Turkey Day stuffing recipe...Have to double 3/4 cup...Three coins...

Happy Thanksgiving and Happy Hanukkah to all out there in cyberspace! The confluence of these two holidays provides plenty of grist for a math blogger but I'm not going there today...

Silly title, perhaps, but to all the experts who will tell us that money is the root of all evil when teaching fraction concepts/skills, uh whatever...

Are there any adults out there who don't want to mess up that Turkey Day recipe and will ask the 'math expert' in the family to verify what doubling three-fourths is?

Three-fourths cup...three quarters...75 cents...double...$1.50...1.5...1 1/2 cups

But, Dave, this won't work for multiplying 5/7 by 2 so this is worthless...

To quote a certain ad,
"Hey, if it works, it's not crazy!"

Seriously, from my perspective, money is as "real" as it gets. A child can be guided to discover the algorithm for multiplying a fraction by a whole number in an endless variety of ways ---
Concrete objects
Fraction pieces and all those other manipulatives
Bar models
Pictures
Simple numerical patterns

But I wouldn't be afraid to use money! Imagine making connections...

How (m^2)/(n^2)=(m/n)^2 is Fundamental to Geometry!

OVERVIEW
The Common Core stresses the importance of students developing a deeper understanding of fundamental concepts and to discover/uncover the interrelatedness of mathematics. The discussion below can be used to demonstrate how a basic law of exponents is tied to the geometry of similar figures.

THE PROBLEM/INVESTIGATION
1) If the sides of 2 squares are in the ratio 2:1, show that their areas are in the ratio 4:1
(a) visually
(b) numerically by examining particular cases
(b)  algebraically

2) If the sides are in the ratio 3:1, do you think the areas will be in the ratio 6:1 or 9:1? Now do parts a-c as in 1).

3) If the ratio of the sides is 3:2 show algebraically that the ratio of the areas is 9:4.

4) Show algebraically that if the ratio of the sides of 2 squares is m:n then the ratio of their areas is (m/n)^2.

Note: How does this result connect to the idea that the area of a square varies directly as the square of its side length?

4) If squares are replaced by circles using radii or diameters in place of "sides" show that the results of questions 1-4 are the same.

How does this result connect to the idea that the area of a circle varies directly as the square of its radius or diameter (or circumference)?

REFLECTIONS
• Squares and circles are of course special cases of similar figures. Beyond this investigation lies the BIG IDEA:

The areas of 2-dim similar figures are proportional to the squares of their linear dimensions.

Note: In 3 dim, we can replace 'area' by what?

• Do you see this as one of the fundamental theorems of Euclidean geometry? Is it sufficiently stressed in textbooks and in the standards? Of course you may not feel as I do about all this!

• So what is the geometry connection to
(m/n)^3 = (m^3)/(n^3)...

'.

Today's MathNotations # is 972...

OVERVIEW AND THE PROBLEM
As posted on Twitter (@dmarain), student's task is to list as many interesting properties/facts about 972 as possible, at least 10.  I posted a couple of possibilities but you may not want to give both of these away...

1) Prod of a perfect sq and a perf cube
2) a perf 5th power × a perf sq
...

REFLECTIONS
• Appropriate grade level? 3rd? 4th? Middle grades?
• 10+ properties way too much? Why do you think I chose that number? Depends on grade level? Abilities?
• What do you see as the principal
learning goals/benefits here? How highly would you rank "Expressing mathematical ideas/number properties in words?"
• We might naturally have students work in teams but I tended to respect that individual student who chose to work alone. Would you make this competitive to stimulate interest or that's not desirable?
• Beyond making a list of properties, what else should be expected here? How important is it for students to check their results? How about independent verification from another team member?

• Allow use of calculator throughout or after a little while?
• How is this different from MAA's "Today's Number is"?
• Anyone recall the last number I asked about a long time ago? [97 and 153!] You may want to click on the link...

Six parking spots were assigned randomly. What is the probability that...

OVERVIEW
No matter how many of these appear on standardized tests, a large per cent of test takers continue to get these wrong. Teachers will teach that unit on combinations, permutations, Multiplication Principle and rules of probability. But learners will still struggle and even if they survive the chapter test, they will probably get the following problem wrong. But some of you may have overcome this...

THE PROBLEM

Six parking spots are assigned randomly to six employees. If Jake and Alex are 2 of these employees, what is the probability they will be assigned to the first 2 spaces?
(A) 1/60 (B) 1/30 (C) 1/15 (D) 1/6 (E) 1/3

Answer: (C)

REFLECTIONS

• It's so simple: (2/6) (1/5) = 1/15  Next...
Of course that would never happen in a classroom!
How much understanding of probability principles and practice is required to feel comfortable with this efficient approach? More importantly when would this approach fail or need to be revised?

