Mental Math for Division
4A · Chapter 5 · Lesson 1 · MATH-4A-C05-L01
Learning goal. Divide by decomposing into convenient multiples and justify quotient invariance for scaled cases.
Before you begin
Recall: Grade 3: single-digit division and remainder; 4A Ch4: multiplication estimates. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
Find 106 × 43 using an area model or partial products.
Show answer & reasoning
Expand (100+6)(40+3) = 4000+300+240+18 = 4558.
Find 175 × 41 using an area model or partial products.
Show answer & reasoning
Expand (170+5)(40+1) = 6800+170+200+5 = 7175.
Find 91 × 34 using an area model or partial products.
Show answer & reasoning
Expand (90+1)(30+4) = 2700+360+30+4 = 3094.
Key vocabulary
- dividend
- number being divided
- divisor
- number of equal groups or group size
- quotient
- result of division
- remainder
- amount left after complete groups
Understand the concept
Mental division works by breaking a quantity into convenient multiples of the divisor. To calculate 168 ÷ 6, split 168 into 120 and 48. Dividing both parts by six gives 20 and 8, so the quotient is 28. This method is partial quotients, not guessing and hoping.
Model it: equal-group bars and partial-quotient boxes. 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 62 ÷ 2 by breaking apart the dividend.
Decompose 62 into convenient multiples of 2; the resulting quotients sum to 31.
Check your understanding 1
Calculate 273 ÷ 7 by breaking apart the dividend.
Show answer & reasoning
Decompose 273 into convenient multiples of 7; the resulting quotients sum to 39.
Worked example 2
Calculate 40 ÷ 4 by breaking apart the dividend.
Decompose 40 into convenient multiples of 4; the resulting quotients sum to 10.
Check your understanding 2
Calculate 58 ÷ 2 by breaking apart the dividend.
Show answer & reasoning
Decompose 58 into convenient multiples of 2; the resulting quotients sum to 29.
Worked example 3
Calculate 161 ÷ 7 by breaking apart the dividend.
Decompose 161 into convenient multiples of 7; the resulting quotients sum to 23.
Check your understanding 3
Calculate 200 ÷ 8 by breaking apart the dividend.
Show answer & reasoning
Decompose 200 into convenient multiples of 8; the resulting quotients sum to 25.
Guided practice
Calculate 150 ÷ 6 by breaking apart the dividend.
Show answer & reasoning
Decompose 150 into convenient multiples of 6; the resulting quotients sum to 25.
Calculate 72 ÷ 4 by breaking apart the dividend.
Show answer & reasoning
Decompose 72 into convenient multiples of 4; the resulting quotients sum to 18.
Calculate 152 ÷ 8 by breaking apart the dividend.
Show answer & reasoning
Decompose 152 into convenient multiples of 8; the resulting quotients sum to 19.
Calculate 87 ÷ 3 by breaking apart the dividend.
Show answer & reasoning
Decompose 87 into convenient multiples of 3; the resulting quotients sum to 29.
Calculate 117 ÷ 3 by breaking apart the dividend.
Show answer & reasoning
Decompose 117 into convenient multiples of 3; the resulting quotients sum to 39.
Independent practice
Calculate 88 ÷ 8 by breaking apart the dividend.
Show answer & reasoning
Decompose 88 into convenient multiples of 8; the resulting quotients sum to 11.
Calculate 74 ÷ 2 by breaking apart the dividend.
Show answer & reasoning
Decompose 74 into convenient multiples of 2; the resulting quotients sum to 37.
Calculate 231 ÷ 7 by breaking apart the dividend.
Show answer & reasoning
Decompose 231 into convenient multiples of 7; the resulting quotients sum to 33.
Calculate 36 ÷ 2 by breaking apart the dividend.
Show answer & reasoning
Decompose 36 into convenient multiples of 2; the resulting quotients sum to 18.
Calculate 66 ÷ 2 by breaking apart the dividend.
Show answer & reasoning
Decompose 66 into convenient multiples of 2; the resulting quotients sum to 33.
Calculate 54 ÷ 2 by breaking apart the dividend.
Show answer & reasoning
Decompose 54 into convenient multiples of 2; the resulting quotients sum to 27.
Calculate 288 ÷ 8 by breaking apart the dividend.
Show answer & reasoning
Decompose 288 into convenient multiples of 8; the resulting quotients sum to 36.
Calculate 152 ÷ 4 by breaking apart the dividend.
Show answer & reasoning
Decompose 152 into convenient multiples of 4; the resulting quotients sum to 38.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
A student solves 156 ÷ 6 by computing 120 ÷ 6 and 36 ÷ 6 and adding. Is this a valid mental strategy? Explain.
Show answer & reasoning
156 is 120 + 36, and both parts divide by 6. The quotients 20 and 6 add to 26.
Calculate 198 ÷ 9 by breaking apart the dividend.
Show answer & reasoning
Decompose 198 into convenient multiples of 9; the resulting quotients sum to 22.
Exit check
Calculate 60 ÷ 4 by breaking apart the dividend.
Show answer & reasoning
Decompose 60 into convenient multiples of 4; the resulting quotients sum to 15.
Calculate 140 ÷ 5 by breaking apart the dividend.
Show answer & reasoning
Decompose 140 into convenient multiples of 5; the resulting quotients sum to 28.
Calculate 102 ÷ 6 by breaking apart the dividend.
Show answer & reasoning
Decompose 102 into convenient multiples of 6; the resulting quotients sum to 17.