Multiplying by a 2-Digit Number — Part 2
4A · Chapter 4 · Lesson 6 · MATH-4A-C04-L07
Learning goal. Connect area models to efficient written two-digit multiplication with carries.
Before you begin
Recall: Grade 3: multidigit × one digit, arrays; 4A Ch3: multiples; Multiplying by a 2-Digit Number — Part 1. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
Find 44 × 25 using an area model or partial products.
Show answer & reasoning
Expand (40+4)(20+5) = 800+200+80+20 = 1100.
Find 58 × 32 using an area model or partial products.
Show answer & reasoning
Expand (50+8)(30+2) = 1500+100+240+16 = 1856.
Find 28 × 35 using an area model or partial products.
Show answer & reasoning
Expand (20+8)(30+5) = 600+100+240+40 = 980.
Key vocabulary
- partial product
- product of one part of each decomposed factor
- distributive property
- multiplication distributed across a sum
- area model
- rectangular representation of a product
- regroup
- exchange ten small units for one larger unit
Understand the concept
The standard two-digit multiplication algorithm is a compact representation of the distributive property. Multiply by the ones digit, then by the tens digit as a multiple of ten. The second partial product must start in the tens place. Combine both rows, regroup if necessary, and estimate to detect place-value errors.
Model it: labeled tens-row and ones-row partial products. Explain what every part of the model represents, then write a matching equation.
Fully worked examples
Read the problem first and predict the answer. Then study how each quantity is represented and calculated.
Worked example 1
Find 101 × 23 using an area model or partial products.
Expand (100+1)(20+3) = 2000+300+20+3 = 2323.
Check your understanding 1
Find 398 × 25 using an area model or partial products.
Show answer & reasoning
Expand (390+8)(20+5) = 7800+1950+160+40 = 9950.
Worked example 2
Find 135 × 59 using an area model or partial products.
Expand (130+5)(50+9) = 6500+1170+250+45 = 7965.
Check your understanding 2
Find 424 × 12 using an area model or partial products.
Show answer & reasoning
Expand (420+4)(10+2) = 4200+840+40+8 = 5088.
Worked example 3
Find 59 × 34 using an area model or partial products.
Expand (50+9)(30+4) = 1500+200+270+36 = 2006.
Check your understanding 3
Find 169 × 48 using an area model or partial products.
Show answer & reasoning
Expand (160+9)(40+8) = 6400+1280+360+72 = 8112.
Guided practice
Find 110 × 49 using an area model or partial products.
Show answer & reasoning
Expand (110+0)(40+9) = 4400+990+0+0 = 5390.
Find 85 × 15 using an area model or partial products.
Show answer & reasoning
Expand (80+5)(10+5) = 800+400+50+25 = 1275.
Find 317 × 55 using an area model or partial products.
Show answer & reasoning
Expand (310+7)(50+5) = 15500+1550+350+35 = 17435.
Find 187 × 17 using an area model or partial products.
Show answer & reasoning
Expand (180+7)(10+7) = 1800+1260+70+49 = 3179.
Find 291 × 33 using an area model or partial products.
Show answer & reasoning
Expand (290+1)(30+3) = 8700+870+30+3 = 9603.
Independent practice
Find 351 × 41 using an area model or partial products.
Show answer & reasoning
Expand (350+1)(40+1) = 14000+350+40+1 = 14391.
Find 243 × 48 using an area model or partial products.
Show answer & reasoning
Expand (240+3)(40+8) = 9600+1920+120+24 = 11664.
Find 295 × 51 using an area model or partial products.
Show answer & reasoning
Expand (290+5)(50+1) = 14500+290+250+5 = 15045.
Find 289 × 53 using an area model or partial products.
Show answer & reasoning
Expand (280+9)(50+3) = 14000+840+450+27 = 15317.
Find 73 × 60 using an area model or partial products.
Show answer & reasoning
Expand (70+3)(60+0) = 4200+0+180+0 = 4380.
Find 406 × 53 using an area model or partial products.
Show answer & reasoning
Expand (400+6)(50+3) = 20000+1200+300+18 = 21518.
Find 372 × 33 using an area model or partial products.
Show answer & reasoning
Expand (370+2)(30+3) = 11100+1110+60+6 = 12276.
Find 138 × 51 using an area model or partial products.
Show answer & reasoning
Expand (130+8)(50+1) = 6500+130+400+8 = 7038.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
A student calculates 69 × 27 by multiplying 69 by the tens digit 2 and the ones digit 7, then adding both results without shifting the tens. Explain the error and give the correct product.
Show answer & reasoning
The tens digit is worth 20, not 2. Correct: 69 × 20 + 69 × 7 = 1863.
Find 214 × 60 using an area model or partial products.
Show answer & reasoning
Expand (210+4)(60+0) = 12600+0+240+0 = 12840.
Exit check
Find 265 × 43 using an area model or partial products.
Show answer & reasoning
Expand (260+5)(40+3) = 10400+780+200+15 = 11395.
Find 148 × 43 using an area model or partial products.
Show answer & reasoning
Expand (140+8)(40+3) = 5600+420+320+24 = 6364.
Find 184 × 52 using an area model or partial products.
Show answer & reasoning
Expand (180+4)(50+2) = 9000+360+200+8 = 9568.