• So is it combinations? Permutations! Do we also need the multiplication principle?
How much experience do students need before being able to write
(2P2) (4P4) ÷ (6P6)

• Those who have experience teaching these will have developed their favorite instructional strategies. Please share!

"LEAST number to select to be CERTAIN of getting at LEAST..." -- Those annoying logic problems!

OVERVIEW

You've seen these on SATs, GREs, etc..
Where exactly does this fit into the Common Core?

Should the underlying Pigeonhole Principle and/or specific strategies/methods be taught to middle or high schoolers?

You can guess my feelings about this...

THE PROBLEM
There are thirty solid-colored scarves in 30 identical unmarked closed boxes, one per box. The 30 scarves consist of:
6 red
7 blue
8 yellow
9 green

What is the LEAST number of boxes that need to be opened to be CERTAIN of getting at LEAST

(A) 2 of the same color [5]
(B) 4 of the same color [9]
(C) one of each color [25]
(D) 4 of each color [28]
(E) 2 green and 2 blue [25]

REFLECTIONS

• Oh those nasty but critical keywords which I wrote in uppercase. Initial instruction must focus on developing meaning for these.

• Those who have had experience teaching this may want to share their strategies. Here are some I've used...

* Act it out with coins in a bag or any concrete objects. Let's say the goal is to get at least one of each color and the student or you selects 4 boxes or objects. I might say, "Do you want to stop? This is surely the LEAST number!" Were hoping students will respond with, "But we can't be sure!"
OR I might grab all of the boxes and say, "Ok, now we are CERTAIN!" Hopefully someone will respond...

* Without getting into the Pigeon Hole Principle I have used what I call the WORST POSSIBLE LUCK (CASE) approach. For example, if the goal is to get one of each color the WORST CASE would be getting the same color repeatedly. Don't like that approach? This is subtle and requires patience, time and several examples. Some learners will grasp it way before others but acting it out with objects and making a game out of it really has made a difference for my students.

Your thoughts?

When Mom was 40, Son was twice as old as Daughter... Age Problems and Singapore Bar Models

THE PROBLEM
When mom was 40, her son was twice as old as her daughter. Now her daughter is 28 and mom is twice as old as her son. How old is mom now?

Answer: 64

REFLECTIONS
Not exactly an authentic real world assessment but there's probably a reason why these are "Problems for the Ages"!! I don't believe harm is done by having students develop these kinds of relationships...

Of course for years I used to teach this using a chart and setting it up algebraically.

For younger students I would encourage a Guess-Test-Revise approach. Say, girls was 3 so boy was 6. Now it's 28-3 = 25 years later, so boy would be 6+25=31 and mom would be 40+25=65 which is a little more than twice 31. Revise to girl was 4, son was 8. 24 years later, girl would be 28, boy would be 32 and mom would be 40+24=64, Bingo!

Ah but children in Singapore perhaps as early as Grade 4 are representing these relationships using unit lengths and bar models. I simply don't have enough experience doing this so those out there with more knowledge please improve upon this so I can learn!

Then
Mom |-----40------|
Son |----||----|
Daughter |----| (some unit length)

Note: [//////] below represents the additional years from Then to Now

Now
Daughter |----| [//////] = [--28--]
Son |----||----| [//////]
Mom |----40---| [//////] = |----||----| [//////] |----||----| [//////] = |----||----||----| [//////] [--28--]
So 40 = 3u + 28 or u = 4 yrs, etc.

I'm limited by my graphics but I believe there are different,  better and more efficient models used in Singapore Math. If you're thinking I retrofitted an algebraic solution to make it look like a bar model you're probably right! I need wiser and more experienced 'bar modelers' to help me!

Mean Equals Median Problem

THE PROBLEM

5, 11, 19, 22, x
If the mean and median of these 5 numbers are equal, determine all possible values of x.

ANSWER
-2, 14.25, 38

REFLECTIONS

• If the question asked for one possible value for x, this could be an SAT Grid-In question of above-average difficulty. I chose to ask for ALL possible values not only to make it more challenging but also to encourage students to probe more deeply.

• Thinking of many strategies how YOU would solve this? But that's not the focus of this blog. After WE figure out how to solve the problem ourselves and, yes, we should try to find more than one method, we should be asking ourselves as educators:

What questions need to be asked to enable our students to solve it for themselves?

What learning environment best facilitates this?

