The Rainbow Fish

Ann was  recently at Frenchville Primary School where she noticed a display based on the Rainbow Fish by Marcus Pfister, one of her favourite series of books.

The  following photograph shows the whole display but look closely at the next photo and you will see that the  scales on each fish are made from fractions of cup cake papers.

How many cup cakes wrappers were needed to make each fish Ann wondered.

The Problem
(Note the first  picture to follow has  scales that are ¼ of the paper and the second photo of scales shows scales that are 1/6  of the paper, select to match your learners.)

My friend, the year 1 teacher, has to buy cup cake papers in green, blue, red, pink, yellow, and silver so that her class can each make a fraction  rainbow fish. Each scale is the same size fraction and each fish will be the same. She  has 23 students in her class.

How many  cup cake papers will she need?

Note: John Van de Walle suggested that students should be encouraged to look at a fraction and visualise the whole from which it was taken so that they could work out what fraction is shown. With this in mind  encourage your students to identify and explain what fraction of the whole they think the scales are before they begin work. Select the class or group size to match the range of your students.

The Spider Party

A couple of weeks ago we showed the Jelly Bean problem from Kathy Lanthois’s Year 7 class and promised more. Well here is the class Spider Party Problem.

Kathy’s class researched the best and most economical spider drinks for the spider party that they were actually going to have. 

Note: this is important. The Spider Party was a real problem not a Mickey Mouse ‘let’s pretend we going to have a party’ problem or even and Ann Baker true story. It took several days of working with surveys and taste tests, proportional reasoning, measuring and costing before the final tightly-budgeted spider-making day. The above work sample is just one of those displayed on the classroom wall. And yes of course the spider party was a huge success and the mathematics located the Spider Drink flavours that meant everyone had the spider they liked the most.

With Halloween just around the corner, Spider Parties must surely be a winner with any class!

A Halloween Problem

Ann was running a demo lesson with the year 3s at Crestmead Primary School this week. Ann told the class that she wanted to make 12 trick or treat bags for a Halloween party but didn’t know what to put in them. The students were quick to identify the quintessential ingredients for the bags. The problem that emerged is shown below along with a few student samples that demonstrate a wide range of strategies.

12 bags

Each bag will have:

6 sour worms

2 sets of vampire teeth

5 bubble gums

4 jelly snakes

10 mini- marshmallows

3 lolly pops

13 little chocolates

How many of each type of sweets will that be?

How many sweets will that be to fill the bags altogether?

Student Work

This problem had just the right amount of desirable difficulty for the class who were engaged readily and who persisted with the problem solving process. As is usual with these problems there was a broad range of approaches allowing students multiple entry points into the problem.

Not all students answered all parts of the problem as can be seen in Hayden’s work sample. Hayden however demonstrated that he had interpreted the problem as a multiplicative situation and linked the multiplications to two different representational forms for multiplication. He is developing a firm foundation for connecting these strategies to the more formal strategies of multiplication. You can also see that Hayden was applying fix up strategies as he worked.

Roania’ work sample shows that she is able to use known multiplication facts flexibly. She has used the distributive property to split 12 into 10 and 2 because she ‘knows her 10s and knows how to double’. The realism of the problem connected to her and is manifested by her idea of presenting each type of sweet in its own box so that it could be used as a shopping list.

Roma’s strategy though not fully correct or complete (she was working on her fix up strategy when time eluded her) focussed on the second question rather than the individual parts. She worked out 43 sweets in each bag and began to carry out a repeated addition with chunking. As she began chunking her answers, place value problems became visible. As formative assessment these types of problematised situations make visible gaps and error patterns that might otherwise go over looked.

The following work sample shows how one student checked the reasonableness of his answers, giving ticks before moving onto the second question, how many altogether. As a work sample that shows the working out and steps involved this one really makes the student’s thinking visible.

And last, the next sample shows counting in 2s, 3s and 5s as well as the use of tallies, with the totals being rearranged to make good use of friendly numbers.

Over all the samples give a snap shot of the range of strategies and levels of development that can be seen in any class. We’d like to say thank you to this class and the teachers involved. The students were AWESOME.

