Math 307

Starting with a number line on which o an d1 have already been plotted, explain where to plot 5/8 and explain why that location fits with the definition of a fraction.

First, divide the line segment between 0 and 1 into 8 pieces of equal length. According to the definition of unit fractions, each of these segments is 1/8 of a unit long. Use tick marks to show where these segments begin and end, but note that it turns out there are only 7 tick marks between 0 and 1. Plot 5/8 at the end of the fifth segment from 0. By the definition of fraction, the length of an interval made from 5 segments, each of which is 1/8 of a unit long, is 5/8 of a unit. SO the location of 5/8 fits with the definition of a fraction.

Ellie solves the subtraction problem 2.5-.13 with toothpicks. She represents2.5 with 2 bundles of ten toothpicks and 5 individual toothpicks, and she represents .13 with 1 bundle of 10 toothpicks and 3 individual toothpicks. Ellie gets the answer 1.2. Is she right? If not, explain why not and discuss how she could use toothpicks correctly.

No, she is not correct. 1 toothpick must represent the same amount when representing both 2.5 and .13. Ellie should let 1 toothpick represents one hundredth in both cases. The 2.5 would be 2 bundles of 100 toothpicks and 5 bundles of 10 tootkpicks, while .13 would be 1 bundle of 10 toothpicks and 3 individual toothpicks. Now ellie should be able to see that she will need to regroup in order to subtract. It might help Ellie to thin in terms of money. 2.5 and .13 can be represented by $2.50 and $0.13. Ellie way is like saying a dime is equal to a penney.

Make a math drawing to show how to organize 19 objects in a way that fits with the base ten system.

See figue 1.5

Describe three ways discussed in the text to represent a decimal such as 1.234. For each way, show how to represent 1.234.

The decimal 1.234 can be represented with bundled objects, as a length, or a number line. To represent 1.234 with bundled toothpicks, let one toothpick represent one thousandth. Then 1.234 is represented with 4 single toothpicks, 3 bundles of ten, 2 bundles of 100 (each which is 10 bundles of 10) and one bundle of a thousand (which is ten bundles of 100). See figure 1.18 representing 1.234 as a length.

Describe where the negative numbers are located on the number line. Then describe how to plot the numbers -2 and -2.41 on a number line in which 0 and 1 have been plotted.

The negative numbers are located to the left of zero on a number line. The number -2 is located 2 units from 0 to the left of 0 (and recall that the distance of 1 unit is the distance between 0 and 1).

Children who have heard of a googol, which is a number that is written as 1 with 100 zeros, will often think it is the largest number. Is it? Is there a largest number?

The number a googol plus one, for example, is greater than a googol. There is no largest number because no matter what number you choose as candidate for the largest number, that number plus on is a larger number.

Why do we line up decimal points before adding or subtracting decimals? What are we really doing?

When we line up decimal points, we are really lining up like places, so that we will add or subtract ones with ones, tenths with tenths, hundredths with hundredths and so on.

Round 6.248 to the nearest tenth. Explain in words why you round the number the way that you do. Use a number line to support your answer.

You are rounding to the nearest tenth, so you must first find the "tenths" that 6.248 lies between. In other words, on a number line on which the tick marks are tenths, you must find the tick marks 6.428 lies between. The 6.248 is between the tenths 6.2 and 6.3. Because of the 4 in the hundreds place, 6.248 is less that 6.25 which is alway between 6.2 and 6.3. Therefore 6.248 is closer to 6.2 than 6.3, so 6.248 rounded to the nearest tenth is 6.2

Round 173.465 to the nearest hundred, ten , one, tenth, and hundredth.

hundred = 200 ten = 170 one = 173 tenth = 173.5 hundredth = 173.47

Ken ordered 3/4 of a ton of gravel. ken wants 1/4 of his order of gravel delivered now and three fourths delivered later. What fraction of a ton of gravel should Ken get delivered now a. make math drawings to help you solve this problem explain why your answer is correct attending carefully to the whole that each fraction is of b.In solving this problem, how does 3/4 appear in a different form?

