Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: cans p with 20% fewer, calculate p-0.20p=p(1-0.20)=p(0.80) (factoring p shows multiply by 0.80); or sales $50 with 20% discount: 50-0.20(50)=50(0.80)=40 (rewrite reveals multiply by 0.80 for 20% decrease). The correct rewriting $0.80p$ shows equivalence by combining like terms p - 0.20p = 0.80p, makes it easiest to compute as 80% of p, and reveals the decrease relationship clearly. Common errors include $1.20p$ confusing decrease with increase, $0.20p$ which is only the reduction, or $p - 20$ subtracting a flat 20 instead of percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, the expression 3x + 12 can be factored as 3(x + 4), pulling out the greatest common factor of 3 from both terms. The correct rewriting is 3(x + 4), which shows equivalence by reverse distributive property and highlights the common factor clearly. A common error is choosing 3(x) + 12 which doesn't fully factor, or x + 4 which ignores the 3, or 3x(12) which multiplies incorrectly to 36x. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in x=2: 3(2+4)=18, 3(2)+12=6+12=18✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).

Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, buying 4 bookmarks at c dollars each, the total c + c + c + c can be combined as 4c, showing it's 4 times the cost per bookmark. The correct rewriting is 4c, which demonstrates equivalence by combining like terms and clearly shows the multiplication relationship. A common error is selecting $c^4$ which exponents instead of multiplies, or 4 + c which adds instead, or c/4 which divides wrongly. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in c=2: 4(2)=8, 2+2+2+2=8✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).

This question tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, a $50 item with 8% tax is 50 + 0.08(50) = 50 + 4 = 54, or rewritten as 50(1 + 0.08) = 50(1.08) = 54, showing the total as 108% of original. The correct rewriting is p + 0.08p = 1.08p, which shows equivalence by combining like terms and reveals the utility of one multiplication for total cost including tax. A common error is p(0.08) only the tax amount, or p + 0.8 adding flat 0.8, or 8p multiplying by 8 incorrectly. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, with a snack pack at $40 with 30% markup, the selling price 40 + 0.3(40) can be rewritten as 40(1 + 0.3) = 40(1.3), making it efficient to calculate as a single multiplication. The correct rewriting is 40(1.3), which demonstrates equivalence by factoring out 40 and shows the utility for quick mental math on the marked-up price. A common error is choosing 40(0.3) which is only the markup amount (forgetting the original), or 0.3(80) which equals the markup but not the total, or 40+0.3 which adds a flat 0.3 instead of percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: 40(1.3)=52, 40+0.3(40)=40+12=52✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).

Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: perimeter of rectangle with sides 5 and 3 is 2×5 + 2×3=2(5+3)=2(8)=16 (factoring 2 shows twice the sum); or length l width w: 2l + 2w=2(l+w) (rewrite clarifies the structure). The correct rewriting 2(l+w) shows equivalence by factoring out 2 from both terms, clearly reveals the meaning as twice the sum of length and width, and simplifies understanding of the perimeter formula. Common errors include l + w which omits the doubling, (2l)(2w) which multiplies instead of adding, or 2(l-w) which subtracts incorrectly. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, for buying 23 water bottles and 77 sports drinks at $7 each, the total 7×23 + 7×77 can be factored as 7(23 + 77) = 7(100), making the calculation straightforward. The correct rewriting is 7(23+77), which shows equivalence by factoring out the common 7 and simplifies to 700 easily. A common error is choosing 7(23×77) which multiplies instead of adds (wrong operation), or (7+23)77 which adds 7 to 23 incorrectly, or 7×(23-77) which subtracts leading to negative. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in values: 7(23+77)=7(100)=700, 7×23 + 7×77=161+539=700✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).

Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). For example, with a video game price p discounted by 25%, the sale price p - 0.25p can be factored as p(1 - 0.25) = 0.75p, showing it's equivalent to multiplying by the remaining 75% for quicker computation. The correct rewriting is 0.75p, which shows equivalence through factoring and highlights the utility of multiplying by the portion that remains after the discount. A common error is selecting 1.25p, which would be for a 25% increase instead of decrease (sign error), or 0.25p which is just the discount amount, forgetting the original minus that (should be 0.75p), or p-25 which subtracts a flat 25 instead of percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if p=100, does 0.75(100)=100-0.25(100)? 75=75✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80).

Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: notebook $6 with 25% off, calculate 6-0.25(6)=6-1.5=4.5, or rewrite: 6-0.25(6)=6(1-0.25)=6(0.75)=4.5 (factoring 6 shows multiply by 0.75); or shirt $20 with 25% off: 20-0.25(20)=20(0.75)=15 (rewrite reveals multiply by 0.75 for 25% discount). The correct rewriting $6(0.75) shows equivalence by factoring out the common factor 6 to get 6(1-0.25), simplifies to multiplying by the portion paid, and makes mental calculation easy as 75% of 6 is 4.5. Common errors include choosing $6(1.25) for an increase instead of decrease, $6(0.25) which is just the discount, or $6 - 25$ which subtracts a flat 25 incorrectly ignoring percentage. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).

Tests rewriting expressions in equivalent forms using properties of operations—factoring, expanding, combining—to simplify problems or reveal relationships. Rewriting purposes: (1) simplify calculation (7×23+7×77 as 7(23+77)=7(100)=700 easier mentally), (2) reveal relationship (a+0.05a=a(1+0.05)=1.05a shows "increase by 5%" means "multiply by 1.05"), (3) combine for clarity (p+0.08p=1.08p shows total with 8% tax). Apply distributive a(b+c)=ab+ac both directions: expanding (multiply out) or factoring (pull out common factor). Example: compute 5×12 + 5×18 as 5(12+18)=5(30)=150 (factoring 5 simplifies addition inside); or 7×23 + 7×77=7(23+77)=7(100)=700 (rewrite makes mental math easy by summing to 100 first). The correct rewriting 7(23+77) shows equivalence by factoring out the common 7 using distributive property, makes mental math easiest by adding 23+77=100 then 7×100=700, and reveals the shared factor. Common errors include (7+23)77 which factors incorrectly, 7(23)+77 which doesn't factor fully, or 7(23-77) which changes addition to subtraction. Strategy: (1) identify operation needed (factor, expand, combine?), (2) apply properties (distributive for expand/factor, commutative/associative for rearranging, combining for like terms), (3) verify equivalence (plug in value: if a=10, does 1.05(10)=10+0.05(10)? 10.5=10.5✓), (4) assess usefulness (which form easier? reveals relationship?). Percent increase/decrease pattern: increase by r% means multiply by (1+r) as decimal (increase by 15% → ×1.15), decrease by r% means multiply by (1-r) (decrease by 20% → ×0.80). Mistakes: forgetting original amount (a+0.1a≠0.1a, =1.1a), sign errors (subtract distributing positive), incomplete operations (partial factoring/distributing).