Mental Math for Multiplication
4A · Chapter 4 · Lesson 1 · MATH-4A-C04-L01
Learning goal. Apply distributive decomposition, doubling/halving when valid, and compensation to multiply mentally.
Before you begin
Recall: Grade 3: multidigit × one digit, arrays; 4A Ch3: multiples. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
What is the greatest common factor of 26 and 45?
Show answer & reasoning
The GCF is the largest whole number dividing both with no remainder: 1.
What is the greatest common factor of 24 and 27?
Show answer & reasoning
The GCF is the largest whole number dividing both with no remainder: 3.
What is the greatest common factor of 24 and 18?
Show answer & reasoning
The GCF is the largest whole number dividing both with no remainder: 6.
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
Mental multiplication uses number structure. Decompose 38 × 6 into (30 × 6) + (8 × 6), then combine partial products. Doubling one factor and halving an even factor preserves the product, as in 25 × 16 = 50 × 8. Always check that both changes were made consistently.
Model it: area decomposition and friendly-factor arrows. 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
Calculate 163 × 9 using a split or friendly number.
Split 163 into 160 + 3. Then multiply each part by 9 and add the partial products.
Check your understanding 1
Calculate 95 × 9 using a split or friendly number.
Show answer & reasoning
Split 95 into 90 + 5. Then multiply each part by 9 and add the partial products.
Worked example 2
Calculate 40 × 5 using a split or friendly number.
Split 40 into 40 + 0. Then multiply each part by 5 and add the partial products.
Check your understanding 2
Calculate 85 × 7 using a split or friendly number.
Show answer & reasoning
Split 85 into 80 + 5. Then multiply each part by 7 and add the partial products.
Worked example 3
Calculate 181 × 9 using a split or friendly number.
Split 181 into 180 + 1. Then multiply each part by 9 and add the partial products.
Check your understanding 3
Calculate 193 × 4 using a split or friendly number.
Show answer & reasoning
Split 193 into 190 + 3. Then multiply each part by 4 and add the partial products.
Guided practice
Calculate 195 × 3 using a split or friendly number.
Show answer & reasoning
Split 195 into 190 + 5. Then multiply each part by 3 and add the partial products.
Calculate 155 × 6 using a split or friendly number.
Show answer & reasoning
Split 155 into 150 + 5. Then multiply each part by 6 and add the partial products.
Calculate 61 × 5 using a split or friendly number.
Show answer & reasoning
Split 61 into 60 + 1. Then multiply each part by 5 and add the partial products.
Calculate 23 × 6 using a split or friendly number.
Show answer & reasoning
Split 23 into 20 + 3. Then multiply each part by 6 and add the partial products.
Calculate 21 × 4 using a split or friendly number.
Show answer & reasoning
Split 21 into 20 + 1. Then multiply each part by 4 and add the partial products.
Independent practice
Calculate 61 × 6 using a split or friendly number.
Show answer & reasoning
Split 61 into 60 + 1. Then multiply each part by 6 and add the partial products.
Calculate 128 × 9 using a split or friendly number.
Show answer & reasoning
Split 128 into 120 + 8. Then multiply each part by 9 and add the partial products.
Calculate 26 × 5 using a split or friendly number.
Show answer & reasoning
Split 26 into 20 + 6. Then multiply each part by 5 and add the partial products.
Calculate 22 × 5 using a split or friendly number.
Show answer & reasoning
Split 22 into 20 + 2. Then multiply each part by 5 and add the partial products.
Calculate 27 × 8 using a split or friendly number.
Show answer & reasoning
Split 27 into 20 + 7. Then multiply each part by 8 and add the partial products.
Calculate 68 × 2 using a split or friendly number.
Show answer & reasoning
Split 68 into 60 + 8. Then multiply each part by 2 and add the partial products.
Calculate 172 × 3 using a split or friendly number.
Show answer & reasoning
Split 172 into 170 + 2. Then multiply each part by 3 and add the partial products.
Calculate 188 × 3 using a split or friendly number.
Show answer & reasoning
Split 188 into 180 + 8. Then multiply each part by 3 and add the partial products.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
A student calculates 28 × 4 using only its tens: 20 × 4. What partial product is missing and what is the correct total?
Show answer & reasoning
The ones part 8 × 4 = 32 must also be included. The complete product is 112.
Calculate 24 × 5 using a split or friendly number.
Show answer & reasoning
Split 24 into 20 + 4. Then multiply each part by 5 and add the partial products.
Exit check
Calculate 26 × 9 using a split or friendly number.
Show answer & reasoning
Split 26 into 20 + 6. Then multiply each part by 9 and add the partial products.
Calculate 23 × 7 using a split or friendly number.
Show answer & reasoning
Split 23 into 20 + 3. Then multiply each part by 7 and add the partial products.
Calculate 21 × 8 using a split or friendly number.
Show answer & reasoning
Split 21 into 20 + 1. Then multiply each part by 8 and add the partial products.