MARS Multiplication and Division

Math Assessment Reasoning and Strategies for Multiplication and Division

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1. 4x3

Doesn't know or guesses

Counts to three 4 times or to four 3 times out loud or on fingers

Uses repeated addition 3+3=6 6+3=9, 9+3=12

Uses a double strategy 4 times is double-double. So double 3 = 6 and double 6 =12

Uses a known fact to reason a new fact: I know 3x3=9 so one more 3 is 9+3=12

Skip counts 3, 6, 9, 12 or 4, 8,12

Knows it

Additional observations:

2. 6x8

Doesn't know or guesses

Counts to 8 six times or to 6 eight times out loud or on fingers and gets 48

Uses repeated addition 8+8=16 16+8=24, 24+8=32 32+8=40 40+8=48

Skip counts 8, 16, 24 ....48

Uses a known fact to reason a new fact: I know 5*8=40 so 6*8 is one more 8, 40 + 8 = 48

Explains 8 times means double double double, says double 6 = 12, double 12 = 24 and double 24 = 48

Uses "half and double" 8*6 or 4*12 or 4*10=40, 4*2=8 together 48

Knows it

Additional observations:

3. 9x18

Doesn't know, does not even guess when asked to do so

Guesses between 100-149, asks why do you think that and write a comment which scaffolding is needed to correct the guess

Rounds 9 to 10 and 18 to 20 and guesses a number between 150-200, asks to guess a bit more precisely and write a comment which scaffolding is needed to guess more accuratly

Needs to write out the column algorithm, makes a mistake. Write a comment which scaffolding the student needs to be able to correct the answer

Needs to write out the column algorithm and gets 162

Imagines column algorithm with regrouping and gets 162

Uses place value to split 18 in tens and ones and multiplies separately 9*10=90, 9*8=72, 90+72=162

Uses compensation: 9x a number is 10x the number - 1x the number: 10*18=180, 180-18=162

Uses compensation: 18x a number is 20x the number - 2x the number: 9*20=180, 2*9=18, 180-18=162

Additional observations:

4. 40x250

Doesn't know, guesses less than 1,000 or more than 100,000

Guesses more than 1,000

Can only do it on paper, writes out all zeros

Imagines the column algorithm

Uses place value to split 250 in 200 and 50, 4x200 = 800 so 40x200 = 8,000 and 4 x 50 = 200 so 40x50 = 2,000 together 10,000

Can use place value and benchmarks: 4*25=100 so 40*25=1000 so 40*250=10,000

Can use place value and benchmarks: 4*250=1000 so 40*250=10,000

Uses place value and the half and double strategy: half of 40 = 20 and double 250 = 500, makes 20*500, and 2*500=1000, 20*500=10,000

Additional observations:

5. 123x3

Can't do it or guesses 126 (adds instead of multiplies), alert the student and ask again

Thinks the answer is 323 (only multiplies the 100) write a comment which scaffolding the learner needs to be able to get the correct answer

Can only do it on paper and correctly gets 369

Imagines the column algorithm and correctly gets 369

Uses place value strategy and separates hundreds, tens and ones: 3x100=300, 3x20=60, and 3x3=9 and adds 300+60+9 = 369

Can do it mentally in one go and correctly gets 369

Additional observations:

6. Estimate 29x31

Can't do it or guesses 60 (adds instead of multiplies) alert the student and ask again

Rounds 30*30=90 due to place value error. Write a comment which scaffolding the student needs to be able to correct the answertotal is $20. Write a comment which scaffolding the student needs to be able to correct the answer

Doesn't round to estimate, can only do it on paper by actual calculation then rounds 899 to 900 (because the question was estimate)

Doesn't round to estimate, imagines the column algorithm for the actual calculation and then rounds 899 to 900 (because the question was estimate)

Knows to round for estimation, rounds 29 to 30 and 31 to 30, 30*30=900

Additional observations:

7. 8÷2

Can't do it, guesses

Holds up 8 fingers with 4 fingers on each hand and says 4

Counts 1, 2, 3, 4, 5, 6, 7, 8 loud or on fingers and says "4"

Uses a double strategy: I know double 4=8 so half of 8 or 8÷2=4

Skip counts by 2 out loud or on fingers, says: 2, 4, 6, 8 so 4

Knows it

Additional observations:

