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…

Linking Cubes

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.

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!

What questions do you have?

Next: Ideas for organizing manipulatives.

Word Problem Wednesday – Jason and Louis

This month’s Word Problem Wednesday problem comes from Primary Mathematics Challenging Word Problems 3.

Jason and Louis picked up a total of 30 cans. For every 2 cans that Jason picked up, Louis picked up 3 cans. How many cans did each boy pick up?

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?

Word Problem Wednesday – Laiza’s Dress

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

Laiza spent 38% of her money on a dress and the rest on a purse. If she spent $114 on the dress, how much did she spend on the purse?

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.

Pei Ling had 3 times as many cards Zandy. Sulaiman had half the number of cards Zandy had. There were a total of 1278 cards.
How many more cards did Pei Ling have than Zandy?

Here’s a solution from Reader Shirley Davis:

 

 

 

 

 

 

 

How did you do?

 

 

Word Problem Wednesday – Pei Ling, Zandy, and Sulaiman

This month’s problem comes 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. This problem is from Raffles Girls School,  Grade 4, and is a Semester 2 assessment, which is the final term of the year.

Pei Ling had 3 times as many cards Zandy.
Sulaiman had half the number of cards Zandy had.
There were a total of 1278 cards.
How many more cards did Pei Ling have than Zandy?

Submit your solutions by the end of the month!


The prior problem was from the Grade 6 STAAR 2013-2017 Released Test questions from lead4ward aligned to the Texas Essential Knowledge and Skills or TEKS.

There are 176 slices of bread in 8 loaves. If there are the same number of slices in each loaf, how many slices of bread are there in 5 loaves?

 

How did you do?

Word Problem Wednesday – Rulers and Bread

Word Problem Wednesday was such a hit, we’re going to continue throughout the year with one problem a month.

This problem popped up in my Medium feed last month:

Algebraic expressions — the return! Guess the Misconception author Craig Barton noted that on a quiz website for test prep in the UK,  only 1 in 3 students could answer this problem correctly. At the time, I was also analyzing the value of model drawing by reviewing released problems from the 6th-grade STAAR tests, so my first thought was, hmm, how would this work as a bar model?

Pretty well, actually. If I know that:

I can find:

The AQA is an independent education charity that offers GCSE testing in the UK. DiagnosticQuestions.com provides multiple choice questions so you can build your own assessment, or use one of their collections.

Check out a bar model solution:

 

Finally, this month’s problem comes from the Grade 6 STAAR 2013-2017 Released Test questions from lead4ward aligned to the Texas Essential Knowledge and Skills or TEKS. It aligns to the standard:

6.4(B) (New) Proportional Reasoning: Apply qualitative and quantitative reasoning to solve prediction and comparison of real-world problems involving ratios and rates.

There are 176 slices of bread in 8 loaves. If there are the same number of slices in each loaf, how many slices of bread are there in 5 loaves?

Submit your solutions by the end of the month!


The prior problem was from the Teacher’s Guide for Primary Mathematics US Edition 5A.

We had a couple of submissions.

Here’s Shirley Davis’ model and algebra combo:

Word Problem Wednesday – Alice, Betty, & Cassie

Word Problem Wednesday was such a hit, we’re going to continue throughout the year with one problem a month.

Our problem this month comes courtesy of a 5th grade teacher who was excited that for the first time, her students understood and easily modeled this problem from the Teacher’s Guide for Primary Mathematics US Edition 5A.

Alice, Betty, and Cassie have $70 altogether. The ratio of Alice’s money to Betty’s money is 1 : 3. Cassie has $10 more than Alice. What is the ratio of Alice’s money to Betty’s money to Cassie’s money?

Submit your solutions by the end of the month!


The last problem was taken from the Dimensions Math 3A Textbook. (Click to learn more about this recently released curriculum):

Shirley Davis shared her algebraic bar model solution:

 

How did you do?

Word Problem Wednesday – Pinecones

Word Problem Wednesday was such a hit, we’re going to continue throughout the year with one problem a month.

Singapore Math, Inc. will be releasing a new series April 25 at the National Council of Teachers of Mathematics Annual Conference. This problem comes from a chapter on two-step word problems from 3A

Mei and Dion together made 11 turtles. Mei made 3 more turtles than Dion.
How many turtles did Mei make?

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


The last problem was taken from Noetic Learning’s problem of the week Sign up to receive their weekly problems.

Robin’s age is 3 times Marcia’s age. Anna is twice as old as Marcia. The sum of their ages is 30. How old is Marcia?

Shirley Davis shared her algebraic bar model solution:

 

How did you do?

 

 

Come see us at NCEA and NCTM

Spring educators’ conference season is upon us and we are thrilled by several opportunities to speak at upcoming events.  The descriptions below are from conference programs.

NCEA 2018 Convention & Expo (April 3 – 5 at the Duke Energy Convention Center in Cincinnati, OH) is the largest private-education association gathering in the nation!

Strip Models, Tape Diagrams, Bar Models, Oh My!
Presenters: Cassy Turner and Beth Curran
Date: Tuesday, April 3, 2018
Time: 1:30 PM – 2:45 PM
Room: 251

Improving students’ problem-solving abilities is a major focus of mathematics education. Model drawing is a powerful tool that students can use to attack complex problems. In this hands-on, minds ­on session, presenters will investigate methods of teaching and assessing tape diagrams for those persnickety word problems, and explore interactive model drawing technology. Walk away with strategies for guiding student learning that you can use tomorrow!