• Do you believe some of your students would reason that the median could only be 11, 19, or x itself? Listen to their discussions. Give them time but if no one is thinking that way we might ask:

Which numbers could NOT be the median? Why can't 5 be the median? 22?

• Beyond solving this problem is an even bigger question:
What must be true about a set of data values if their mean (avg) equals their median? We know there are sufficient conditions like:
The data is normally distributed about the mean, in which case, the mean, median and mode coincide.
But this is NOT NECESSARY!

In the above example, you might want to ask students to examine the case where x=14.25:

"What is the mean of the numbers if x is NOT in the set? Why does this make sense?

[If you have an 80 average and one of your scores was exactly 80, what would happen to the mean if you removed that score?]

So what does it 'mean' if the mean and median of a set of data are equal?

Now that's an open-ended assessment question!

'Half as many as' vs 'Twice as many as' Revisited

OVERVIEW
Hundreds of math posts over the last 6+ years and my most popular post is still by far "There are twice as many girls as boys...".
http://mathnotations.blogspot.com/2008/08/there-are-twice-as-many-girls-as-boys.html

Still hard to do word problems without coping with the vagaries of language and grammar in particular.

Contrast phrases such as

'Half OF the girls' and 'Half AS MANY Girls as' OR

'Twice as many X's as Y's'

These are confusing enough for native English speakers. Imagine the terror if you're not! Phrases like these don't respect anyone!

THE PROBLEM

Attendance at a stadium broke down as follows...
Of the children, half as many girls as boys
Of the adults, half as many women as men
The number of females was what fractional part of the total attendance?

(A) 1/2  (B) 1/3  (C) 1/4  (D) 1/6  (E) cannot be determined from info given

Answer: B

REFLECTIONS

1. Although the title of the post suggests the focus was on interpreting the phrase "as many as", there are some significant underlying ratio concepts here.

2.  As I discussed in my earlier post, I've observed that students do better with the semantics if they first decide which is the larger of two quantities. Thus "half as many girls as boys" hopefully suggests that there are fewer girls but my experience is that some are simply blocked by the sentence structure and will guess randomly or say nothing. Remember we can always go back to concrete numerical relationships:

"Ok say there are 12 boys.
If there are half as many girls, then how many girls?"

Note how I not only used numbers but I also inverted the problem by giving the number of boys first. Yes we are also teachers of reading! Will students do this on their own? If trained!!

3. From this point there are many solution paths and I would definitely allow upper  elementary or middle school students to play with this for a few minutes in their groups. There's no rushing this process.

We can simply explain our method to them but this is only a part of their learning. Of course that's my opinion and there are many out there who would cringe at this. The eternal battle between The Direct Instructionists and the Constructivists! Those are just labels about which I care little. Whatever works... Since I can speak from 40 yrs of experience I know what worked for my students...
Besides I've already debated this ad infinitum and ad nauseam with some of the best. No one ever changes their mind!

4.  There are some subtle part:whole ratio concepts embedded here. Isn't it tempting to pick choice (E) here because we're not given the Adult:Child ratio. But the question asked for the ratio of the combined female to the total. It is very instructive to see this algebraically.

5. 'Plugging in' convenient numbers for the subgroups in this problem makes it accessible to 4th-5th graders. Organization is very helpful. I use a tree model to represent this kind of data but most do not do this.  Say there are 2 girls, 4 boys; 5 women, 10 men. Then the number of females (7) is still one-third of the total (21)!

6. Hopefully my readers will suggest more efficient ratio methods, Singapore models, other algebraic representations, etc... OR no comments at all!

Divide 3 Pizzas Equally Among 4 Children -- Developing Division Series Common Core

OVERVIEW

Note: You may want to first read #7 below in Reflections to clarify the suggested approach.

Some of you may be averse to using pizzas for concept development of division, fractions, ratios,... I've seen criticism of pizzas for awhile!

Similarly some educators don't like using money to develop fractions and decimals.

Here's my philosophy --
Whatever works...
I don't believe children's minds will be permanently damaged from pizzas and money!

THE PROBLEM
[Appropriate Grade Levels: 2+]

Directions to groups: Show at least 4 ways to share the 3 paper pizzas as described in the title.

Then write in words the part of one whole pizza each child gets.

REFLECTIONS...
1) Anything missing here in building division/fraction concepts?
I believe so!

BEFORE THE CHILDREN START SHARING PIECES,
WHAT QUESTION COULD WE ASK TO BUILD FRACTION SENSE, ESTIMATION, etc.,??

One possibility...
Teacher: Before you start drawing lines, choose from one of these:
(Write your choice on your paper in the next 5 secs)
(A) Each child will get more than one whole pie.
(B) Each child will get less than one whole pie.
OK, now defend your answer to your partners in the next minute.
Anyone change their first answer?
So the majority chose (B). Why?

