Friday, October 31, 2014

Implement The Core: No *Mean* Tricks!

Halloween Twitter Problem

          1 ___4
M🎃an treats/child? M🎃dian?

(1) This question fits where in Common Core? Grade levels?
(2) What questions could you ask before calculation to develop number sense/conceptual thinking?
Some ideas...
Why is this sometimes referred to as a frequency table?
Which is easier to determine -- mean or median? OR
If  frequency = 4 kids for all # of treats, mean = ? Mental Math!!
Explain to your partner why mean > median.


Wednesday, October 22, 2014

A Dose of Reality -- My Latest Common Core Rant

I'm reproducing my comment to the post, "Who Needs Algebra?"on Mr. Honner's outstanding blog...
I strongly recommend you  read all of his excellent pieces. The current one is compelling for all math educators not to mention the public...


First of all requiring an in-depth conceptual understanding of algebra for all students shows complete insensitivity to special needs students and their longsuffering teachers and parents. Sure just modify the curriculum for them. Go ahead. Show me exactly what that looks like and those who are pontificating the loudest come with me on the front lines of these classrooms and put your money where your mouth is.

Now for the rest…
Students should be expected to struggle much more than has been required of them for the past 3 decades. I've supported Common Core long before that name was coined because I believed not having uniform standards across the states was unethical and promotes inequalities for children. That belief is unwavering. However I've never believed all children should be subjected to a deluge of high-stakes assessments from the age of 8 or 9. Particularly when it takes 5-10 years for any new curriculum to "set". Particularly when teachers need extensive preservice and inservice training. Particularly when full released versions of these assessments have not yet been made public by PARCC or SBAC.

IMO, the rush to assess is purely politically driven and our leaders should be ashamed of themselves. In the name of accountability our children are needless guinea pigs. That is unconscionable. Sone of our best teachers are frustrated to the point that they might walk away from the profession they love. And that would be a real tragedy. The efficacy of the Core is dependent on our classroom leaders. If we lose the best of the best, we will all lose. Wake up before it's too late. Sadly that time may have passed…

Just How Common is our Core?

Borrowing a problem from the comments in the excellent blog CorkboardConnections. Hope that's ok...


Mdm Shanti bought 1/3 as many chocolates as sweets. She gave each of her neighbours' children 4 chocolates and 3 sweets, after which she had 6 chocolates and 180 sweets left.

(a) How many children received the chocolates and sweets?
(b) how many sweets did she buy?

ans: 18 children; 234 sweets.


This is the questions our 12 year old do for their National exams.. is this type of questions easier or tougher than your core maths ?


Dave MarainOctober 21, 2014 at 7:02 AM
My thoughts...
1. Unless Singapore Math materials are being used, US students could only solve this with algebra. For example, let y=# of children,etc. Students trained in Singapore Math might consider a "bar model" approach.
2. Problems of this level of complexity are unusual in US texts. Most 7th graders here are in prealgebra. This type of question would fit into 1st year algebra but I haven't yet seen many problems requiring this level of reasoning.
3. My instinct is that many of our **secondary** students would struggle with this! That's easy enough for teachers to verify.
4. Yes, Common Core has raised the bar but the proof will be in the difficulty of the problems students are expected to solve. If 12 year olds in your country are expected to solve this question on a National Exam then they must have been exposed to similar questions in their classes. In my opinion, we are not there yet...


1. I hope you'll take exception to my comments above and prove me wrong by copying a page from a current COMMON CORE 7th-10th grade text. A page of problems similar to this one. Similar not only in content but in **difficulty**. An algebra problem tied to ratio concepts. In yesteryear, Dolciani would have problems like:

Determine a fraction in lowest terms with the property that that when the numerator and denominator are each increased by 2 the result is 4/5 (this one is easy; Mary P. Dolciani had harder ones!)

2. Some of my faithful readers are far more proficient with Singapore bar model methods). I tried it, it worked but I personally felt it wasn't worth the effort for me. Algebra seemed more natural. If you see a straightforward model solution, pls share!

3. What do you see as the complications in the problem above. The stumbling blocks for  some of your students? Remember the commenter is talking about a 12 year old, a 7th grader...

I asked myself if my 11 yr old grandson will be ready to tackle this next year? I think so if he's exposed to similar problems.

And that's the whole point of this post. Higher expectations are necessary but are they sufficient?

Monday, October 20, 2014

Round your answer to nearest cent: $1.29 or $1.30?

Tweeted (@dmarain) the above a couple of days ago. Moderate reaction so far which I find fascinating since I've done my own "random" survey...

6th gr student calculates an *exact* answer of $1.29. Directions read "round ans to nearest cent." Student writes $1.30 in the answer box on the test. Teacher notes $1.29  was correct but the answer in box was wrong. No credit for problem...


