## Ask the Experts: What manipulatives do you suggest for my grade level?

Singapore Mathematics instruction – or, really, just good math instruction – will have students working through three phases of learning, referred to as the Concrete-Pictorial-Abstract approach. In order to teach following this approach, you need to start at the concrete level. Jean Piaget, a Swiss psychologist, believed that in order for students to be able to visualize and abstract mathematics they first must manipulate materials. He called this the concrete operational phase of learning.

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…

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

1st – 2nd 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.

3rd 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.

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

### Base-Ten Blocks and Place Value Discs

Find Base Ten Blocks here and Place Value Discs here.

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

2nd 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.

### Paper Strips and Squares of Equal Size

Cut them from paper found in the recycle bin.

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

2nd 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.

### Number Cards (Playing Cards) and Dice

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!

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## Word Problem Wednesday – Dogs and Ropes

“The Internet Is Losing It Over This Second Grade Math Problem,” reads the headline from an article posted online by msn.com. The article goes on to support the student’s mother’s conclusion that, “this isn’t exactly a question most 8-year-olds would understand.”

The problem reads, “There are 49 dogs signed up to compete in the dog show. There are 36 more small dogs than large dogs signed up to compete. How many small dogs are signed up to compete?”

Yes, this is that “new” math. This is the math that second graders will need to succeed as adults. Gone are the days when correctly completing 20 addition or subtraction problems is enough. Problem-solving and logical thinking is what employers are looking for. So, yes, this is a challenging problem, but not a problem we should be avoiding in our schools simply because the internet says it’s too hard.

I agree there is a flaw in this problem, but it’s not in the problem itself, it’s in the numbers that were chosen. Fortunately, the numbers are the least important part of solving a word problem. That stands worthy of repeating. The numbers are the LEAST important part of solving a word problem.

So, what is then? Visualization and comprehension!

Students need to visualize the problem and then represent it with models or pictures. This is why teaching bar modeling is so very important in the early grades.

Here’s a video that shows how easy it is to solve this problem if you focus on visualization first.

Any good teacher will follow up a lesson with practice, so here is your Word Problem Wednesday for February.

This problem was taken from Challenging Word Problems 2, a supplement to the Primary Mathematics series:

##### The total length of two ropes is 36 in. One rope is 4 in. longer than the other. What is the length of the longer rope?

Submit your solutions and we’ll post all interesting strategies.

The previous problem came from i-Excel Heuristic and Model Approach Primary 5 by Li Fanglan published by FAN-Math:

##### Bob’s Bikes sold 96 bikes during the week and 1/4  of what was left on the weekend. After that, Bob still had 1/2 of his bikes left. How many bikes did Bob have at first?

Dedicated readers submitted the following solutions, first an image from Shirley Davis:

And a video from Kristine Simonson, who has been using some of the problems with her fourth graders:

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## Structuring the Math Day

One of the questions I get most often is:

##### How do I use the materials with my Singapore Math curriculum and fit it all into an hour math block?

First off, kudos to your school for setting aside an hour math block for your youngest learners! Through math instruction, students will gain the skills and thought processes necessary to solve problems. Math needs to be given a priority in the schedule. Following is one of my favorite quotes from Dr. Yeap Ban Har, author, and contributor to several Singapore Math style curriculum.

“We are not teaching math. We are teaching thinking through the medium of math.”

##### What should I include in my lessons?
• Ongoing cumulative review
• Direct instruction
• Guided practice
• Independent practice
##### How much time should I spend on each component?

10 minutes – Ongoing Cumulative Review
20 minutes – Direct Instruction
30 minutes – Guided and Independent Practice

##### What does each component consist of?
###### Ongoing Cumulative Review (10 minutes)

According to Steven Leinwand, in his book Accessible Mathematics: 10 Instructional Shifts that Raise Student Achievement, in every classroom there should be signs of:

A deliberate and carefully planned reliance on ongoing, cumulative review of key skills and concepts.

As you teach concepts, you will want to include them in your ongoing cumulative review. With such an emphasis on mental math strategies and the development of number sense, mental math should play a major role in your daily review.

Mental Math can be practiced through the use of:

Direct Instruction (20 minutes)

• Teacher directed (follow the plan in the Teacher’s Guide)
• Through student exploration (also known as, an Anchor Task)

Guided Practice (30 minutes combined with Independent Practice)

• Textbook problems can be worked:
• Whole group answering problems on individual whiteboards,
• With partners working through problems together, or
• Individually

Independent Practice (30 minutes combined with Guided Practice)

• Workbook problems
• As home enjoyment
• As classwork
• Fluency practice

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## Take a Moment to Pause and Reflect

The last few weeks of school can be a challenging time. You might find yourself rushing to finish projects, complete curriculum and bring closure to the school year.  In the midst of all of this end-of-year frenzy, many will ask students to pause and think about all they have accomplished and what they still have to learn, as a means of self-reflection.

Shouldn’t we ask the same of ourselves? Wherever you may be in your Singapore Math journey, reflection can be a valuable tool for continuing to improve your practice.

As you wrap up the year, take a moment to reflect on the following.

### Celebrate the successes.

Maybe a particular lesson or unit stands out to you as one in which your students made the most progress or you felt the most confident with your instruction. Maybe you were able to reach that struggling student by approaching the concept in a different way or for the first time in recent years, your students enjoyed math. At the close of one of my lessons, I recall a fourth-grade student asking me if all our math classes could be like this one. Her comment caused me to pause and think about what we had done that she found so engaging and fun and I made a point to model more lessons in that way.