8. 24÷4

Can't do it, guesses

Counts all 24 in groups of 4: 1-4, 5-8, 9-12, 13-16, 17-20, 21-24 and says 6

Skip counts up by 4 to 24 and says 6

Uses multiples of 5: split 24 in 20. There are 5 groups of 4 in 20 and 24 is one more 4, so 5+1=6

Uses multiplication facts and connection between multiplication and division: 6x4=24 so 24÷4=6

Just knows it

Additional observations:

9. 70÷2

Doesn't know, guesses

Estimates 30 or 40. Write a comment which scaffolding the student needs to be able to correct the answer

Can only do it on paper using partial long division (Big Seven method)

Can only do it on paper using traditional long division

Imagines the column division and says 35

Uses partitioning 70=60+10 half of 60 is 30, half of 10 is 5, together 35

Uses partitioning 70=50+20, half of 50 is 25, half of 20 is 10, together 35

Knows it is 35 but can't explain how it is done

Additional observations:

10. 75÷5

Doesn't know or guesses

Skip counts up by 5, makes tally marks or counts on fingers, answer 15

Can only do it on paper using partial long division (Big Seven method)

Can only do it on paper using traditional long division

Splits 75 according to place value: 70 : 5 = 14, 5 : 5 = 1 total answer 15

Splits 75 in 50 and 25: 10 fives in 50 and 5 fives in 25, so total 10+5=15 fives in 75

Knows it is 15 but can't explain how it is done

Additional observations:

11. 91÷7

Doesn't know it or guesses

Skip counts up by 7, makes tally marks or counts on fingers, answer 13

Reasoning it must be more than 10 because 10x7=70 or 10 sevens in 70. Write a comment which scaffolding the student needs to be able to correct the answer

Can only do it on paper using partial long division (Big Seven method)

Can only do it on paper using traditional long division

Imagines the long division algorithm: 7 divides into 9 1x, 9-7=2, bring down the 1 to get 21 ÷ 7 = 3 so 10+3=13

Uses multiplication and place value: 10*7=70, 91 is 21 more, that is 3 sevens, so 10+3=13

Additional observations:

12. 588÷6

Can't do it, guesses below 90 or more than 100

Uses place value knowledge to guess between 90 and 100 by using 9x6=54 and 90x6=540 so answer must be more than 90. 100x6 = 600 that is more than 588 so the answer is less than 100, but does not know the answer. Write a comment which scaffolding the student needs to be able to correct the answer

Can only do it on paper using partial long division (Big Seven method) to get 98

Can only do it on paper using traditional long division to get 98

Imagines column algorithm: 6 divides into 58 9 times, 58-54=4 bring down the 8 to get 48, 6 divides into 48 8 times so 98

Multiplies up: 9x6=54 so 90x6=540, 588 is 48 more, that is 8 sixes more, answer 90+8=98

Uses place vale and compatible numbers 54 : 6 = 9 so 540 : 6 = 90. 588 is 48 more so another 8 sixes, the answer is 90 + 8 = 98

Rounds 588 to 600 and divides by 6 = 100, than compensates for the real number 588 which is 12 or two sixes less, so answer is 98

Uses a benchmark number 100x6=600, 588 is 12 less than 600, so answer 100-2=98

Additional observations:

13. How do you estimate 763÷24

Can't do it, guesses without explanation.

Rounds to tens and answers 760 ÷ 20 but the guess is less than 35 or more than 40

Rounds to tens and answers 760 ÷ 20 and the guess is 35 through 40

Rounds to hundreds and tens and estimates 800 ÷ 20 but the guess is less than 35 or more than 40

Rounds to hundreds and tens and estimates 800 ÷ 20 but the guess is 35 through 39

Rounds to hundreds and tens and estimates 800 ÷ 20 and the guess is 40

Uses 25 as a benchmark and knows there are four 25's in a 100, but gets to an estimate of less than 25 or more than 30

Uses 25 as a benchmark and knows there are four 25's in a 100, so 7x4 = 28 in 700 and gets to an estimate of 25 through 30

Uses 25 as a benchmark and knows there are four 25's in a 100, so 7x4 = 28 in 700 and adds the two 25's that are in the 63 and estimates 28+2=30

Additional observations: