**There are twice as many nails as screws in a bin. If 510 of the nails and 75 of the screws are used, there will be the same number of nails as screws. How many nails were in the bin to start with?**

Submit your solutions by the end of the month!

Last month’s Word Problem Wednesday problem was from Math in Focus Grade 3.

How did you do?

The author, Sara Kronstain, has almost a decade of experience guiding elementary and middle school mathematicians to become critical thinkers and problem solvers. She teaches fifth-grade math and is the Kindergarten-6th Grade Math Department Chair at St. Anne’s-Belfield School in Charlottesville, Virginia.

When I was in school, my math classes were typical of what one would expect a “traditional” math class to look like. I remember sitting in my elementary and middle school classes, watching as my teachers modeled problem after problem. The class would listen and then practice many of the same types of problems in our notebooks. While this type of teaching may achieve the immediate goal of learning a mathematical procedure, it does not guide students to reach an integral part of learning mathematics: problem-solving (Cai & Lester, 2010).

Singapore Math is comprised of a framework with problem-solving being the center of learning mathematics. This framework is built around five key components – metacognition, process, attitudes, skills, and concepts – all being of equal importance in developing mathematical problem solving in students. Whereas traditional math classes may place primary importance on developing skills and concepts in students, the additional three components of metacognition (self-regulation of learning), process (reasoning, making connections, and applying knowledge), and attitudes (perseverance, confidence, interest) are all key to developing critical thinking and problem solving skills in students (Ministry of Education Singapore, 2006).

A typical Singapore Math lesson is taught with a concrete-pictorial-abstract approach. Where many of my lessons as a math student began in the abstract stage (solving equations), the concrete and pictorial stages allow students to create and solidify their own understanding of a topic. The concrete stage refers to using hands-on materials to model a mathematical situation. The pictorial stage consists of diagrams and other visuals, thus building students’ learning in a tangible way (Maths No Problem!, 2018). The concrete and pictorial stages allow students to understand why math works the way it does before learning the procedure of how to solve using an algorithm.

Most Singapore Math lessons begin with an anchor task, allowing students to explore these three stages. The anchor task is a question that allows students the chance to deeply explore a topic and develop multiple methods for solving a problem (Ban Har, 2013). Let us say, for example, a group of fifth graders were posed the problem, “The distance of a race is 3km. Lily ran two-fifths of the distance. How many kilometers did Lily run?” Students would be given the opportunity to freely explore this question by using manipulatives such as fraction bars, fractions circles, or paper (for folding) along with writing materials. Here are a few examples of possible student responses to this question:

Add ⅖ + ⅖ + ⅖**. **Students may use fraction bars, fraction circles, or pictures. Students become familiar with the phrase “3 groups of ⅖”.

Three boxes are each split into fifths. Two of each of the fifths in all three boxes are shaded in. The shaded parts are added together.

A bar with the length of 3 wholes can be split into five parts. Each part has a value of ⅗. Then add ⅗ + ⅗ .

Three boxes are split into 5 equal groups, first by placing one half in each group. Then, split the leftover half into five parts (tenths). Each group will have one half of a whole and one tenth of a whole. Combine two groups by adding two halves to two tenths.

In this example, the repeated addition method reinforces addition with fractions, while the last method has students thinking about and manipulating fractions in a much more complex way. Thinking back to the five key components of Singapore Math, students in this example are refining their process of learning operations with fractions by making connects across operations. It is powerful that these responses are coming from students, as they are building their understanding of math through collaboration with their peers. In sharing methods, listening to other’s methods, and processing others methods, students are also developing their metacognition. This question could also be modified and challenge students to problem solve in an even deeper way. “What if the total distance was ½ km? What if the total distance was 3 ½ km?” Students can then go back to the concrete, pictorial, and abstract stages and continue to build on their problem-solving abilities.

At the end of the day the primary purpose of this math lesson, or any math lesson for that matter, is not simply to learn how to multiply fractions by a whole number. The most important takeaways are the critical thinking, questioning, collaboration, and problem-solving that happens among students. Teachers are not preparing students to go out into a world where they will simply be asked to recite an algorithm. While a goal is for each child to develop a deep love of math, the biggest hope is that students learn to ask questions, logically think through problems, and make sense of the world around them.

