Wrapping up Investigation 3

Today we are wrapping up Investigation 3 in our Accentuate the Negative book.  We've been exploring multiplication and division with positive and negative numbers.  We've been able to discover and write algorithms (rules/plans/steps) for performing these tasks.

 

Multiplication

positive * positive = positive

negative * negative = positive

positive * negative = negative

and since multiplication is commutative, the order doesn't matter!

 

Division

positive / positive = positive

negative / negative = positive

negative / positive = negative

positive / negative = negative

 

Now let's use our rules to practice some problems in ACE #22.  If you need help with the fractions, check out Math.com.

http://math.com/school/subject1/lessons/S1U4L4GL.html

 

#23: There is more than one correct answer to these problems.  Fill in the blanks to make the number sentences true.  For example, one possible answer to 23a is 5*6 = 30...but now you can't use that one!! : )

 

#24: You need graph paper for this one.  This problem is a review all the way back to Variables & Patterns.  One way to do this problem is to make a table of time & temperature.  For example...

 

Time           Temperature

12noon        75

1 pm           73

2 pm

...

10 am

 

OR, you might be able to skip the table and do some multiplication to find the answers to a & b!

 

Now, for part c, make a graph of this information.  Time is the independent variable.  When it says use noon Tuesday as time zero, that means its point will be (0, 75).  The coordinate for 1 pm would be (1, 73) and so on. 

 

For part d, we want to know the rate of change.  Remember, a rate compares two quantities with different units.  Examples of rates are 20 miles per hour, 5 gallons per minute, etc.  What is the RATE in this question?  How does the pattern in the table tell you the rate??

 

That's it!  Post your questions! 

Subtracting Integers

We've spent the last few days working on subtraction of integers...it can be a confusing process, right?  Its hard to think about what subtracting a negative number even means!  So, what we tried to do last night and today is figure out how subtraction can be turned into addition...because adding is soooo much easier!  Seriously!

 

By looking at our models from 2.3, we discovered a couple patterns:

 

subtracting a negative = adding a positive

subtracting a positive = adding a negative

 

I hope you remembered to bring your yellow foldable home this weekend, because we are gathering lots of good rules on it!  We have the rules for adding integers already!

 

To sum it up: don't subtract!  Add the opposite!!

 

Ex: +2 - -5 ===== +2 + +5

because the opposite of -5 is +5!

 

Just don't ever change the first number (like you can't change the amount of money in your pocket.  It is what it is!)

 

Homework this weekend...ACE #10, 11, 13, 15, 16

 

For ACE 10 and 11, you don't actually have to solve anything.  Use your reasoning skills to make decisions about the answer. 

 

For example, 10a...which will give a greater answer?

+5280 + -768   OR   +5280 - -768

 

Well, since I know I can change the second problem into addition by adding the opposite, I am really comparing

+5280 + -768   OR   +5280 + +768

 

The second one will give me a greater result because adding a positive increases the value.  Adding a negative takes away.

 

For ACE 11, think about what you know about absolute value.

 

For 13, it may be helpful to sketch the algebra tiles/dots, but it is not required. 

For 15 and 16, take it step by step.  Don't try to add/subtract all the numbers at once.  Do it two at a time and be sure your sign (+ or -) is correct before you add/subtract the next number. 

 

For example 15a...

+3 + -3 = 0 (because they were opposites!)

then take the 0 and add -7

0 + -7 = -7

 

Post your questions...if someone asks a question and you have a method or answer, please post that, too!

 

Have a nice long weekend!  And Happy Valentine's Day!  Mrs. S

 

 

Adding Integers

We're moving ahead in Accentuate the Negative!  We've discussed opposite numbers and comparing and ordering integers using a number line.  One of the keys to our work this week was on our warm up today:

 

The SUM of a number and its opposite is...ZERO!

 

In class, we practiced using the algebra tiles to model addition problems.  Remember, red = negative.  You could also use pink or orange.  Use any other color for positive numbers (I like to use black, blue, green or yellow). 

 

You should have been able to finish problem 2.1 (page 24) in class.  Be careful not to spend too much time coloring.  Make your sketches quickly and move on.  They are only to help you find the answer and figure out HOW addition of integers works. 

 

When you are asked to write or use an ALGORITHM think about the steps you took to solve a problem, or the method you used. 

 

D2 may be a challenge because it involves fractions.  Recall from 6th grade that in order to add fractions, you need a common denominator.  So, rewrite the fraction 2/3 as something over 6.  How do you find the "something" AKA, the new numerator?  Think about scale factor! : ) 

 

Okay, onto the homework...page 32, ACE Qs # 1, 2, 3, 30, 37

 

#1 and 2: Feel free to use a chip model or number line to help you.  This is your chance to practice your skills!  At home you could use heads & tails of coins as chips, or anything else two-sided that you find around the house.

