Purple Math Age Problem

I've been assessing Pamela's word problem solving abilities and I'm pleased with what she can do with simple one step problems requiring whole-part thinking. In the process of assessing her, I've been transitioning her to the Singapore Math bar model concept that I explained on Wednesday. First, I assessed if she recognized whole and parts in ordinary situations that she understands. Pamela loves making her own lemonade, so this set-up was easy for her to do. I gave her this:

She easily set-up a whole-part model.

I gave her several problems like this and she set every single one up correctly. Then I gave her addition (unknown whole) and subtraction (unknown part) word problems based upon bar models she set up. Here is the one for lemonade problem:

Lemon juice has 11 calories. Lemonade has 165 calories. Sugar has 154 calories. How many calories does water have?

She correctly recorded the numbers in the model and figured out that water has 0 calories.

Once I was satisfied Pamela understood these problems, I gave her problems with distractors such as the following:

Mom baked the crust for 10 minutes and the filling for 55 minutes. She read a book for 15 minutes. How long did it bake?

She spotted the distractors right away and began crossing them out. Not all problems had them. I think from now on, I will include distractors occasionally so that she realizes she doesn't have to use every piece of information in a problem. Now, I plan to assess how well she applies the bar model graph to these concepts. I have a feeling we will be charting new territory in some cases: comparison, change, remainder, equal, excess value, repeated value, constant difference, constant quantity, and constant total.

The reason why I am so excited about helping Pamela learn to think in pictorial models is that they can solve many word problems covered in algebra without using letters and numbers. Once she understands them pictorially, I suspect the transition to letters and numbers will be easier for her. Here is an age problem I found at Purple Math. Notice that Purple Math sets up a system of equations and solves it. Now compare their method to a pictorial method, that is not quite Singapore Math and not quite Jacob's Elementary Algebra.

In January of the year 2000, I was one more than 11 times as old as my son William. In January of 2009, I was 7 more than 3 times as old as him. How old was my son in January of 2000

We will represent William's age in 2000 with an empty box:

2009 is 9 years later, and you can represent his age in this way:

In 2000, his mother was 1 more than 11 times his age, which is 11 empty boxes. You would add 1 to that to represent her age:

To represent her age 9 years later, increase the 1 to 10:

This table organizes all the information except for the last relationship we will analyze next:

In 2009, his mother was 7 more than 3 times as old as his age in that same year. So, you have to triple that age by writing the empty box and 9 3 times and add 7 to that:

We also know her age in 2009 was 11 empty boxes and 10 from the table we set up. All we need to do now is rearrange boxes and numbers until it works:

Rearrange the first line to match up the elements better. Compose 9, 9, 9, and 7 to make 34 and decompose that to 10 and 24.

Separate 24 into 8 equal parts yields 3.

That means an empty box is the same thing as 3. William was 3 in 2000. His mother was 34. In 2009, he was 12 and she was 43. Her age in 2009 (43) is 7 more than 3 times his age, or 7 + 3x12.

Now, here is the challenge for anyone wishing to try. Can you solve the other next age problem at Purple Math through pictures? If you email me your work, I will include it in my next math post.

In three more years, Miguel's grandfather will be six times as old as Miguel was last year. When Miguel's present age is added to his grandfather's present age, the total is 68. How old is each one now?

Pictorial Ways of Solving Algebraic Problems

A farmer had twice as many ducks as chickens. After the farmer sold 413 ducks and 19 chickens died, he has half as many ducks as chickens. How many ducks does he have now?

I have been studying ways to solve a simple problem in algebra pictorially rather than the traditional methods as I think through how to teach Pamela how to do this one day. Since I know many folks scared off by high school math occasionally pop into my blog, I was wondering if this makes more sense to you than the traditional method which I list at the end of this post. This pictorial method incorporates two strategies: (1) working backward from the answer and (2) incorporating the techniques taught in the first four chapters of Jacob's Elementary Algebra. Pamela has been using an empty box for the unknown for years, based on how Making Math Meaningful teaches whole-part thinking.

Ducks
We want to know the number of ducks the farmer has now, which is our unknown, represented by the empty box.

We know that the farmer used to have 413 more ducks than he has now because he sold that many. So, at the beginning of the problem, the farmer had whatever number he has now and the 413 he sold.
 

Chickens
We want to know the number of chickens the farmer has now. We know that he has half as many ducks as he does chickens. If he has 8 ducks, he would have 16 chickens. That means that the number of chickens is double the number of ducks. Whatever the number of ducks is now, the number of chickens is twice the amount.

We know that the farmer used to have 19 more chickens than he has now because that many chickens died. So, at the beginning of the problem, the farmer had whatever number of chickens he has now and the 19 that died.