Making too big a deal of this? After all "rounded to nearest cent" means "round to nearest hundredth". So $1.29 is already rounded to the nearest cent whereas $1.30 is rounded to the nearest tenths or dime, right? Adults know that, right? Students should know, right? Certainly higher-achieving HS students know that, right? Hmm...

Maybe you should try your own informal survey. Let me know...

Saturday, October 18, 2014

Implement The Core -- Opposite Corners of a Square

Twitter Problem 10-18-14
If (a,b),(-a,-b) are opposite vertices of a square, show that its area=2(a^2+b^2)
EXTENSION: What if (a,b),(-a,-b) are adjacent?
(1) What do you believe will challenge your geometry students here? The abstraction? "Show that"?
(2) Predict how many of your students would "complete the rectangle" by  incorrectly drawing sides || to the axes?
(3) Even if not an assessment question, is it a good strategy to "plug in" values for a&b? This is worthy of more dialog IMO...
(4) How many of your students would question the lack of restrictions on a&b? Would most place (a,b) in 1st quadrant without thinking? So why doesn't it matter!
(5) Is it worth asking students to learn the formula "one-half diagonal squared" for the area of a square?  I generally don't promote a lot of memorization but this one is useful!
Answer to extra question: 4(a^2+b^2).
Ask your students to explain visually why this area is TWICE the area of the original square!

Thursday, October 16, 2014

Implement The Core: Arithmetic Patterns & Generalizations in Middle School Math

As tweeted on 10-16-14...

Pattern #1

Explore on calculator...
Keep going!


Pattern #2

Keep going!

Describe, extend,generalize!

Is 407×9=3663 unrelated?


(1) But these are just math curiosities, Dave. They don't really tie into the Common Core, do they? Well, doesn't multiplying by 11 connect nicely to the Distributive Property:
352×11=352×(10+1)=3520+352 etc.
How about 9?

(2) My goal has always been to expose our students to engaging and meaningful mathematics. But deeper conceptual understanding results from going beyond the "Oh, I get the pattern!" response. That's where the "describe, extend, generalize" and group dialog come into play. Not to mention our guidance!

(3) Students are always intrigued by the mystery and wonder of 9 and 11. How ironic that these 2 numbers put together will forever have a negative connotation for our society. It's important for our  students to understand that many of the "tricks" involving these numbers are directly linked to their juxtaposition to 10, the base of our number system. In base 8, for example, 7 would display many of these properties!

(4) The more inquisitive students can research palindromes like 3663.

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Monday, October 13, 2014

Implement The Core: 'Dates' and 'Figs' - Middle School Investigation

Twitter Problem @dmarain...

Yesterday's date here in the US was 10-12-14: an arithmetic sequence.

(a) List the other 5 such dates this year

(b) List them for 2015 & 2016

(c) Observations & Explanations

In your group make at least 5 observations and/or conjectures. Explain/prove or show they are false.


(1) Observation: There are fewer such dates in 2016 than in 2015.
Possible Explanation: In 2015, the months are the 7 odd numbers  from 1 through 13; in 2016, the months are the 6 evens from 2-12 . There is no 14-15-16.
Note:Would the same be true for all even years from 2014 on?

(2) Conjecture: The middle number in the date is the average (arithmetic mean) of the other 2.
Possible Explanation for Algebra Students: The 3 numbers can be expressed as n,n+k,n+2k. The average of n and n+2k is (2n+2k)/2 or n+k.

(1) Ask students about the phrase "here in the US".
(2) Arithmetic sequences a middle school topic in the Common Core? What about patterns and linear relationships?
(3) Rich discussion of odds and evens
(4) Connections to Geometry: Find a "Pythagorean Triple" date!
Does anyone find the recycled arguments against Common Core Math as ironic and sad as I do? 1960? 1990? 2014? 'Dejá vu all over again' as Yogi would say. Meanwhile students in other countries are bemused as they pass us in the fast lane...

Saturday, October 11, 2014

Implement The Core: f(3)=5,f(5)=5 and much more

Twitter Problem 10-11-14
f is a linear function with f(3)=5 and f(5)=3. f(0)=?
(1) The title has an error and omits the critical linear condition. Note that f(5)=3 not f(5)=5.
(2) The Mathematics Practice Standards ask us to extend student thinking, make connections and go beyond the superficial qualities of a problem.
My hope is that you will see the Twitter problem as a  door marked ENTER not EXIT...
How do we do this?
One possibility is to ask our students to generalize. Note that the responsibility is shifted from us to them. We can guide this by prompting with: "Suppose f(a)=b..."
Here's one possible generalization:
f is linear, f(a)=b,f(b)=a. Show that f(0)=a+b
Since the answer to the Twitter problem is 8, do you believe some of your students will make the connection from 8=3+5?
Yes, this is time-consuming. Some of the best food requires slow cooking! (Sorry for all the metaphors...)
(3) Would you also want your students to relate f(a)=b and f(b)=a to the reflection relationship between the points (a,b) and (b,a)?

