As you learned in an earlier unit, you can find the decimal equivalent by dividing the numerator by the denominator.

37-40: Look back at the numbers in #35-36.

41-43: You will need to find the area of the rectangle first. Knowing that and knowing that the other figures have the same area as it does, think about how you would make the figure in #41 from the first rectangle – how long would n be? For #42, the area of a parallelogram is found by multiplying the length of the base by the height – you will need to work backwards. For #43, the area of a triangle is found by multiplying the length of the base by the height and dividing by 2 – you will need to work backwards.

Investigation 4

Problem 4.1

Use the example on the bottom of page 50 to help you. Also, keep in mind that 1% of a dollar is 1 cent, so (for instance) 5% of every dollar would be 5 cents. Also remember that the tax is only what is added on; the total cost includes both the original cost and the tax. For part D, you are given what 6% and 5% are in each case, so use these to find out what 1% of the total cost would be in each case. Then, how can you find 100% knowing what 1% is?

Problem 4.2

Remember that the tax is only what is added on; the total cost (without a tip) includes both the original cost and the tax. To solve these problems, use the example on the bottom of page 50 to help you. Also, keep in mind that 1% of a dollar is 1 cent, so (for instance) 5% of every dollar would be 5 cents. The total cost for each meal with a tip includes the cost of the food, the tax, and the tip. For part D, you will need to work backward. For example, in D2, if $1.00 is the 20% tip after rounding up to the nearest nickel or dime, what is $1.00 20% of? To find out: if $1.00 is 20% of the cost, how much is 1% of the cost? Knowing this, find 100%, or the actual cost. Then, remember that Omar rounded up to get the $1.00 tip, so estimate a total cost that is slightly less than the answer you got.

Problem 4.3

Remember that a discount is only the amount that is taken away from the original price. The sale price is the original price minus the discount. Then tax is added onto this new amount. For part B, you will need to find the cost of 6 CD singles at $3.45 each, then find 20% of this cost and take it away from the total, then find 6.5% of this new cost (for the tax) and add it on to find the total cost. Then, find the cost of 1 CD and 1 single; find 6.5% of this, and add it on. Then you can compare the two costs and answer the question. To solve all of these problems, use the example on the bottom of page 50 to help you. Also, keep in mind that 1% of a dollar is 1 cent, so (for instance) 5% of every dollar would be 5 cents. For part D, it may help you to write a fraction: discount/original cost; then find the percent to which this fraction is equivalent (need an equal fraction with a denominator of 100).

Selected ACE questions – not all in order

**For any/all of these problems, remember:

• Use the example on the bottom of page 50 to help you. Also, keep in mind that 1% of a dollar is 1 cent, so (for instance) 5% of every dollar would be 5 cents.
• To find any percent of a number, one strategy is to find 1% by dividing the number by 100; then multiply this by the size of the percent.
• Remember that tax (and/or a tip) is only what is added on to the original cost; the total cost includes both the original cost and the tax (and/or tip). A tip is often calculated after the tax is added on.
• Remember that a discount is only the amount that is taken away from the original price. The sale price is the original price minus the discount. Then tax is added onto this new amount.

2, 5-6, 9-10, 12, 14, 20, 27-29: See note above.

3: It may help you to write a fraction: difference in cost/higher cost. What percent is this equal to?

4: See note above. For part c, knowing that 24 students represent 12%, how many represent just 1%? You can multiply this answer by 100 to find the total number of sixth graders.

7: See note above. For part c, think about how you can break $325 into amounts that are shown on the card.

8: See note above. For part b, remember what the discount was – what is left?

11: It may help to write a fraction: (difference in price from old to new)/original price. What fraction and percent is this equal to?

13: What fraction out of 100 is equal to 3/10?

15: The first equation tells us that there are twelve 0.2πs in 2.4. Could there be more 0.5πs in 2.4 than this? Or would there be fewer? Why? (Hint: what fraction is 0.5 equal to?)

16: Divide 0.25 by 0.8. One way to do this is to use fractions: 25/100 divided by 8/10 (or 80/100), which is the same as 25 divided by 80.

17: Write 15% as a fraction out of 100. What decimal is this equal to?

18: 25% is equal to 1/4. Which of these fractions is equal to 1/4?

19: What is the total of these percents of the whole pizza?

21-22: Note how many of the smaller pieces in each problem are equal to one of the larger pieces. Use this to find out how many pieces there are altogether in each strip. Then you can write the fractions needed (think of the strips like number lines or rulers, where the mark represents the amount of the strip up to that point).

23-26: Remember that in two equal fractions, you can multiply or divide the numerator and denominator of one by the same number to get the other.

30: The February rate is 10% higher than the January rate. What is 10% higher than 4?

31-32: Use the numerator of each given fraction to divide the section between it and 0 into this many pieces. Then, count up from 0 to label each mark, and be sure to label the fraction that is equal to 1. You may have to add on to what is already there in order to do this.

33: Think about the difference between 100% and 120%. How many sets of this are in 100%? In 120%? Use this to divide the strip into sections and mark where 100% would be.

34: See note above, but keep in mind that you must find the first sale price first, and then take the new discount off of this sale price.

