MATHGet Something on the Table

“A blank stare solves nothing”

What is the smallest possible value for the product of 2 integers that differ by 7?

2 9 2 9 18 ProductDiffer by 7

1 8 1 8 8

-3 4 3 4 12

-7 0

-6 1

7 0 0

6 1 6

“2 integers that differ by 7?” What 2 do I use? Use any! Just get started!

Let’s pick two more

8 is less than 18 … Now I see the problem!

Let’s keep going!

The Plan?

To understand a question like this, I find it’s easiest to just try something. This helps me see what the problem is asking:

Three distinct lines, all contained within a plane, separate that plane into distinct regions. What are all of the possible numbers of distinct regions of the plane that could be separated by

any such three lines?

Let’s draw a box and draw three lines …

Three lines drawn like this create 4 regions …

I start to see the problem, and draw the

lines differently …

This way gives me 6 regions!

How else can I draw the lines?

7 regions!

The Plan?

What’s the problem asking? It’s hard to say. Draw a box and draw three lines.

When the choir is arranged in rows of 5 people each, the last row is one person short. When the choir is arranged in rows of 6 people each, the last row is still one person short. What is the least possible number of people in the choir?

(29; 30; 56; 60; 99)

X X X X X

X X X X X

X X X X O

X X X X X

X X X X X

X X X X O

X

X

X

5 Chairs / Row 6 Chairs / Row

There can be 14 or 19 or 24 or 29 or 34 ... people in

the band.

There can be 17 or 23 or 29 or 35

... people in the band.

There could be 29 people in the band.

The Plan?What’s the problem asking? There’s 5 chairs in each row (all filled by people), except for the last row (where one chair is empty). Draw this …

The choir is rearranged. Now there’s 6 chairs in each row (all filled by people), except for the last row (where one chair is empty). Draw this …

I start to see the problem …

Melissa had 3 fewer apples than Marcia. Then, she gave 2 apples to Marcia. Now

how many fewer apples does Melissa have than Marcia?

MELISSA MARCIAMelissa has 3 few apples than Marcia

Melissa gave 2 to Marcia

Melissa has 7 fewer apples than Marcia.

The Plan?

Forget formulas – use real things.

Gary has turtles, cats, and birds for pets. The number of birds he has is 4 more than the number of turtles, and the

number of cats is 2 times the number of birds. Of the following, which could be

the total number of Gary’s pets?(14; 18; 20; 22; 26)

The Plan?

Try a single turtle. What does this mean?

Turtles Cats Birds Total

1 5 10 16

2 6 12 20

nope but now I've got the feel of the problem

Yes!

When x is divided by 7, the remainder is 4. What is the remainder when 2x is divided

by 7?

The Plan?

Look for an easy number to get started.

11 7 1 4remainder

22 7 3 1remainder

Suppose I let x = 11This is what the problem asked for –

In fact, that’s why I chose 11!

Now answer the problem!

Bus X travels 40 miles per hour for 2 hours; Bus Y travels 60 miles per hour for 1.5 hours. What is the difference, in miles, between the number of miles traveled by

Bus X and the number of miles traveled by Bus Y ?

Bus X:40 mph for 2 hours

Bus Y:60 mph for 1.5 hours

The Plan?If the problem says “suppose you have a triangle”, draw a triangle. If the problem says “suppose you have two buses”, label two buses!

Forget formulas. Just draw a timeline.

60 m1 hr

40 m1 hr

30 m.5 hr

40 m1 hr

80 Miles

90 Miles

Sales for a business were 3 million dollars more the second year than the first, and sales for the third year were double the sales for the second year. If sales for the third year were 38 million dollars, what were sales, in millions of dollars, for the

first year?

year 1

16

year 2

19

year 3

38

What do we know?If sales for the third year were 38 million

dollars …

What else?sales for the third year were double the sales

for the second year.

What else? million dollars more the second year than

the first,year.

The Plan?If the problem says “suppose you have a triangle”, draw a triangle. If the problem says “suppose you have three years”, label three years!

Forget formulas. Fill in what you know.

An industrial cleaner is manufactured using only the 3 secret ingredients A, B, and

C, which are mixed in the ratio of 2:3:5, respectively, by weight. How many pounds

of secret ingredient B are in a 42-pound (net weight) bucket of this cleaner?

2 3 5

A B C Total

10

4 6 10 20

8 12 20 40

2x 3x 5x 10x

10 42x

Total is 42

4.2x 8.4 12.6 21 42

The Plan?If the problem says “suppose you have a triangle”, draw a triangle. If the problem says “suppose you have three ingredients”, label three ingredients!

Where do I start? Since the ratio is 2:3:5, why not start with that?

Not enough – let’s double everything.

Still not enough – let’s double everything!

Close. I need 42. I won’t be able to get it with whole numbers, clearly. But I also see the nature of the problem now.

The 2 diagrams below show a circle of radius 1 inch with shaded sectors of angle x°, for 2 different

values of x. One of the following is the graph in the standard (x,y)

coordinate plane of the area, y, of a shaded sector with angle x°, for all values of x between 0 and 360.

Which is that graph?

angle

0

part of circle

shaded?0

90 1/4

180 1/2

360 whole thing

The Plan?

This is a hard one – wrestle with what they’re asking for.Read the question carefully. Look at the two circles. What are they asking? A hint: if x is small, how much of the circle does it “carve out”? How about as x gets large?sking?

Now I have an idea what to graph …

The lead of a screw is the distance the screw advances in a straight line when the screw is

turned 1 complete turn. If a screw is 2½ inches long and has a lead of ⅛ inch, how many complete turns would get it all the

way into a piece of wood?

11 "

8turn

22 "

8turns

4 14 " "

8 2turns

88 " 1"

8turns

16 2"turns 1

20 2 "2

turns

12 "

21

2 "2

1"

8 1 128 21 x

12 28

x

5

8 2

x

2 40x

20x

The Plan?

If the problem says “suppose you have a triangle”, draw a triangle. If the problem says “suppose you have a screw going into wood”, draw a screw going into wood!” Make sure to label everything.What does the problem say about this screw going into the wood?

If xy = 144, x + y = 30, and x > y, what is the value of x – y?

20

25

24

10

5

6

30

30

30

x y x + y xy

200

150

144

The Plan?

Sometimes it’s easiest to just try some numbers and see what works …

Nope – too much. I need two different numbers.

Still too much, but closer!

Bingo!

Now make sure you answer the right question!

18x y