Monday, September 22, 2014

Implementing The Core: This is not a parenthetical remark...

-3a^2+4b+c; a=-3,b=-2,c=-1
Step One:  -3(  )^2+4(  )+(  )

Note that I'm recommending this BEFORE the numbers go in!

Do you share my belief in the critical role of (  ) in evaluating algebraic expressions?


Are you thinking this is too much detail and most students don't need to do this?

And I haven't even gotten to replacing -7-3 by (-7)+(-3)!

Sunday, September 21, 2014


For Teachers/Parents/Students

Subscribe now to start immediately receiving CCSS/SAT/ACT Twitter Challenge Problems with DETAILED Solutions, Extensions, Strategies and Common Core Implementation Ideas!

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Tuesday, September 16, 2014

CCSS: One Less Than a Million - How Many Nines? Grade 2? 4? 6? 8?

Implementing The Core - Raising The Bar
One less than a million. How many 9's?
Developmentally inappropriate for 7 year olds?
***What questions should we be asking to develop this kind of arithmetic reasoning?***
What should children be writing on their paper to make conjectures about numerical patterns?
[1] less than 10: 9 [1 nine, 1 zero]
[1] less than 100: 99 [2 nines, 2 zeros]
How can this be EXTENDED to challenge the child who's ready for higher-order thinking?
Note that I didn't say *OLDER* children!
One less than a trillion? How many nines?
One less than a googol?
One less than 10^n?
One MORE than a billion? How many 1's?
One MORE than 10^10? What is the SUM OF THE DIGITS?
Discover a general rule and have them memorize it?
It's OK. I'm not expecting comments. I'm just planting seeds. Up to you to consider, modify, plant, add nutrients and illumination, watch growth
OR ignore all of this!

Monday, September 15, 2014

Typical 2nd Gr Assessment Questions and Your Thoughts...

Imagine that. I'm not promoting my problems/solutions!

As posted on my twitter account ( today...

Here are a couple of typical questions your 2nd grade child/student may be working on...

(Clock shows 1:00)
In 1/2 hour, it will be ___.

Mathematical Practices Reflections...
Why do you believe some children would struggle with this?
Possible teacher/parent interventions?

Write arrow rule. Fill in missing frames..

Mathematical Practices Reflections...
After child demonstrates proficiency, what can teacher/parent do to raise the bar? OR
Are you thinking this is ambitious enough for a 7 yr old?

Plugging in to avoid the algebra? Today's CCSS/SAT Twitter Problem

1/|8x-4| > 1
Possible value for x?
(SAT-type grid-in question)

"Plug in" - Easy?
Graphing calculator?

How do you think I devised this problem?

Want solution? Uh, you know what to do...

Sunday, September 14, 2014

More Solutions to Twitter CCSS/SAT Questions

The following is part of what everyone on my Twitter Problems mailing list has been receiving every day or two for the past 3 weeks. For free... You have 2 weeks left to sign up. Free...

As you can see I go beyond the answers. Way way beyond...
Just as the Math Practices of CCSS suggest we do...

1. I walked my daily path 25% slower than usual and took 5 min longer. How  many min does it usually take?

Let R=usual rate (mi/min); T=usual time (min)
One can infer that distances are equal from the phrase "daily path". Using D=RT and equating:
R's " cancel" leaving
15/4=1/4•T or T=15.

If slower rate is k•R (0<k<1) and extra time is m min then
kR(T+m)=RT --->
km=T(1-k) --->
Test it: k=3/4,m=5 ---> T=5((3/4)/(1/4))=5•3=15
Special case: If k=1/2 then T=m or if one walks half as fast trip will take double the usual time!

2. Test this "rule":
3 more than the square of an odd integer is a mult of 4.
Now prove it!
Devise a rule if "more" is repl'd by "less!"

An odd integer can be expressed as 1 more than even or 2n+1.
"3 more than the square of an odd integer" translates to
3+(2n+1)^2 = 3+4n^2+4n+1=4(n^2+n+1), a mult of 4.

Three less than the square of an odd becomes
(4n^2+4n+1)-3 = 4n^2+4n-2 which represents 2 less than a multiple of 4.
Thus "three less than the square of an odd" cannot be a multiple of 4 and in fact will always leave a remainder of 2. Why?

Saturday, September 13, 2014

Sample Solutions to Recent Twitter CCSS/SAT Problems

The following is copied from solutions I sent today to my mail list of those who have opted for free solutions for the rest of September...

Yes, I've been giving these away for weeks now. Hard to believe anyone would do this? There must be a catch, right?

I will continue this until the 30th then am considering a low fee subscription for the rest of the school year.

Subscribers will get detailed solutions which include strategies, big ideas, extensions, etc. Further I may include additional problems which will not appear on Twitter or this blog.

To sign up, provide all pertinent info in the Blogger Contact Form in the sidebar.

1. Rectangle has integer sides and area=96
(a) How many possible perimeters?
(b) Greatest perim? Least? L=? W=?

(a) 6
(b) Greatest perim:194; Least:40

Solution: 96 has 6 pairs of factors ---
Each pair has a different sum so there are 6 possible perimeters.
The greatest and least possible occur in the extreme cases, i.e., when the factors are farthest and closest apart. This is generally true.

Note: If integer condition is removed there would be no greatest perimeter and the least would be 16√6, a square!

2. Data:4,6;mean:5
(a) Avg diff from mean= ((4-5)+(6-5))/2=?
(b) v=((4-5)^2+(6-5)^2)/2=?
(c) √v=?
(d)Repeat for 3,7
Common name for √v?

(a) 0
Note: This is always true -- the avg difference or deviation from the mean is zero! This is why we square the differences to measure deviation!
(b) v=1
Note: The avg of the squared "deviations" from the mean is called the variance.
(c) √v=1
Note: The square root of the variance is called the standard deviation!
(d) For 3,7 ---
Mean is still 5
√v=2, the stand dev.
√v gives a measure of how dispersed the data is from the mean...

Friday, September 12, 2014

Six more Free Twitter CCSS/SAT questions...

Free Twitter CCSS/SAT Problems with complete solutions sent to your inbox ending in 18 days on 9-30-14. You know what you have to do...

1. List in order the ten 5-digit pos int containing 3 nines & 2 eights?
Explain connection to 5C2.
Is this open-ended?

2. The median of 100 different integers is 100. If the numbers are in increasing order and the 50th # is 83 what is the 51st #? Explain/show...

3. Rectangle has integer sides and area=96
(a) How many possible perimeters?
(b) Greatest perim? Least? L=? W=?

4. Data:4,6;mean:5
Avg diff from mean= ((4-5)+(6-5))/2=?
Repeat for 3,7
Common name for √v?

5. From the brilliant #xkcd...

Sneeze droplet: 200 million germs. Hand sanitizer kills 99.99%. How many live? No calculator-15 seconds!

6. Least positive odd integer with 8 factors?
Strategy: Make it ____
2 factors:3
4: 15 or 3×5