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Standards involving the distributive property and sample tasks
Distribution of the Distributive Property
Dr. Roger Fischer
EMAT Project Facilitator
Montana State University
June 2016
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Standards involving the distributive property and sample tasks
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O VERVIEW
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Standards involving the distributive property and sample tasks
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Standards involving the distributive property and sample tasks
A UDIENCE P OLLING
https://www.polleverywhere.com/my/polls
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Standards involving the distributive property and sample tasks
M ULTIPLICATION FACTS
Grade 3: 3.OA.5
Apply properties of operations as strategies to
multiply.
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Standards involving the distributive property and sample tasks
S AMPLE TASK
from Kling and Bay-Williams (2015)
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A REA M ODELS
Grade 3: 3.MD.7C
Use area models to represent the distributive property
in mathematical reasoning.
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S AMPLE TASK
From Illustrative Mathematics (2016d)
1. How many circles are there in all? Write down a number
sentence that shows how you might have thought about it.
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Standards involving the distributive property and sample tasks
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S AMPLE TASK
From Illustrative Mathematics (2016d)
1. How many circles are there in all? Write down a number
sentence that shows how you might have thought about it.
2. Alonso said he figured out how many shaded circles there
were first and then how many unshaded circles there were
second. Once he knew how many of each, he added them
together to find the total. Write a number sentence Alonso
could have used that shows his reasoning.
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S AMPLE TASK
From Illustrative Mathematics (2016d)
1. How many circles are there in all? Write down a number
sentence that shows how you might have thought about it.
2. Alonso said he figured out how many shaded circles there
were first and then how many unshaded circles there were
second. Once he knew how many of each, he added them
together to find the total. Write a number sentence Alonso
could have used that shows his reasoning.
3. Jennifer said “I just saw 3 rows of 8 circles.” Write a
number sentence that Jennifer could have used that shows
her reasoning.
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Standards involving the distributive property and sample tasks
M ENTAL D IVISION
Grade 4: 4.NBT.6
Find whole-number quotients and remainders with
up to four-digit dividends and one-digit divisors,
using strategies based on place value, the properties
of operations, and/or the relationship between
multiplication and division. Illustrate and explain the
calculation by using equations, rectangular arrays,
and/or area models.
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S AMPLE TASK
From Illustrative Mathematics (2016a)
Jillian says “I know that 20 times 7 is 140 and if I take away 2
sevens that leaves 126. So 126 ÷ 7 = 18.
1. Is Jillian’s calculation correct? Explain.
2. Draw a picture showing Jillian’s reasoning.
3. Use Jillian’s method to find 222 ÷ 6.
S OLUTION:
https://www.illustrativemathematics.org/
content-standards/4/NBT/B/6/tasks/1774
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Standards involving the distributive property and sample tasks
N UMBER T HEORY
Grade 6: 6.NS.4
Use the distributive property to express a sum of two
whole numbers 1-100 with a common factor as a
multiple of a sum of two whole numbers with no
common factor.
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S AMPLE TASK
From Illustrative Mathematics (2016b)
Nina was finding multiples of 6. She said, “18 and 42 are both
multiples of 6, and when I add them, I also get a multiple of 6.
18 + 42 = 60
Explain to Nina why adding two multiples of 6 will always
result in a multiple of 6.
S OLUTION:
https://www.illustrativemathematics.org/
content-standards/tasks/257
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Standards involving the distributive property and sample tasks
M ORE N UMBER T HEORY
Grade 6: 6.EE.3
Apply the. . . distributive property to the expression
3(2 + x) to produce the equivalent expression 6 + 3x;
apply the distributive property to the expression
24x + 18y to produce the equivalent expression
6(4x + 3y).
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M ORE N UMBER T HEORY
Grade 6: 6.EE.3
Apply the. . . distributive property to the expression
3(2 + x) to produce the equivalent expression 6 + 3x;
apply the distributive property to the expression
24x + 18y to produce the equivalent expression
6(4x + 3y).
‘
I
Discuss with your peers any possible connections between
this standard and the last sample task.
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Standards involving the distributive property and sample tasks
“T WO N EGATIVES M AKE A P OSITIVE ”
Grade 7: 7.NS.2a
Understand that multiplication is extended from
fractions to rational numbers by requiring that
operations continue to satisfy the properties of
operations, particularly the distributive property,
leading to products such as (−1)(−1) = 1 and the
rules for multiplying signed numbers.
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S AMPLE TASK
From Illustrative Mathematics (2016c)
5×5
25
5×4
20
5×3
15
5×2
10
5×1
5
5×0
0
5×(−1)
-5
5×(−2)
-10
5×(−3)
-15
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Standards involving the distributive property and sample tasks
S AMPLE TASK
From Illustrative Mathematics (2016c)
5×5
5 × (5 − 0)
5×5−5×0
5×4
5 × (5 − 1)
5×5−5×1
5×3
5 × (5 − 2)
5×5−5×2
5×2
5 × (5 − 3)
5×5−5×3
5×1
5 × (5 − 4)
5×5−5×4
5×0
5 × (5 − 5)
5×5−5×5
5 × (−1)
55 × (5 − 6)
5 × 5 − 5 × (−1)
5 × (−2)
5 × (5 − 7)
5 × 5 − 5 × (−2)
5 × (−3)
5 × (5 − 8)
5 × 5 − 5 × (−3)
5 × (−4)
5 × 5 − 5 × −4
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Standards involving the distributive property and sample tasks
L OOK FOR AND M AKE U SE OF S TRUCTURE
Grade 8: 8.EE.7b
Solve linear equations with rational number
coefficients, including equations whose solutions
require expanding expressions using the distributive
property and collecting like terms.
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L OOK FOR AND M AKE U SE OF S TRUCTURE
1. Solve the equation 9x + 9 = 180 for x.
2. Solve the equation 7y + 7 = 140 for y
3. Solve the equation 12z + 12 = 240 for z
Find a way to solve each of these equations using the
distributive property. How are these equations similar to
Nina’s question about multiples of 6?
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Standards involving the distributive property and sample tasks
T HANKS FOR COMING !
Dr. Roger Fischer
Mathematics Instructor
Gallatin College – Montana State University
201 Hamilton Hall
Bozeman, MT 59717
[email protected]
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R EFERENCES I
Illustrative Mathematics. (2016a, June). 4.NBT: Mental division
strategy. Retrieved June 9, from
https://www.illustrativemathematics.org/
content-standards/4/NBT/B/6/tasks/1774
Illustrative Mathematics. (2016b, June). 6.NS: Adding multiples.
Retrieved June 9, from
https://www.illustrativemathematics.org/
content-standards/tasks/257
Illustrative Mathematics. (2016c, June). 6.NS: Why is a negative
times a negative always a positive? Retrieved June 9, from
https://www.illustrativemathematics.org/
content-standards/7/NS/A/2/tasks/1667
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Standards involving the distributive property and sample tasks
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R EFERENCES II
Illustrative Mathematics. (2016d, June). 6.RP: Unit conversions.
Retrieved June 10, 2016, from
https://www.illustrativemathematics.org/
content-standards/tasks/1992
Kling, G., & Bay-Williams, J. (2015). Three steps to mastering
multiplication facts. Teaching Children Mathematics, 21(9),
548–559.