Friday, October 10, 2014

Implement the Core-- A binomial activity with connections

Twitter Problem 10-13-14

List the different trinomials which result from assigning 1,2,3,5 to a,b,c,d in all possible ways.
List as follows:,

Explain why there are 12 possibilities!


1) Do you think this type of activity will facilitate factoring? OR factoring involves different skills/reasoning?

2) Activities which connect algebra to other content areas like discrete math (combinations, multiplication principle,etc) are fundamental to the Common Core. While students are practicing multiplication of binomials ( a lower-level algorithm) they are also exercising higher-order reasoning. Do you feel this is overly ambitious for students who struggle with distributive property?

3) Students need to understand that listing the 12 possibilities is not the same as **EXPLAINING WHY** there are 12!

You might challenge them to explain the flaw in the following reasoning:

There are 4 choices for 'a'.
Then 3 remaining choices for 'b, so there are 12 assignments for a,b. For each of these there are two assignments for c, etc. Thus there are (4)(3)(2)(1) = 24 outcomes.

Tuesday, October 7, 2014

Implement The Core:9 rolls of quarters,40 quarters per roll


#CCSS #Grade3-6

9 rolls of quarters
40 quarters in a roll
Total Value?

Sample Student Solutions/Approaches


Student 2
9×40 quarters=9×(10×4) quarters=
90×(4 quarters)=90×$1=$90

1) Which approach is more likely to be used by a 4th grader? 5th? 6th?

2) Which method is more commonly demonstrated in the text or by the teacher?

3) How do you get Student 1 to include UNITS/LABELS like
9 rolls × 40 quarters/roll × $0.25 etc???

4) How many children in Grades 3-6 intuitively use 2nd approach and can solve it mentally?

Of course no one out there is thinking:

"Well those are the 'smart' kids. We don't even have to teach them a method. Besides, their way of thinking is just not accessible to the rest of the group and would only confuse them. So I wouldn't even bring it up..."


Monday, October 6, 2014

Twitter CCSS Algebra Challenge Problem

Higher level of difficulty here. You may want to give this as a team challenge...

If P,Q are consecutive integer values, P<Q, for some integer value of x, what is the greatest possible integer value of a?

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Implement The Core: Delving Deeper into Quadratic Functions

Twitter Problem 10-6-14

The graphs of f(x)=x^2 and g(x)=k-f(x), k>0, intersect at pts A and B. Show that AB=√(2k)


1) To me, Implementing the Core means challenging our students with problems that go beyond the "standard" textbook exercise.

2) In your opinion, what makes this question difficult?

The definition of g(x) in terms of f?
The coordinate concepts?
The algebra (simplifying √, etc)?

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Sunday, October 5, 2014

Implement The Core with RE2PECT!

Twitter Problem 10-5-14

Avg × At Bats = Hits
0.310 × 11177 = 3465
Note: To make this more accurate we would need to ____ the number of walks, sacrifices???

1) Avg=0.310, At Bats=11177, Hits=x
Write equation, solve.

2) Write 2 more problems, solve!

Thursday, October 2, 2014

A Triangle Classic -- So many congruent parts but not enough...

Twitter Problem 10-2-14

∆ABC: AC=6,BC=9,AB=4
∆DEF: DE=6,DF=9
If angle BAC is congruent to angle EDF, EF=?


1. Not that easy to find examples where two NONCONGRUENT triangles have 5 pairs of congruent parts!

2. This might drive home the meaning of important terms like corresponding parts, included vs non-included angles, etc

3. I used this example in the classroom to help students avoid jumping to conclusions! Geometry teachers love the play on words with "ASS-U-ME" but this example may be more about SSA and its reversal!

4. SAS is both a congruence and a similarity theorem/postulate. Once the student draws the diagram and labels corresponding parts carefully the similarity should become clearer. But do you think some students would match up the congruent sides before looking for proportional parts? Let me know if you use this at some point.

5. You might challenge your students to devise other pairs of similar triangles that have 2 pairs of congruent sides and three pairs of congruent angles. Maybe they'll notice the 2:3 ratio *within* each triangle as well as the 2:3 ratio between the triangles! Lots of interesting relationships there. Like 6 is the mean promotional (aka geometric mean) between 4 and 9!