Summarize the Quantity of Ducks and Chickens Before and After

We were given one more relationship: at the beginning of the problem, he had twice as many ducks as chickens. If he had 8 ducks, then he had 4 chickens. That means the number of ducks at the beginning was twice the number of chickens. Twice the number of chickens at the beginning would be double of what is in the table, or


We can rewrite twice the number of chickens as,


The number of ducks at the beginning of the problem is the same as double the number of chickens,


We can decompose 413 in a way to help us see the answer:
413 = 375 + 38 = 125 + 125 + 125 + 38



That means 125 goes into the empty box, which is the number of ducks that the farmer has now.

To check our work, we can plug 125 into the empty boxes into the table. The number of chickens now is twice what was in the empty box, 2 x 125 or 250. The number of ducks at the beginning was the empty box and 413, 125 + 413 or 538. The number of chickens at the beginning was the twice the empty box and 19, or 2x125 + 19 or 269.

Does this make sense? At the beginning of the problem, the number of ducks (538) is twice the number of chickens (269). At the end of the problem, the number of ducks (125) is half the number of chickens (25). All relationships make sense!

Other Methods:
I found this problem at a Singapore Math blog, which offers multiple problem solving strategies that are more pictorial than the traditional method. The author linked to a more thorough explanation of Singapore's model method that you might enjoy.

Traditional Method:
Let c be the number of chickens and d be the number of ducks at the beginning of the problem. Since there are twice as many ducks as there are chickens at the beginning,
d = 2c

Since the farmer sold 413 ducks, the number of ducks at the end of the problem is the expression,
d - 413

Since 19 chickens died, the number of chickens at the end of the problem is the expression,
c - 19

We also know that now there are half as many ducks as chickens, so we can write an equation for this relationship:
d - 413 = ½(c - 19)

Substituting the first relationship between ducks and chicks, we can now solve for the number of chickens at the beginning of the problem.
2c - 413 = ½(c - 19)

2(2c - 413) = 2[½(c - 19)]

4c - 826 = c - 19

4c - 826 - c = c - 19 - c

3c - 826 + 826 = -19 + 826

3c = 807

c = 269

If the number of the chickens at the beginning of the problem was 269, the number of ducks was twice that, or 538. If the number of ducks at the beginning of the problems was 538, the number of ducks at the end is 413 less than that, or 125.

To check our work, if the number of chickens at the beginning of the problem was 269, the number of chickens at the end is 19 less than that, or 250. Since 250 is twice 125, the number of chickens is now twice the number of ducks.

You Know You're a Geek When You're IM'ing about Factoring Polynomials

Two posts in two days? Pass the smelling salts!

I've been busy writing what amounts to a fifty-page research paper on the teaching of mathematics. It has been eating up much of my time and distracting me from my blog, and I do it gladly. On top of that, Google has decided to delete old videos next month, forcing me to transfer my blog videos to You-Tube and download my private ones. It's a good excuse to attach labels and revamp the blog. The timing stinks!

Wednesday, I tutored an intelligent young woman taking college algebra. She was struggling with factoring polynomials, so we spent over an hour working through problems. My primary goal in tutoring math students is to put them on a search for meaning. If they understand why they are doing what they are doing, they will be more likely to remember it. Even if they forget, meaning helps them reason their way through a problem. If I sniff any hint of wavering, I will ask them why something is true.

My friend had to solve 9x⁵ - 9x³ = 0. She had no problems with step one, dividing both sides of the equation by 9 to get x⁵ - x³ = 0 and knew to pull out x³. When asked what x⁵ divided by x³ was, she made the fatal mistake. She raised her eyebrows and questioned, "X squared?"

Me: "Are you sure about that?"

Her: "Our teacher said you subtract."

Me: "Did she explain why?"

Her: "No. She's old school. She just tells us how to do it. She doesn't have time to explain why."

Sigh. She is smart. She is perfectly capable of understanding why. People who disrespect motivated students enough not to explain why bug me. So, we headed down the path of meaning, peeling back her uncertainty until we reached something solid.

Me: "What does x⁵ mean?"

Her: "You times x by itself 5 times. You know, x times x times x times x times x."

Me: "Good. Any time I'm unsure about a procedure, I start thinking about meaning. If you freeze on a test and forget whether to add, subtract, multiply or divide, you can always fall back on meaning. Write it out the long way."

So, she wrote x⁵ ÷ x³ = (x ∙ x ∙ x ∙ x ∙ x) ÷ (x ∙ x ∙ x). Then I showed her how that is just like saying x ∙ x ∙ (x ÷ x) ∙ (x ÷ x) ∙ (x ÷ x). Then her face lit up, "Oh! Then you can cancel and get 0."

Believe it or not, that is a misunderstanding because we throw around words without meaning and precision and end up confusing students. I responded, "No. Lots of students do that. Let's go back go meaning. What does x divided by x mean?"

Her face went blank. Yes, I know I'm a pain, but this is important! Math makes sense when taught properly. So, I peeled the onion back further. I said, "Sometimes, it is easier to think about numbers. What does 5 divided by 5 mean?"

