Math 208: Test 1 Spring Date:

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1 Math 28: Test 1 Spring 28 Name: Date: INSTRUCTIONS: Point values are listed by each question for a total of points possible. Be sure to include concise explanations along with your solutions. You may use the backs of the pages if needed. At any time you may include drawings to help solve and/or explain the problem. No calculators are allowed. 1. (pts.) Write,4,1.14 in expanded form using powers of ten. Explain why your answer is correct.,4,1.14 =,, + 4, +, = The number,4,1.14 is equal to three million (since there is a in the millions place) plus four hundred thousand (since there is a 4 in the hundred thousands place) plus six thousand (since there is a in the ten thousands place and a in the thousands place) plus one (since there are zeros in the hundreds and tens place, but a 1 in the ones place) plus one tenth (since there is a 1 in the tenths place) plus four hundredths (since there is a 4 in the hundredths place). Because one million is equal to1, I wrote 1 for,,. Because one hundred thousand is equal to1, I wrote 4 1 for 4,. Because one thousand is equal to1, I wrote 1 for,. Because one is equal to1, I wrote 1 1 for 1. Because one tenth is 1 equal to1 1 2, I wrote 1 1 for.1. Because one hundredth is equal to1 2, I wrote 4 1 for (pts.)write 1 in base four. Draw pictures and/or use words to describe your solution. In base four, 1 extra = 1, 1 group = 4, 1 gog = 1, and 1 gogog = 4. Since 4 is more than 1, there are no gogogs in 1. However, there is one gog in 1 with 1 leftover. The 1 leftovers can be arranged into groups and extra. So, 1 = 1 gog + groups + extra. Thus, 1 ten = 1 four. Visually, we can see that 1 arranged into groups and gogs looks like this: 1 gog + groups + extra. We can also write 1 = = 1 four. 2 1 ten

2 . (pts.) Given the number line below, label the rest of the tick marks correctly. What decimal is represented by the length between two consecutive tick marks? How did you determine this? The decimal represented by the length between two consecutive tick marks is.1. Because the length between.1 and.11 is.1, I divided.1 by 1 (the number of tick marks from.1 to.11) to find that the length between two consecutive tick marks is.1. I labeled each tick mark by starting with the one after.1 and adding.1 to the number on the previous tick mark. Since =.11, I knew I was right. 4. (4pts.) Carrie says the following decimal numbers are listed in increasing order:.1,.2,.,.4,.,.,.7,.8,.9,.1,.11,.12,,.98,.99,.1. How should the numbers be ordered? Write a paragraph to help her understand why her ordering is incorrect. Carrie has fallen into the usual trap of just putting the whole numbers in order behind the decimal point. In fact,.1,.1, and.1 are all the same number. If we write each number with three digits by adding zeros, Carrie will see that.1 =.1 =.1 and that.11 =.11 is less than.2 =.2. We can re-write the numbers in increasing order as:.1,.11,.12,.2,.,.4,.,.,.7,.8,.9.98,.99.

3 . (8pts.) Alex says 7.2 > 7.8. Is he correct? Explain why or why not in two different ways. Alex is not correct. (1) Using a number line, we can see that 7.8 lies to the right of 7.2 and thus 7.8 > (2) We can attach zeros to the right of the decimal as it does not change the size of the number. Thus, 7.8 = 7.8. Now we can see that seven and eighty hundredths is greater than seven and twenty hundredths. Thus, 7.8 > (4pts.) Round 14.4 to the nearest tenth. Explain in words why you round the decimal number the way you do. We round 14.4 to nearest tenth by looking at the numeral immediately to the right of the tenths place, which in our case is the numeral 4 in the hundredths place. Since 4 is less than, we don t have to round up. So, 14.4 rounded to the nearest tenth is 14..

4 7. (8pts.) You showed Jay the picture shown below to help him understand the meaning of the fraction. But Jay doesn t get it. He says the picture shows, not. 4 i) What must you clarify in order to interpret the picture as showing? We must clarify and tell Jay that the whole is one rectangle, which looks like this:. ii) What is another way to show to Jay, other than with parts of objects? We can also think of fractions as numbers and use a number line to show The following number line illustrates. to Jay. 1/ 2/ / 4/ / / (pts.) Draw a number line and plot, 1,, and 14 on your number line in such a way that each number falls on a tick mark. Explain how you chose the length between each tick mark. 1/ 2/ / 4/ 7/ 8/ 9/ 2 11/ 12/ 1/ Because 1, 1 / 14/, and 14 are made up of pieces of size between two consecutive tick marks 1. 1, I decided to make the length

5 9. (8pts.) Write equivalent fractions for and 4 that have the same denominator. Explain your processes using words and pictures and explain why the fractions are in fact equivalent. In terms of pictures, when you give two fractions common denominators you are starting with two pictures whose wholes are the same size and you are simply making the pieces in the two pictures the same size. For this example, I chose to use pieces that are 1/2 of the whole because if I divide a ¼ piece into five pieces I get pieces of size 1/2. If I divide a 1/ piece into four pieces, I get pieces of size 1/2. The pictures below help clarify why ¾ = 1/2 and / = 12/2, the reason being that two fractions are equivalent if they represent the same part of the same whole. one whole /4 = 1/2 / = 12/2 1. (pts.) Heidi says that the picture shown below shows that Explain carefully the error in Heidi s reasoning. 1 1 > by comparing the areas. 2 Heidi is making the usual mistake of confusing the wholes. The whole she is using for the fraction 1/ is bigger than the whole she pictures for the fraction ½. It is true that in her picture the shaded area on the left is larger than the one on the right.