Ban Har, Yeap. (2013, June 13). Singapore Math at the Blake School, Hopkins, MN. Retrieved from __http://banhar.blogspot.com/search?q=anchor+task__

Cai, Jinfa, & Lester, Frank. (2010, April 8). Problem Solving. *National Council of Teachers of Mathematics.* Retrieved from __https://www.nctm.org/Research-and-Advocacy/Research-Brief-and-Clips/Problem-Solving/#brief__

Kaur, Berinderjeet. (2018, March 29). Building the Maths house: Singapore’s curriculum framework. *Oxford Education Blog*. Retrieved from __https://educationblog.oup.com/secondary/maths/building-the-maths-house-singapores-curriculum-framework__

Ministry of Education Singapore. (2006). *Mathematics Syllabus Primary*. Retrieved from __https://www.moe.gov.sg/docs/default-source/document/education/syllabuses/sciences/files/2007-mathematics-%28primary%29-syllabus.pdf__

Maths No Problem! (2018). *Concrete Pictorial Abstract.* Retrieved from __https://mathsnoproblem.com/en/the-maths/teaching-methods/concrete-pictorial-abstract/__

Imagine starting your day in first grade with a question about favorite holiday treats. Students can answer the question and instantly you have meaningful data that can be organized into a tally chart, picture graph, or bar graph for students to analyze. Or, students can build a bar graph with post-it notes as they make their choices. Then, spend some time analyzing the results.

Ask 5th graders if they traveled over Thanksgiving break. If so, how far? Now use this data to find mean, median, and mode, or to create a histogram for students to analyze. Or, chart the temperatures over the course of a couple of weeks and use this data to create a line graph.

Third and fourth graders could tally the number of candles in their homes for the holidays and use this data to create a line plot. Fourth graders can use their line plots to explore finding the median.

Planning a holiday party? Survey the students on what should be served and what activities should be included. Students can present the findings in a graph and use the results to determine how much and what needs to be donated or purchased to make the party a success.

The holidays are a great time to share family traditions. Why not use that information to meet some graphing and data analysis standards?

For other ideas to keep students engaged in learning read Mental Math Breaks from December 2017.

**Farmer Fred makes 4 quarts of orange juice on Monday. He makes 2 quarts more orange juice on Tuesday than on Monday. He makes 2 more quarts on Wednesday than on Tuesday. He carries on making 2 more quarts of orange juice every day than the day before. In how many days will he make a total of 80 pints of orange juice?**

Submit your solutions by the end of the month!

Last month’s problem was from Primary Mathematics Challenging Word Problems 3:

How did you do?

Math Champions is happy to be returning to Palm Springs on November 2nd and 3rd to work with educators from Southern California. Cassy and Beth will be presenting three sessions this year, supporting the theme of Mathematical Journeys to Empower All Students.

Join us on Friday at 1:30 pm for **Ready, Set, Play!: Practicing Number Sense with Games**. Engage in tried-and-true math games that support the development of number sense and place value. Leave with ideas and materials to take right back and use immediately in your classrooms.

Continue your learning on Saturday at 8:30 am with **Navigating Word Problems with Models**. We’ll investigate methods of teaching and assessing tape diagrams for those persnickety word problems with hands-on materials. We’ll look at strategies to introduce model drawing to both beginning and struggling learners.

Come back again on Saturday at 10:30 am for **Using Mental Math Strategies to Deepen Number Sense**. Learn what we mean by mental math, explore strategies, and experience how to practice mental math in your classrooms. Having a deep sense of number will empower and build confidence in your learners.

Not registered? No problem! Registration is currently open.

Attend one of our sessions and identify yourself as a blog follower to receive a gift of thanks. We hope to see you there!

I highly recommend that you dedicate a shelf or area of your classroom to math materials. It’s equally important for students to choose the most appropriate tool, as it is for them to use them. Having materials out for students at all times will allow for that.

One of the joys of my job is that I get to visit schools and classrooms across the country. So, I will share with you some organizational tips that I have gathered from my journeys.

There are three schools of thought (no pun intended) when it comes to organizing manipulatives; individual kits, group kits or community tubs. You may find it helpful to use a combination of the three, depending on the item.

I’ll mention a couple of manipulatives specifically here.

Many teachers prefer to organize discs into student kits. The idea being that students will have easy access to the discs for lessons with minimal time getting discs out and cleaning them up.

This option works great if you have enough discs for each student to have 20 of each place value; 20 ones, 20 tens, 20 hundreds, etc. Students are expected to keep these baggies or boxes of discs in their desks.

Pros: Easy access

Cons: Relies on students to maintain the correct number of discs in their kits. (I was that teacher who couldn’t stand the fact that there was one ten disc on the floor at the end of the lesson that seemed to belong to no one!)

Like student kits, you’ll need 20 of each place value in each kit. With group kits, you don’t need as many total discs. The idea here is that students will use discs with a partner or in small groups. These kits can be stored in a community tub and pulled out for use during lessons.

Pros: Easy access

Cons: See above. (Which kit does this disc belong to?!?)

In this case, discs are organized by place value into tubs. So, you would have a tub of ones, a tub of tens, and so on. In each tub, you can keep a set of small cups (Dixie cups work well) for students to take a scoop of the discs when needed. Clean up is a snap. Students simply dump the cups of discs back into the correct place value tub.

