Tuesday, May 24, 2011

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?


Ellen said...

You are an extraordinary teacher, Tammy!!!

The Glasers said...

Thanks, El. I guess all those years of guiding plebes through three semesters of calculus taught me to make things simple!

Stranded said...

Hey Tammy
Do you make these pictorials?? They are awesome. What do you use to make them?

The Glasers said...

I use a spreadsheet program to make them (putting the shapes in each cell). Then I cut and paste it into paintbrush, crop it, and save the image.

Stranded said...

brilliant. Thanks.

The Glasers said...

Care to submit an answer to the last question?

Bueller, Bueller, anyone?????

Anonymous said...

OK Tammy, Here it goes. I don't know how to make the boxes or circled numbers in this comment box, so I will write out the words. I think this is the method you showed, but I might have reverted back to traditional algebra...
Let Miguel's age last year be an empty box.
So Miguel's age this year is a box and a 1.
Grandfather's age in 3 years is 6 boxes.
Miguel's age plus his grandfather's age this year equals 68 equals the box and a 1 and 6 more boxes minus 3.
So combining all of that, their ages this year = 68 = 7 boxes minus 2.
So 70 = 7 boxes.
So each box is 10.
Miguel is 11 this year (box and 1).
Grandfather is 57 (6 boxes minus).