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.

A new version of Symmetricon

In an earlier blog, we wrote about the visible learning effect of our Symmetricon app. Originally, we had planned for the app to have three levels but as we came to review the initial program, we felt that to include Level 3 would make it just too hard. Well, how wrong can you be?

Because of the way in which the app enables the players to ‘get it’ really quickly, we have frequently been asked if it would be possible to make it a bit more challenging. Hence the new version, in which there is a Level 3, which is based on a hexagonal design.

Writing about this also gives us a chance to point you to the feature that allows you to send your design to Facebook or a class blog or to email it to a class email address. Go to the 'Help file' and scroll to the bottom where you can select which of the social media features you want to use. That way you can accumulate a whole spectrum of symmetricon designs, which can enliven the walls of your classroom at a time when symmetry has become a hot topic.

And don’t forget that you can download a lesson plan with suggestions for using Symmetricon in the classroom.

Teaching ratio using pizza

Last week Ann ran a three day transition project across Parafield Gardens High School and its feeder primary schools. The teachers involved were open to ideas and sharing in the interests of improving understanding and continuity between primary and high school maths and pedagogy.

Each day, Ann ran a demonstration lesson which the teachers then unpacked with student work samples to guide them. On the third day, Ann took the teachers into a year 7 class to give the students a problematised situation involving ratio. The class teacher was a bit alarmed as she hadn't introduced ratio yet.

“All the better for our purposes,” thought Ann, “...we should see students struggle with the problem!” The following problem is loosely true.

The Problem

Thomas has surveyed the 18 teachers here today to find their pizza preferences.
He says that ham and pineapple, supreme and vegetarian were selected in the following ratio, 3:2:1, and that everyone would be happy if they had three eighths of a pizza. Amanda says she is going to get six pizzas two of each flavour. 

I think this is going to be a poor decision. Please work out what our pizza order should be.

The Sting

We are on a limited budget so need to buy the least number of pizzas. They are $7.50. What should we order to get the best deal?

As it turned out, the students got straight into making sense of the problem, often by drawing, as the following example shows.

Although the students had not been formally introduced to ratio, they made a great job of interpreting what the question meant. In less time than Ann thought possible, the problem was solved and the sting-in-the-tail was required. Here the students brought their own understanding of what would constitute a ‘good deal’ and many of them focused on minimising wastage.

Learning fractions out of cutting salad ingredients - it works!

Last week I was working with the year 1/2 class at Torrensville Primary School and the students were given a salad problem as we considered healthy foods.

The problem: Last night my neighbour's little boy came to visit so we made some salad.

• We chopped the veggies.
• We chopped 5 baby costs lettuces in half.
• We chopped 4 tomatoes into halves
• We chopped 3 carrots into quarters.
• We chopped 2 celery sticks into eighths.

After we had eaten the salad, Kai asked how many pieces we had made with all that chopping. I said I would ask my students to work it out for him.

The following work samples are just a few of the many solutions that the students found.

In the last sample, you can see four plates after the first working out. That little girl finished so easily that she needed the sting in the tail!

• We put the salad equally shared onto four plates - so what was on each plate?

Look closely and you will see that two lettuce halves were cut again and a quarter put onto each plate.

The students needed no assistance getting started and one child called out "Eighths means eight pieces doesn't it?" That was the only hint as such. It is amazing how easily students work with fractions if we let them work intuitively rather than try to teach adult-centred strategies.

Try it with your students and email us the results - john@naturalmaths.com.au. Have a great week!