Unit 3
A student asks if (0/6) is in its simplest form. What is the correct response?
(0/6) is not in its simplest form. A fraction (a/b) is in simplest form if, and only if, b>0 and GCD(a,b)=1. Notice that GCD(0,6)=6. The simplest form of (0/6) is (0/1).
Amy says that dividing a number by (1/2) is the same as taking half of a number. How do you respond
(1/2) of a number x is equivalent to (x/2), but dividing a number x by (1/2) is equivalent to 2x.
What is a key idea in connecting whole number place value and decimal fraction place value?
10 to 1 multiplicative relationship between values of any two adjacent positions
Which of the following is important to do before students learn the formal algorithms?
Address misconceptions
Which of the following is a good explanation for how to add fractions?
Add equal-sized parts—finding a common denominator can help to solve the problem
Why is it possible to have an increase of 157% in price but not a 157% decrease in price?
An increase of 157% means that the new price is equal to 2.57 times the original price, which is possible, while a 157% decrease in price would mean that the new price was lower than the original price by more than the original price, which is impossible because prices cannot be negative.
When using base-ten materials in developing decimal concepts what is an important idea to be realized?
Any piece could be effectively chosen as the ones piece
A student says, "My answer must be wrong—my answer got bigger." Which of the following responses will best help the student understand why the answer got bigger?
Ask them to explain the meaning of 8 ÷ 2, using a cutting ribbon as a context, and then ask them to re-explain to you using 8 ÷ (1/2), still using cutting ribbon as a context.
Which of the following best describes how to teach multiplication involving a whole number and a fraction?
A "fraction times a whole number" and a "whole number times a fraction" are conceptually different, so they should be taught separately.
Two classes were given a same test. In class A, 20 out of 25 students passed, and in class B, 32 of 40 passed. One of students in class B claimed that the classes did equally well. How could you explain the student's reasoning?
Because the fraction of students passing in each class was (4/5), the argument could be made that they did equally well on the test.
The way we write fractions with a top and bottom number is a convention. What method focuses on making sense of the parts rather than the symbols?
Begin by using words (i.e., one-fourth)
All the following are recommendations for effective fraction computation instruction except:
Carefully introduce procedures
Which of the following strategies would you like students to use when determining which of these fractions is greater (7/8) or (5/6)?
Compare how far from 1
All the following are representative of how algebraic thinking is integrated across the curriculum except:
Composing and decomposing shapes
All of the methods below would work to support students' knowledge about what is happening when multiplying a fraction by a whole number except:
Compute with a calculator
Locating a fraction on a number line can be challenging but is very important. Which is a common error that students make in working with the number line?
Count the tick marks that appear without noticing any missing ones
The teachers have identified three manipulatives to use when teaching fractional concepts. Each teacher intended to select one manipulative to show each fraction model. Which teacher succeeded in selecting manipulatives for each type?
Denise selected tangrams, color tiles, and number lines.
What is the primary reason to not focus on specific algorithms for comparing two fractions?
Developing number sense about relative size of fractions is less likely
The goal is to rename a fractional amount. What is the concept that requires the use of many contexts and models?
Equivalent fractions
What is a problem with learning only designated (standard) algorithms for fraction operations?
Follow a procedure in a short term, but not retain
Fred says that to multiply 30.2•2.7, he just multiplies 302•27=8154, and then because 30•3=90, he records the answer as 81.54. What is your response?
Fred's method works. For any decimal multiplication problem, first multiply the numbers represented by the given digits without trying to place the decimal point in the resulting product. Then rounding the decimal numbers to the nearest integer and multiplying them determines the approximate integer non-decimal value that the product should have, and indicates the position of the decimal point in the product found using the numbers without initially placing the decimal point.
Which of the following analyzes how the pattern is changing with each new element in the pattern?
Geometric growing patterns
All of the following statements are research-based recommendations for teaching and learning about fractions except one. Which one?
Give greater emphasis to specific algorithms for finding common denominators
Research findings support all of the following fraction teaching ideas but one. Which of the following is the unsupported method?
Give students area models that are already partitioned and ask them to record the fractional amount shaded.
Noah says that dividing a number by 6 is the same as multiplying it by (1/6). He wants to know if he is right, and if so, why. How do you respond?
He is correct. To divide by a number, multiply by the reciprocal of the number. In this case, to divide by 6, multiply by (1/6).
