Deck 1: Reasoning About Quantities

Question 1

Give an example of a quantity. What is a possible value for your example?

Answer 1

A possible example is the length of this room. A possible value is 20 feet.

Question 2

Name five quantities that you have dealt with so far today.

Answer 2

Possible answers include: distance from home to class; time spent traveling from home to class; amount of gasoline purchased; amount of milk drunk at breakfast; amount of money spent on a Starbucks coffee; etc.

Question 3

Name three quantities that relate to you, and tell how they are measured.

Answer 3

Possible answers include: weight (in pounds or kilograms); height (in inches); arm span (inches); shoe size (standard sizes for shoes); waist size (in inches); etc.

Question 4

Would student motivation be difficult or easy to quantify? Explain. Tell how you might go about quantifying student motivation in this class.

Question 5

How can the term "big" be quantified?

Question 6

Give two quantities that one could have in mind when he/she says, "This has been a good day."

Question 7

List at least five relevant quantities that are involved with this problem situation. For each quantity, if the value is given, write it next to the quantity. If the value is not given, write the unit you would use to measure it.

Question 8

Carry out a quantitative analysis of the following problem situation by answering each of the questions that follow.
Jennie got on the freeway at 2:00 PM, using the entrance closest to her home, and traveled at 55 mph to the College Avenue exit, where she turned off at 2:12 PM. Her roommate Cassie had finished her morning classes and was headed home at about the same time. Cassie entered the freeway from the College Avenue entrance at 2:08 PM and traveled to the home exit at 60 mph. At what time did Cassie arrive at the exit ramp to go home?

A) What quantities here are critical?
B) What quantities here are related?
C) For what quantities do I know the value?
D) For what quantities do I need to know the value?

Question 9

Consider this problem situation:
The school cafeteria is ready to serve two kinds of sandwiches, tuna and ham, and two kinds of pizza, pepperoni and vegetarian. There are 48 servings of pizza prepared. There are eight more tuna sandwiches prepared than there are servings of pepperoni pizza. There are four fewer ham sandwiches prepared than there are servings of vegetarian pizza. Altogether, how many sandwiches are prepared?

A) List eight quantities involved in this problem.
B) Sketch a diagram to show the relevant sums and differences in this situation.
C) Solve the problem.

Question 10

Considering the following problem situation:
Two trains leave from different stations and travel toward each other on parallel tracks. They leave at the same time. The stations are 217 miles apart. One train travels at 65 mph, and the other travels at 72 mph. How long after they leave their stations do they meet each other?
List six quantities in the problem (note that you are not asked to solve this problem). If a value is given, write it next to the quantity. If no value is given, write an appropriate unit of measure.

Question 11

Carry out a quantitative analysis of the following problem situation by answering each of the questions that follow, and then solve the problem:
A butcher had two pieces of bologna, A and B, with A weighing 3 1/3 times as much as B. After the butcher cut 1.8 pounds off A, A was still 2 1/3 times as heavy as B. How many pounds does piece B weigh?

A) What quantities here are critical?
B) What quantities here are related?
C) For what quantities do I know the value?
D) For what quantities do I need to know the value?
E) Use a diagram to find the weight of B, in pounds.

Question 12

Consider the following problem situation:
Two boats simultaneously left a pier and traveled in opposite directions. One traveled at a speed of 18 nautical miles per hour and the other at 22 nautical miles per hour. How far apart were they after 2.5 hours?
List five relevant quantities that are involved in this problem. For each quantity, if a value is given, write it next to the quantity. If value is not given, write the unit you would use to measure it and its value if possible.

Question 13

My brother and I go to the same school. My brother takes 50 minutes to walk to school, and I take 40 minutes. If he gets a 49-minute head start one day, can I catch him before he gets to school? Explain without referring to any short-cut in your explanation. (Hint: Do not do a lot of calculation.)