Metric Measurement Week

This week is National Metric Week in America where despite the fact that they still use Imperial measures such as feet and inches, there are some people who think that metric measures are the way to go. One thing led to another and the next thing we were discussing the developmental sequence that leads to deep understanding of linear measurement. As we were talking about that we decided to blog about our Linear Measurement Series, Books1, 2 and 3 which we researched, trialled in classrooms and produced in response to the poor attempts at the standardised test question that required students to mark the bubble that said how long the given line was. Right next to the line and correctly aligned was a broken ruler. Most students shaded the 9 cm bubble but the correct answer was 5 cm. That is almost twice the length. How come students: 

• could not measure with a broken ruler and 
• (perhaps more importantly) didn’t have any kind of visual spatial alarm bells going off in their heads telling them that no way was the line 9 cm.

From the research that we looked at it became clear that teachers needed more guidance about how to teach linear measurement than their basic training provided. The Linear Measurement books ensure that students have worthwhile experiences (no one in real life measures lines on a page for a living) and that those experiences should include: iteration, transitivity, conservation and estimation skills.

A developmental sequence

Each of the books has a Top 5 that explains exactly what aspect of linear measurement is being developed. The activities in the books are focussed on the Top 5 and, as an example, here is Activity 8 which focuses on the estimation concepts that are part of the above Top 5.

Using technology to teach measurement

The Three Snakes measurement app also grew out of this research. The app focuses on looking at the starting point and the finishing point when making comparisons. Often students only look at the end point as with the standardised test mentioned earlier. The Three Snakes app also provides opportunities for students to apply reasoning skills to comparison problems focussing on the comparative language of measurement, short, shorter, shortest, long, longer, longest.

Showcasing South Australian Teachers

Ann is really enjoying her work with South Australian teachers. Recently Kathy Lanthois of Darlington Primary School invited Ann into her Year 7 classroom to see what her class had been up to in maths. What a term her class must have had and no wonder her students look forward to maths so much. Today we are showcasing her Jelly Bean project. Her class had been learning about fractions, decimals and percents and so, to see what they had learned and could apply, the Jelly Bean Maths activity was set up.

Each group was given a packet of Jelly Beans to investigate. Their mission was to find out about the distribution of colours in a packet and also to research favourite colours with a goal of informing the manufacturer of their findings and recommendations. The hidden agenda of course was to find out how and whether the students would apply their new understanding of decimals and percents. What a great formative and/or summative assessment task!

The following work sample was one of many showcased on the classroom wall. Sorry we only have one photo - should have taken loads.

It shows, as did the others, that the students worked out what fraction, decimal and percent of the packet each colour represented. They could translate between those forms and offer explanations of their thinking.

We love the free form layout, all different, the use of colour and details that personalise the maths on each poster. Can we start a BAN THE GREY, BORING, MATHS movement? Maybe then there would be fewer disengaged maths students around.

Watch this space for more exciting maths from this class.

Halloween is coming!

How about a few themed Halloween Maths Books to create some fun. Here are some of the books that we love at Halloween.

Adler, David(2011) Mystery Maths: A First Book of Algebra Random House.

As it says, this is a first look at algebra and simple equations with pronumerals. It is easy to follow and good fun for 10 years and up.

Armstrong-Ellis, Carey (2012) Ten Creepy Monsters Harry N. Abrams

This is a simple count back book for 4, 5 and 6 year olds. Students could create their own Halloween monster count back sequences.

Axelrod, Amy (1999) Pigs Go to Market: Halloween Fun with Maths and Shopping

 A book about shopping for Halloween and sharing the treats, fairly or unfairly!

Gunnufson, Charlotte (2013) Halloween Hustle Two Lions

I included this thinking it would be fun to create repeating dance patterns from the sequences in the book, age 5 and up

Jane, Pamela (2011) Little Goblins Ten Harper Collins

This is a counting book which could be used to challenge students to work  out just how many monsters and spooks there were in the forest that night, ages 5 and up.

O’Connell (2000) Ten Timid Ghosts Scholastic

A counting back book, suitable for ages 4, 5, and 6.

Savage, Stephen (2013) Ten Orange Pumpkins Dial

A counting back book, suitable for ages 4, 5, and 6.

Williams, Simon (2013) Ten Hooting Owls Scholastic

Follow the owls in this counting book - all the way back to the nest! Keep a look-out on every page for hidden numbers to find.

Yates, Philip (2003) Ten Little Mummies Puffin

This book counts down to 1 and then there is a surprise showing that the subtraction 10 - 9 can be undone as the 9 mummies reappear. An early introduction to inverse operations.