Unit 6 Multiplying And Dividing Multi Digit Numbers — Unit Plan

1.  

   

2. Sample responses:
   • There will always be 3 triangles because adding a new floor will not change the number of triangles.
   • The total number of small orange blocks will change by adding 2 more each time. The floors will always be two blocks wide, so adding a new floor will always mean adding 2 more blocks.
   • The total number of small orange blocks will always be even because you’re adding 2 blocks each time.
   • The total number of pattern blocks will always be odd because you keep adding only 2 more blocks.

1. square, square, square, square, circle, square, square, square, square, circle, square, square, square, square, circle
2. 5, 10, 15, 20, 25
3. Square. Sample response: The circles are multiples of 5, and 42 is not a multiple of 5.

1. 8, 16, 24, 32, 40
2. No. Sample response: 105 doesn’t have an even digit in the ones place.

1. 100, 200, 400, 800, 1600
2. Sample responses:
   1. The digit in the hundreds place starts as odd, then stays even. It’s just like when you double other numbers, it might start odd, but it’ll be even after you double it because you’re making 2 equal groups.
   2. There’s always a 0 in the ones place and tens place because when you double 0, it’s 0. 

216 seats. Sample responses:

• Eight rows of 20 is 160, and 8 rows of 7 is 56. $160 + 56 = 216$
• Eight rows of 30 is $8 \times 30$, which is 240. Because there are 27 seats per row and not 30 seats per row, I subtracted $8 \times 3$ or 24 from 240, which gives 216.
• I know $2 \times 27$ is 54, so $4 \times 27$ is twice 54 or 108, and $8 \times 27$ is twice 108, which is 216.

1. 3,192. Sample responses: 
   • $(4 \times 700) + (4 \times 90) + (4 \times 8) = 2,800 + 360 + 32 = 3,192$
   • I know that 798 is 2 less than 800. So 4 groups of 798 is $4 \times 2$ less than $4 \times 800$ or 8 less than 3,200, which is 3,192.
2. 22,912. Sample response:
   • $(8 \times 2,000) + (8 \times 800) + (8 \times 60) + (8\times 4) = 16,000 + 6,400 + 480 + 32 = 22,91 2$

Each student cleans 16 desks. Sample responses:

• $6 \times 10 = 60$ and $6 \times 6 = 36$ so $6 \times 16 = 96$.
• $60 \div 6 = 10$, $36 \div 6 = 6$, and $10 + 6 = 16$.

Sample response: It was helpful when we were working with smaller numbers and we didn’t have to decompose blocks. It wasn’t helpful when I was trying to work with larger numbers.

• I know that $132 = 100 + 32$. I also know that $100 \div 4 = 25$ and $32 \div 4 = 8$, so $132 \div 4$ is the sum of 25 and 8, which is 33.
• The large square represents 1 hundred and can be decomposed into 10 tens. Now, we have 13 tens. Twelve of the tens can be put into 4 groups of 3 tens. The last ten can be decomposed into 10 ones. There are now 12 ones, or 4 groups of 3 ones. Three tens and 3 ones is 33.
• I know that $120 \div 4 = 30$ and $12 \div 4 = 3$, so $132 \div 4 = 33$.

86. Sample responses:

400 is 80 groups of 5 and 30 is 6 groups of 5. Adding the groups of 5—80 and 6—gives the quotient.

\begin{align} 400\div 5&= 80\\ 30 \div 5 &= 6\\ \overline {\hspace{5mm}430 \div 5} &\overline{\hspace{1mm}= 86\phantom{000000}}\end{align}

289. Sample response: partial quotients 200, 80, 9

• 194 is not a multiple of 6. I know that $6 \times 30 = 180$, and 194 is 14 away from 180. Because 14 is not a multiple of 6, 194 is also not a multiple of 6.
• Six is not a factor of 194. I divided 194 by 6 and got 32 with a remainder of 2. If Mai counted correctly, she would have called out 192 and then 198.

She will not. Sample response: 140 is a multiple of 7 because $7\times2 =14$. So, $7\times20 = 140$. 149 is the same as $140 + 9$, so there would be a remainder of 2.

1. No. Sample response:
   • The paper for this year’s banner has an area of 3,456 square inches, because $96 \times 36 = 3,456$. Last year’s banner had an area of 2,304 square inches, because $48 \times 48 = 2,304$, so Han will need a bigger space to hang the new banner.
2. The difference is 1,152. $3,456 - 2,304 = 1,152$

1. No. Sample response:
   • Even if she drives 300 km a day, she’d only travel 2,400 km in 8 days, so she can’t travel 2,654 km with less than 300 km per day.
   • $285 \times 8 = 2,280$. At 285 km per day, she’d only travel 2,280 km in 8 days.
   • $2,654 = 8 \times 337 + 6$. This means she’d need to travel at least 338 km a day to get to her destination in 8 days.
2. 595 km. Sample response: After the third day, she had $2,654 - 1,087$ or 1,567 km left. After the fourth day she has 972 km left, so she must have traveled 595 km, because $1,567 - 972 = 595$.

1. 345,589 children under 18
2. Estimate: $110,000 + 180,000 + 50,000 = 340,000$. There are about 340,000 people under 18, which is close to the actual number 345,599 calculated.