35-38: Remember that in two equal fractions, you can multiply or divide the numerator and denominator of one by the same number to get the other. This should work between any pair of fractions in each of these tasks.

40: What fraction is 25% equal to? Can you find this fraction of 3 cups? What would the new total be?

Investigation 5

Problem 5.1

Use the example on page 62 to help you. Remember that when determining the percent discount, you can write a fraction: discount/original price (NOT new price/original price); then find the fraction out of 100 to which this fraction is equal. Do not confuse the discount itself (amount of money) with the percent of the discount! For part C, if $24.75 is 25% of the original cost, what can you multiply it by to find 100% of the original cost? For part D, the same idea works – what do you have to multiply 15% by to get 100%?

Problem 5.2

For part A, note that both the tip and tax, in this case, are calculated based on the original cost of the food. So, what percent will be added to the cost of the food? Knowing this, think: $60 is ___% of the cost of the food. (The blank would be 100% plus the percent that is added on.) If $60 is this percent, how can you find just 100% (which would be the cost of the food)? One way would be to divide $60 by the percent in the blank to find 1%, then multiply by 100 to find 100%. Use a similar strategy to solve part B. For part C, find 20% off, then 10% off of this new amount; then find 30% off the original. Are these the same? (If you need help with how to do this, see the notes at the start of the ACE questions for Inv. 4 above.)

Problem 5.3

Remember that there are 360 degrees in a circle. To find a given percent of this, one way is to divide 360 by 100 to find the degrees equal to 1%; then multiply by the size of the percent. You will need a protractor to actually create the angles in the graphs (and possibly a compass or other tool to draw the circles).

Selected ACE questions, not all in order

**For any/all of these problems, remember:

• Use the examples on the bottom of page 50 or 62 to help you.
• Keep in mind that 1% of a dollar is 1 cent, so (for instance) 5% of every dollar would be 5 cents.
• To find any percent of a number, one strategy is to find 1% by dividing the number by 100; then multiply this by the size of the percent.
• To find a missing percent, write a fraction with the part/whole; then find the fraction out of 100 to which this fraction is equal.
• Remember that tax (and/or a tip) is only what is added on to the original cost; the total cost includes both the original cost and the tax (and/or tip). A tip is often calculated before the tax is added on.
• Remember that a discount is only the amount that is taken away from the original price. The sale price is the original price minus the discount. Then tax is added onto this new amount.

1-10, 16-20: See notes above.

11: For part a, imagine that 30 more events have been held and each person attended all of them. This would add 30 to the number of events attended and to the total events held during each personπs membership in the group. To answer this question, use these new numbers to find the percent of events each person attended. (See notes above.) For part b, use your answers from part a to find these percents of 120. (See notes above.)

12: $91 is ____% of the total that the gift can cost. (This blank is 100% plus the percent for the tax.) Knowing this, you can find 1% of the cost of the gift by dividing $91 by the size of the percent in the blank; then multiply by 100 to find 100% (or the highest possible cost of the gift).

13: Use a strategy similar to the one described for #12.

14: For part a, there are two problems to solve. First, $60 is ___% of the total that she can spend. (This blank is 100% plus the percent for the tax.) Knowing this, you can find 1% of the cost by dividing $60 by the size of the percent in the blank; then multiply by 100 to find 100% (or the highest possible cost). However, with her coupon, she can actually spend more. Since the coupon is for a 20% discount, how much of the cost will she pay? (100%-20%) So your answer is ___% of the actual total that she can spend. Knowing this, you can find 1% of the cost by dividing the first answer by the size of the percent in the blank; then multiply by 100 to find 100% (or the highest possible cost). For part b, the $60 now is (100%-20%) of the total price of the items she can buy. So, $60 is ___% of the total. Knowing this, you can find 1% of the cost by dividing $60 by the size of the percent in the blank; then multiply by 100 to find 100% (or the highest possible cost).

15, 21-22: Remember that there are 360 degrees in a circle. To find a given percent of this, one way is to divide 360 by 100 to find the degrees equal to 1%; then multiply by the size of the percent. You will need a protractor to actually create the angles in the graphs (and possibly a compass or other tool to draw the circles). For 15c, remember that the area of a whole circle is found by multiplying the radius by itself and then by pi (about 3.14). Then, use the percents in the table to find the percent of the area that is ≥Morning Only≤ in each graph. (See notes above.) For 21, you will first need to find a fraction or percent to represent the dentists recommending sugarless gum. For 22a, you can estimate the percents shown on the graph. For 22b, which graph do you think makes it most clear that most people moved for housing-related reasons?

23-28: For the equations, remember that with two equal fractions, you can multiply or divide the numerator and denominator of one by the same number to get the other. This should work between any pair of fractions in each of the equations. For #26 and 28, it might help to solve #27 first; then use this answer to think about how to answer 26 and 28.

29-30: One strategy to solve these problems is to divide the given number by the size of the percent to find 1%; then multiply this answer by 100 to find the missing number (which would be 100%).

32: For part a, you have to get back to 100%. What percent of 200% is 100%? For part b, you have to get back to 100%. What percent of 50% is 100%? (Be careful!) For part c, what would you have to multiply 1 1/2 (or 3/2) by to get 1? Use a similar strategy for part d.

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