A. in figure 2.49 one ton of gravel is represented by a rectangle, and ken's order is represented by 3/4 of the rectangle. Then Ken's order is divided into four equal parts and of those parts one is darkly shaded. This darkly shaded amount is 3/16 of the original rectangle representing 1 ton. Therefore, one fourth of Ken's order is 3/16 of a ton. b. in this problem the 3/4 of a ton of gravel appears as 3x4/4×4 = 12/16

If one serving of juice gives you 3/2 of your daily value of vitamin C, how much of your daily value of vitamin C will you get in 2/3 of a serving of juice? Make a math drawing that helps you solve this problem. Use the drawing to help explain your solution. Use our definition of a fraction to describe the fractions in this problem, drawing attention to the whole that each fraction is of. What are the different wholes in this problem?

Because one serving got juice provides 3/2 of the daily value of vitamin C, one serving of juice represents 3 parts, each of which is 1/2 of the daily value of vitamin C. The daily value of vitamin C is represented in two of those parts, as shown in figure 2.18. Those 2 parts are each 1/3 a serving of juice. So if you drink 2/3 a serving of juice, you will get the full daily value of vitamin C.

The distance between two cities is described as 1500 miles. Should you assume that this is the exact distance between the two cities? If not, what could the distance possibly be?

Because the reported distance has zeros in the tens and ones places, assume the distance has been rounded to the nearest hundred, Therefore, the distance between two cities is probably not 1500 miles; it could be anywhere in the range between 1450 miles and 1550 (but less than 1550).

Describe how to organize 100 toothpicks in a way that fits with the structure of the base ten system, Explain how your organization reflects the structure of the base-ten system and how it fits with the way we write the number 100.

First, bundle all the toothpicks into bundles of 10. Then gather those bundles of 10 into a single bundle. This repeated bundling of ten groups is the basis of the base-ten system. the 1 in 100 stands for this large 1 bundle of 10 bundles of 10.

You have bundles of toothpicks like the ones shown in figure 1.27 and you wan tot use these bundled toothpicks to represent a decimal. List at least 3 decimals that you could use thee bundles to represent and explain you're answer in each case.

If one toothpick represents 1, then figure 1.27 represents 214. The following table shows several other possibilities. 100 > 21,400 10> 2140 1 >214 .1>21.4 .01> 2.14 .001> .214

Hermione has a potion recipe that calls for 4 drams of snake oil. She wants to make 2/3 of the potion. Rather than calculate 2/3 of 4, she measures 2/3 of each dram of snake oil and uses that amount in her potion. Use pictures and the meaning of fractions to explain why this is valid.

If we represent the 4 dram of snake live oil with drawings of 4 jars, then we can divide the jars into three parts and shade 2 out of the three parts of each one to represent 2/3 of each. The unshaded parts, then represent the 1/3 that she will not be using. therefore, the shaded part represents 2/3 total.

Use a math drawing to explain why 4/5= 4x3/5x3 = 12/15

If you have a rod that is divided into 5 equal parts, and 4 are shown shaded, then if you divide each of the 5 parts into 3 smaller parts, the shaded amount will then consist of 4 x 3 smaller pieces, and the whole rod will consist of 5 x 3 small parts. Thus, the shaded part of the rod can be described both as 4/5 of the pie and as 4/5= 4x3/5x3 = 12/15 of the pie. Therefore, 4/5 and 12/15 represent the same number.

If a young child can correctly say the number word list to five - "one, two, three, four, five" - will the child necessarily be able to determine how many bears are in a collection of 5 toys bears that are lined up in a row? Discuss why or why not.

No, the child might not be able to determine that there are five bears in the collection because the child might not be able to make a one-to-one correspondence between the number words 1,2,3,4,5 and the bears I.e. the child might point twice to one of the bears and count two umbers for that bear , or the child might skip over a bear while counting.

If a young child can correctly say the number list to five - "one, two,three, four, five" - and point one by one to each bear in a collection of 5 toy bears while saying number words, does the child necessarily understand that there are 5 bears in the collection? Discuss why or why not.