 

#3: Follow the arrows to know where to begin and what to add.  For example, 3a starts at +15.  Then they add -35 and end up at -20.  So the number sentence would read:

 

15 + -35 = -20

 

Some of you will write 15-35 = -20.  Is this the same thing?  We'll discuss this more towards the end of the week!

 

#30 is just like problem 48 from last week (page 20, we used a worksheet).  In fact, you might want to set up a table like a checkbook register to help you organize your work for this problem.  If you set up your table and show your steps like this, you may use a calculator to assist you.  I would like to see a balance at each point in the problem. 

 

Example:

 

Transaction                Amount             Balance

start                                                      0

rent                            -1800                 -1800

payment for bikes        -2150                 -3950

pymt for office equip.   -675

 

Finally, #37...you are adding numbers to -15.  Trial and error might help you solve this one.

 

For example, 37a:  What could you add to -15 that would equal a number greater than 0?  Would 10 work?  -15 + 10 is only -5...so NO.  We need to add something more than 10...how about 15?  Well, that would be adding the opposite, right?  -15 + 15 = 0.  So that doesn't work...need something still bigger than 15...get it????

 

Come in tomorrow with COMPLETED work, TRY YOUR BEST, and write down your QUESTIONS! : )

 

By the way, Punxsutawney Phil saw his shadow...6 more weeks of winter.  Boo!  Good thing he's right only about 40% of the time! 

 

Happy Groundhog Day!  Mrs. S

Comparing & Scaling Test Tomorrow!

Our test is tomorrow...but you are ready!  All classes did a great job with our review problems today.  Let me give you some hints on tonight's homework to help with your final review.

 

4.3 Part A

Remember you can set up ratios in different ways.  You could write miles/feet or feet/miles.  Other examples would be miles per gallon or gallons per mile...dollars per ounce or ounces per dollar.

 

Part B

Three enchiladas have 705 calories...use this ratio to start your proportion.

 

3 enchiladas / 705 calories...remember "/" means "divided by" or "over"

 

Now figure out how many calories in the 240 enchiladas Jack ate last year (your second ratio will have 240 and a variable in it.  Label your numbers and line them up!).  They want you to describe your strategy for solving this proportion, so you have to put into words what you & your calculator did.  Did you divide?  Multiply?  Use scale factor?  Simplify?  Unit rate?  Something else?  Think about all the strategies we've used this week and made notes about on our index cards.  Can you use this strategy all the time or only on certain problems? 

 

Finally, for B5, we want to know the number of calories in ONE enchilada...go back to the original proportion, but change your 240 to 1...this is just like finding the unit rate!  Calories per enchilada!

 

OK, now I'm totally hungry for Mexican food for dinner!  Yum!

 

Back to math...let's do Part C...

First, figure out how many kids are in the school.  Your fractions in part 1 should all have this total number of kids as the denominator.

 

58+76+38=???

 

Now, in C2, use these three fractions to set up three proportions.  The goal is to figure out what PART of the 35 TOTAL kids on student council will come from each grade.  For example, the 6th grade proportion would look like this:

 

58 sixth graders/172 total students = x sixth graders/35 total students on student council

 

Solve the proportion for x to find the number of sixth graders on the student council. (Sorry I don't have a way to type the proportions with a horizontal division bar.  I hope its not too confusing with the / for division.)  Round your answers to the nearest whole student!  (No partial people, please!)

 

In C3, they change the number of student council members from 35 to 37.  How does this change the proportions we just wrote?  How does this change the number of representatives from each grade?

 

Let's move on to ACE Questions.  We're looking at 5-14, so all of page 56.

 

#5: Everything is based on Denzel's average of 10 made/15 attempted.  Use this as the first ratio for your proportions in parts a-d.  Keep the free throws MADE in the numerator and the free throws ATTEMPTED in the denominator (labeled & lined up!).

 

#6-13: These are just like the workbook problems we've been doing in class.  Use your strategies to solve for x.  In #6, think about what number times .8 would equal 12.5...just like when we do the ".8*___ = 12.5" when we're looking for scale factor.  Did you divide 12.5 by .8?  Hooray! : )

 

#14  You gotta love a good multiple choice question!  Process of elimination will get rid of one option right away...you have to have 20 delegates total!

 

Now, think about how many students there are total...

618+378+204=???

Set up some proportions...

 

Number of students from North/Total number of students = delegates from North/20 total students on the council

 

Do the same for Central and South...they come out to be decimals, so round to the nearest whole student.  It doesn't come out exact, but one choice is the closest.  Remember process of elimination! 

 

OK...leave me QUESTIONS in the comments.  I'll check back later tonight and answer them.  You all are doing great!  Stay focused and positive...remember to "label & line up," use proportions, scale factor, all that good stuff.  : )

 

B-A-N-A-N-A-S!  Mrs. S

Help has arrived!

Need some help with tonight's homework?  This is great practice for tomorrow's test!

 

First, page 32...