Pros: No more mystery missing discs! Very quick set up. (No more evenings spent counting out discs while watching TV.)

Cons: Requires a bit more practice with the procedure of gathering and returning discs to the correct tub.

Linking cubes are a multi-functional manipulative that each classroom should have. For a class of about 20 students, you’ll want to have at least 400 individual cubes. That’s enough for each student to have a set of 20 when needed for instruction. If you’re using them for modeling area or multiplication arrays, you might want double that amount.

You’ll want to put at least 20 in each kit.

Pros: Ease of access.

Cons: Whose cube is this?!?

If you are keeping your cubes in tubs, for ease of passing out and cleaning up, organize them in rods of 10, preferably by color. That way you can quickly pass out 2 rods (or more) to each student or partner group.

Pros: Fewer materials in student desks. No more mystery cubes.

Cons: Need to establish procedures for keeping cubes in rods of ten. (Easy, peasy!)

Other manipulatives should be in tubs on a shelf in the classroom available to all students at all times!

If you have any organizational tips from your classroom that you’d like to share, please send us a comment.

We were thrilled to welcome teachers, coaches, specialists and administrators from 18 states to our Jumpstart your Singapore Math® Instruction workshops this summer.

We are so very grateful that you took time from your summer to join us….And we are delighted that you found it valuable!

*-Keith Grifffin, 1 ^{st} and 2^{nd} grade Math Specialist, City Academy School, St. Louis, MO*

*As an administrator, this training was invaluable to my understanding of the Singapore approach to teaching math!*

*-Melanie Stivers, 5-8 Principal, Springfield Christian School, Springfield, IL*

*This is the best training I’ve been to. Every minute was enjoyable and educational. I feel better going into the school year and am excited to teach the Singapore way. It was life changing and *mind blowing*!*

*-Penny Hagerman, Interventionist, 3-5, Vanguard Classical School West, Aurora, CO*

*Truly appreciated the lesson planning information. The teacher’s guide does not have enough information to assist teachers with teaching strategies. I feel I can teach better and help my students better understand and build on the concepts. Awesome Class!*

Clayborne Education – Charlottesville, VA

Augustine Christian Academy – Tulsa, OK

Liberty Common School – Fort Collins, CO

Mounds Park Academy – Saint Paul, MN

We will announce details regarding 2019 Workshops soon. If you would like to receive notice of upcoming workshops and are not already on our email list, please complete our Training Needs Survey or give us a call.

So, what do you need to teach concretely? A complete list of recommended materials by grade level can be found here.

Really, though, with a few basic items you can get started…

Find linking cubes here.

Kindergarten – used for counting with one-to-one correspondence, measuring with non-standard units, and for modeling basic addition and subtraction situations.

1^{st} – 2^{nd} grade – used for place value understanding, to model story problems and mental math strategies, for measurement with non-standard units, building array models for multiplication, and for beginning bar modeling.

3^{rd} grade – used to model part-whole and comparison word problems involving addition, subtraction, multiplication, and division, for building array models for multiplication and division, and for modeling area.

4^{th} grade and up – used to model word problems for multiplication, division, and ratio, and to model area and volume.

Find Base Ten Blocks here and Place Value Discs here.

1^{st} grade – Base-Ten Blocks are used to model place value for numbers to 100

2^{nd} grade and up – Place Value Discs are used as a more abstract (and manageable) model for place value understanding for numbers from thousandths to millions, and for modeling and developing a conceptual understanding of the four standard algorithms. Base-Ten blocks can continue to be used for those students needing a one-to-one representation.

Cut them from paper found in the recycle bin.

1^{st} and 2^{nd} grade – used to model fractions of a whole.

2^{nd} grade and up – used to model the four operations of fractions with the same size whole and for modeling part-whole and comparison word problems.

Find number cards on our resources page or pick up some playing cards at your local dollar store. Dice can be found here.

All grades – for playing games and making math fun!

Get creative and have fun building your inventory of math manipulatives!

Submit your solutions by the end of the month!

Last month’s problem was from Dimensions Math 6A:

Here’s a solution from Reader Shirley Davis:

How did you do?

This month’s problem comes from Dimensions Math 6A and highlights the unitary method of solving problems:

Submit your solutions by the end of the month!

Last month’s problem was from the website TestPapersFree.com, which provides past copies of continual and semestral assessments from Singapore Primary Schools. This is a great resource if you’re looking to see questions directly from Singapore classrooms. The problem is from Raffles Girls School, Grade 4, and is a Semester 2 assessment, which is the final term of the year.

How many more cards did Pei Ling have than Zandy?

Here’s a solution from Reader Shirley Davis:

How did you do?