EDUC 170
Problem Set #2
Name: ________________________________________
DIRECTIONS:
This assignment should be completed individually. If you have questions, contact me. If suspected of academic misconduct, you will be notified, your grade will be withheld and the case will be sent to the Dean’s Representative.
This assignment is out of 40 points; per the syllabus it will be worth 15% of your final grade. Evaluation criteria are provided in the Rubric. Please read it before you complete the assignment.
There are 8 required parts of this problem set (A through H). Problems B, D & H are typically the ones that require the most time; please plan accordingly.
Choose any of the following ways to complete the Problem Set:
Print the problem set or request a printed copy and write/draw directly onto it. Scan and return via Canvas.
Answer the questions on plain paper, recreating drawings where you need to. Scan and return via Canvas.
Use the drawing tool embedded in Microsoft Word under the draw menu to draw on the Word document. Save and return via Canvas.
Cut and paste scans/photos into this Word document. Save and return via Canvas.
In this assignment, “symbols” means numbers & equations – e.g., 5 + 6. “Models” means diagrams like an area model or pictures of Base-10 blocks.
A half point will be deducted any time you err in using the equal sign.
Don’t read too much into how much space is given for each item; that’s often just about formatting. Write as much as you feel you need to in order to respond fully to the prompts.
Analyzing Student Thinking: Multiplication [4 points]
For this task, you will watch video of a student solving 7 x 8 using a “number sense strategy.” You can access the video at the following link: http://smarturl.it/CM4.8.
[A1] Represent the student’s strategy using symbols and briefly describe what the student is doing with words.
[A2 & A3] Represent what is happening mathematically in the student’s strategy in two ways:
[A2a] using an equal groups model
[A2b] using an area model or array
Each part of the student’s strategy should be clearly visible in the model.
[A2] Equal groups model
[A3] Area model or array
[A4] Show a different strategy children might use to solve that problem and label it as a direct modeling, counting & adding, or number fact strategy.
Analyzing Student Thinking: Multiplication [6 points]
[B1] A 4th-grade student, Kayla, is solving 16 x 25. Here is what she has written and what she tells you:
“I split the 16 into 10 + 6, and the 25 into 20 + 5. Then I multiplied the parts. I multiplied 10 times 20, which is 200, and 6 times 5, which is 30. 200 plus 30 is 230, so I thought the answer should be 230. But that can’t be right, because 10 x 25 is 250, so 16 x 25 should be more. I DON’T GET IT!
Explain where Kayla went wrong and how she can build on what she has already done correctly to arrive at the correct solution. Use a model (diagram) in your explanation. Use color or arrows/labels to show the connections between the symbols and the model.
[B2] A different 4th-grade student, Ryan, tried to solve 16 x 25 using the standard US algorithm. Here’s what he’s written.
Explain where Ryan went wrong and show how he can build on/modify what he’s done to arrive at the correct solution. In your explanation, make connections to a model and/or a different multiplication strategy that helps show how Ryan can build on/modify his thinking.
C. Representing Multiplication Strategies Using Area Models [6 points]
Use a complete and clear area model to represent each of these “invented” multiplication strategies for solving the problem 18 x 15. Represent each part of the strategy in your model. If you prefer, you may do your area models on graph paper and attach it. (You can also use a digital tool.)
Verbal & Symbolic
Area Model
[C1] I broke up the 15 into 10 and 5. I knew that 18 times 10 is 180. I couldn’t figure out 18 x 5, so I broke up the 18 into 10 and 8. 10 times 5 is 50 and 8 times 5 is 40.
[C2] I know that 18 is 3 times 6, and I know that 6 times 15 is 90. So I did 6 x 15 is 90, and then I just multiplied 90 by 3.
[C3] I multiplied 20 x 15 to get 300 and then subtracted 2 x 15 = 30, because I’d added two groups of 15.
[D] Justifying a Multiplication Strategy [5 points]
During a discussion, one of your 5th-graders, Ebony, shares her strategy for solving 18 x 15.
“I changed 18 x 15 to an easier problem. I halved the 18 and doubled the 15. Half of 18 is 9, and double 15 is 30. That means that 18 x 15 is the same amount as 9 x 30, and 9 x 30 is just 270.”
Ebony then makes a general claim about multiplication:
“When you’re multiplying two numbers, you can halve one of them and double the other, and the product stays the same.”
Your students start trying to justify Ebony’s claim. You expect your students to create the following types of justifications:
Justification by examples
Reasoning by story context
Reasoning by visual model + explanation
Choose TWO of those types of justifications and show it would look like to justify Ebony’s claim correctly and completely using that type of justification. Name the type of justification you’re using. For an additional point, connect your justifications.
Justification Type 1: _______________________________
Justification Type 2: _______________________________
[E] Analyzing Division Problems & Strategies [3 points]
[E1] Your fourth graders are solving these two problems about soda machines.
Problem 1: One soda machine in the teacher’s lounge holds 156 cans of Coke. How many
6-packs of Coke is that? (A 6-pack holds 6 cans of Coke.)
Problem 2: A different soda machine in the teacher’s lounge also holds 156 cans of soda, but it has 6 different flavors. It has the same number of cans of each flavor. How many cans of each flavor does this soda machine hold?
Both problems represent 156 6. What’s the key difference between them?
[E2] Here’s the work from two of your students, Indra & Ruby. They forgot to label their work.
Strategy A
Strategy B
Which strategy is probably for Problem 1 and which is probably for Problem 2? Explain your thinking.
F. Developing Multi-Digit Division Strategies [8 points]
[F1] Write a realistic story problem that you would model using the equation .
[F2 & F3] Solve your story problem for using 2 different efficient strategies or algorithms other than the standard US long-division algorithm.
You may use the “Big 7” algorithm once, but not multiple times. Direct modeling, skip-counting, and repeated addition/subtraction do not count as efficient strategies.
Read F4 before you do this, so that you leave space to complete it.
[F2] First strategy:
[F3] Second strategy:
[F4] Choose one of the division strategies (F2 or F3) and, above, “explain conceptually” how you can use it to solve your story problem using words and by adding units to the quantities in your strategy.
[G] Making Sense of Remainders in Division [4 points]
[G1& G2] For each problem, circle the answer that makes the most sense given the context and question. No explanation is needed.
[G1] Books are on sale for $8 each. I have $20 in my pocket. How many books can I buy?
2 remainder 4 2½ 3 2 4
[G2] Principal Talbert is ordering buses to take 175 students and teachers from Dunlap Elementary School on a field trip to the theater. Each bus holds 50 people. How many buses does Principal Talbert need to order?
3 remainder 42 3½ 3 4 25
[G3] Write a division story problem that represents for which the answer 3 makes the most sense.
[H] Standard US Long Division Algorithm [4 points]
Hae Min used the standard US long division algorithm to solve the following problem. Her work is below.
Draw a base-10 block model of the division problem 705 ÷ 3
Explain what the highlighted 3, 2, 6, 10 in the symbols below represent by referring your diagram of base-10 blocks/picture of bills AND by using the story context (i.e., money and people). Aim to connect all three: the story, the symbols, and your model by using labels, arrows, and/or color.
Mihajlo, Huiyu and Aditi made $705 at their yard sale. If they want to split the money equally among the 3 of them, how much money will each person get?
EDUC 170, Winter 2016
EDUC 170 1
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