In the real-world decimal fractions are rarely those with exact equivalents to common fractions. Students need to wrestle with the magnitude of decimal fractions. Identify the activity below that addresses magnitude.
Identify what 7.396 is close to 7, 7(1/2), or 8
A student says that (3/8)>(2/3) because 3>2 and 8>3. How would one help this student?
If the student writes each fraction over a common denominator, it is easier to compare them. Thus, (3/8)=(9/24) and (2/3)=(16/24). Then (2/3)>(3/8), because (16/24)>(9/24).
A student says that (3/8)>(2/3) because 3>2 and 8>3. How would one help this student?
If the student writes each fraction over a common denominator, it is easier to compare them. Thus, (3/8)=(9/24) and (2/3)=(16/24). Then (2/3)>(3/8), because (16/24)>(9/24).
Jill claims that for positive fractions, ab+ac=ab+c because the fractions have a common numerator. How do you respond?
Jill is confusing this problem with adding fractions with like denominators. She should use (a/b)+(a/c)=(ac + ab/bc).
Jillian says she learned that 65÷7=9 R2, but she thinks that writing 65÷7=9 (2/7) is much better. How do you respond?
Jillian is correct if a fractional answer is needed.
Identify which statement below would not be considered a common or limited conception related to fractional parts?
Knowing that answers can be left as fractions rather than writing them as mixed numbers
Explain why subtraction of terminating decimals can be accomplished by lining up the decimal points, subtracting as if the numbers were whole numbers, and then placing the decimal point in the difference.
Lining up the decimal points acts like using place value.
Why are estimation skills important in dividing decimals?
Many of the estimation techniques that work for whole numbers also work for decimals. Estimation is important in order to determine whether an answer obtained by long division is reasonable.
David worked a 32-hour week at $10.25/hr. Mentally compute his salary for the week and explain how you did it
Multiply 10 and 32. Then add to that the product of 0.25 and 32.
Multiplication of decimals is poorly understood for many reasons. Identify the misunderstanding that relates to whole number multiplication.
Multiplying makes the product larger
Kara spent (1/2) of her allowance on Saturday and (1/3) of what she had left on Sunday. Can this situation be modeled as (1/2)−(1/3)? Explain why or why not.
No, because she spent one-third of her remaining half on Sunday. This is modeled by (1/3)•(1/2). So she had (1/2)−[(1/3)•(1/2)] remaining after Sunday.
Bente says to do the problem 8 (1/7) ÷4 (3/7) you just find 8÷4=2 and (1/7)÷(3/7)=(1/3) to get 2(1/3). How do you respond?
No. Bente is thinking (a+b/c+d)=(a/c)+(b/d), which is incorrect. Convert the mixed numbers into improper fractions. To divide two fractions, multiply the dividend by the reciprocal of the divisor. The correct answer is 1(26/31).
Suppose a large pizza is divided into 3 equal size pieces and a small pizza is divided into 4 equal size pieces and you get 1 piece from each pizza. Does (1/3)+(1/4) represent the amount you received? Explain why or why not.
No. The pizzas are not the same size, therefore, (1/3)+(1/4) or (7/12) does not accurately represent how much pizza you received.
When a positive decimal less than 1 is multiplied by a decimal greater than 1, will the result always be greater than 1? Explain.
No. The product can be any positive number. Consider the product 1.5•0.5=0.75
What form of algebraic reasoning is the heart of what it means to do mathematics?
Noticing generalizations and attempting to prove them true
Research recommends that teachers use one of the following to support students' understanding that fractions are numbers and they expand the number system beyond whole numbers.
Number lines
There are multiple contexts that can guide students understanding of fractions. Which of the following would involve shading a region or a portion of a group of people?
Part-whole
Which of the following best describes the relationship between iterating and partitioning?
Partitioning is finding the parts of a whole, whereas iterating is counting the fractional parts.
Arithmetic and algebra are closely connected. Identify the reason below that best describes why?
Place value and operations are generalized rules; a focus on algebraic thinking can help students make connections across problems and strengthen understanding.
Which instructional method does not support purposeful teaching of mathematical properties?
Providing opportunities for students to name and match properties to examples
Estimation is particularly important for students who have learned the rules of computation but cannot decide about?
Reasonable answers
Which of the following can be presented to students that will open opportunities for them to generalize?
Set of related problems
Dani says that if we have (3/4)•(2/5), we could just do (3/5)•(2/4)=(3/5)•(1/2)=(3/10). Is she correct? Explain why.