Think about how what you did made a difference and plan to do more of that in the upcoming year.

### Acknowledge the struggles.

Recognize the areas within your instruction that were a challenge for you. Maybe your students really struggled with a particular concept or maybe you left a lesson feeling like you needed a do-over. Now is the time to pinpoint those stumbling blocks and think about what support you might need to improve your instruction in those areas. It was my first year teaching a Singapore Math curriculum. With hesitation, I approached a lesson on problem-solving using the bar model. It was a complete flop. Reflecting on it, I realized I hadn’t allowed myself enough time to practice with the bar model and develop the confidence I needed to answer student’s questions on the fly. So, I made a plan to practice bar modeling with Thinking Blocks over the summer. (Thinking Blocks is one of our favorite tools for model drawing. Find an interactive desktop version at MathPlayground.com and four iPad apps on the iTunes store.)

Accepting your struggles and using them as a means to grow professionally, will strengthen your instruction and ultimately, improve your practice.

### Set a goal for next year.

Did you try something new this year that you would like to incorporate in more of your lessons? Is there an area within the content that you would like to improve your understanding and confidence? Do you have an idea about how to approach a concept differently? Take a moment to imagine what you want the year ahead to look like and set a goal to make it happen.

As you pack up your room and head off into summer bliss, know that you will return to the new school year with an awareness of all that you gained by reflecting on your practice.

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## Make Summer Practice Fun With Games!

Congratulations! You are in the final stretch of the school year. Teachers, students, and parents are feeling excited about the carefree summer months ahead and potentially anxious about keeping students’ math skills fresh.

You will inevitably receive inquiries from parents asking how they can help their children over break.

When given the choice, rarely will a child choose to do paper-pencil workbook work over playing a game. These student-approved games require minimal materials and are easy to play at home.  They promote number sense and fluency and, as an added bonus, kids love to play them!

#### What should students practice?

Following is a list of skills to practice for Kindergarten through 3rd grade. Because the skills build, a student should practice the skills from the grade level just completed and the grade level below. Students in 4th grade and above would also benefit from practicing all the skills listed.

• Kindergarten: Number combinations to make 10
• First Grade: Recall from memory addition and subtraction within 20, addition and subtraction within and up to 100 using mental math strategies.
• Second Grade: Recall from memory multiplication and division facts for 2, 3, 4, 5 and 10.
• Third Grade: Recall from memory multiplication and division facts for 6, 7, 8 and 9, 2-digit by 1-digit multiplication, division of a 2-digit number by a 1-digit number

#### What should parents have on hand to make math fun over the summer?

All you need to play the games below is a deck of cards! (Dice are good, too.)

#### What could students play?

Here are three of our favorite games.  All can be adapted to practice across grade levels.

#### Rock-Paper-Scissors-Math!

Number of players: 2 (more for a challenge)
Materials: None
Play:
This game resembles Rock-Paper-Scissors.  When players say “Math!” each shoots out 1-5 fingers on one hand. The first player to find the sum of both
their fingers and their partner’s fingers wins the round. For example, I shoot out 3 fingers and my partner shoots out 5 fingers. The first to say 8 wins.

Variations:

• Use both hands to add up to 20
• Use one hand and multiply the fingers for facts up to 5 x 5. For example, I shoot out 3 fingers and my partner shoots out 5 fingers. The first to say 15 wins.
• Use 2 hands and multiply for facts up to 10 x 10.
• Play with more players to add or multiply multiple numbers for a challenge

#### Salute!

Number of players: 3 (more for a challenge)
Materials: Deck of cards with face cards removed, ace is 1
Play:
Split the deck in half and give each pile to 2 of the players. The third player is the Caller.  When the Caller says, “Salute!” the players each flip a card from their pile and place it on their forehead to salute each other.  Each player can see the others card, but not their own. The Caller tells them the sum of the 2 players cards. The first player to tell the number of their own card wins the round. Players can switch jobs after each round or when the pile of cards is depleted.

Variations:

• The Caller tells the product of the 2 players cards.
• With 4 players (one Caller and 3 players) students can practice sums or products of 3 numbers.
• Need more challenge? In the picture, the Caller has squared our numbers before adding them together and Melanie and Cassy are trying to find their own number.

#### Greatest Sum!

Players: 2 or more
Material: Deck of cards with face cards and 10 removed, ace is 1
Play:

This game is played like the card game of War. Shuffle the deck and place it in the center.  Each player chooses 2 cards off the top of the deck and finds the sum of their cards. The player with the greatest sum wins the cards for the round. Play continues until the deck is depleted or until time is called. This game is great for practicing mental math strategies, but can also be used to practice traditional algorithms with paper and pencil (see the variations).

Variations:

• Players choose 2 cards and find the difference. The player with the smallest difference wins.
• Players choose 2 cards and find the product. Greatest product wins.
• Each player chooses 3 cards and creates a 2-digit and a 1-digit number and finds the sum or difference. The player with the greatest sum or least difference is the winner.
• Each player chooses 3 cards and creates a 2-digit and a 1-digit number and finds the product. The player with the greatest product is the winner.
• Each player chooses 3 cards and creates a 2-digit and a 1-digit number and finds the quotient and remainder when the 2-digit number is divided by the 1-digit number. The player with the greatest remainder wins.
• Each player chooses 4 cards and creates two 2-digit numbers and finds the sum or difference. The player with the greatest sum or least difference in the winner.

Adding the element of competition to practicing basic facts makes it more fun for all. Let us know if you have a favorite game you’d like to share.

Looking for more ideas? See: “Summer Math” Suggestions to Boost Student Understanding”

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