No, the child might not understand that the last number word that is said while counting the bears tells how many there are in the entire collection.

Give examples of decimal numbers that cannot be represented with bundles of toothpicks- even if you had as many toothpicks as you wanted.

Realistically, we would be hard-pressed to represent numbers with more than 4 nonzero digits with toothpicks. Even 999 would be hard to represent. However, there are some numbers whose expanded form can't be represented by toothpicks (in the manner described in the text) - even if you had as many toothpicks as you wanted. For example, consider .33333333333.... where the 3s go on forever. Which place value would you pick to be represented by one toothpick? If you did pick such a place, you would have to represent the places to the right by tenths of a toothpick, hundredths of a toothpick, thousandths of a toothpick, and so on forever, in order to represent this number. As an aside, here is a surprising fact: We can represent .33333333 with toothpicks but in a different way. (not by bundling so as to show the places in a decimal number). Namely, we can represent it with one third of a toothpick because it so happens that one third is 0.333333333... which you can see by dividing 1 by 3.

Describe how to represent .0278 with bundles of small objects in a way that fits with and shows the structure of the decimal system. In this case, what does each small object represent?

Represent .0278 as 2 bundles of 100 objects (each of which is 10 bundles of 10), 7 bundles of 10 objects, 8 individual objects. In this case, each individual objects must represent one ten-thousandth , since 0.0278 is 2 hundredths, and 7 thousandths, and 8 ten-thousandths.

ROUND 39,995 to the nearest ten. Explain why and use a number line.

Since you are rounding to the nearest ten, you must find the tens that 39,995 lies between, or on a number line where the tick marks are tens, you must find the tick marks 39,995 lies between. The number 39,995 lies directly between 3,990 and 40,000. By the "round 5 up convention", we round up to 40,000.

What problem in the history of mathematics did the creation of the decimal system solve?

The base-ten system solved the problem of having to invent more and more new symbols to stand for larger and larger numbers. By using the base-ten system, and place value, every counting number can be written using only the ten digits 0,1,2,3,4,5,6,7,8,9.

Explain why -1.2 < -.89 without using a number line

To explain why -1.2 <-.89 consider having -1.2 as owing $1.20 and having -.89 as owing 89 cents. If you owe $1.20 you have less than if you owe 89 cents. Therefore, having -1.2 is having less than -.89

Using the example 1/4 + 5/6, explain why we must give fractions common denominators in order to add them.

When we add 1/4 and 5/6, we first give a fraction a common denominator so they have like parts. We can interpret 1/4 + 5/6 as the total amount of pie you have if you start with 1/4 a pie and then get 5/6 of another, equivalent same sized pie. But fourths and sixths are pieces of different size, so we must express both kinds of pie pieces in terms of like pieces. If we divide each of the four fourths of a pie into 3 pieces and each of the 6 sixths of the other pie into 2 pieces, then 1/4 of the first pie becomes 3/12 and 5/6 of the second pie becomes 10/12. Now both fractions of pie are expressed in terms of twelfths of a pie, as seen in figure 3.26. Therefore, the total amount of pie you have consists of 3+10=13 pieces, and these pieces are twelfths of a pie. Thus, you have 13/12 of a pie, which is 1 1/2. Numerically, this process of subdividing parts looks like this:

Kelsey has 3/5 of a chocolate bar. She wants to give some of her chocolate bar to Jenelle what section of her chocolate bar should Kelsey give Jenelle so that Kelsey will be left with one half of the original chocolate bar? a. make math drawings to help you solve this problem explain why your answer is correct attending carefully to the whole that each fraction is of In solving this problem, how do 3/5 and 1/2 appear in different forms?

a. If the original chocolate bar is divided into 10 equal parts, then the amount of chocolate that Kelsey has is 6/10 of the original chocolate. The amount that Kelsey wants to keep is 5/10 of the original chocolate; so Kelsey should give one part of her six parts to Janelle. b. In solving this problem 3/5 becomes 3x2/5x2= 6/10 and 1/2 becomes 1x5/2x5=5/10