Common Core Standards: Math
Expressions and Equations 7.EE.B.3
3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
The pitcher shuffles the dirt on the mound, tossing the baseball lightly into her mitt while she considers the situation. The catcher gives her the signal for Hit 'em with a fastball, but she knows better: the batter she's facing will be expecting that. She squints into the sun and shakes her head no. Then the catcher flashes the How 'bout a curveball? sign. Again, our heroine shakes her head almost imperceptibly. Too easy—this is one clever batter at the plate. Finally, the catcher does a complicated little hand-signal that means Change-up, and the pitcher grins, nods, and lets the ball fly. Stee-riiiike!
Baseball players use a variety of different strategies to obliterate their opponents, and multi-step word problems are the same way—each one's different, and we need to decide which math-pitch will help us defeat them. Sadly, chanting, "Hey batter, batter," doesn't seem to faze them.
This standard is sort of a catch-all for all the different operations, properties, conversions, and tools your students can employ in their epic battle against real-world story problems, so it's a good idea to throw in a lot of review. You'll want to include increasingly complicated rational numbers, chock-full of decimals and weird fractions. Spend some extra time going over addition, subtraction, multiplication, and division of rational numbers (positive and negative), and your students should be able to knock this standard out of the park.
Speaking of which, you wouldn't play baseball with shoulder pads and a volleyball net. Similarly, the tools your students use will depend on the problem. Charts and tables are usually an awesome idea when they've got tons of different values to compare (especially with money problems). For visual learners, drawing a picture or diagram can be massively helpful, particularly when there's a bunch of different physical objects involved. And equations are always a great way to keep things neat 'n' simple when we've got a variable or unknown quantity in the mix. (We'll dive deeper into this in the next standard.)
Another big focus in this standard is that students should use their own logic to make sure their answers make sense in the real-world context of the problem. We want 'em to make quick, approximate deductions like a Victorian gentleman-detective.
There are a few different ways students can double-check that their answers are roughly in the right ballpark: rounding, front-end estimation, and clustering are all solid options. (And remind your students never to forget the importance of units, too!)
Rounding: Round stuff to the nearest easy number. If you're solving 97.34 × 807.339, round both terms to get 100 × 800 = 80,000. So you know your actual answer should be close-ish to 80,000.
Front-End Estimation: Ignore everything except the highest place value. If we're trying to find our hourly pay when we made $843.25 in a 38-hour work week, divide $800 by 40 hours instead to get a rough estimate.
Clustering: If all your quantities are decently close to one another in value, use the same average number for each term to get an approximate answer. Let's say Jim drove 238.7 miles on Monday, 244.1 miles on Tuesday, 240.6 miles Wednesday, 237.6 miles Thursday, and 242.9 miles Friday. If Jim wants to know how far he's driven on his road trip this week, just round 'em all to 240 miles and do 240 × 5 = 1200 miles.
Yep, we're literally teaching them how to guess more accurately. They'll still need to get an exact answer, of course, but estimating is a really, really helpful method for students to check themselves (before they wreck themselves). We're in the big leagues now.
- GED Math 2.2 Expression and Equations
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