Last week, in Number Sense 023, we took our toy base four number system, figured out its multiplication table, and did a multiplication problem. In California, and most of the rest of the United States, there are standards for teaching mathematics. One of California's standards (and I imagine all states with standards have one similar) is
Communicate precisely about quantities, logical relationships, and unknown values through the use of signs, symbols, models, graphs, and mathematical terms.
We see a lot of modeling in this series, on a more or less informal basis (have you ever met a formal goat?) This week, I'd like to take a look at ways to model mathematics problems using simple diagrams developed from the number line.
Paul is 8 years old. Nick is 2 years younger than Paul. How old is Nick?
Typical elementary word problem: two facts involving numbers, and a question. Let's put it on a number line:
Paul is 8 years old.
Nick is two years younger than Paul. If we subtract two years from Paul's age, we get Nick's age.
How old is Nick? We don't know yet. This is the number we have to figure out. Since we don't have a known number to model on our number line with an arrow of fixed length, we will use something else to represent our unknown number. We will use a goat.
We can simplify this model. One simplification is using simple rectangles instead of arrows. (We can give the goat a rectangle, too, filling the gap between zero and the red rectangle.)
We can actually do without the number line
Finally, we can replace the goat with his initial b, (this is Billy Goat) and label the other two rectangles with the appropriate number
We can see from this model that the two rectangles on the bottom are the same length as the rectangle on top. This means
This is a model of an addition fact family. The other facts in this family are
That last fact give us an arithmetic calculation we can do to find the value of b, which is Nick's age. (Yes, I know you are WAAAY ahead of me here, having solved this problem in your head ages ago. What I am really showing you is what went on in your head, probably to quickly for you to notice it, and giving you a way to show your thinking in a diagram on paper.)
Jessie rode her bicycle 12 miles more than Sam did last week. Sam only rode his bicycle 22 miles. How many miles did Jessie ride?
This time, Jessie's total miles is the goat. Since Billy Goat wandered off to eat some Kudzu, we will ask Sally Goat to help us out.
Either of the first two facts in the fact family will let us figure out s, the total number of miles Jessie rode her bike last week.
Next week we will see what we can do with a train leaving from Boston passing a train leaving from New York.
Have fun in the comments.