Multiplying by a 2-Digit Number — Part 1
4A · Chapter 4 · Lesson 5 · MATH-4A-C04-L06
Learning goal. Expand both factors and combine four partial products in a diagram.
Before you begin
Recall: Grade 3: multidigit × one digit, arrays; 4A Ch3: multiples; Multiplying by a Multiple of 10. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
Calculate 222 × 50.
Show answer & reasoning
50 represents 5 tens. Multiply by 5, then scale by 10 to obtain 11,100.
Calculate 402 × 20.
Show answer & reasoning
20 represents 2 tens. Multiply by 2, then scale by 10 to obtain 8,040.
Calculate 183 × 90.
Show answer & reasoning
90 represents 9 tens. Multiply by 9, then scale by 10 to obtain 16,470.
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
Two-digit multiplication combines four partial products. For 24 × 13, split 24 into 20+4 and 13 into 10+3. The four rectangle areas are 200, 60, 40, and 12; together they make 312. This diagram also explains the two shifted rows in column multiplication.
Model it: four-region rectangle area model. 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 54 × 19 using an area model or partial products.
Expand (50+4)(10+9) = 500+450+40+36 = 1026.
Check your understanding 1
Find 37 × 67 using an area model or partial products.
Show answer & reasoning
Expand (30+7)(60+7) = 1800+210+420+49 = 2479.
Worked example 2
Find 57 × 19 using an area model or partial products.
Expand (50+7)(10+9) = 500+450+70+63 = 1083.
Check your understanding 2
Find 30 × 41 using an area model or partial products.
Show answer & reasoning
Expand (30+0)(40+1) = 1200+30+0+0 = 1230.
Worked example 3
Find 20 × 54 using an area model or partial products.
Expand (20+0)(50+4) = 1000+80+0+0 = 1080.
Check your understanding 3
Find 46 × 22 using an area model or partial products.
Show answer & reasoning
Expand (40+6)(20+2) = 800+80+120+12 = 1012.
Guided practice
Find 62 × 25 using an area model or partial products.
Show answer & reasoning
Expand (60+2)(20+5) = 1200+300+40+10 = 1550.
Find 48 × 39 using an area model or partial products.
Show answer & reasoning
Expand (40+8)(30+9) = 1200+360+240+72 = 1872.
Find 65 × 55 using an area model or partial products.
Show answer & reasoning
Expand (60+5)(50+5) = 3000+300+250+25 = 3575.
Find 21 × 20 using an area model or partial products.
Show answer & reasoning
Expand (20+1)(20+0) = 400+0+20+0 = 420.
Find 64 × 16 using an area model or partial products.
Show answer & reasoning
Expand (60+4)(10+6) = 600+360+40+24 = 1024.
Independent practice
Find 21 × 27 using an area model or partial products.
Show answer & reasoning
Expand (20+1)(20+7) = 400+140+20+7 = 567.
Find 40 × 36 using an area model or partial products.
Show answer & reasoning
Expand (40+0)(30+6) = 1200+240+0+0 = 1440.
Find 27 × 28 using an area model or partial products.
Show answer & reasoning
Expand (20+7)(20+8) = 400+160+140+56 = 756.
Find 63 × 36 using an area model or partial products.
Show answer & reasoning
Expand (60+3)(30+6) = 1800+360+90+18 = 2268.
Find 27 × 42 using an area model or partial products.
Show answer & reasoning
Expand (20+7)(40+2) = 800+40+280+14 = 1134.
Find 25 × 53 using an area model or partial products.
Show answer & reasoning
Expand (20+5)(50+3) = 1000+60+250+15 = 1325.
Find 46 × 29 using an area model or partial products.
Show answer & reasoning
Expand (40+6)(20+9) = 800+360+120+54 = 1334.
Find 15 × 60 using an area model or partial products.
Show answer & reasoning
Expand (10+5)(60+0) = 600+0+300+0 = 900.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
A student calculates 50 × 32 by multiplying 50 by the tens digit 3 and the ones digit 2, then adding both results without shifting the tens. Explain the error and give the correct product.
Show answer & reasoning
The tens digit is worth 30, not 3. Correct: 50 × 30 + 50 × 2 = 1600.
Find 16 × 26 using an area model or partial products.
Show answer & reasoning
Expand (10+6)(20+6) = 200+60+120+36 = 416.
Exit check
Find 56 × 42 using an area model or partial products.
Show answer & reasoning
Expand (50+6)(40+2) = 2000+100+240+12 = 2352.
Find 22 × 42 using an area model or partial products.
Show answer & reasoning
Expand (20+2)(40+2) = 800+40+80+4 = 924.
Find 30 × 31 using an area model or partial products.
Show answer & reasoning
Expand (30+0)(30+1) = 900+30+0+0 = 930.