She is correct since (3/4)•(2/5)=(3•2/4•5)=(3•2/5•4)=(3/5)•(1/2)=(3•1/5•2)=(3/10).
A student claims that division always makes things smaller so 11÷(1/4) cannot be 44 because 44 is greater than the number 11 she started with. How do you respond?
She is incorrect. The expression 11÷(1/4) is equivalent to finding how many fourths there are in 11. In this case there are 44 fourths in 11.
Which model below would not provide a clear illustration of equivalent fractions?
Show an algorithm of multiplying the numerator and denominator by the same number
A student claims that 0.407=0.47. How do you respond?
Since 0.407=(407/1000), 0.47=(47/100)=(470/1000), and 407≠470, the decimals are not equal.
A student claims that 0.75 is greater than 0.9 because 75 is greater than 9. How do you respond?
Since 0.75=(75/100), 0.9=(9/10)=(90/100), and 90>75, 0.9 is greater than 0.75.
Identify the example below that represents a relational-structural approach for the problem 8 + 4 = n + 5
Since 4 is one more than 5 on the other side, that means n is one less than 8
Providing students with many contexts and visuals is essential to their building understanding of equivalence. More examples of linear situations are needed to make comparisons more visible. Which of the following would not be best to model on a number line?
Slices of pizza eaten
What activity described below would guide students understanding of halves, thirds, fourths, and eighths as decimal fractions?
Students shade in a 10 x 10 grid to illustrate a familiar fraction
Which of the following options would be misleading for student understanding of fractions?
Tell students that fractions are different from whole numbers, so the procedures are also different.
Which of the following is shown through research to be a common error or misconception when students are comparing or ordering decimals?
The decimal that is the shortest is the largest.
Two equal amounts of money were invested in two different stocks. The value of the first stock increased by 17% the first year and then decreased by 17% the second year. The second stock decreased by 17% the first year and increased by 17% the second year. Was one investment better than the other? Explain your reasoning
The first and second investments are equally good because they have the same value after two years.
A student writes (15/53)<(1/3) because 3•15<53•1. Another student writes (15/53)=(1/3). Where is the fallacy?
The first student's approach is correct. What the second student has done is to treat the problem as if it had been (1/5) (5/3) =13, when in reality, the problem is (15/53). One cancels factors not digits.
A student writes (15/53)<(1/3) because 3•15<53•1. Another student writes (15/53) *5's marked out*=(1/3). Where is the fallacy?
The first student's approach is correct. What the second student has done is to treat the problem as if it had been (15/53)=(1/3), when in reality, the problem is (15/53). One cancels factors, not digits.
Explain why 50 cents is one-half of a dollar, yet 30 minutes is one-half of an hour. Why should these one-halfs not be equal?
The fractions are each (1/2) and are equivalent, but in context with the different bases, they represent different quantities.
Explain why 5 cents is one-twentieth of a dollar, yet 3 minutes is one-twentieth of an hour. Why should these one-twentieths not be equal?
The fractions are each (1/20) and are equivalent, but in context with the different bases, they represent different quantities.
Explain why you think a sign on a copying machine reading ".05¢ a copy" is put up by mistake.
The price .05¢ seems very inexpensive. The intended price is probably 5 cents a copy.
A student writes a(bc)=(ab)(ac). How do you respond?
The student is applying the distributive property of multiplication over addition to multiplication. The student is incorrect in his/her extension of the distributive property.
A student claims that the equation 3x=5x has no solution because 3≠5. How do you respond?
The student is incorrect. The equation has one solution, x=0.
A student argued that a pizza cut into 12 pieces was more than a pizza cut into 6 pieces. How would you respond?
The student was probably thinking that more pieces meant more pizza. The amount of pizza did not change and only the number of pieces changed.
Make a statement about a person or an environment and use fractions in each. Explain why your statements are true. Choose the correct answer below.
There are three birds on a wire, two of them are blackbirds; hence (2/3) of the birds are blackbirds. This is true because the fraction (2/3) represents the ratio of blackbirds to the total number of birds.
All the following are reasons that data and algebra are good topics to integrate except:
There isn't enough time in the year to address everything, so it is more efficient to teach these two together.
The benefits of using a rectangular area to represent multiplication of fractions include all the following except which?
They are easy for students to draw.
What statement below best describes functions?
