Module 7 Problem Solving

Posted on: 17th May 2023


Module 7 Problem Solving Set Module - VDW Chapter 16 and 17 and Principles to Action 1. Explain how the line-up-the-decimals rule for adding and subtracting can be developed through practice with estimation. (2 points) 2. Give an example explaining how, in many problems, multiplication and division with decimals can be replaced with estimation and whole number methods. (3 points) 3. Choose one of the activities from VDW chapter 16 (list the name of the activity as given in the text) and explain why you think it would be a good lesson to build conceptual understanding. Determine (name it specifically) and justify the level of cognitive demand, based on the Smith and Stein work. Justify your decision. (5 points) 4. Write a review and reflection from THREE of the Effective Teaching Practices (find them listed in the table on page 10 in Principles to Action). Please don’t write about Supporting Productive Struggle, or Elicit and Use Evidence of Student Thinking as we wrote about those last week. Offer an overview of the big ideas linking the work from this course and your understanding of the three practices that you choose to discuss. This is a place for you to talk about your personal growth as a mathematician and a math instructor. How can you implement these practices in your classroom? What questions do you still have? After studying these ideas, will you make adjustments to your instruction? What do you hope your students walk away with at the end of a year of instruction with you? This should be 3-5 paragraphs. (15 points) Ratios, Proportions, and Proportional Reasoning Please solve the following problems. Model your thinking using diagrams, pictures, double number lines, bar model, or another appropriate model. I’d like to see your thinking as you solve, so simply offering a solution is not enough. The problems are from NCTM’s Putting Essential Understanding of Ratio and Proportion into Practice Grades 6-8. Each item is worth 2 points 1. Sands of Time Jamie solved the following problem for homework: Three-fifths of the sand went through a sand timer in 18 minutes. If the rest of the sand goes through at the same rate, how long does it take all the sand to go through the timer? Jamie got 30 minutes for her answer. Show two different ways that Jamie could have solved the problem. 2. Red Gumball or White? That Is the Question! Taufique has two bags, each containing some red and some white gumballs. In bag A, there are 10 red and 15 white gumballs. In bag B, there are 6 red and 8 white gumballs. a. If Taufique reaches in without looking, from which bag is he more likely to pull out a red gumball? b. Draw a diagram to represent how you solved the problem, and explain it. 3. Pizza Presto: Which Chef Is Faster? In the Fastest Pizza Maker contest, Keala made three pizzas in five minutes. Casey made four pizzas in nine minutes. If both chefs make pizzas at the same rate as they did in the Fastest Pizza Maker Contest, who do you think will make a pizza faster in a Keala vs. Casey competition? 4. Off the Top of Your Head, Which Is Larger? Use only mental arithmetic to determine which is larger and avoid using an algorithm. Please reason about the numbers. (an explanation, without a model, is fine for this one) or 14 29 15 30 5. Relating Reciprocals to 1 Which is closer to 1. Again, reason about these numbers. Don’t convert to a decimal, don’t find a common denominator, or use another algorithm (an explanation, without a model, is fine for this one) or 5 6 6 5 6. How are you? I know you have a lot on your plate and I’d like to check in again. I appreciate all that you are doing and am thankful for your commitment to your education so that you can be the best for your students. (Please feel free to skip this questionknow that I am seriously asking how you are)

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Module 7 Problem Solving

Question 1

Adding and subtracting decimals requires mathematicians: to arrange numbers so that decimal points lie under one vertical line, add extra zeros to numbers towards the right side of the decimal point to ensure that each number possesses the same number of digits. As a rule, whole numbers and decimal parts can be added or subtracted separately, just like fractions. Decimals are obtained through the division of numerators by denominators. Therefore, estimation is an essential practice that aids in rounding to convenient numbers if the task response does not demand an exact answer.

Question 2

Nancy has $60 and wants to buy fruits that cost $0.80 each. Many fruits can she buy?

Amount: $60.

Cost of one Fruit: $0.80.

Round off each cost to the nearest whole number

= $1.0.

The number of fruits to purchase:

= (60/1)

= About 60 fruits.

Question 3

          According to Smith and Stern. (2011), educators need to adjust the cognitive demand in a learning environment to enable learners to solve varied forms of classroom tasks throughout the learning process. I identified the group discussions where learners can work in pairs to share mathematical ideas and problem-solving skills. Mathematics entails complex tasks designed for learners to explore specific mathematical concepts such as algebraic expressions. I believe that group discussion enhances cognitive demands such as memorization and tackling mathematical tasks.

Question 4

According to Huinker. (2018), teaching and learning form part of substantial processes that impact the individuals’ minds and behavior. The changing trends across the globe demand effective approaches to teaching and learning to guarantee learners’ maximum knowledge acquisition; therefore, keeping learners to comply with technological advancement provokes creativity. Teaching and learning are aligned to principles for action to advance the learning rendered at educational institutions. The practices embraced should also take into account the interests and abilities of every learner. Active engagement and interaction among learners and other stakeholders in the teaching fraternity bring unity and commitment to day-to-day tasks in the entire learning process.

One of the practical teaching practices standards in the learning process is the demand for quality performances, timely positive feedback, and enhancing motivation for learners to learn. My personal growth as a mathematician began at elementary schools, where teachers demanded incredible mathematics performances (Huinker, 2018). I remember receiving several motivation gifts from my former mathematics teacher due to outstanding performances in all subjects. Mathematics is a subject that demands commitment, motivation, and hard work; therefore, as a mathematics instructor, rewarding students’ better performance will ignite commitment to quality performances within them (Huinker, 2018).  Also, offering timely feedback to learners is an effective practice that enables them to make relevant adjustments in their academic matters. The feedback should be encouraging, brief, and precise.

Proportions, Proportional Reasoning, and Ratios

Each Item is worth two points

#1. Sands of Time

(3/5) full = 18 minutes

(5/5) full =?

(5/5 x 18) ÷ (3/5) = 18 x (5/3)

18 x (5/3) = 30 minutes.


If 3 out of 5 proportions of sand took 18 minutes to pass through the sand timer

One proportion of sand takes (18/3) = 6 minutes

All sands will go through the sand timer in 5 proportions = 5 x 6

                                                                         = 30 minutes.

#2. Red Gumball or White? That Is the Question!

a). Bag B


#3. Pizza Presto: Which Chef Is Faster?

Kaela = 3 Pizzas in 5 minutes

Casey = 4 Pizzas in 9 minutes

             Rate of Work per Minute

Kaela = 3/5

Casey = 4/9

Kaela or Casey

(3/5 x 45) or (4/9 x 45)

27 or 20

Kaela makes a Pizza faster than Casey.

#4. Off the Top of Your Head, which is Larger?

If the difference between denominators and numerators of given fractions are the same, then the fraction with a bigger denominator is the more significant fraction. Therefore, 15/30 is the larger fraction as compared to 14/29.

#5. Relating Reciprocals to 1

Answer: 5/6 or 6/5. While relating reciprocals to 1, if the difference between the numerators and denominators are the same, then the fraction closer to 1 is the one with a larger numerator. Therefore, the closer fraction to one is 6/5 as compared to 5/6.



Huinker, D. (2018). Principles to Actions: Moving to a framework for teaching mathematics. Teaching Children Mathematics25(3), 133-136.

Smith, Margaret S., and Mary Kay Stein. 2011. 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA: National Council of Teachers of Mathematics.

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