Another blank stare. The way we teach math focuses on doing, not thinking deeply. I explained, "Dividing means putting objects into equal groups. Suppose you had to share 5 cookies with 5 people. How many cookies would each person get?"

Her: "Oh, 1!"

Me: "What if you shared 10 cookies equally with 10 people?"

Her: "They'd each get 1."

Me: "What if you shared a million cookies equally with a million people?

Her: "They'd get 1!"

Me: "Now, let's get back to x divided by x. What does x mean?"

Her: "I don't know."

My friend answered correctly without realizing it. I explained to her that we use x to represent a number we don't know. It's a placeholder that means a number that we don't know. Having placeholders allows us to set up relationships between known and unknown numbers and manipulate them to figure out the unknowns or refine those relationships. I added, "We have a number of objects and the same number of people. We'll call that number x. If we have x pencils to give to x people, how many pencils would each person get?"

Her: "It's 1."

Me: "What is x divided by x?"

Her: "It has to be 1."

Then, everything fell into place, and she understood:

x⁵ ÷ x³ = (x ∙ x ∙ x ∙ x ∙ x) ÷ (x ∙ x ∙ x)

= x ∙ x ∙ (x ÷ x) ∙ (x ÷ x) ∙ (x ÷ x)

= x² ∙ 1 ∙ 1 ∙ 1

= x²

We had to peel the onion for only a few more glitches. My friend said she had a much better understanding. It disappoints me to know how much rote, meaningless instruction is happening in the math world.

Grace in Geometry

Lisa Cadora's blog post on Grace and Learning got me to thinking about David and his checkered past with math. She described the frustrations of teaching herself to crochet a cool, hip accessory and how much more gracious we are with ourselves than with our students. She concludes,

Charlotte Mason said that the only education is self-education. Did she see that grace is necessary for learning, and that we are most graceful with ourselves? If so, maybe it’s not only that we as teachers must create gracious, grace-ful conditions, environments and relationships in which our students can learn, but that we must bring them to be gracious to themselves.

My husband has two engineering master's degrees, and I have one in statistics. For many years, I thought the math gene had skipped my fifteen-year-old, neurotypical son, David. His temperament is very much like that of my father, who has never met a math problem he liked. Teaching David elementary school math frustrated us both. In hindsight, I think I was part of the problem. I think sometimes, if I had shown more grace, we would have shed fewer tears. Fortunately, he finds algebra and geometry a breeze. Was it maturity and a leap in abstract thinking or a more gracious attitude from me?

I think grace in learning might be related to masterly inactivity (wise letting alone). Elements of masterly inactivity include "authority, good humor, confidence, both self-confidence and confidence in the children," which I lacked because I assumed David would always struggle with math like my father. I stopped looking him as a unique person and saw him as a mirror image of my father because they have so many personality traits in common.

Charlotte Mason believed that we should be gracious enough to let children take personal initiative in their work (page 37-38):

In their work, too, we are too apt to interfere with children. We all know the delight with which any scope for personal initiative is hailed, the pleasure children take in doing anything which they may do their own way; anything, in fact, which allows room for skill of hand, play of fancy, or development of thought. With our present theories of education it seems that we cannot give much scope for personal initiative. There is so much task-work to be done, so many things that must be, not learned, but learned about, that it is only now and then a child gets the chance to produce himself in his work. But let us use such opportunities as come in our way.

On the flip side of this coin, we hurt our children by letting them get so frustrated that they develop the habit of tears. I think we also must keep in mind scaffolding, being alert when to step in and support the child and when to step out a la masterly inactivity. The geometry problem above is a great example. David had to figure out the measurement of each angle in the problem, based upon the diagram and information provided. He had to apply the definitions of bisected angles and right angles, the relationship between vertical and supplementary angles, and the sum of interior angles for triangles (180 degrees) and quadrilaterals (360 degrees). What made this problem difficult is that one wrongly calculated angle would create a domino effect of errors.

Applying masterly inactivity, I left David to his own devices. He worked his way through the calculations and figured out the angles for about five shapes before coming to me because the problem stopped making sense. Then, I switched to scaffolding and congratulated him for recognizing when he was stuck. I studied his work and noticed an error. I erased all of the mistakes and highlighted what was correct, and he went back to work. He went back and forth with me several times, getting frustrated at himself for his errors. Rather than joining him in his vent, I told him about Lisa's blog post about giving yourself grace when making mistakes. I even emailed it to him later in the day. I reassured him that the problem really was challenging and got him back on track.

In the last round, he made another little mistake and I decided to put all of the formulas into a spreadsheet to make sure I was on the right track, too. As I built the spreadsheet, I realized how complicated the problem was. At that point, I was so thankful to have read Lisa's post that morning and let grace win the day.