Question 14

My sister can walk from school to home in 40 minutes. I can walk from school to home in 30 minutes. But today I stayed for some extra help, and my sister was already 2/5 of the way home when I started.
If I walk at my usual speed, can I catch my sister before she gets home?
If YES, exactly what fraction of the trip have we covered when I catch her?
If NO, exactly what fraction of the trip have I covered when my sister gets home?
In either case, write enough (words, numbers, drawings) to make your thinking clear.

Question 15

The big dog weighs five times as much as the little dog. The little dog weighs 2/3 as much as the medium-sized dog. The medium-sized dog weighs 9 pounds more than the little dog. How much does the big dog weigh?

A) List three quantities associated with this problem. If possible, give the associated value.
B) Draw a diagram to represent the quantities in this problem.
C) This diagram was provided by a fifth-grader. Tell why it is not helpful.

D) Solve the problem and explain your solution process.

Question 16

The label on a can of chicken broth claims that its weight is 1.4 kg. Use your metric knowledge to tell how many milligrams this would be.

Question 17

The larger the unit of measure used to express the value of a specific quantity, the larger its numeric value will be.

Question 18

Give an estimate of the following and explain how you did it.
The length of this line in metric units: ____________________________

Question 19

List the following measurements in order from greatest to least:
8 m, 359 cm, 66 m, 3912 mm, 5198 cm, 7 m

Question 20

How many grams are there in centigrams?

A) 1 × 10^-2
B) 100
C) 0.001
D) 1 × 10²

Question 21

If a Pascal is some unit of measure, use your knowledge of metric prefixes to complete the following:
4 kiloPascals = _____ Pascals

Question 22

Name a metric unit that is analogous to a quart. Which is LARGER?

Question 23

Name a metric unit that is analogous to a yard. Which is LARGER?

Question 24

Size 1 Pampers fit babies who weigh 4 to 6 kg. Maggie weighs 11 pounds. Will size 1 Pampers fit her? Justify your answer and use a sketch or diagram to show the relationship between kilograms and pounds. (A kilogram is about 2.2 pounds.)

Question 25

What are some advantages to using the metric system of measurement?

Question 26

Some children, when asked to solve a story problem, try different operations on the numbers and then decide which one seems to give the best answer. What is the danger of solving problems in this way?

Question 27

Many teachers teach "key words" for solving word problems. What are the limitations of this strategy?

Question 28

Use a diagram to solve the following problem:
Jesse collects stamps. He now has 444 stamps. He has three times as many stamps from European countries as he does from Asian countries. How many of his stamps are from European countries?

Question 29

Use a diagram to solve the following problem:
Silvia and Jesus are buying a new table and new chairs for their dining area. Chairs with armrests are $45; those with no armrests are $8.50 cheaper. The table is four times as much as a chair without armrests. If they buy a table and six chairs, two with armrests and four without, what is the total price they pay?

Question 30

Use a diagram to solve the following problem:
Dylan lives 8 miles from campus. Connor lives 2 miles farther away from campus than Dylan does. If each drives a car to campus, how many miles altogether do Dylan and Connor drive to and from campus?

Question 31

Use a diagram to solve the following problem:
A combinded third- and fourth-grade elementary school classroom has 29 students. There are seven more third-graders than there are fourth-graders. How many students are there in each grade?

Question 32

Draw a diagram to represent the quantities in the following problem. Solve the problem and explain your solution process.
​
Jacqui, Karen, and Lynn all own several bottles of nail polish. When they line them up, Jacqui has 12 more bottles than Karen and Karen has three times as many as Lynn. Together, they have 124 bottles of nail polish. How many bottles does each person have?

Question 33

Your elementary student brings in a picture and story about a fat cat brought to an animal shelter weighing a hefty 18.1 kilograms. You want to help them understand how heavy that cat is.

A) How much did the cat weigh in pounds and ounces?
B) What personal benchmark might be useful in understanding this weight?

Question 34

Haley and Tyler are solving the following story problem:
"One kind of candy costs $3.50 per pound. How much will 0.42 pound cost?"
Haley thinks you should subtract, and Tyler thinks you should divide.

A) What are these children likely to be thinking?
B) Is either correct? Explain your reasoning.