They describe a relationship between two variables and may be linear or not.
Students need to be acquainted with various visual models to help them think flexibly of quantities in terms of tenths and hundredths. Which example below would help students understand the decimal fraction (65/100) in terms of place value?
This decimal fraction could be thought of as 6 tenths and 5 hundredths
When adding fractions with like denominators it is important for students to focus which key idea?
Units are the same
Teaching fractions involves using strategies that may not have been part of a teacher's learning experience. What is a key recommendation to teachers from this chapter?
Use multiple representations, approaches, explanations, and justifications
It is important for students to experience and explore percent relationships in realistic contexts. Three of the statements below are guidelines to follow for presenting percents. Identify the one that does not support best practices.
Use the following sentences in their solutions "____ is ____ percent of _____"
Algebraic thinking includes several characteristics. Which of the following statements is not a part of algebraic thinking?
Using manipulatives to reason about situations
Which of the following is not representative of the current thinking about arithmetic and algebra in the elementary classroom?
Variables are not appropriate for elementary-age students; a box is a more concrete representation.
Mathematical modeling is appropriate for investigating real challenges. Which of these examples requires some mathematical modeling?
What would be the better deal, buy-one-get-one half off, 25% off, buy-two-get-one-free?
Is a product of two positive decimals less than 1 always less than each of the decimals? Justify your answer.
Yes. Suppose the decimals are x and y. Since terminating decimals are rational numbers, inequalities for decimals have the same properties as inequalities among rationals. If 0<x<1, multiply both sides of the inequality by y to get 0<xy<y. Similarly, 0<xy<x.
Which of the following result in equal quotients? a. 20÷440 b. 2÷4.4 c. 0.2÷4.4 d. 200÷4400
a, c, d
Mixed numbers:
can be changed into fractions or "improper" fractions and added.
A good teaching option for developing a full understanding of computation with decimals is to focus on:
concrete models, drawings, place value knowledge, and estimation.
An important concept in working with repeating patterns is for the student to identify the:
core of the pattern
The role of the decimal point in a number is to:
designate the units position.
Conceptualizing the symbol for equal as a balance can support students' understanding of:
equality or inequality.
A common misconception with set models is:
focusing on the size of the subset rather than the number of equal sets.
Writing fractions in the simplest terms means to write it so:
fraction numerator and denominator have no common whole number factors.
Children as early as first grade can explore functional thinking by using:
input-output activities.
All of the materials below can be used to represent an area model of decimal fractions except:
meter stick.
Mathematical models are useful in both real life and mathematics because:
models such as equations, graphs, and tables can be used to analyze empirical situations, to understand them better, and to make predictions.
Determine if the following pairs are equal by changing both to the same denominator. (17/24) and (26/27)
not equal
Using contextual problems with fraction division works in providing students with an image of what is being:
shared or partitioned.
A critical aspect of understanding divisions of fractions is:
the divisor is the unit.
A fraction by itself does not describe the size of the whole. A fraction tells us only:
the relationship between part and whole.
Students need experiences with variables that vary, and pairs of variables that covary, early in the elementary curriculum. It is important to emphasize the:
variable stands for the number of.
Establishing a culture where students are making their own conjectures develops their skills at justification. Which of the following would foster this culture?
Always, sometimes or never mathematical statements
Physical models provide the main link between fractions, decimals, and percents. Identify the one model that is suitable for all three because they all represent the same idea.
Base-ten models
Base-ten models, the rational number wheel with 100 markings around the edge, and a 10-by-10 grid are all models for linking which three concepts together?
Fractions, decimals, and percents
If 61% of the girls in a class wanted to have a prom and 70% of the boys wanted a prom, is it possible that only 52% of the students in the class wanted a prom? Explain your answer
No, it is not possible. Since more than 52% of boys and more than 52% of girls wanted a prom, more than 52% of all students must have wanted to have a prom.
Explain whether 1 day can be expressed as a terminating decimal part of a 365-day year.
No, the prime factorization of 365 contains primes other than 2 or 5.
According to your textbook, which of the following trios of real-world situations represent common uses for estimating percentages?
Tips, taxes, and discounts
One recipe calls for 1 (1/2) cups of milk and a second recipe call for 1 (3/4) cups of milk. If you only have 4 cups of milk, can you make both recipes? Why?
Yes, 1 (1/2) cups plus 1 (3/4) cup is less than 4 cups of milk