2) Do you believe for young children it is important to give them the time for these kinds of activities?

3) What and how many prior division experiences do you think children need before tackling 3 divided by 4? Do you believe some children are ready from the beginning?

4) I always find an original approach from listening to children. If you choose to use this activity, what 2-3 methods do you think most children would use?

Do you think a few children will immediately divide each pie into quarters? What might they say if you ask them why they did that?

5) How would you enable the TRANSFER of learning from concrete manipulatives to WORDS (verbal representation) to the SYMBOLIC?
Sample...
So you have been dividing 3 pies equally among 4 children.
Let's say that together...
Now let's write 3 pies ÷ 4 children =
Three-_______ of ------ for ----- child =
3/4 pie/child

There's so much more to this. These are just a couple of suggestions which you could modify and improve upon!

6) How would you assess their understanding during the lesson? At the end of the lesson? The next day? On a written assessment or would you consider an individual performance assessment (Show me how you would...)
How do you think PARCC would assess the important ideas here?

7) Of course many of you are thinking:

Why not simply show them 3×8÷4=6 since most pizzas are cut into 8 slices? 

Well, I'm suggesting we introduce this to 7-8 yr olds before they learn multiplication and division. Secondly, the children are given BLANK circles to encourage creative open-ended approaches. I want them to experiment with dividing the circles into halves, quarters, eighths, etc...

Remember, as always, I'm writing these for my fellow educators to reflect on their practice, now in the context of the Mathematical Practices of the Common Core.

Been doing this now for almost 7 years on this blog and invariably these kinds of posts are viewed but not commented upon. Why do you think that is? Obviously a rhetorical question!

TEACHING DIVISION: I said to make 2 equal groups not groups of 2!

Ever experience this in the classroom working with elementary/middle school students, particularly with LLD students?  I believe to some degree most of us have some language processing issues, often connected to auditory processing issues and complicated by attentional deficits.

I recently observed this with a few students and my thought was that there is a developmental stage which precedes this question. Next time you're teaching division and you encounter this you might want to step back and assess the child's understanding of
NUMBER OF GROUPS vs
NUMBER IN EACH GROUP

A misunderstanding here could be a barrier to conceptual development of multiplication and division.

One strategy is to show the child 4 groups of 3 counters either with objects (preferably) or on paper. Keep the groups separated. On paper one could simply loop them.  Physically, you could put the counters in separate containers or put some kind of ring around them.

"So, how many groups do you see?
How many in each group?"

Most should get this but, if not, you know there's a  language barrier which must be addressed.

Now what? Ready for division questions? Not quite...
Have them now CONSTRUCT groups physically, then on paper.
For example have them "make" 3 groups of 4 from the original setup.

Some youngsters need considerable practice and reinforcement.

What do you think? Have you had similar experiences? Found another way to cope? A different theory about the underlying problem? Remember I'm just sharing my observations and conclusions. They're just mine...

International call costs x¢ per min for 1st n min, then y¢ for...

I can't believe I'm actually posting again. This won't last!

OVERVIEW

Developing abstract reasoning needs to start early and often and it's founded on a strong foundation of arithmetic/quantitative reasoning. That is, children normally learn to generalize from several concrete numerical examples before patterning takes place. Seems too obvious? Well, at what point would you expect a majority of your algebra students to do this successfully?

THE PROBLEM

Phone carrier charges for an international call are x¢ per minute for first n minutes, then y¢ per min for each additional minute or part thereof. Write an expression for the cost, in dollars, of a call lasting z minutes, z>n. Assume n,z are positive integers.

REFLECTIONS
Appropriate level of difficulty for Algebra 1? Algebra 2? Only for the 'honors' kids?

Too much expense of time to get students to be able to do this? Not worth the effort?

How many of this type are your algebra students normally exposed to? From the text? From teacher-constructed materials?

How many of these would most students need to practice to be proficient?

What % of your 'average' middle schoolers could solve this quantitatively, i.e., if all variables were replaced by numbers?

What are some of your favorite instructional strategies for these kinds of 'literal' word problems? By the way, this is the primary reason for my posting this!

What do you see as the main challenges to student performance on these kinds of assessment questions? Do you place 'understanding the question' high on this list?

Which of the 8 Common Core Mathematical Practices come into play here? By the way, do you have your own 'laminated' copy of these practices in front of you at all times! Here's the link:
http://www.corestandards.org/Math/Practice

Sorry, the 'answer' will have to come from someone who comments! I always assumed one of my students or a group would solve the problem...