Using Mental Math Strategies to Deepen Number Sense
Presenters: Beth Curran and Cassy Turner
Date: Thursday, April 5, 2018
Time: 11:15 AM – 12:30 PM
Room: 251

Number sense = mental math. Participants will actively explore mental math strategies used throughout the elementary grades. Engaging in mental math activities allows students to develop a relational understanding of numbers and their magnitude. Students begin to see numbers as being made up of parts and develop an understanding of how numbers can be composed and decomposed for mental calculations. Discourse around mental math allows students to expand their toolbox of strategies for solving problems and to evaluate strategies and answers for efficiency and reasonableness.

NCTM Annual Meeting and Exposition 2018 (April 25 – 28 at the Walter E. Washington Convention Center in Washington, D.C.) is the premier math education event of the year!

Do Not Invert and Multiply! Building the Bridge to Algebra Through Fractions Tasks
Date:  Friday, April 27, 2018
Time:  1:30 PM – 2:30 PM
Room:  159 AB

Join Cassy Turner, Beth Curran, and Allison Coates as they work through hands-on tasks for fractions. We’ll investigate how the progression of fractions problems helps students build mastery of algebraic concepts such as naming unknown quantities, writing expressions, and laying the foundation for solving for x.

Using Anchor Tasks to Engage Learners: Deepening Understanding through Exploration and Discourse
Date:  Saturday, April 28, 2018
Time:  8:00 AM – 9:00 AM
Room:  146 B

Participants will engage in active math lessons and learn how to use learning objectives to create anchor tasks that spark student interest and allow students of all levels to build on prior knowledge, explore concepts with concrete materials and engage in productive discourse to deepen conceptual understanding with a focus on problem-solving. Cassy Turner and Beth Curran will lead this interactive workshop

Beginning Bar Model Boot Camp: Getting Started with Model Drawing
Date:  Saturday, April 28, 2018
Time:  9:45 AM – 11:00 AM
Room: 144 ABC

Improving students’ problem-solving abilities is a major objective of Common Core and state standards, and model drawing is a powerful tool that students can use to attack complex problems. Join Cassy Turner and Beth Curran to investigate methods of teaching and assessing tape diagrams for those persnickety word problems, and explore interactive model drawing technology.

Throwback Thursday Extra-Test Prep

As the standardized testing season approaches, we present readers a special edition of Throwback Thursday featuring one of our more popular posts. Here, Beth Curran addresses common questions and misconceptions on the topic of Test Preparation. As a teacher, I encouraged my students to welcome their annual opportunity to “show what they know.”

 


Test Prep: Is it really necessary?

Originally published February 16, 2017

For many, Spring brings with it those two dreaded words: standardized tests.

Whether your school is required to take PARCC, Smarter Balanced, state mandated standards-based tests or ERBs, you inevitably will want to make sure your students are prepared.  Many teachers will plan to block out two to three weeks prior to the testing dates to review and teach content that may not have been covered, but is this interruption to instruction necessary?

It’s estimated that students and teachers lose an average of 24 hours of instructional time each year administering and taking standardized tests.  This doesn’t include time taken out of the instructional day for test prep so that number may even be quite higher.

Q: But, I need to review to make sure my students remember concepts taught at the beginning of the year.

 A: Not if you have been teaching to mastery.

Teaching math with a mastery-based program that is rich in problem-solving may all but eliminate the need for any test prep or review.  If your students have a solid foundation in the basics and have practiced applying that knowledge to solving problems throughout the school year, then nothing a standardized test can throw at them should be unachievable. With a cohesive curriculum, where concepts build on each other, your students have essentially been revisiting concepts throughout the year. So, trust in what your students have learned and skip the review.

Q: What about going over topics that I haven’t covered yet?

A: How much success have you had cramming for an exam?

If material is thrown at students for the sake of a test a few things can happen.

  • Students won’t retain information. If students have not been given enough time to progress through the concrete-representational-abstract phases of learning, they will likely not be able to recall concepts or apply those concepts to the unfamiliar situations they might encounter on the standardized test.
  • Students will be stressed out. They will feel the pressure (that unfortunately, you are likely feeling as well) to get a good score on the test. Learning becomes just something to do for a test.
  • You will get false positive results. Have you ever had the teacher in the next grade up comment that students couldn’t remember a concept that you know you taught? Or, better yet, had test scores reflect learning, but students couldn’t perform at the next grade level? That can be a result of concepts being taught too quickly.

So, rather than block out a few weeks to cram in topics that you haven’t covered, try integrating them into other areas of your day. Do some data analysis in morning meeting. Add some questions about telling time to your calendar activities. Play with measurement and geometry during recess (The weather is getting nice, right?).

If you follow the sequence in your well-thought-out curriculum and teach some of those missing concepts after testing, it’s ok. Your students will experience those concepts in an order that makes sense and will be able to make connections, apply their thinking and master those concepts. That mastery will stay with them into the next year and will be reflected on upcoming standardized tests.

After all, we don’t stop teaching after standardized tests.  Well… that’s probably a topic for another post.

photo courtesy of Alberto G.

Word Problem Wednesday – Robin, Marcia, & Anna’s Ages

Word Problem Wednesday was such a hit, we’re going to continue throughout the year with one problem a month.

This problem was taken from Noetic Learning’s problem of the week and builds on the problem with the ropes from last month.  Sign up to receive their weekly problems.

Robin’s age is 3 times Marcia’s age. Anna is twice as old as Marcia. The sum of their ages is 30. How old is Marcia?

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

 


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?

 

Dedicated reader, Shirley Davis submitted the following solution:

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