 

2a: Be sure you are comparing the short side to the short side and the long side to the long side...it looks like they tried to flip around these rectangles to confuse you!

 

2b: Same thing here...one of the trapezoids is upside down!  Doesn't matter if you do B or C first.  Just be sure you don't take the numbers for a ratio from two different figures.  For example, is I was finding C, I would probably set up my ratios like this:

 

1.5/2.5 = c/10

 

Notice the first ratio's numbers come from the smaller figure and the second ratio's numbers come from the larger figure.  In both cases, the numerators are the short sides and the denominators are the long sides.  Everything lines up with its corresponding parts. 

 

Remember our four step plan:

1. Draw a picture (done for us!)

2. Write the ratios (just did!)

3. Find the scale factor...don't skip that step!!!

4. Use the scale factor to find the missing measurement ("do the same to the bottom as the top")

4b. Don't forget the units!

 

So back to 2b...to find the scale factor, ask yourself, "Self, what number times 2.5 would give me 10??"  In other words, 2.5 * ____ = 10.  How do you find the number that goes in the blank??  10 / 2.5 = ___ = the scale factor!

 

Did you get 4??  I hope so!

 

Now, "do the same to the bottom as the top"...multiply the scale factor, 4 by 1.5...did you get 6?  C = 6!

 

Use a calculator for 2c if you want...just be sure to SHOW YOUR RATIOS & SCALE FACTOR! : )

 

If you know the four step plan and how to use it, it won't matter if the numbers are funny/complicated/crazy!  The steps work for EVERY problem!

 

On to page 38...numbers 3 & 4 might be tricky.  But the SAME FOUR STEP PLAN WORKS!!  You can do this! 

 

A couple other things to know for tomorrow...do you know how to find area & perimeter of a quadrilateral?  Can you do problems like we've had on the warm up all week with triangles?  Post your questions (please include your first name and class period) and I'll check back around 8:30 pm. 

 

Give 100% effort and you'll do awesome!  Mrs. S

Similar Polygons

Tonight's homework is workbook page 29-30.  This is from the blue "Additional Skills & Practice Workbook" that you should have in your math binder.

 

For page 29, find the scale factor from one figure to the other and then use it to find the missing measurement.  For example, on #3, to get the scale factor, we are looking for some number ("n") that we can multiply by 14 to get 20.  (Remember to compare short side to short side, long side to long side.)  So, we could say that 14n = 20.

 

If 14n = 20, we can find n by dividing: 20/14

20/14 = 10/7...let's leave it like that for a minute and say our scale factor is 10/7.

 

Now, multiply the scale factor by 6...

 

6/1 * 10/7 = 60/7 = about 8.57...and that's the value of x!

 

On page 30, as long as you multiply or divide BOTH the numerator and denominator by the same thing, you'll get equivalent fractions.  For example, #1 is 3/10

 

3/10 = 6/20 (multiplied both numbers by 2)

3/10 = 9/30 (multiplied both numbers by 3)

3/10 = 30/100 (multiplied both numbers by 10)

 

You need three equivalent fractions for each problem.

 

For 9-12, you can find the decimal equivalent by dividing the numerator by the denominator.  Then, multiply the decimal by 100 to get the percent.

 

For example, #9: 3/5...three divided by five...three on the inside, five on the outside...

 3/5 = .6

.6 *100 = 60%!

Stretching & Shrinking Weekend Homework

Hey guys, hope you are enjoying the weekend! Don't worry, this homework shouldn't take long. It is just a few problems to make sure we understand the concepts in Investigation 3.

You are doing ACE questions 7 and 9-14. They start on page 47. Here are some tips.

7a. There is one pair of similar rectangles, one pair of similar triangles and one pair of similar parallelograms. Remember that similar means same shape, different size. You are looking for the pairs that have been stretched/shrunk by the same factor on the length and width.

7b. Since they want the scale factor from the smaller shape to the larger shape, your answers should all be greater than 1. Remember that a scale factor greater than 1 makes the figure bigger. Scale factors less than 1 make the figure smaller. Scale factor =1 keeps the figure the same, or congruent.

9-14. First find the scale factor from the smaller triangle to the larger triangle. Use this scale factor to help you find the missing side lengths and perimeter. Did you bring home your notes from Friday? If not (or if you were bowling, at the fundraiser party, absent, etc), here they are again.

Scale factor helps us find the following measurements in similar figures:

• Side lengths: multiply the original side lengths by the scale factor to find the side lengths of the similar figure.
• Perimeter: multiply the original perimeter by the scale factor to find the perimeter of the similar figure.
• Area: the scale factor squared (multiplied by itself) tells how the area changes.
• Angle measurements: in similar figures, corresponding angles are congruent.

One more thing that will help: all the angles in a triangle add up to 180 degrees. Remember that from 6th grade??

Hope this helps! Post questions in the comments and I'll check back over the weekend. Don't forget to include your first name and period in your comment. Thanks! Mrs. S